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+{"text":"\n\\section{Introduction}\n\\label{introduction}\n\\textls[-25]{{Since the first direct observation of gravitational waves (GWs), carried out by the interferometers of the LIGO\/VIRGO collaboration \\cite{abbott2016}, gravitational radiation has proved to be a precious messenger of information for astrophysics, allowing not only to confirm the predictions of general relativity \\cite{einstein1916,depa2022} but also to study extraordinary phenomena, such as the merger of neutron stars \\cite{abbott2017}, and to discover several black holes \\cite{abbott2020}. However, gravitational astronomy is still in its infancy and to date, only the tip of the iceberg of the GW spectrum has been scratched. Additionally, low-frequency GWs are not accessible by ground-based interferometers, which are currently only sensitive to high-frequency GWs (in the range 10--$10^{4}$ Hz) emitted in the final stages of the coalescence of compact objects \\cite{maggiore2008a}. To explore the low-frequency side of the GW spectrum (in the range $10^{-5}$--1 Hz), space-based gravitational interferometers, such as Laser Interferometer Space Antenna (LISA) \\cite{lisa2017}, are required and, in addition, for very low-frequency GWs (in the range $10^{-10}$--$10^{-6}$ Hz), pulsar timing arrays (PTAs) must be used, which are currently the only instruments sensitive to them \\cite{moore2015}.}\n\nPTAs are arrays of extremely regular millisecond pulsars (MSPs), constantly monitored by radio telescopes to measure the variations in the times of arrival (ToAs) of the emitted radio pulses, commonly referred to as timing residuals (TRs). Since GWs can influence the radio pulse ToAs, their signature can be found by studying the TRs \\cite{estabrook1975,sazhin1978,detweiler1979}, once cleaned from other effects that can affect them. At present, the main purpose of PTAs is to reveal the gravitational wave background (GWB) due to the superposition of GWs emitted by the supermassive black hole binary systems (SMBHBs) that, according to the current cosmological model, populate the Universe \\cite{maggiore2008b}. The GWB should induce spatially correlated TRs in all the MSPs of PTAs \\cite{mashhoon1982}. Since this correlation should take the form described by the Hellings and Downs function \\cite{hellings1983}, the latter is considered the smoking gun for the GWB \\cite{maiorano2021a}. Although from the data collected by the main PTA collaborations (the European Pulsar Timing Array (EPTA) \\cite{desvignes2016}, the Indian Pulsar Timing Array (InPTA) \\cite{joshi2018}, the North American Nanohertz Observatory for Gravitational Waves (NANOGrav) \\cite{arzoumanian2018}, and the Parkes Pulsar Timing Array (PPTA) \\cite{reardon2016}, which join their efforts as the International Pulsar Timing Array (IPTA) \\cite{verbiest2016}) it has been possible to obtain strong constraints on both the frequency and the amplitude of the GWB, and despite the emergence from them of a common red noise compatible with that expected for the GWB, the smoking gun is still missing, and the first GW detection by PTAs has not yet been claimed. Furthermore, the first detection of continuous GW emitted by a SMBHB seems even more distant, as it may be subject to knowledge of the GWB.\n\n\\textls[-25]{This article is motivated by the exigency to look for additional instruments and tools for detecting very low-frequency GWs \\cite{maiorano2021b}. This necessity is made more urgent by recent research works, which show the limits and criticalities of the investigation principles adopted up to now~\\cite{zic2022}. For this purpose, the potential of the surfing effect for the detection of the continuous GWs emitted by a SMBHB has been analyzed. Although the surfing effect is present in some works on PTAs, the literature to date is lacking in articles dedicated to its discussion in the context of general relativity, and in which a possible concrete application is presented. For this reason, this paper is structured as follows: Section \\ref{sec:1} is dedicated to a detailed description of the surfing effect, Section \\ref{sec:2} describes an application of the surfing effect to the SMBHB candidate \\linebreak PKS 2131--021, in Section \\ref{sec:3} the signatures of the GWs possibly emitted by the SMBHB candidate \\linebreak PKS 2131--021, have been searched in the averaged narrowband and wideband post-fit whitened TRs obtained by NANOGrav collaboration on the MSP J2145--0750, and finally, the main conclusions are presented Section \\ref{sec:4}.}}\n\n\\section{An Overview on the Surfing Effect}\n\\label{sec:1}\nThe surfing effect occurs when the angle $\\theta$ seen by an observer on Earth between the travel direction of the radio pulses emitted by a MSP and the travel direction of the GWs emitted by a single GW source is sufficiently small. In this case, the radio pulses behave as if they were surfing the GWs, producing an increased value of the pulsar TRs. That can be shown by considering the function $R\\left(t,n_i\\right)$ which describes the pulsar TRs induced by the GWs emitted by a single source \\cite{estabrook1975, sazhin1978, detweiler1979}:\n\\begin{equation}\n\\label{residualsfunction}\nR\\left(t,n_i\\right)=\\sum_{A=+,\\times}F^A\\left[r^A_e\\left(t,n_i\\right) -r^A_p\\left(t,n_i\\right)\\right]\n\\end{equation}\n\\textls[-25]{where $t$ is the time, $n_i$ is the versor oriented toward the source of the GWs\\endnote{Sometimes, in the literature, the versor oriented along the travel direction of the GWs $\\Omega_i$ is considered in place of $n_i$. \\linebreak Since the two versors point toward opposite directions, the relation between them is $\\Omega_i=-n_i$.}, $A$ is the GW polarization state index, which is $+$ in the case of the plus-polarization state, and $\\times$ in the case of the cross-polarization state, and $F^A$ is the antenna pattern function, defined as\\endnote{In this paper, the Einstein notation, which indicates the sum over repeated indices, has been adopted.}:\n\\begin{equation}\n\\label{antennafunction}\nF^A=\\frac{1}{2}\\frac{p^ip^je_{ij}^A}{(1-n_ip^i)}\n\\end{equation}\n\nHere, $p^i$ is the versor oriented toward the MSP, and $e_{ij}^A$ is the GW polarization tensor:\n\\begin{equation}\n\\label{polarizationplus}\ne_{ij}^+=\\begin{pmatrix}\n1& 0& 0 \\\\ \n0& -1& 0 \\\\ \n0& 0& 0 \n\\end{pmatrix}\n\\end{equation}\n\\begin{equation}\n\\label{polarizationscross}\ne_{ij}^\\times=\\begin{pmatrix}\n0& 1& 0 \\\\ \n1& 0& 0 \\\\ \n0& 0& 0 \n\\end{pmatrix}\n\\end{equation}\n\n\\textls[-35]{Moreover, $r^A_e\\left(t,n_i\\right)$ and $r^A_p\\left(t,n_i\\right)$ are the Earth and the pulsar terms defined respectively by\\endnote{In this paper, the geometrical units c=G=1 have been adopted.}:}\n\\begin{equation}\n\\label{earthterm}\nr^A_e\\left(t,n_i\\right)=\\int^t_0\\dd{t}h^A_e\\left(\\omega_e t\\right)\n\\end{equation}\n\\begin{equation}\n\\label{pulsarterm}\nr^A_p\\left(t,n_i\\right)=\\int^t_0\\dd{t}h^A_p\\left(\\omega_p t-\\omega_p t_p\\left(1-n_ip^i\\right)\\right)\n\\end{equation}\nwhere $\\omega_e$ and $\\omega_p$ are the GW angular frequency, evaluated at the Earth and MSP positions, respectively, $t_p$ is the time distance from the MSP to the Earth, and $h^A_e\\left(\\omega_e t\\right)$ and $h^A_p\\left(\\omega_p t-\\omega_p t_p\\left(1-n_ip^i\\right)\\right)$ are the perturbations on the flat space-time metric induced by the GWs, evaluated at the Earth and MSP positions, respectively\\endnote{For an exhaustive derivation of Equation \\eqref{residualsfunction} see ref. \\cite{maggiore2008b}.}.\n\nEquation \\eqref{residualsfunction} is particularly useful to describe the pulsar TRs induced by the GWs emitted by a circularized SMBHBs in the continuous emission regime. In the literature, this assumption is often made in order to simplify the description of the principles on which PTAs are based. That is reasonable because, according to the most widely accepted models, SMBHBs form during the collision of galaxies by dynamic friction, which, as it is widely known, tends to circularize their orbits \\cite{bonetti2020}.\n\nIn this case, it is convenient to adopt a coordinate system $Oxyz$ with its origin on the Earth\\endnote{Such a choice would induce spurious TRs, with a period of one year, due to the motion of the Earth relative to the Solar System Barycenter. Although accounting for that is crucial from the experimental point of view and is done during the data processing phase, this issue can be safely ignored in this theoretical context.}, so that:\n\\begin{equation}\n\\label{nversor}\nn_i\\equiv\\left(0,0,1\\right)\n\\end{equation}\n\\begin{equation}\n\\label{pversor}\np^i\\equiv\\left(\\sin{\\theta_p}\\cos{\\phi_p},\\sin{\\theta_p}\\sin{\\phi_p},\\cos{\\theta_p}\\right)\n\\end{equation}\n\\textls[-35]{where $\\theta_p$ is the angle between the versor $p^i$ and the positive $z$-axis, and $\\phi_p$ is the angle between the $x$-axis and the projection on the $xy$-plane of the versor $p^i$ (see Figure \\ref{fig:reference}). With that choice, the product $n_ip^i$, which appears in Equations \\eqref{antennafunction}, \\eqref{earthterm}, and \\eqref{pulsarterm}, is simply $\\cos\\theta_p$.}\n\n\\begin{figure}[H]\n\\hspace{-30pt}\n\\includegraphics[scale=0.65]{reference.png}\n\\caption{Scheme of the adopted $Oxyz$ coordinate system, chosen so that the versor $n_i$ is aligned with the $z$-axis. The black circle pair indicates the SMBHB, pointed by the versor $n_i$, while the white circle indicates the MSP, pointed by the versor $p_i$. In the figure, the distances of the SMBHB and the MSP from the origin $O$, where the observer is placed, are not in scale.}\n\\label{fig:reference}\n\\end{figure}\n\nThe antenna pattern function in Equation \\eqref{antennafunction} can be rewritten in this coordinate system, taking into account the Equations \\eqref{polarizationplus} and \\eqref{polarizationscross}, for both the polarization states:\n\\begin{equation}\n\\label{antennaplus}\nF^+=\\frac{1}{2}\\frac{\\sin^2\\theta_p}{1-\\cos\\theta_p}\\cos(2\\phi_p)\n\\end{equation}\n\\begin{equation}\n\\label{antennacross}\n F^\\times=\\frac{1}{2}\\frac{\\sin^2\\theta_p}{1-\\cos\\theta_p}\\sin(2\\phi_p)\n\\end{equation}\n\nMoreover, the perturbations on the flat space-time metric induced by the GWs evaluated at the Earth and MSP positions are \\cite{perrodin2018}:\n\\begin{equation}\n\\label{hearthterm}\nh^A_e\\left(\\omega_e t\\right)=\\mathcal{F}^A\\mathcal{A}\\sin{\\left(\\omega_e t+\\alpha^A\\right)}\n\\end{equation}\n\\begin{equation}\n\\label{hpulsarterm}\nh^A_p\\left(\\omega_p t-\\omega_p t_p\\left(1-\\cos\\theta_p\\right)\\right)=\\mathcal{F}^A\\mathcal{A}\\sin{\\left(\\omega_p t-\\omega_p t_p\\left(1-\\cos\\theta_p\\right)+\\alpha^A\\right)}\n\\end{equation}\nwhere $\\mathcal{F}^A$ is a factor that accounts for the orbital inclination angle $\\iota$ of the SMBHB \\cite{sesana2010}:\n\\begin{equation}\n\\label{inclinationplus}\n\\mathcal{F}^+=1+\\cos^2{\\iota}\n\\end{equation}\n\\begin{equation}\n\\label{inclinationcross}\n\\mathcal{F}^\\times=-2\\cos{\\iota}\n\\end{equation}\nand $\\mathcal{A}$ is the perturbation amplitude, given by:\n\\begin{equation}\n\\label{amplitude}\n\\mathcal{A}=2\\frac{\\mathcal{M}^{5\/3}}{D}\\left(\\frac{\\omega}{2}\\right)^{2\/3}\n\\end{equation}\nwhere $\\mathcal{M}$ is the chirp mass of the SMBHB, $D$ is the luminosity distance, and $\\omega$ is the angular frequency of GWs, and $\\alpha^A$ is the initial phase, with $\\alpha^+=0$ and $\\alpha^\\times=\\pi\/2$. Note that all the quantities in Equation \\eqref{amplitude} are red-shifted \\cite{perrodin2018}.\n\nThe difference between the Earth term, in Equation \\eqref{earthterm}, and the pulsar term, \\linebreak in Equation \\eqref{pulsarterm}, can be determined for both the polarization states by taking into account Equations \\eqref{hearthterm}--\\eqref{amplitude}, obtaining:\n\\begin{equation}\n\\label{differenceplus}\n\\begin{split}\n&r^+_e\\left(t,n_i\\right) -r^+_p\\left(t,n_i\\right)=-\\frac{\\mathcal{A}}{\\omega_e}\\left(1+\\cos^2{\\iota}\\right)\\left[\\cos{\\left(\\omega_e t\\right)}-1\\right]+\\\\\n&+\\frac{\\mathcal{A}}{\\omega_p}\\left(1+\\cos^2{\\iota}\\right)\\left[\\cos{\\left(\\omega_p t-\\omega_p t_p\\left(1-\\cos\\theta_p\\right)\\right)}-\\cos{\\left(-\\omega_p t_p\\left(1-\\cos\\theta_p\\right)\\right)}\\right]\n\\end{split}\n\\end{equation}\n\\begin{equation}\n\\label{differencecross}\n\\begin{split}\n&r^\\times_e\\left(t,n_i\\right) -r^\\times_p\\left(t,n_i\\right)=-\\frac{\\mathcal{A}}{\\omega_e}\\left(-2\\cos{\\iota}\\right)\\left[-\\sin{\\left(\\omega_e t\\right)}\\right]+\\\\\n&+\\frac{\\mathcal{A}}{\\omega_p}\\left(-2\\cos{\\iota}\\right)\\left[-\\sin{\\left(\\omega_p t-\\omega_p t_p\\left(1-\\cos\\theta_p\\right)\\right)}+\\sin{\\left(-\\omega_p t_p\\left(1-\\cos\\theta_p\\right)\\right)}\\right]\n\\end{split}\n\\end{equation}\n\nThen, the full expression of the function $R\\left(t,n_i\\right)$ is found by substituting the Equations \\eqref{antennaplus}, \\eqref{antennacross}, \\eqref{differenceplus} and \\eqref{differencecross} in Equation \\eqref{residualsfunction}. The result obtained above can be simplified by introducing some additional working hypotheses. First of all, the SMBHB can be assumed to be face-on, which means that $\\iota=\\pi\/2$. In this case, as can be seen from Equations \\eqref{inclinationplus} and \\eqref{inclinationcross}, $\\mathcal{F}^+=1$ and $\\mathcal{F}^\\times=0$. Secondly, since only one MSP is under consideration, the angle $\\theta_p$ coincides with the angle $\\theta$, so $\\theta_p=\\theta$, the coordinate system is rotated to have $\\phi_p=0$, and $t_p$ is replaced by $T$ to simplify the notation. In this case, as can be seen by Equations $\\eqref{antennaplus}$ and \\eqref{antennacross}, $F^+=\\sin^2\\theta\\left[2\\left(1-\\cos\\theta\\right)\\right]^{-1}$ and\n$F^\\times=0$. Thirdly, the SMBHB can be assumed to be very far from its coalescence, as most of the SMBHBs observable with PTAs should be. Under this assumption, the variation of the angular frequency of the GWs~\\cite{shapiro1983} during the interval of time $\\Delta t=T\\left(1-\\cos\\theta\\right)$, which denotes the time delay between the Earth epoch and the pulsar epoch \\cite{jenet2004}, can be neglected, so $\\omega_e\\approx\\omega_p$ and the angular frequency of GWs can be denoted by just $\\omega$. Using all these assumptions, one gets:\n\\begin{equation}\n\\label{compactresiduals}\n\\begin{split}\n&R\\left(t,\\theta\\right)=-\\frac{1}{2}\\frac{\\sin^2\\theta}{1-\\cos\\theta}\\frac{\\mathcal{A}}{\\omega}\\left[\\cos{\\left(\\omega t\\right)}-1\\right]+\\\\&-\\frac{1}{2}\\frac{\\sin^2\\theta}{1-\\cos\\theta}\\frac{\\mathcal{A}}{\\omega}\\left[-\\cos{\\left(\\omega t-\\omega T\\left(1-\\cos\\theta\\right)\\right)}+\\cos{\\left(-\\omega T\\left(1-\\cos\\theta\\right)\\right)}\\right]\n\\end{split}\n\\end{equation}\nwhere the dependence on $n_i$ has been replaced by the dependence on $\\theta$. As can be seen from Equation \\eqref{compactresiduals}, the way the function $R\\left(t,\\theta\\right)$ depends on the angle $\\theta$ makes its use for the detection of GWs rather problematic. For example, it is interesting to note that $R\\left(t,\\theta\\right)$ has a discontinuity for $\\theta=0$, which can be removed by placing $R\\left(t,0\\right)\\equiv\\lim_{\\theta\\to 0} R\\left(t,\\theta\\right)=0$. Moreover, for values of $\\omega$ and $T$ of the same order of magnitude of the average angular frequency of the GWs emitted by SMBHB observable by PTAs and of the average time distance of the MSPs included in PTAs, respectively, the function $R\\left(t,\\theta\\right)$ oscillates extremely rapidly with $\\theta$.\n \nSince the function $R(t,\\theta)$ carries all the information about the GWs emitted by the SMBHB, it is crucial to have a clear understanding of its behavior. The easiest way to do this is by considering, instead of $R\\left(t,\\theta\\right)$, the function $S(\\zeta,\\eta)$, which, using the substitutions:\n\\begin{equation}\n\\label{sostitutionzeta}\n\\zeta\\equiv\\chi\\frac{t}{T}\n\\end{equation}\n\\begin{equation}\n\\label{sostitutioneta}\n\\eta\\equiv\\chi\\left(1-\\cos\\theta\\right)\n\\end{equation}\nwhere $\\chi\\equiv\\omega T$, $\\zeta\\in\\left[0,\\infty\\right[$ for $t\\in\\left[0,\\infty\\right[$ and $\\eta\\in\\left[0,2\\chi\\right]$ for $\\theta\\in\\left[0,\\pi\\right]$, can be defined by:\n\\begin{equation}\n\\label{sfunctiondef}\nS\\left(\\zeta,\\eta\\right)\\equiv2\\frac{\\chi}{\\mathcal{A}T}\\left|R\\left(\\zeta,\\eta\\right)\\right|\n\\end{equation}\n\nThe function defined in Equation \\eqref{sfunctiondef} can be written more explicitly as:\n\\begin{equation}\n\\label{sfunction}\nS\\left(\\zeta,\\eta\\right)=\\left(2-\\frac{\\eta}{\\chi}\\right)\\left|\\cos\\zeta-1-\\cos\\left(\\zeta-\\eta\\right)+\\cos(-\\eta)\\right|\n\\end{equation}\n\\textls[-25]{where the first factor in Equation \\eqref{sfunction} has taken out the absolute value since it is always positive in the defined domain. The TRs induced by GWs are enhanced for the values which maximize the function $S\\left(\\zeta,\\eta\\right)$ in Equation \\eqref{sfunction}. The function $S\\left(\\zeta,\\eta\\right)$ is a periodic function characterized by an amplitude that depends only on $\\eta$ through the first linear factor:}\n\\begin{equation}\n\\label{samplitude}\na\\left(\\eta\\right)\\equiv 2-\\frac{\\eta}{\\chi}\n\\end{equation}\nand its shape depends on both the variables $\\zeta$ and $\\eta$ through the second oscillating factor:\n\\begin{equation}\n\\label{soscillating}\nf\\left(\\zeta,\\eta\\right)\\equiv\\left|\\cos\\zeta-1-\\cos\\left(\\zeta-\\eta\\right)+\\cos(-\\eta)\\right|\n\\end{equation}\nso that:\n\\begin{equation}\n\\label{shortsfunction}\nS\\left(\\zeta,\\eta\\right)=a\\left(\\eta\\right)f\\left(\\zeta,\\eta\\right)\n\\end{equation}\n\nEquation \\eqref{shortsfunction} allows to find the global maximum of the function $S\\left(\\zeta,\\eta\\right)$ by studying the maxima of the functions $a\\left(\\eta\\right)$ and $f\\left(\\zeta,\\eta\\right)$. The function $a\\left(\\eta\\right)$ is maximized when: \n\\begin{equation}\n\\label{firstfactor}\na\\left(\\eta\\right)=2\\text{ for }\\eta=0\n\\end{equation}\nthat occurs when:\n\\begin{equation}\n\\label{surfingeffect}\n\\theta=0\n\\end{equation}\n while the function $f\\left(\\zeta,\\eta\\right)$ is maximized when: \n\\begin{equation}\n\\label{secondfactor}\nf\\left(\\zeta,\\eta\\right)=4\\text{ for }\\left(\\zeta,\\eta\\right)=\\left(\\pi+2\\pi n_\\zeta,\\pi+2\\pi n_\\eta\\right)\n\\end{equation}\nthat occurs when:\n\\begin{equation}\n\\label{timeposition}\nt=\\frac{\\pi T}{\\chi}\\left(1+2n_\\zeta\\right)\\equiv t_{n_\\zeta}\n\\end{equation}\n\\begin{equation}\n\\label{angularposition}\n\\theta=\\arccos\\left(1-\\frac{\\pi}{\\chi}\\left(1+2n_\\eta\\right)\\right)\\equiv\\theta_{n_\\eta}\n\\end{equation}\n\nTherefore, the results in Equations \\eqref{surfingeffect} and \\eqref{angularposition} imply that, for each value of $t$, the global maximum of the function $R\\left(\\zeta,\\eta\\right)$ can be found placing\n$n_{\\eta}\\equiv 0$. Then, since for $n_{\\eta}= 0$ results $\\theta_0\\ll 1$, this property is referred to as surfing effect, and $\\theta_0=\\arccos\\left(1-\\pi\/\\chi\\right)$ is the surfing effect angle.\n\n\\section{An Application of the Surfing Effect to the Case of the Supermassive Black Hole Binary Candidate PKS 2131--021}\n\\label{sec:2}\nThe existence of SMBHBs is not only predicted by most of the hierarchical structure formation models \\cite{sesana2013} but also supported by observational evidence. In some rare cases, the inferred orbital parameters suggest that, if the data analysis is correct, the SMBHB should emit GWs observable with PTAs. In these cases, as has already been conducted with the SMBHB candidate in the Seyfert galaxy $3$C $66$B, the data collected by PTAs can be used to rule out the GW emission \\cite{jenet2004,depa2004,arzoumanian2020a}.\n\nAn interesting recent study on the blazar PKS 2131--021 opens the possibility for an application of the surfing effect \\cite{oneill2022}. In fact, according to the data analysis of the blazar PKS 2131--021 radio emission, it can be identified as an SMBHB candidate located at a redshift $z=1.285$ \\cite{wu2007}, with an observed orbital period of $1760.4^{+5.3}_{-5.3}$ d and a non-red-shifted chirp mass $\\lesssim 5.4 \\times 10^9 M_\\odot$ (see ref. \\cite{oneill2022}). Adopting the most recent values for cosmological constants determined by the Planck collaboration \\cite{planck2020}, the luminosity distance of the SMBHB candidate is $D=9.2$ Gpc, while the angular frequency and the amplitude of the expected GW emission are $\\omega=8.262^{+0.025}_{-0.025}\\times 10^{-8}$ Hz and $\\mathcal{A}\\lesssim 2.4\\times 10^{-15}$, respectively.\n\nAn optimal strategy to confirm these results, by taking advantage of the surfing effect, is to perform a single-pulsar search of continuous GWs focusing on the MSP with the smallest angular distance from the SMBHB candidate PKS 2131--021 among the ones currently included in PTAs. For this reason, the best MSP is the MSP J2145--0750, which lies at the angle $\\theta=0.1156$, corresponding to about $6.6234^{\\circ}$, from the SMBHB candidate PKS 2131--021 \\cite{gaia,deller2019}. The MSP J2145--0750 is a well-studied MSP, lying in a binary system with a white dwarf, at a distance of $0.62^{+0.00}_{-0.02}$ kpc \\cite{deller2019} from the Earth and timed for about $12.5$~years by the NANOGrav collaboration \\cite{alam2021a, alam2021b}. The choice of the MSP J2145--0750 is also convenient because it allows neglecting the GW angular frequency variation between the Earth and the pulsar terms due to the SMBHB PKS 2131--021 orbital evolution. In fact, by considering the equation describing the GW angular frequency variation \\cite{shapiro1983}:\n\\begin{equation}\n\\label{frequencyevolution}\n\\dot{\\omega}=\\frac{12}{5}2^{1\/3}\\mathcal{M}^{5\/3}\\omega^{11\/3}\\left(1+z\\right)\n\\end{equation}\nand substituting in the integral of Equation \\eqref{frequencyevolution} the time delay between the Earth epoch and the pulsar epoch, which is $\\Delta t = 13.5\\pm 0.4$ yr, the GW angular frequency variation between the Earth and the pulsar terms turns out to be $\\Delta\\omega\\lesssim0.042\\times 10^{-8}$ Hz.\n\nWith all these data in hand, Equation \\eqref{angularposition} can be used to determine the function $S\\left(\\zeta,\\eta\\right)$ and then to find out the surfing effect angle (see Figure \\ref{fig:sfunction}).\n\\begin{figure}[H]\n\n\\includegraphics[scale=0.85]{sfunction.png}\\hfill\n\\caption{Plot of the $S\\left(\\zeta,\\eta\\right)$ function relative to the SMBHB candidate PKS 2131--021 and the MSP J2145--0750. The axes on the plane indicate the values assumed by the $\\zeta$ and $\\eta$ variables defined in Equations \\eqref{sostitutionzeta} and \\eqref{sostitutioneta}, respectively, expressed in radians. The vertical axis and the color, described by the color bar, indicate the values assumed by $S\\left(\\zeta,\\eta\\right)$, defined in Equation \\eqref{sfunctiondef}, dimensionless.}\n\\label{fig:sfunction}\n\\end{figure}\nOnce $t$ has been chosen to satisfy Equation \\eqref{timeposition}, by setting, for example, $n_\\zeta\\equiv0$, it is possible to define the function $R\\left(\\theta\\right)\\equiv R\\left(t_0,\\theta\\right)$, useful for verifying whether the angular separation $\\theta$ between the MSP and the SMBHB candidate is such as to maximize the TRs. In this case, $\\omega=8.262^{+0.025}_{-0.025}\\times 10^{-8}$ Hz and $T=2024^{+16}_{-65}$ yr, therefore $\\chi=5274^{+58}_{- 186}$, and at the angle $\\theta_0=0.0345$, corresponding to about $1.9778^{\\circ}$, one has $R\\left(\\theta_0\\right)\\lesssim 0.116$ $\\upmu$s. Even if the actual angle between the SMBHB candidate PKS 2131--021 and the MSP J2145--0750, which is $\\theta=0.1156$, is larger than $\\theta_0$, it can be found that this is very close to $\\theta_5=0.1145$, which identifies the sixth peak of the function $R\\left(\\theta\\right)$. In fact, for $\\theta=0.1156$ results $R\\left(\\theta\\right)\\lesssim0.104$ $\\upmu$s (see Figures \\ref{fig:seangle} and \\ref{fig:seanglezoom}).\n\\begin{figure}[H]\n\\hspace{-5pt}\n\\includegraphics[scale=0.53]{seangle.png}\\hfill\n\\caption{{Plot} \n of the $R\\left(\\theta\\right)$ function relative to the SMBHB candidate PKS 2131--021 and the MSP J2145--0750. The horizontal axis indicates the values assumed by the $\\theta$ variable, expressed in radians. The vertical axis indicates the values assumed by $R\\left(\\theta\\right)$, expressed in nanoseconds. The orange dot indicates the value assumed by the $R\\left(\\theta\\right)$ function when $\\theta$ coincides with the actual angle between the SMBHB candidate PKS 2131--021 and the MSP J2145--0750, which is $\\theta=0.1156$.}\n\\label{fig:seangle}\n\\end{figure}\n\\begin{figure}[H]\n\\hspace{-5pt}\n\\includegraphics[scale=0.53]{seanglezoom.png}\\hfill\n\\caption{Zoomed plot of the $R\\left(\\theta\\right)$ function relative to the SMBHB candidate PKS 2131--021 and the MSP J2145--0750. The horizontal axis indicates the values assumed by the $\\theta$ variable, expressed in radians. The vertical axis indicates the values assumed by the $R\\left(\\theta\\right)$ function, expressed in nanoseconds. The orange dot indicates the value assumed by $R\\left(\\theta\\right)$ when $\\theta$ coincides with the actual angle between the SMBHB candidate PKS 2131--021 and the MSP J2145--0750, which is $\\theta=0.1156$.}\n\\label{fig:seanglezoom}\n\\end{figure}\nWe emphasize that this is a lucky coincidence and allows one to say that if the chirp mass of the SMBHB candidate PKS 2131--021 coincides with the estimated upper limit, the TRs induced by the GWs emitted by the SMBHB candidate PKS 2131--021 on the MSP J2145--0750 might be observable even at the current PTA sensitivity level due to the surfing effect. In addition, as can be easily deduced from the Figures \\ref{fig:seangle} and \\ref{fig:seanglezoom}, if $\\theta$, whether small or large, is in a node of the function $R\\left(\\theta\\right)$, where it vanishes, in the TRs of the considered MSP there will be no trace of the influence of GWs. Eventually, it is necessary to stress the fact that the larger is $\\theta$, the higher the precision required for $\\chi$. In fact, as it is shown in \\mbox{Figure \\ref{fig:seanglezoomerror}}, for the same value of $\\theta$ the function $R\\left(\\theta\\right)$ can assume sensibly different values within the uncertainty on $\\chi$. However, if $\\theta$ is in the first peak, the role played by the error on $\\chi$ is marginal.\n\\begin{figure}[H]\n\\hspace{-5pt}\n\\includegraphics[scale=0.53]{seanglezoomerror.png}\\hfill\n\\caption{Comparative zoomed plot of the $R\\left(\\theta\\right)$ function relative to the SMBHB candidate PKS 2131--021 and the MSP J2145--0750. The horizontal axis indicates the values assumed by the $\\theta$ variable, expressed in radians. The vertical axis indicates the values assumed by the $R\\left(\\theta\\right)$ function, expressed in nanoseconds. The green and red dotted lines indicate the values assumed by the $R\\left(\\theta\\right)$ function for $\\chi=5274-186$ and $\\chi=5274+58$, respectively. The orange dot indicates the value assumed by $R\\left(\\theta\\right)$ when $\\theta$ coincides with the actual angle between the SMBHB candidate PKS 2131--021 and the MSP J2145--0750, which is $\\theta=0.1156$. The green and red dots indicate the values assumed by $R\\left(\\theta\\right)$ when $\\theta$ coincides with the actual angle between the SMBHB candidate PKS 2131--021 and the MSP J2145--0750, which is $\\theta=0.1156$, for $\\chi=5274-186$ and $\\chi=5274+58$, respectively.}\n\\label{fig:seanglezoomerror}\n\\end{figure}\n\n\\section{A Look to NANOGrav Data}\n\\label{sec:3}\nThe NANOGrav collaboration has recently published the $12.5$ yr Data Set,\nwhich is publicly available (see ref. \\cite{nanoweb}), making it possible to examine the TRs of the MSP J2145--0750. In particular, the datasets useful for searching GWs are the averaged narrowband \\cite{alam2021a} and wideband \\cite{alam2021b} post-fit whitened TRs (see Figures \\ref{fig:narrow1} and \\ref{fig:wide1}) since they are characterized by a low weighted root mean squared value $\\sigma_w$, which has been evaluated by using the following definition:\n\\begin{equation}\n\\label{wrms}\n\\sigma_w = \\sqrt{\\frac{\\sum_{i}^n{\\frac{\\left(x_i-\\Bar{x}\\right)^2}{\\sigma_i^2}}}{\\sum_{i}^n(\\frac{1}{\\sigma_i^2})}}\n\\end{equation}\nwhich is valid for any set of quantities $x_1, x_2, \\ldots, x_n$ characterized by the errors $\\sigma_1, \\sigma_2, \\ldots, \\sigma_n $ and by an average $\\Bar{x}$. In this case, using Equation \\eqref{wrms} it has been found $\\sigma_w=0.333$ $\\upmu$s for narrowband, and $\\sigma_w=0.276$ $\\upmu$s for wideband.\n\\begin{figure}[H]\n\\hspace{-5pt}\n\\includegraphics[scale=0.53]{narrow1.png}\\hfill\n\\caption{Plot of the averaged narrowband post-fit whitened TRs of the MSP J2145--0750, which are from the $12.5$ yr Data Set published by the NANOGrav collaboration. The horizontal axis indicates the time, expressed in years. The vertical axis indicates the averaged narrowband post-fit whitened TRs, expressed in~nanoseconds.}\n\\label{fig:narrow1}\n\\end{figure}\\vspace{-6pt}\n\\begin{figure}[H]\n\n\\includegraphics[scale=0.53]{wide1.png}\\hfill\n\\caption{Plot of the wideband post-fit whitened TRs of the MSP J2145--0750, which are from the $12.5$ yr Data Set published by the NANOGrav collaboration. The horizontal axis indicates the time, expressed in years. The vertical axis indicates the wideband post-fit whitened TRs, expressed in~nanoseconds.}\n\\label{fig:wide1}\n\\end{figure}\nIn Figures \\ref{fig:narrow1} and \\ref{fig:wide1}, the origin of time is at Modified Julian Date $53267$, corresponding to 19 September $2004$, from which the observation started. Equation \\eqref{compactresiduals} implies that the TRs induced by the GWs emitted by a SMBHB are periodic, with an angular frequency that is the same as the angular frequency $\\omega$ of the GWs. This periodic signature can be searched in TRs using periodic analysis algorithms, such as the Lomb-Scargle (LS) algorithm \\cite{lomb1976,scargle1982}.\n\nThe LS periodograms of the TRs plotted in Figures \\ref{fig:narrow1} and \\ref{fig:wide1} have been obtained and plotted in Figures \\ref{fig:narrow2} and \\ref{fig:wide2}, respectively.\n\\begin{figure}[H]\n\n\\includegraphics[scale=0.53]{narrow2.png}\\hfill\n\\caption{Plot\nof the LS periodogram of the averaged narrowband post-fit whitened TRs of the MSP J2145--0750. The black-dotted lines indicate the angular frequencies found by the LS algorithm. The red-dotted line indicates the angular frequency $\\omega=8.262^{+0.025}_{-0.025}\\times 10^{-8}$ expected for the GWs possibly emitted by the SMBHB candidate PKS 2131--021. The gray shaded area indicates uncertainty on $\\omega$ at the $3\\sigma$ confidence level. The horizontal axis indicates the GW angular frequency, expressed in Hertz. The vertical axis indicates the normalized amplitude, dimensionless.}\n\\label{fig:narrow2}\n\\end{figure}\n\\vspace{-12pt}\n\\begin{figure}[H]\n\n\\includegraphics[scale=0.53]{wide2.png}\\hfill\n\\caption{Plot\nof the LS periodogram of the wideband post-fit whitened TRs of the MSP J2145--0750. The black-dotted lines indicate the angular frequencies found by the LS algorithm. The red-dotted line indicates the angular frequency $\\omega=8.262^{+0.025}_{-0.025}\\times 10^{-8}$ expected for the GWs possibly emitted by the SMBHB candidate PKS 2131--021. The gray shaded area indicates uncertainty on $\\omega$ at the $3\\sigma$ confidence level. The horizontal axis indicates the GW angular frequency, expressed in Hertz. The vertical axis indicates the normalized amplitude, dimensionless.}\n\\label{fig:wide2}\n\\end{figure}\nIn both cases (see Figures \\ref{fig:narrow2} and \\ref{fig:wide2}), among the periodicities with the highest normalized amplitude, the LS algorithm finds one associated with an angular frequency close to \\linebreak $\\omega=8.262^{+0.025}_{-0.025}\\times 10^{-8}$ Hz, expected for GWs possibly emitted by the SMBHB candidate PKS 2131--021. Specifically, it finds $\\omega=8.006\\times 10^{-8}$ Hz in the case of the narrowband dataset and $\\omega=8.0139\\times 10^{-8}$ Hz in the case of the wideband dataset. At this point, a shuffling test was conducted to verify that these values are not artifacts and emerge from true periodicities in the datasets. To this aim, several datasets were created by shuffling the starting datasets, and, for each of the shuffled datasets, the LS periodogram was obtained. After doing so, the periodogram associated with the starting dataset was compared with each periodogram associated with the shuffled datasets to establish which is characterized by the largest normalized amplitude at the frequency value under consideration. \\linebreak This procedure was iterated 10,000 times, with the result that the former normalized amplitude was larger with respect to the latter the $\\simeq$87\\% of cases for narrowband and the $\\simeq$93\\% of cases for wideband.\n\nAlthough it can be read as an encouraging result, it is important to stress that the two periodograms have been obtained using the complete datasets, provided by the NANOGrav collaboration. However, from Figures \\ref{fig:narrow1} and \\ref{fig:wide1}, it can be noted that, in the first observation years, the cadence of TRs was sporadic and irregular, and the error bands were significantly large than in the last observation years. Therefore, it may be convenient to reanalyze the datasets after applying some filters. Based on the previous considerations, one possibility is to remove both the TRs falling in the first six observation years and those associated with an uncertainty larger than $3\\sigma_w$ (see Figures \\ref{fig:narrow3} and \\ref{fig:wide3}). Applying this filter also assures that, as shown in ref. \\cite{oneill2022}, the dataset is limited to a time interval from 2010 onwards when the SMBHB candidate should have been in the continuous emission regime.\n\\begin{figure}[H]\n\n\\includegraphics[scale=0.53]{narrow3.png}\\hfill\n\\caption{Plot of the averaged narrowband post-fit whitened TRs of the MSP J2145--0750, filtered by removing the TRs associated with an uncertainty larger than $3\\sigma_w$ and the TRs before the sixth year of observation. The horizontal axis indicates the time, expressed in years. The vertical axis indicates the averaged narrowband post-fit whitened TRs, expressed in nanoseconds.}\n\\label{fig:narrow3}\n\\end{figure}\n\\vspace{-12pt}\n\\begin{figure}[H]\n\n\\includegraphics[scale=0.53]{wide3.png}\\hfill\n\\caption{Plot of the wideband post-fit whitened TRs of the MSP J2145--0750, filtered by removing the TRs associated with an uncertainty larger than $3\\sigma_w$ and the TRs before the sixth year of observation. The horizontal axis indicates the time, expressed in years. The vertical axis indicates the averaged narrowband post-fit whitened TRs, expressed in nanoseconds.}\n\\label{fig:wide3}\n\\end{figure}\nThe LS periodograms of the TRs plotted in Figures \\ref{fig:narrow3} and \\ref{fig:wide3} have been obtained and plotted in Figures \\ref{fig:narrow4} and \\ref{fig:wide4}, respectively.\n\\begin{figure}[H]\n\n\\includegraphics[scale=0.53]{narrow4.png}\\hfill\n\\caption{Plot\nof the LS periodogram of the averaged narrowband post-fit whitened TRs of the MSP J2145--0750, filtered by removing the TRs associated with an uncertainty larger than $3\\sigma_w$ and the TRs before the sixth year of observation. The black-dotted lines indicate the angular frequencies found by the LS algorithm. The red-dotted line indicates the angular frequency $\\omega=8.262^{+0.025}_{-0.025}\\times 10^{-8}$ expected for the GWs possibly emitted by the SMBHB candidate PKS 2131--021. The gray shaded area indicates uncertainty on $\\omega$ at the $3\\sigma$ confidence level. The horizontal axis indicates the GW angular frequency, expressed in Hertz. The vertical axis indicates the normalized amplitude, dimensionless.}\n\\label{fig:narrow4}\n\\end{figure}\n\\vspace{-12pt}\n\\begin{figure}[H]\n\\includegraphics[scale=0.53]{wide4.png}\\hfill\n\\caption{Plot\nof the LS periodogram of the wideband post-fit whitened TRs of the MSP J2145--0750, filtered by removing the TRs associated with an uncertainty larger than $3\\sigma_w$ and the TRs before the sixth year of observation. The black-dotted lines indicate the angular frequencies found by the LS algorithm. The red-dotted line indicates the angular frequency $\\omega=8.262^{+0.025}_{-0.025}\\times 10^{-8}$ expected for the GWs possibly emitted by the SMBHB candidate PKS 2131--021. The gray shaded area indicates uncertainty on $\\omega$ at the $3\\sigma$ confidence level. The horizontal axis indicates the GW angular frequency, expressed in Hertz. The vertical axis indicates the normalized amplitude, dimensionless.}\n\\label{fig:wide4}\n\\end{figure}\nIn both cases (see Figures \\ref{fig:narrow4} and \\ref{fig:wide4}), the LS algorithm allows finding the periodicity with the highest normalized amplitude associated with an angular frequency even closer to $\\omega=8.262^{+0.025}_{-0.025}\\times 10^{-8}$ Hz. Specifically, it is found $\\omega=8.147\\times 10^{-8}$ Hz in the case of the narrowband dataset, and $\\omega=8.205\\times 10^{-8}$ Hz in the case of the wideband dataset. Even if these values are so close to the angular frequency of the GW emission expected for the SMBHB candidate PKS 2131--021, this is not enough for claiming its detection. Further, more sophisticated analyses must be conducted in order of ensuring the solidity of this result and the nature of this periodicity.\n\n\\section{Conclusions}\n\\label{sec:4}\nIn this article, a possible investigation method to search for continuous GWs, based on the surfing effect, has been proposed. The surfing effect, described in Section \\ref{sec:1}, is usually ignored in most of the studies on the GWB since it is not relevant when the entire array of MSPs is used for its detection \\cite{hellings1983,lentati2015,arzoumanian2020,goncharov2021}. However, when performing a single-pulsar search of GWs emitted by a circularized SMBHBs in the continuous emission regime, it has to be taken into consideration. In fact, in this case, as shown in Sections \\ref{sec:1} and \\ref{sec:2}, the global maximum of the function $R\\left(t,\\theta\\right)$, which describes the TRs induced by GWs, corresponds, at any time, to an angular separation between the SMBHB and the MSPs almost null. However, it is important to keep in mind that, at any time, the function $R\\left(t,\\theta\\right)$ is characterized by very rapid angular oscillations, the more frequent the greater the $\\chi$ factor, as shown in Figures \\ref{fig:seangle} and \\ref{fig:seanglezoom}. Therefore, the function $R\\left(\\theta\\right)$ represents a sort of map, which can be used to verify if a MSP is more or less suitable for performing a single-pulsar search of continuous GWs, so MSPs that fall on one of the peaks will be preferred, and those that fall into one of the nodes will be discarded.\n\nAs an example, in Sections \\ref{sec:2} and \\ref{sec:3}, the case of the SMBHB candidate PKS 2131--021 was considered. This system, based on the orbital parameters determined by the radio analysis of its emission \\cite{oneill2022}, should be an emitter of GWs satisfying the hypotheses made in Section \\ref{sec:1} for surfing effect. By a lucky coincidence, as shown in \\mbox{Figures \\ref{fig:seangle} and \\ref{fig:seanglezoom}}, among the MSPs currently included in PTAs, the angularly closest MSP to the SMBHB candidate PKS 2131--021, which is the MSP J2145--0750, is such that its angular separation falls into a peak of the function $R\\left(\\theta\\right)$. This makes it worthy of attention since in its TRs a periodic signature induced by the GWs emitted by the SMBHB candidate PKS 2131--021 may be present. For this reason, in Section \\ref{sec:4}, the LS algorithm was used to search for such periodic signature. The periodograms plotted in Figures \\ref{fig:narrow2} and \\ref{fig:wide2} have been obtained from the complete datasets, provided by the NANOGrav collaboration \\linebreak (see Figures \\ref{fig:narrow1} and \\ref{fig:wide1}), while the periodograms plotted in Figures \\ref{fig:narrow4} and \\ref{fig:wide4} have been obtained from the filtered datasets (see Figures \\ref{fig:narrow3} and \\ref{fig:wide3}), discussed in Section \\ref{sec:3}. Interestingly, the LS algorithm finds in each dataset a periodicity associated with an angular frequency very close to the angular frequency of the GW emission expected for the SMBHB candidate PKS 2131--021. Although this result does not constitute proof of the SMBHB candidate PKS 2131--021 GW emission, it is certainly worthy of further investigation. A more detailed and rigorous analysis of the MSP J2145--0750 TRs, using the NANOGrav $12.5$ yr Data Set in conjunction with the IPTA Data Release 2 \\cite{perera2019}, will be proposed in a future paper entirely dedicated to searching for GWs potentially emitted by the SMBHB candidate PKS 2131--021. There, it will also be discussed the use of other MSPs not too angularly far from the SMBHB candidate PKS 2131--021, as well as the possible improvements that will be brought on this kind of search by next-gen PTAs, such as Square-Kilometer Array (SKA) \\cite{johnston2007,kramer2015,weltman2020,padmanabhan2022}.\n\n\n\\vspace{6pt} \n\\authorcontributions{Conceptualization, M.M. and F.D.P.; Formal analysis, M.M. and F.D.P.; Writing---original draft, M.M.; Writing---review \\& editing, M.M. and F.D.P.; Supervision, A.A.N. All authors have read and agreed to the published version of the manuscript.}\n\n\\funding{This research received no external funding.}\n\\dataavailability{The data used in this paper were collected by the NANOGrav collaboration and are publicly available.\n} \n\\acknowledgments{We warmly acknowledge Andrea Possenti (INAF-OAC) for the enlightening discussions and comments on the paper. We acknowledge the support of the Theoretical Astroparticle Physics (TAsP) and Euclid projects of the Istituto Nazionale di Fisica Nucleare (INFN).}\n\\conflictsofinterest{The authors declare no conflict of interest.}\n\\abbreviations{Abbreviations}{\nThe following abbreviations are used in this manuscript:\\\\\n\n\\noindent \n\\begin{tabular}{@{}ll}\nEPTA~~~~~~~~~~ & European Pulsar Timing Array\\\\\nGW & Gravitational Wave\\\\\nGWB & Gravitational Wave Background\\\\\nINAF & Istituto Nazionale di Astrofisica\\\\\nINFN & Istituto Nazionale di Fisica Nucleare\\\\\nInPTA & Indian Pulsar Timing Array\\\\\n\n\\end{tabular}}\n\n\\abbreviations{}{%\n\\noindent\n\\begin{tabular}{@{}ll}\nLIGO & Laser Interferometer Gravitational-Wave Observatory\\\\\nLISA & Laser Interferometer Space Antenna\\\\\nMDPI & Multidisciplinary Digital Publishing Institute\\\\\nMSP & Millisecond Pulsar\\\\\nNANOGrav & North American Nanohertz Observatory for Gravitational Waves\\\\\nOAC & Osservatorio Astronomico di Cagliari\\\\\nPPTA & Parkes Pulsar Timing Array\\\\\nPTA & Pulsar Timing Array\\\\\nSMBHB & Supermassive Black Hole Binary\\\\\nSKA & Square Kilometre Array\\\\\nTAsP & Theoretical Astroparticle Physics\\\\\nToA & Time of Arrival\\\\\nTR & Timing Residual\\\\\n\\end{tabular}}\n\\begin{adjustwidth}{-\\extralength}{0cm}\n\\printendnotes[custom]\n\\reftitle{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\chapter{Mathematica codes}\n\\label{appendFloquet}\n\\justifying\n\n\n\\section{Electric and magnetic fields}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\mathit{r}=\\sqrt{x^2+y^2+z^2};}\\\\\n\\pmb{\\gamma =\\frac{2\\ell ^2}{\\sqrt{4\\ell ^2t^2+\\left(\\mathit{r}^2-t^2+\\ell ^2\\right)^2}};}\\\\\n\\pmb{\\omega _1=\\gamma \\frac{x}{\\ell };}\\\\\n\\pmb{\\omega _2=\\gamma \\frac{y}{\\ell };}\\\\\n\\pmb{\\omega _3=\\gamma \\frac{z}{\\ell };}\\\\\n\\pmb{\\omega _4=\\gamma \\frac{\\mathit{r}^2-t^2-\\ell ^2}{2\\ell ^2};}\\\\\n\\pmb{J_-[\\text{f$\\_$}] \\text{:=} \\frac{1}{\\sqrt{2}}(\\text{c$\\alpha $} D[f,\\beta ] - \\text{c$\\beta $} D[f,\\alpha ]);}\\\\\n\\pmb{\\text{Il}_-[\\text{f$\\_$}] \\text{:=} \\frac{1}{\\sqrt{2}}(\\text{c$\\alpha $} D[f,\\text{c$\\beta $}] - \\beta  D[f,\\alpha ]);}\\\\\n\\pmb{\\text{expi$\\tau $} = \\frac{(\\ell +I*t)^2+\\mathit{r}^2}{\\sqrt{4\\ell ^2t^2+\\left(\\mathit{r}^2-t^2+\\ell ^2\\right)^2}};}\\\\\n\\pmb{\\text{Sin$\\tau $}[\\text{n$\\_$}]\\text{:=}\\text{ComplexExpand}\\left[\\text{Im}\\left[\\text{expi$\\tau $}^n\\right]\\right]\\text{\/\/}\\text{FullSimplify};}\\\\\n\\pmb{\\text{Cos$\\tau $}[\\text{n$\\_$}]\\text{:=}\\text{ComplexExpand}\\left[\\text{Re}\\left[\\text{expi$\\tau $}^n\\right]\\right]\\text{\/\/}\\text{FullSimplify};}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{ClearAll}[Y];}\\\\\n\\pmb{Y[\\text{j$\\_$},\\text{m$\\_$},\\text{n$\\_$}]\\text{:=}0\\text{\/;}(j<0 \\|m>j \\| m<-j \\| n>j \\| n<-j);}\\\\\n\\pmb{Y[\\text{j$\\_$},\\text{m$\\_$},\\text{n$\\_$}]\\text{:=} }\\\\\n\\pmb{Y[j,m,n]=\\text{Simplify}\\left[\\left(\\sqrt{\\frac{2 \\pi ^2}{2j+1}}\\right)^{-1}\\sqrt{\\frac{2^{2 j - m - n}(j+m)!(j+n)!}{(2j)!(2j)!(j-m)!(j-n)!}}\\text{Nest}\\left[\\text{Il}_-,\\text{Nest}\\left[J_-,\\alpha\n^{2 j},j-n\\right],j-m\\right]\\text{\/.}\\right.}\\\\\n\\pmb{\\left.\\left\\{\\alpha \\to  \\omega _1+I \\omega _2,\\text{c$\\alpha $}\\to  \\omega _1-I \\omega _2,\\beta \\to  \\omega _3+I \\omega _4,\\text{c$\\beta $}\\to\n \\omega _3-I \\omega _4\\right\\}\\right];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{Z_+[\\text{j$\\_$},\\text{m$\\_$},\\text{n$\\_$}]\\text{:=}\\sqrt{(j-n)(j-n+1)\/2}Y[j,m,n+1];}\\\\\n\\pmb{Z_3[\\text{j$\\_$},\\text{m$\\_$},\\text{n$\\_$}]\\text{:=}\\sqrt{(j+1)^2-n^2}Y[j,m,n];}\\\\\n\\pmb{Z_-[\\text{j$\\_$},\\text{m$\\_$},\\text{n$\\_$}]\\text{:=}-\\sqrt{(j+n)(j+n+1)\/2}Y[j,m,n-1];}\\\\\n\\pmb{X[1][\\text{j$\\_$},\\text{m$\\_$},\\text{n$\\_$}]\\text{:=}Z[1][j,m,n]=\\text{Cos$\\tau $}[2j+2]\\left(Z_+[j,m,n]+Z_-[j,m,n]\\right)\/\\sqrt{2};}\\\\\n\\pmb{X[2][\\text{j$\\_$},\\text{m$\\_$},\\text{n$\\_$}]\\text{:=}Z[2][j,m,n]=\\text{Cos$\\tau $}[2j+2]I\\left(-Z_+[j,m,n]+Z_-[j,m,n]\\right)\/\\sqrt{2};}\\\\\n\\pmb{X[3][\\text{j$\\_$},\\text{m$\\_$},\\text{n$\\_$}]\\text{:=}Z[3][j,m,n]=\\text{Cos$\\tau $}[2j+2]Z_3[j,m,n];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{HopfRanadaX}[1]=-\\frac{1}{8}\\text{Sin$\\tau $}[2];}\\\\\n\\pmb{\\text{HopfRanadaX}[2]=-\\frac{1}{8}\\text{Cos$\\tau $}[2];}\\\\\n\\pmb{\\text{HopfRanadaX}[3]=0;}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{ClearAll}[e];}\\\\\n\\pmb{e[\\text{a$\\_$}]\\text{:=}}\\\\\n\\pmb{e[a]=}\\\\\n\\pmb{\\text{Simplify}[}\\\\\n\\pmb{\\frac{\\gamma ^2}{\\ell ^3}\\left(t*R[a]d[t]-\\frac{1}{2}\\text{Sum}\\left[\\left(t^2-\\mathit{r}^2+\\ell ^2\\right)\\text{KroneckerDelta}[a,k]d[R[k]],\\{k,1,3\\}\\right]-\\right.}\\\\\n\\pmb{\\text{Sum}[R[a]R[k]d[R[k]],\\{k,1,3\\}]-\\text{Sum}[\\ell *\\text{LeviCivitaTensor}[3][[a,j,k]]R[j]d[R[k]],}\\\\\n\\pmb{\\{k,1,3\\},\\{j,1,3\\}])\\text{\/\/.}\\{R[0]\\to t,R[1]\\to x,R[2]\\to y,R[3]\\to z\\}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{Evec}[\\text{X1$\\_$},\\text{X2$\\_$},\\text{X3$\\_$}]\\text{:=}\\text{Module}[\\{X,\\mathcal{A},A,F\\},}\\\\\n\\pmb{X[1]\\text{:=}\\text{X1};}\\\\\n\\pmb{X[2]\\text{:=}\\text{X2};}\\\\\n\\pmb{X[3]\\text{:=}\\text{X3};}\\\\\n\\pmb{\\mathcal{A}\\text{:=}\\text{FullSimplify}[\\text{Sum}[X[a]e[a],\\{a,1,3\\}]];}\\\\\n\\pmb{A[0]\\text{:=}\\mathcal{A}\\text{\/.}\\{d[t]\\to 1,d[x]\\to 0,d[y]\\to 0,d[z]\\to 0\\};}\\\\\n\\pmb{A[1]\\text{:=}\\mathcal{A}\\text{\/.}\\{d[t]\\to 0,d[x]\\to 1,d[y]\\to 0,d[z]\\to 0\\};}\\\\\n\\pmb{A[2]\\text{:=}\\mathcal{A}\\text{\/.}\\{d[t]\\to 0,d[x]\\to 0,d[y]\\to 1,d[z]\\to 0\\};}\\\\\n\\pmb{A[3]\\text{:=}\\mathcal{A}\\text{\/.}\\{d[t]\\to 0,d[x]\\to 0,d[y]\\to 0,d[z]\\to 1\\};}\\\\\n\\pmb{F[\\text{u$\\_$},\\text{v$\\_$}]\\text{:=}D[A[u],Y[v]]-D[A[v],Y[u]];}\\\\\n\\pmb{\\{\\text{Simplify}[F[0,1]],\\text{Simplify}[F[0,2]],\\text{Simplify}[F[0,3]]\\}]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{ClearAll}[\\text{EMfields}];}\\\\\n\\pmb{\\text{EMfields}[\\text{j$\\_$},\\text{m$\\_$},\\text{n$\\_$},\\text{L$\\_$}]\\text{:=}\\text{EMfields}[j,m,n,L]=}\\\\\n\\pmb{\\text{Module}[\\{R,\\mathcal{A},A,F\\},}\\\\\n\\pmb{R[0]=t;}\\\\\n\\pmb{R[1]=x;}\\\\\n\\pmb{R[2]=y;}\\\\\n\\pmb{R[3]=z;}\\\\\n\\pmb{\\mathcal{A}[\\text{a$\\_$}]\\text{:=}\\text{Simplify}[X[a][j,m,n]*e[a]\\text{\/.}\\{\\ell \\to L\\}];}\\\\\n\\pmb{\\text{FullGauge}=\\text{Sum}[\\mathcal{A}[a],\\{a,1,3\\}];}\\\\\n\\pmb{A[0]\\text{:=}\\text{FullGauge}\\text{\/.}\\{d[t]\\to 1,d[x]\\to 0,d[y]\\to 0,d[z]\\to 0\\};}\\\\\n\\pmb{A[1]\\text{:=}\\text{FullGauge}\\text{\/.}\\{d[t]\\to 0,d[x]\\to 1,d[y]\\to 0,d[z]\\to 0\\};}\\\\\n\\pmb{A[2]\\text{:=}\\text{FullGauge}\\text{\/.}\\{d[t]\\to 0,d[x]\\to 0,d[y]\\to 1,d[z]\\to 0\\};}\\\\\n\\pmb{A[3]\\text{:=}\\text{FullGauge}\\text{\/.}\\{d[t]\\to 0,d[x]\\to 0,d[y]\\to 0,d[z]\\to 1\\};}\\\\\n\\pmb{F[\\text{u$\\_$},\\text{v$\\_$}]\\text{:=}D[A[u],R[v]]-D[A[v],R[u]];}\\\\\n\\pmb{\\text{Ef}[1]\\text{:=}F[0,1]\\text{\/\/}\\text{Simplify};}\\\\\n\\pmb{\\text{Ef}[2]\\text{:=}F[0,2]\\text{\/\/}\\text{Simplify};}\\\\\n\\pmb{\\text{Ef}[3]\\text{:=}F[0,3]\\text{\/\/}\\text{Simplify};}\\\\\n\\pmb{\\text{Bf}[1]\\text{:=}F[3,2]\\text{\/\/}\\text{Simplify};}\\\\\n\\pmb{\\text{Bf}[2]\\text{:=}F[1,3]\\text{\/\/}\\text{Simplify};}\\\\\n\\pmb{\\text{Bf}[3]\\text{:=}F[2,1]\\text{\/\/}\\text{Simplify};}\\\\\n\\pmb{\\{\\{\\text{Ef}[1],\\text{Ef}[2],\\text{Ef}[3]\\},\\{\\text{Bf}[1],\\text{Bf}[2],\\text{Bf}[3]\\}\\}]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{Ef}[\\text{j$\\_$},\\text{m$\\_$},\\text{n$\\_$},\\ell \\_]\\text{:=}\\text{EMfields}[j,m,n,\\ell ][[1]]\\text{\/\/}\\text{Re}\\text{\/\/}\\text{ComplexExpand}\\text{\/\/}\\text{Simplify};}\\\\\n\\pmb{\\text{Bf}[\\text{j$\\_$},\\text{m$\\_$},\\text{n$\\_$},\\ell \\_]\\text{:=}\\text{EMfields}[j,m,n,\\ell ][[2]]\\text{\/\/}\\text{Re}\\text{\/\/}\\text{ComplexExpand}\\text{\/\/}\\text{Simplify};}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{En}[\\text{j$\\_$},\\text{m$\\_$},\\text{n$\\_$}]\\text{:=}\\frac{1}{2}\\left(\\text{Sum}\\left[\\text{Ef}[j,m,n,1][[a]]^2+\\text{Bf}[j,m,n,1][[a]]^2,\\{a,1,3\\}\\right]\\right)\\text{\/\/}\\text{FullSimplify}}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{ClearAll}[\\text{HRfields}];}\\\\\n\\pmb{\\text{HRfields}[\\text{L$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{R,\\mathcal{A},A,F\\},}\\\\\n\\pmb{R[0]=t;}\\\\\n\\pmb{R[1]=x;}\\\\\n\\pmb{R[2]=y;}\\\\\n\\pmb{R[3]=z;}\\\\\n\\pmb{\\mathcal{A}[\\text{a$\\_$}]\\text{:=}\\text{Simplify}[\\text{HopfRanadaX}[a]*e[a]\\text{\/.}\\{\\ell \\to L\\}];}\\\\\n\\pmb{\\text{FullGauge}=\\text{Sum}[\\mathcal{A}[a],\\{a,1,3\\}];}\\\\\n\\pmb{A[0]\\text{:=}\\text{FullGauge}\\text{\/.}\\{d[t]\\to 1,d[x]\\to 0,d[y]\\to 0,d[z]\\to 0\\};}\\\\\n\\pmb{A[1]\\text{:=}\\text{FullGauge}\\text{\/.}\\{d[t]\\to 0,d[x]\\to 1,d[y]\\to 0,d[z]\\to 0\\};}\\\\\n\\pmb{A[2]\\text{:=}\\text{FullGauge}\\text{\/.}\\{d[t]\\to 0,d[x]\\to 0,d[y]\\to 1,d[z]\\to 0\\};}\\\\\n\\pmb{A[3]\\text{:=}\\text{FullGauge}\\text{\/.}\\{d[t]\\to 0,d[x]\\to 0,d[y]\\to 0,d[z]\\to 1\\};}\\\\\n\\pmb{F[\\text{u$\\_$},\\text{v$\\_$}]\\text{:=}D[A[u],R[v]]-D[A[v],R[u]];}\\\\\n\\pmb{\\text{Ef}[1]\\text{:=}F[0,1]\\text{\/\/}\\text{Simplify};}\\\\\n\\pmb{\\text{Ef}[2]\\text{:=}F[0,2]\\text{\/\/}\\text{Simplify};}\\\\\n\\pmb{\\text{Ef}[3]\\text{:=}F[0,3]\\text{\/\/}\\text{Simplify};}\\\\\n\\pmb{\\text{Bf}[1]\\text{:=}F[3,2]\\text{\/\/}\\text{Simplify};}\\\\\n\\pmb{\\text{Bf}[2]\\text{:=}F[1,3]\\text{\/\/}\\text{Simplify};}\\\\\n\\pmb{\\text{Bf}[3]\\text{:=}F[2,1]\\text{\/\/}\\text{Simplify};}\\\\\n\\pmb{\\{\\{\\text{Ef}[1],\\text{Ef}[2],\\text{Ef}[3]\\},\\{\\text{Bf}[1],\\text{Bf}[2],\\text{Bf}[3]\\}\\}]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{EfHR}[\\ell \\_]\\text{:=}\\text{HRfields}[\\ell ][[1]]\\text{\/\/}\\text{Re}\\text{\/\/}\\text{ComplexExpand}\\text{\/\/}\\text{Simplify};}\\\\\n\\pmb{\\text{BfHR}[\\ell \\_]\\text{:=}\\text{HRfields}[\\ell ][[2]]\\text{\/\/}\\text{Re}\\text{\/\/}\\text{ComplexExpand}\\text{\/\/}\\text{Simplify};}\\)\n\\end{doublespace}\n\n\n\\vspace{12pt}\n\\section{Field lines}\n\\vspace{2pt}\n\nThis code is taken from the following open access platform:\\\\\n\\href{https:\/\/mathematica.stackexchange.com\/questions\/687\/id-like-to-display-field-lines-for-a-point- charge-in-3-dimensions}{https:\/\/mathematica.stackexchange.com\/questions\/687\/ \\\\ id-like-to-display-field-lines-for-a-point- charge-in-3-dimensions}\n\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{fieldSolve}\\text{::}\\text{usage}=}\\\\\n\\pmb{\\text{$\\texttt{\"}$fieldSolve[f,x,x0,$\\backslash $!$\\backslash $($\\backslash $*SubscriptBox[$\\backslash $(t$\\backslash $), $\\backslash $(max$\\backslash\n$)]$\\backslash $)] symbolically takes a vector field }}\\\\\n\\pmb{\\text{f with respect to the vector variable x, and then finds a vector curve r[t] starting at }}\\\\\n\\pmb{\\text{the point x0 satisfying the equation dr\/dt=$\\alpha $ f[r[t]] for t=0...$\\backslash $!$\\backslash $($\\backslash $*SubscriptBox[$\\backslash\n$(t$\\backslash $), }}\\\\\n\\pmb{\\text{$\\backslash $(max$\\backslash $)]$\\backslash $). Here $\\alpha $=1\/$|$f[r[t]]$|$ for normalization. To get verbose output add debug=True\n}}\\\\\n\\pmb{\\text{to the parameter list.$\\texttt{\"}$};}\\\\\n\\pmb{}\\\\\n\\pmb{\\text{fieldSolve}[\\text{field$\\_$},\\text{varlist$\\_$},\\text{xi0$\\_$},\\text{tmax$\\_$},\\text{debug$\\_$}: \\text{False}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{xiVec},\\text{equationSet},t\\},}\\\\\n\\pmb{\\text{If}[\\text{Length}[\\text{varlist}]\\neq \\text{Length}[\\text{xi0}],\\text{Print}[\\text{$\\texttt{\"}$Number of variables must equal number of\ninitial \n}}\\\\\n\\pmb{\\text{  conditions $\\backslash $nUSAGE:$\\backslash $n$\\texttt{\"}$}<>\\text{fieldSolve}\\text{::}\\text{usage}];\\text{Abort}[]];}\\\\\n\\pmb{\\text{xiVec}=\\text{Through}[\\text{varlist}[t]];}\\\\\n\\pmb{\\text{equationSet}=\\text{Join}[\\text{Thread}[\\text{Map}[D[\\#,t]\\&,\\text{xiVec}]==}\\\\\n\\pmb{\\text{Normalize}[\\text{field}\\text{\/.}\\text{Thread}[\\text{varlist}\\to \\text{xiVec}]]],\\text{Thread}[(\\text{xiVec}\\text{\/.}t\\to 0)==\\text{xi0}]];}\\\\\n\\pmb{\\text{If}[\\text{debug},}\\\\\n\\pmb{\\text{Print}[\\text{Row}[\\{\\text{{``}Numerically solving the system of equations$\\backslash $n$\\backslash $n{''}},}\\\\\n\\pmb{\\text{TraditionalForm}[(\\text{Simplify}[\\text{equationSet}]\\text{\/.}t\\to \\text{t})\\text{\/\/}\\text{TableForm}]\\}]]];}\\\\\n\\pmb{\\text{Map}[\\text{Head},\\text{First}[\\text{xiVec}\\text{\/.}\\text{Quiet}[\\text{NDSolve}[\\text{equationSet},\\text{xiVec},\\{t,0,\\text{tmax}\\}]]],2]]}\\\\\n\\pmb{}\\\\\n\\pmb{\\text{fieldLinePlot}[\\text{field$\\_$},\\text{varList$\\_$},\\text{seedList$\\_$},\\text{opts}:\\text{OptionsPattern}[]]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{sols},\\text{localVars},\\text{var},\\text{localField},\\text{plotOptions},\\text{tubeFunction},\\text{tubePlotStyle},}\\\\\n\\pmb{\\text{postProcess}=\\{\\}\\},\\text{plotOptions}=\\text{FilterRules}[\\{\\text{opts}\\},\\text{Options}[\\text{ParametricPlot3D}]];}\\\\\n\\pmb{\\text{tubeFunction}=\\text{OptionValue}[\\text{{``}TubeFunction{''}}];}\\\\\n\\pmb{\\text{If}[\\text{tubeFunction}\\text{=!=}\\text{None},\\text{tubePlotStyle}=\\text{Cases}[\\text{OptionValue}[\\text{PlotStyle}],\\text{Except}[\\text{$\\_$Tube}]];}\\\\\n\\pmb{\\text{plotOptions}=\\text{FilterRules}[\\text{plotOptions},\\text{Except}[\\{\\text{PlotStyle},\\text{ColorFunction},}\\\\\n\\pmb{\\text{ColorFunctionScaling}\\}]];}\\\\\n\\pmb{\\text{postProcess}=\\text{Line}[\\text{x$\\_$}]:\\to \\text{Join}[\\text{tubePlotStyle},\\{\\text{CapForm}[\\text{{``}Butt{''}}],}\\\\\n\\pmb{\\text{Tube}[x,\\text{tubeFunction}\\text{@@@}x]\\}]];}\\\\\n\\pmb{\\text{If}[\\text{Length}[\\text{seedList}[[1,1]]]\\neq \\text{Length}[\\text{varList}],}\\\\\n\\pmb{\\text{Print}[\\text{$\\texttt{\"}$Number of variables must equal number of initial conditions\n}}\\\\\n\\pmb{\\text{   $\\backslash $nUSAGE:$\\backslash $n$\\texttt{\"}$}<>\\text{fieldLinePlot}\\text{::}\\text{usage}];\\text{Abort}[]];}\\\\\n\\pmb{\\text{localVars}=\\text{Array}[\\text{var},\\text{Length}[\\text{varList}]];}\\\\\n\\pmb{\\text{localField}=\\text{ReleaseHold}[\\text{Hold}[\\text{field}]\\text{\/.}\\text{Thread}[\\text{Map}[\\text{HoldPattern},}\\\\\n\\pmb{\\text{Unevaluated}[\\text{varList}]]\\to \\text{localVars}]];}\\\\\n\\pmb{\\text{Show}[}\\\\\n\\pmb{\\text{ParallelTable}[}\\\\\n\\pmb{\\text{ParametricPlot3D}[\\text{Evaluate}[\\text{Through}[\\#[t]]],\\{t,\\#[[1,1,1,1]],\\#[[1,1,1,2]]\\},}\\\\\n\\pmb{\\text{Evaluate}@\\text{Apply}[\\text{Sequence},\\text{plotOptions}]]\\&[}\\\\\n\\pmb{\\text{fieldSolve}[\\text{localField},\\text{localVars},\\text{seedList}[[i,1]],\\text{seedList}[[i,2]]]]\\text{\/.}\\text{postProcess},}\\\\\n\\pmb{\\{i,\\text{Length}[\\text{seedList}]\\}]]];}\\\\\n\\pmb{}\\\\\n\\pmb{\\text{Options}[\\text{fieldLinePlot}]=\\text{Append}[\\text{Options}[\\text{ParametricPlot3D}],\\text{{``}TubeFunction{''}}\\to \\text{None}];}\\\\\n\\pmb{}\\\\\n\\pmb{\\text{SyntaxInformation}[\\text{fieldLinePlot}]=\\{\\text{{``}LocalVariables{''}}\\to \\{\\text{{``}Solve{''}},\\{2,2\\}\\},}\\\\\n\\pmb{\\text{{``}ArgumentsPattern{''}}\\to \\{\\_,\\_,\\_,\\text{OptionsPattern}[]\\}\\};}\\\\\n\\pmb{}\\\\\n\\pmb{\\text{SetAttributes}[\\text{fieldSolve},\\text{HoldAll}];}\\)\n\\end{doublespace}\n\n\\vspace{12pt}\n\\section{Rotation of indices}\n\\vspace{2pt}\n\n\\subsection*{j=0 case}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{rules0}=\\{}\\\\\n\\pmb{Y[\\text{n$\\_$}]:\\to  0\\text{\/;}\\text{Not}@(n\\text{==}0),}\\\\\n\\pmb{\\delta [\\text{a$\\_$}]\\cdot (\\text{b$\\_$} \\text{c$\\_$}):\\to  b \\delta [a]\\cdot c\\text{\/;}\\text{NumericQ}[b], \\text{(* linearity on numbers *)}}\\\\\n\\pmb{\\delta [\\text{a$\\_$}]\\cdot (\\text{b$\\_$} + \\text{c$\\_$}):\\to  \\delta [a]\\cdot b +\\delta [a]\\cdot c,\\text{(* linearity on the sum *)}}\\\\\n\\pmb{\\delta [\\text{a$\\_$}]\\cdot \\text{b$\\_$}:\\to  0\\text{\/;} \\text{NumericQ}[b],\\text{(* 0 on numbers *)}}\\\\\n\\pmb{\\delta [\\text{a$\\_$}]\\cdot e[\\text{b$\\_$}]:\\to 2\\text{Sum}[\\text{LeviCivitaTensor}[3][[a,b,c]]e[c],\\{c,3\\}],\\text{(* action on the forms *)}}\\\\\n\\pmb{\\delta [\\text{a$\\_$}]\\cdot (\\text{b$\\_$} \\text{c$\\_$}):\\to  (\\delta [a]\\cdot b) c+b(\\delta [a]\\cdot c),\\text{(* product rule *)}}\\\\\n\\pmb{\\delta [1]\\cdot Y[\\text{n$\\_$}]:\\to  Y[n+1]+Y[n-1],\\text{(*} \\text{action} \\text{on} \\text{harmonics}, \\text{calculated} \\text{by} \\text{hand}\n\\text{*)}}\\\\\n\\pmb{\\delta [2]\\cdot Y[\\text{n$\\_$}]:\\to  I(Y[n-1]-Y[n+1]),}\\\\\n\\pmb{\\delta [3]\\cdot Y[\\text{n$\\_$}]:\\to  -2I n Y[n]}\\\\\n\\pmb{\\};}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{Z_+[\\text{n$\\_$}]\\text{:=}\\sqrt{(-n)(-n+1)\/2}Y[n+1]; }\\\\\n\\pmb{Z_3[\\text{n$\\_$}]\\text{:=}\\sqrt{(1)^2-n^2}Y[n];}\\\\\n\\pmb{Z_-[\\text{n$\\_$}]\\text{:=}-\\sqrt{(n)(n+1)\/2}Y[n-1];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{A_-=\\text{Simplify}@\\text{Sum}\\left[Z_-[n]\\Lambda [n],\\{n,-1,1\\}\\right]\\text{\/\/.}\\text{rules0}; \\text{(* Combining solutions *)}}\\\\\n\\pmb{A_+=\\text{Simplify}@\\text{Sum}\\left[Z_+[n]\\Lambda [n],\\{n,-1,1\\}\\right]\\text{\/\/.}\\text{rules0};}\\\\\n\\pmb{A[3]=\\text{Simplify}@\\text{Sum}\\left[Z_3[n]\\Lambda [n],\\{n,-1,1\\}\\right]\\text{\/\/.}\\text{rules0};}\\\\\n\\pmb{A[1]=\\frac{\\left(A_++A_-\\right)}{\\sqrt{2}};}\\\\\n\\pmb{A[2]=\\frac{I*\\left(-A_++A_-\\right)}{\\sqrt{2}};}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{Acomp}[\\text{a$\\_$},\\text{n$\\_$}]\\text{:=}\\partial _{\\Lambda [n]}A[a]\\text{(*} \\text{Defining} A_a{}^{\\text{mn}} \\text{*)}}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{variation}[\\text{arot$\\_$}]\\text{:=}\\text{(*} \\text{Fully} \\text{opening} D_a\\left(\\Lambda _{\\text{mn}}A_a{}^{\\text{mn}}e^a\\right)\n\\text{*)}}\\\\\n\\pmb{\\text{Sum}[\\text{d$\\Lambda $}[\\text{arot},n]\\text{Acomp}[a,n]e[a]+\\Lambda [n](\\delta [\\text{arot}]\\cdot \\text{Acomp}[a,n])e[a]}\\\\\n\\pmb{+\\Lambda [n]\\text{Acomp}[a,n](\\delta [\\text{arot}]\\cdot e[a]),\\{n,-1,1\\},\\{a,3\\}]\\text{\/\/.}\\text{rules0}; }\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{eqnsd$\\Lambda $}[\\text{arot$\\_$},\\text{bform$\\_$}]\\text{:=}\\text{Simplify}@D[D[\\text{variation}[\\text{arot}],e[\\text{bform}],Y[0]]];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{d$\\Lambda $list}=\\text{Flatten}@\\text{Table}[\\text{d$\\Lambda $}[a,n],\\{a,3\\},\\{n,-1,1\\}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{alleqnsd$\\Lambda $}=\\text{Flatten}@\\text{Table}[\\text{eqnsd$\\Lambda $}[a,b]==0,\\{a,3\\},\\{b,3\\}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{Solve}[\\text{alleqnsd$\\Lambda $},\\text{d$\\Lambda $list}] }\\)\n\\end{doublespace}\n\n\\subsection*{j=1\/2 case}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{rules}=\\{}\\\\\n\\pmb{Y[\\text{m$\\_$},\\text{n$\\_$}]:\\to  0\\text{\/;}\\text{Not}@((m\\text{==}0.5\\lor m\\text{==}-0.5)\\land (n\\text{==}0.5\\lor n\\text{==}-0.5)),}\\\\\n\\pmb{\\delta [\\text{a$\\_$}]\\cdot (\\text{b$\\_\\_$} \\text{c$\\_$}):\\to  b \\delta [a]\\cdot c\\text{\/;}\\text{NumericQ}[b], \\text{(* linearity on numbers\n*)}}\\\\\n\\pmb{\\delta [\\text{a$\\_$}]\\cdot (\\text{b$\\_\\_$} + \\text{c$\\_\\_$}):\\to  \\delta [a]\\cdot b +\\delta [a]\\cdot c,\\text{(* linearity on the sum *)}}\\\\\n\\pmb{\\delta [\\text{a$\\_$}]\\cdot \\text{b$\\_$}:\\to  0\\text{\/;} \\text{NumericQ}[b],\\text{(* 0 on numbers *)}}\\\\\n\\pmb{\\delta [\\text{a$\\_$}]\\cdot e[\\text{b$\\_$}]:\\to 2\\text{Sum}[\\text{LeviCivitaTensor}[3][[a,b,c]]e[c],\\{c,3\\}],\\text{(* action on the forms *)}}\\\\\n\\pmb{\\delta [\\text{a$\\_$}]\\cdot (\\text{b$\\_\\_$} \\text{c$\\_\\_$}):\\to  (\\delta [a]\\cdot b) c+b(\\delta [a]\\cdot c),\\text{(* product rule *)}}\\\\\n\\pmb{\\delta [1]\\cdot Y[\\text{m$\\_$},\\text{n$\\_$}]:\\to  -I(Y[m+1,n]+Y[m,n+1]+Y[m-1,n]+Y[m,n-1]),}\\\\\n\\pmb{\\text{(*} \\text{action} \\text{on} \\text{harmonics}, \\text{calculated} \\text{by} \\text{hand} \\text{*)}}\\\\\n\\pmb{\\delta [2]\\cdot Y[\\text{m$\\_$},\\text{n$\\_$}]:\\to  Y[m-1,n]+Y[m,n-1]-Y[m+1,n]-Y[m,n+1],}\\\\\n\\pmb{\\delta [3]\\cdot Y[\\text{m$\\_$},\\text{n$\\_$}]:\\to  -2I(m+n)Y[m,n]}\\\\\n\\pmb{\\};}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{Z_+[\\text{m$\\_$},\\text{n$\\_$}]\\text{:=}\\sqrt{(1\/2-n)(1\/2-n+1)\/2}Y[m,n+1]; }\\\\\n\\pmb{Z_3[\\text{m$\\_$},\\text{n$\\_$}]\\text{:=}\\sqrt{(1\/2+1)^2-n^2}Y[m,n];}\\\\\n\\pmb{Z_-[\\text{m$\\_$},\\text{n$\\_$}]\\text{:=}-\\sqrt{(1\/2+n)(1\/2+n+1)\/2}Y[m,n-1];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{A_-=\\text{Simplify}@\\text{Sum}\\left[Z_-[m,n]\\Lambda [m,n],\\{m,-1\/2,1\/2\\},\\{n,-3\/2,3\/2\\}\\right]\\text{\/\/.}\\text{rules}; }\\\\\n\\pmb{\\text{(* Combining solutions *)}}\\\\\n\\pmb{A_+=\\text{Simplify}@\\text{Sum}\\left[Z_+[m,n]\\Lambda [m,n],\\{m,-1\/2,1\/2\\},\\{n,-3\/2,3\/2\\}\\right]\\text{\/\/.}\\text{rules};}\\\\\n\\pmb{A[3]=\\text{Simplify}@\\text{Sum}\\left[Z_3[m,n]\\Lambda [m,n],\\{m,-1\/2,1\/2\\},\\{n,-3\/2,3\/2\\}\\right]\\text{\/\/.}\\text{rules};}\\\\\n\\pmb{A[1]=\\frac{\\left(A_++A_-\\right)}{\\sqrt{2}};}\\\\\n\\pmb{A[2]=\\frac{I*\\left(-A_++A_-\\right)}{\\sqrt{2}};}\\\\\n\\pmb{\\text{Acomp}[\\text{a$\\_$},\\text{m$\\_$},\\text{n$\\_$}]\\text{:=}\\partial _{\\Lambda [m,n]}A[a]\\text{(*} \\text{Defining} A_a{}^{\\text{mn}} \\text{*)}}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{variation}[\\text{arot$\\_$}]\\text{:=}\\text{(*} \\text{Fully} \\text{opening} D_a\\left(\\Lambda _{\\text{mn}}A_a{}^{\\text{mn}}e^a\\right)\n\\text{*)}}\\\\\n\\pmb{\\text{Sum}[\\text{d$\\Lambda $}[\\text{arot},m,n]\\text{Acomp}[b,m,n]e[b]+\\Lambda [m,n](\\delta [\\text{arot}]\\cdot \\text{Acomp}[b,m,n])e[b]+}\\\\\n\\pmb{\\Lambda [m,n]\\text{Acomp}[b,m,n](\\delta [\\text{arot}]\\cdot e[b]),\\{m,-1\/2,1\/2\\},}\\\\\n\\pmb{\\text{     }\\{n,-3\/2,3\/2\\},\\{b,3\\}]\\text{\/\/.}\\text{rules}; }\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{eqnsd$\\Lambda $}[\\text{arot$\\_$},\\text{bform$\\_$},\\text{m$\\_$},\\text{n$\\_$}]\\text{:=}\\text{Simplify}@D[D[\\text{variation}[\\text{arot}],e[\\text{bform}],Y[m,n]]];}\\\\\n\\pmb{\\text{(*} \\text{Taking} \\text{the} \\text{term} \\text{of} D_a \\text{with} Y_{\\text{mn}} \\text{and} e^b \\text{*)}}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{d$\\Lambda $list}=\\text{Flatten}@\\text{Table}[\\text{d$\\Lambda $}[a,m,n],\\{a,3\\},\\{m,-1\/2,1\/2\\},\\{n,-3\/2,3\/2\\}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{alleqnsd$\\Lambda $}=\\text{Flatten}@\\text{Table}[\\text{eqnsd$\\Lambda $}[a,b,m,n]==0,\\{a,3\\},\\{b,3\\},\\{m,-1\/2,1\/2\\},}\\\\\n\\pmb{\\{n,-1\/2,1\/2\\}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sol}=\\text{Solve}[\\text{alleqnsd$\\Lambda $},\\text{d$\\Lambda $list}][[1]]}\\)\n\\end{doublespace}\n\n\\subsection*{j=1 case}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{rules}=\\{}\\\\\n\\pmb{Y[\\text{m$\\_$},\\text{n$\\_$}]:\\to  0\\text{\/;}}\\\\\n\\pmb{\\text{Not}@((m==-1\\lor m==0\\lor m==1)\\land (n==-1\\lor n==0\\lor n==1)),}\\\\\n\\pmb{\\delta [\\text{a$\\_$}]\\cdot (\\text{b$\\_\\_$} \\text{c$\\_$}):\\to  b \\delta [a]\\cdot c\\text{\/;}\\text{NumericQ}[b], \\text{(* linearity on numbers\n*)}}\\\\\n\\pmb{\\delta [\\text{a$\\_$}]\\cdot (\\text{b$\\_\\_$} + \\text{c$\\_\\_$}):\\to  \\delta [a]\\cdot b +\\delta [a]\\cdot c,\\text{(* linearity on the sum *)}}\\\\\n\\pmb{\\delta [\\text{a$\\_$}]\\cdot \\text{b$\\_$}:\\to  0\\text{\/;} \\text{NumericQ}[b],\\text{(* 0 on numbers *)}}\\\\\n\\pmb{\\delta [\\text{a$\\_$}]\\cdot e[\\text{b$\\_$}]:\\to 2\\text{Sum}[\\text{LeviCivitaTensor}[3][[a,b,c]]e[c],\\{c,3\\}],\\text{(* action on the forms *)}}\\\\\n\\pmb{\\delta [\\text{a$\\_$}]\\cdot (\\text{b$\\_\\_$} \\text{c$\\_\\_$}):\\to  (\\delta [a]\\cdot b) c+b(\\delta [a]\\cdot c),\\text{(* product rule *)}}\\\\\n\\pmb{\\delta [1]\\cdot Y[\\text{m$\\_$},\\text{n$\\_$}]:\\to  -I*\\sqrt{2}(Y[m+1,n]+Y[m,n+1]+Y[m-1,n]+Y[m,n-1]),}\\\\\n\\pmb{\\text{(*} \\text{action} \\text{on} \\text{harmonics}, \\text{calculated} \\text{by} \\text{hand} \\text{*)}}\\\\\n\\pmb{\\delta [2]\\cdot Y[\\text{m$\\_$},\\text{n$\\_$}]:\\to  \\sqrt{2}(Y[m-1,n]+Y[m,n-1]-Y[m+1,n]-Y[m,n+1]),}\\\\\n\\pmb{\\delta [3]\\cdot Y[\\text{m$\\_$},\\text{n$\\_$}]:\\to  -2I(m+n)Y[m,n]}\\\\\n\\pmb{\\};}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{Z_+[\\text{m$\\_$},\\text{n$\\_$}]\\text{:=}\\sqrt{(1-n)(1-n+1)\/2}Y[m,n+1]; }\\\\\n\\pmb{Z_3[\\text{m$\\_$},\\text{n$\\_$}]\\text{:=}\\sqrt{(1+1)^2-n^2}Y[m,n];}\\\\\n\\pmb{Z_-[\\text{m$\\_$},\\text{n$\\_$}]\\text{:=}-\\sqrt{(1+n)(1+n+1)\/2}Y[m,n-1];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{A_-=\\text{Simplify}@\\text{Sum}\\left[Z_-[m,n]\\Lambda [m,n],\\{m,-1,1\\},\\{n,-2,2\\}\\right]\\text{\/\/.}\\text{rules};}\\\\\n\\pmb{\\text{(* Combining solutions *)}}\\\\\n\\pmb{A_+=\\text{Simplify}@\\text{Sum}\\left[Z_+[m,n]\\Lambda [m,n],\\{m,-1,1\\},\\{n,-2,2\\}\\right]\\text{\/\/.}\\text{rules};}\\\\\n\\pmb{A[3]=\\text{Simplify}@\\text{Sum}\\left[Z_3[m,n]\\Lambda [m,n],\\{m,-1,1\\},\\{n,-2,2\\}\\right]\\text{\/\/.}\\text{rules};}\\\\\n\\pmb{A[1]=\\frac{\\left(A_++A_-\\right)}{\\sqrt{2}};}\\\\\n\\pmb{A[2]=\\frac{I*\\left(-A_++A_-\\right)}{\\sqrt{2}};}\\\\\n\\pmb{\\text{Acomp}[\\text{a$\\_$},\\text{m$\\_$},\\text{n$\\_$}]\\text{:=}\\partial _{\\Lambda [m,n]}A[a]\\text{(*} \\text{Defining} A_a{}^{\\text{mn}} \\text{*)}}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{variation}[\\text{arot$\\_$}]\\text{:=}\\text{(*} \\text{Fully} \\text{opening} D_a\\left(\\Lambda _{\\text{mn}}A_a{}^{\\text{mn}}e^a\\right)\n\\text{*)}}\\\\\n\\pmb{\\text{Sum}[\\text{d$\\Lambda $}[\\text{arot},m,n]\\text{Acomp}[b,m,n]e[b]+\\Lambda [m,n](\\delta [\\text{arot}]\\cdot \\text{Acomp}[b,m,n])e[b]+}\\\\\n\\pmb{\\Lambda [m,n]\\text{Acomp}[b,m,n](\\delta [\\text{arot}]\\cdot e[b]),\\{m,-1,1\\},\\{n,-2,2\\},\\{b,3\\}]\\text{\/\/.}\\text{rules}; }\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{eqnsd$\\Lambda $}[\\text{arot$\\_$},\\text{bform$\\_$},\\text{m$\\_$},\\text{n$\\_$}]\\text{:=}\\text{Simplify}@D[D[\\text{variation}[\\text{arot}],e[\\text{bform}],Y[m,n]]];}\\\\\n\\pmb{\\text{(*} \\text{Taking} \\text{the} \\text{term} \\text{of} D_a \\text{with} Y_{\\text{mn}} \\text{and} e^b \\text{*)}}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{d$\\Lambda $list}=\\text{Flatten}@\\text{Table}[\\text{d$\\Lambda $}[a,m,n],\\{a,3\\},\\{m,-1,1\\},\\{n,-2,2\\}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{alleqnsd$\\Lambda $}=\\text{Flatten}@\\text{Table}[\\text{eqnsd$\\Lambda $}[a,b,m,n]==0,\\{a,3\\},\\{b,3\\},\\{m,-1,1\\},\\{n,-1,1\\}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sol}=\\text{Solve}[\\text{alleqnsd$\\Lambda $},\\text{d$\\Lambda $list}][[1]]}\\)\n\\end{doublespace}\n\n\n\\vspace{12pt}\n\\section{Trajectories}\n\\vspace{2pt}\n\n\\subsection*{Single particle}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{RelSoln}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_,\\text{pos0$\\_$}?\\text{ListQ},\\text{vel0$\\_$}?\\text{ListQ},\\kappa\n\\_]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{r,v,\\Gamma ,\\text{initcond},\\text{eom},\\text{vars}\\},}\\\\\n\\pmb{r[\\text{t$\\_$}]=\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\};}\\\\\n\\pmb{v[\\text{t$\\_$}]=\\{\\text{v1}[t],\\text{v2}[t],\\text{v3}[t]\\};}\\\\\n\\pmb{\\text{initcond}=\\{r[0]==\\text{pos0},v[0]\\text{==}\\text{vel0}\\};}\\\\\n\\pmb{\\Gamma [\\text{t$\\_$}]\\text{:=}\\frac{1}{\\sqrt{1-\\text{v1}[t]^2-\\text{v2}[t]^2-\\text{v3}[t]^2}};}\\\\\n\\pmb{\\text{eom}=\\left\\{\\partial _tr[t]==v[t],\\partial _t(\\Gamma [t]v[t])==\\right.}\\\\\n\\pmb{\\kappa  \\text{Simplify}[((\\text{Ef}[\\text{config}[[1]],\\text{config}[[2]],\\text{config}[[3]],\\ell ]}\\\\\n\\pmb{+v[t]\\times \\text{Bf}[\\text{config}[[1]],\\text{config}[[2]],\\text{config}[[3]],\\ell ])}\\\\\n\\pmb{\\text{\/.}\\{x\\to \\text{r1}[t],y\\to \\text{r2}[t],z\\to \\text{r3}[t]\\})]\\};}\\\\\n\\pmb{ \\text{vars}=\\text{Flatten}@\\{r[t],v[t]\\};}\\\\\n\\pmb{\\text{    }\\text{NDSolve}[\\text{Flatten}@\\text{Join}[\\text{eom},\\{\\text{initcond}\\}],\\text{vars},\\{t,-\\text{tf},\\text{tf}\\},}\\\\\n\\pmb{\\text{Method}\\text{-$>$}\\{\\text{{``}EquationSimplification{''}}\\text{-$>$}\\text{{``}Residual{''}}\\}]]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{Soln}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_,\\text{pos0$\\_$}?\\text{ListQ},\\text{vel0$\\_$}?\\text{ListQ},\\kappa\n\\_]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{r,v,\\text{initcond},\\text{eom},\\text{vars}\\},}\\\\\n\\pmb{r[\\text{t$\\_$}]=\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\};}\\\\\n\\pmb{v[\\text{t$\\_$}]=\\{\\text{v1}[t],\\text{v2}[t],\\text{v3}[t]\\};}\\\\\n\\pmb{\\text{initcond}=\\{r[0]==\\text{pos0},v[0]\\text{==}\\text{vel0}\\};}\\\\\n\\pmb{\\text{eom}=\\left\\{\\partial _tr[t]==v[t],\\partial _t(v[t])==\\right.}\\\\\n\\pmb{\\kappa  \\text{Simplify}[((\\text{Ef}[\\text{config}[[1]],\\text{config}[[2]],\\text{config}[[3]],\\ell ]+v[t]\\times }\\\\\n\\pmb{\\text{Bf}[\\text{config}[[1]],\\text{config}[[2]],\\text{config}[[3]],\\ell ])\\text{\/.}\\{x\\to \\text{r1}[t],y\\to \\text{r2}[t],z\\to \\text{r3}[t]\\})]\\};}\\\\\n\\pmb{ \\text{vars}=\\text{Flatten}@\\{r[t],v[t]\\};}\\\\\n\\pmb{\\text{    }\\text{NDSolve}[\\text{Flatten}@\\text{Join}[\\text{eom},\\{\\text{initcond}\\}],\\text{vars},\\{t,-\\text{tf},\\text{tf}\\},}\\\\\n\\pmb{\\text{Method}\\text{-$>$}\\{\\text{{``}EquationSimplification{''}}\\text{-$>$}\\text{{``}Residual{''}}\\}]]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{RelPlots}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\text{pos0$\\_$}?\\text{ListQ},\\text{vel0$\\_$}?\\text{ListQ},\\kappa\n\\_:1]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{TempSoln}=\\text{RelSoln}[\\text{tf},\\text{config},\\ell ,\\text{pos0},\\text{vel0},\\kappa ]\\},}\\\\\n\\pmb{\\{\\text{Plot}[\\text{r1}[t]\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{r2}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{r3}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{v1}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{v2}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{v3}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],}\\\\\n\\pmb{\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]\\text{\/.}\\text{Line}\\text{-$>$}\\text{Tube},}\\\\\n\\pmb{\\left.\\left.\\text{Plot}\\left[\\sqrt{\\text{v1}[t]^2+\\text{v2}[t]^2+\\text{v2}[t]^2}\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to\n\\text{All}\\right]\\right\\}\\right]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{Plots}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\text{pos0$\\_$}?\\text{ListQ},\\text{vel0$\\_$}?\\text{ListQ},\\kappa\n\\_:1]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{TempSoln}=\\text{Soln}[\\text{tf},\\text{config},\\ell ,\\text{pos0},\\text{vel0},\\kappa ]\\},}\\\\\n\\pmb{\\{\\text{Plot}[\\text{r1}[t]\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{r2}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{r3}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{v1}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{v2}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{v3}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],}\\\\\n\\pmb{\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},}\\\\\n\\pmb{\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}],}\\\\\n\\pmb{\\left.\\left.\\text{Plot}\\left[\\sqrt{\\text{v1}[t]^2+\\text{v2}[t]^2+\\text{v2}[t]^2}\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to\n\\text{All}\\right]\\right\\}\\right]}\\)\n\\end{doublespace}\n\nHopf--Ran\\~{a}da\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{RelSolnHR}[\\text{tf$\\_$},\\ell \\_,\\text{pos0$\\_$}?\\text{ListQ},\\text{vel0$\\_$}?\\text{ListQ},\\kappa \\_]\\text{:=}\\text{Module}[\\{r,v,\\Gamma\n,\\text{initcond},\\text{eom},\\text{vars}\\},}\\\\\n\\pmb{r[\\text{t$\\_$}]=\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\};}\\\\\n\\pmb{v[\\text{t$\\_$}]=\\{\\text{v1}[t],\\text{v2}[t],\\text{v3}[t]\\};}\\\\\n\\pmb{\\text{initcond}=\\{r[0]==\\text{pos0},v[0]\\text{==}\\text{vel0}\\};}\\\\\n\\pmb{\\Gamma [\\text{t$\\_$}]\\text{:=}\\frac{1}{\\sqrt{1-\\text{v1}[t]^2-\\text{v2}[t]^2-\\text{v3}[t]^2}};}\\\\\n\\pmb{\\text{eom}=\\left\\{\\partial _tr[t]==v[t],\\partial _t(\\Gamma [t]v[t])==\\kappa  \\text{Simplify}[((\\text{EfHR}[\\ell ]+v[t]\\times \\text{BfHR}[\\ell\n])\\right.}\\\\\n\\pmb{\\text{\/.}\\{x\\to \\text{r1}[t],y\\to \\text{r2}[t],z\\to \\text{r3}[t]\\})]\\};}\\\\\n\\pmb{ \\text{vars}=\\text{Flatten}@\\{r[t],v[t]\\};}\\\\\n\\pmb{\\text{    }\\text{NDSolve}[\\text{Flatten}@\\text{Join}[\\text{eom},\\{\\text{initcond}\\}],\\text{vars},\\{t,-\\text{tf},\\text{tf}\\},}\\\\\n\\pmb{\\text{Method}\\text{-$>$}\\{\\text{{``}EquationSimplification{''}}\\text{-$>$}\\text{{``}Residual{''}}\\}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{SolnHR}[\\text{tf$\\_$},\\ell \\_,\\text{pos0$\\_$}?\\text{ListQ},\\text{vel0$\\_$}?\\text{ListQ},\\kappa \\_]\\text{:=}\\text{Module}[\\{r,v,\\text{initcond},\\text{eom},\\text{vars}\\},}\\\\\n\\pmb{r[\\text{t$\\_$}]=\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\};}\\\\\n\\pmb{v[\\text{t$\\_$}]=\\{\\text{v1}[t],\\text{v2}[t],\\text{v3}[t]\\};}\\\\\n\\pmb{\\text{initcond}=\\{r[0]==\\text{pos0},v[0]\\text{==}\\text{vel0}\\};}\\\\\n\\pmb{\\text{eom}=\\left\\{\\partial _tr[t]==v[t],\\partial _t(v[t])==\\kappa  \\text{Simplify}[((\\text{EfHR}[\\ell ]+v[t]\\times \\text{BfHR}[\\ell ])\\right.}\\\\\n\\pmb{\\text{\/.}\\{x\\to \\text{r1}[t],y\\to \\text{r2}[t],z\\to \\text{r3}[t]\\})]\\};}\\\\\n\\pmb{ \\text{vars}=\\text{Flatten}@\\{r[t],v[t]\\};}\\\\\n\\pmb{\\text{    }\\text{NDSolve}[\\text{Flatten}@\\text{Join}[\\text{eom},\\{\\text{initcond}\\}],\\text{vars},\\{t,-\\text{tf},\\text{tf}\\},}\\\\\n\\pmb{\\text{Method}\\text{-$>$}\\{\\text{{``}EquationSimplification{''}}\\text{-$>$}\\text{{``}Residual{''}}\\}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{RelPlotsHR}[\\text{tf$\\_$},\\ell \\_:1,\\text{pos0$\\_$}?\\text{ListQ},\\text{vel0$\\_$}?\\text{ListQ},\\kappa \\_:1]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{TempSoln}=\\text{RelSolnHR}[\\text{tf},\\ell ,\\text{pos0},\\text{vel0},\\kappa ]\\},}\\\\\n\\pmb{\\{\\text{Plot}[\\text{r1}[t]\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{r2}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{r3}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{v1}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{v2}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{v3}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],}\\\\\n\\pmb{\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]\\text{\/.}\\text{Line}\\text{-$>$}\\text{Tube},}\\\\\n\\pmb{\\left.\\text{Plot}\\left[\\sqrt{\\text{v1}[t]^2+\\text{v2}[t]^2+\\text{v2}[t]^2}\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to\n\\text{All}\\right]\\right\\}}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsHR}[\\text{tf$\\_$},\\ell \\_:1,\\text{pos0$\\_$}?\\text{ListQ},\\text{vel0$\\_$}?\\text{ListQ},\\kappa \\_:1]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{TempSoln}=\\text{SolnHR}[\\text{tf},\\ell ,\\text{pos0},\\text{vel0},\\kappa ]\\},}\\\\\n\\pmb{\\{\\text{Plot}[\\text{r1}[t]\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{r2}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{r3}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{v1}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{v2}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{v3}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],}\\\\\n\\pmb{\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},}\\\\\n\\pmb{\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}],}\\\\\n\\pmb{\\left.\\text{Plot}\\left[\\sqrt{\\text{v1}[t]^2+\\text{v2}[t]^2+\\text{v2}[t]^2}\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to\n\\text{All}\\right]\\right\\}}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\subsection*{Multi-particle (line)}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{GenerateLine}[\\text{direc$\\_$}?\\text{ListQ},\\text{distance$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{list}=\\{\\{0,0,0\\}\\},\\text{elem}\\},}\\\\\n\\pmb{\\text{Do}[\\text{elem}=i \\text{distance} \\text{Normalize}[\\text{direc}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{list},\\text{elem}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{list},-\\text{elem}],\\{i,\\text{npts}\\}];}\\\\\n\\pmb{\\text{list}}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\nStarting at rest and positioned on the line\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosLine}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\kappa \\_,\\text{direc$\\_$}?\\text{ListQ},\\text{distance$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}=\\text{GenerateLine}[\\text{direc},\\text{distance},\\text{npts}],\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{RelSoln}[\\text{tf},\\text{config},\\ell ,\\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},}\\\\\n\\pmb{\\text{PlotRange}\\to \\text{All}]\\text{\/.}\\text{Line}\\text{-$>$}(\\text{Tube}[\\#,.005]\\&)],\\{i,\\text{Length}@\\text{listpos}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ListLinePlot3D}[\\text{listpos},\\text{PlotStyle}\\to \\text{RGBColor}[0,0.4,0.6]]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosLine2}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\kappa \\_,\\text{direc$\\_$}?\\text{ListQ},\\text{distance$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}=\\text{GenerateLine}[\\text{direc},\\text{distance},\\text{npts}],\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{Soln}[\\text{tf},\\text{config},\\ell ,\\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},}\\\\\n\\pmb{\\text{PlotRange}\\to \\text{All}]\\text{\/.}\\text{Line}\\text{-$>$}(\\text{Tube}[\\#,.005]\\&)],\\{i,\\text{Length}@\\text{listpos}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ListLinePlot3D}[\\text{listpos},\\text{PlotStyle}\\to \\text{RGBColor}[0,0.4,0.6]]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\nHopf--Ran\\~{a}da\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosLineHR}[\\text{tf$\\_$},\\ell \\_:1,\\kappa \\_,\\text{direc$\\_$}?\\text{ListQ},\\text{distance$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}=\\text{GenerateLine}[\\text{direc},\\text{distance},\\text{npts}],\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{RelSolnHR}[\\text{tf},\\ell ,\\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]}\\\\\n\\pmb{\\text{\/.}\\text{Line}\\text{-$>$}(\\text{Tube}[\\#,.005]\\&)],\\{i,\\text{Length}@\\text{listpos}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ListLinePlot3D}[\\text{listpos},\\text{PlotStyle}\\to \\text{RGBColor}[0,0.4,0.6]]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosLineHR2}[\\text{tf$\\_$},\\ell \\_:1,\\kappa \\_,\\text{direc$\\_$}?\\text{ListQ},\\text{distance$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}=\\text{GenerateLine}[\\text{direc},\\text{distance},\\text{npts}],\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{SolnHR}[\\text{tf},\\ell ,\\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]}\\\\\n\\pmb{\\text{\/.}\\text{Line}\\text{-$>$}(\\text{Tube}[\\#,.005]\\&)],\\{i,\\text{Length}@\\text{listpos}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ListLinePlot3D}[\\text{listpos},\\text{PlotStyle}\\to \\text{RGBColor}[0,0.4,0.6]]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\nStarting at origin with velocities along the line\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsVelLine}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\kappa \\_,\\text{direc$\\_$}?\\text{ListQ},\\text{distance$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listvel}=\\text{GenerateLine}[\\text{direc},\\text{distance},\\text{npts}],\\text{listplots}=\\{\\},\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{RelSoln}[\\text{tf},\\text{config},\\ell ,\\{0,0,0\\},\\text{listvel}[[i]],\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Hue}[i\/10],\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{Hue}[i\/10],\\text{Arrowheads}[0.03],}\\\\\n\\pmb{\\text{Arrow}[\\text{Tube}[\\{\\{0,0,0\\},\\text{listvel}[[i]]\\}]]\\}]],\\{i,\\text{Length}@\\text{listvel}\\}];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsVelLine2}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\kappa \\_,\\text{direc$\\_$}?\\text{ListQ},\\text{distance$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listvel}=\\text{GenerateLine}[\\text{direc},\\text{distance},\\text{npts}],\\text{listplots}=\\{\\},\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{Soln}[\\text{tf},\\text{config},\\ell ,\\{0,0,0\\},\\text{listvel}[[i]],\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Hue}[i\/10],\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{Hue}[i\/10],\\text{Arrowheads}[0.03],}\\\\\n\\pmb{\\text{Arrow}[\\text{Tube}[\\{\\{0,0,0\\},\\text{listvel}[[i]]\\}]]\\}]],\\{i,\\text{Length}@\\text{listvel}\\}];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\nHopf--Ran\\~{a}da\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsVelLineHR}[\\text{tf$\\_$},\\ell \\_:1,\\kappa \\_,\\text{direc$\\_$}?\\text{ListQ},\\text{distance$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listvel}=\\text{GenerateLine}[\\text{direc},\\text{distance},\\text{npts}],\\text{listplots}=\\{\\},\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{RelSolnHR}[\\text{tf},\\ell ,\\{0,0,0\\},\\text{listvel}[[i]],\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Hue}[i\/10],\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{Hue}[i\/10],\\text{Arrowheads}[0.03],}\\\\\n\\pmb{\\text{Arrow}[\\text{Tube}[\\{\\{0,0,0\\},\\text{listvel}[[i]]\\}]]\\}]],\\{i,\\text{Length}@\\text{listvel}\\}];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsVelLineHR2}[\\text{tf$\\_$},\\ell \\_:1,\\kappa \\_,\\text{direc$\\_$}?\\text{ListQ},\\text{distance$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listvel}=\\text{GenerateLine}[\\text{direc},\\text{distance},\\text{npts}],\\text{listplots}=\\{\\},\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{SolnHR}[\\text{tf},\\ell ,\\{0,0,0\\},\\text{listvel}[[i]],\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Hue}[i\/10],\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{Hue}[i\/10],\\text{Arrowheads}[0.03],}\\\\\n\\pmb{\\text{Arrow}[\\text{Tube}[\\{\\{0,0,0\\},\\text{listvel}[[i]]\\}]]\\}]],\\{i,\\text{Length}@\\text{listvel}\\}];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\subsection*{Multi-particle (circle)}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{CirclePoints3D}[\\text{direcN$\\_$},\\text{npts$\\_$}]\\text{:=} \\text{Module}[\\{\\text{list}=\\text{Table}[\\{0,0,0\\},\\text{npts}]\\},}\\\\\n\\pmb{\\text{Which}[\\text{direcN}\\text{==}1,\\text{Do}[\\text{list}[[n,2]]=\\text{CirclePoints}[\\text{npts}][[n,1]];}\\\\\n\\pmb{\\text{list}[[n,3]]=\\text{CirclePoints}[\\text{npts}][[n,2]],\\{n,1,\\text{npts}\\}],}\\\\\n\\pmb{\\text{direcN}\\text{==}2,\\text{Do}[\\text{list}[[n,1]]=\\text{CirclePoints}[\\text{npts}][[n,1]];}\\\\\n\\pmb{\\text{list}[[n,3]]=\\text{CirclePoints}[\\text{npts}][[n,2]],\\{n,1,\\text{npts}\\}],}\\\\\n\\pmb{\\text{direcN}\\text{==}3,\\text{Do}[\\text{list}[[n,2]]=\\text{CirclePoints}[\\text{npts}][[n,2]];}\\\\\n\\pmb{\\text{list}[[n,1]]=\\text{CirclePoints}[\\text{npts}][[n,1]],\\{n,1,\\text{npts}\\}]];}\\\\\n\\pmb{\\text{list}}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{circle3D}[\\text{centre$\\_$}:\\{0,0,0\\},\\text{radius$\\_$}:1,\\text{normal$\\_$}:\\{0,0,1\\},\\text{angle$\\_$}:\\{0,2 \\text{Pi}\\}]\\text{:=}}\\\\\n\\pmb{\\text{Composition}[\\text{Line},\\text{Map}[\\text{RotationTransform}[\\{\\{0,0,1\\},\\text{normal}\\},\\text{centre}],\\#]\\&,}\\\\\n\\pmb{\\text{Map}[\\text{Append}[\\#,\\text{Last}@\\text{centre}]\\&,\\#]\\&,\\text{Append}[\\text{DeleteDuplicates}[\\text{Most}@\\#],\\text{Last}@\\#]\\&,}\\\\\n\\pmb{\\text{Level}[\\#,\\{-2\\}]\\&,\\text{MeshPrimitives}[\\#,1]\\&,\\text{DiscretizeRegion},}\\\\\n\\pmb{\\text{If}}\\\\\n\\pmb{][\\text{First}@\\text{Differences}@\\text{angle}\\text{$>$=}2 \\text{Pi},\\text{Circle}[\\text{Most}@\\text{centre},\\text{radius}],}\\\\\n\\pmb{\\text{Circle}[\\text{Most}@\\text{centre},\\text{radius},\\text{angle}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\nStarting at rest and located on the circle\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosCircle}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\kappa \\_,\\text{direcN$\\_$},\\text{r$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}= \\text{CirclePoints3D}[\\text{direcN},\\text{npts}],}\\\\\n\\pmb{\\text{TempSoln},N=\\{\\{1,0,0\\},\\{0,1,0\\},\\{0,0,1\\}\\}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{RelSoln}[\\text{tf},\\text{config},\\ell ,r \\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]}\\\\\n\\pmb{\\text{\/.}\\text{Line}\\text{-$>$}\\text{Tube}],\\{i,\\text{Length}@\\text{listpos}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{RGBColor}[0,0.4,0.6],}\\\\\n\\pmb{\\text{Tube}[\\text{circle3D}[\\{0,0,0\\},r,N[[\\text{direcN}]]],0.001]\\}]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosCircle2}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\kappa \\_,\\text{direcN$\\_$},\\text{r$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}= \\text{CirclePoints3D}[\\text{direcN},\\text{npts}],}\\\\\n\\pmb{\\text{TempSoln},N=\\{\\{1,0,0\\},\\{0,1,0\\},\\{0,0,1\\}\\}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{Soln}[\\text{tf},\\text{config},\\ell ,r \\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},}\\\\\n\\pmb{\\text{PlotRange}\\to \\text{All}]\\text{\/.}\\text{Line}\\text{-$>$}\\text{Tube}],\\{i,\\text{Length}@\\text{listpos}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{RGBColor}[0,0.4,0.6],}\\\\\n\\pmb{\\text{Tube}[\\text{circle3D}[\\{0,0,0\\},r,N[[\\text{direcN}]]],0.001]\\}]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\nHopf--Ran\\~{a}da\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosCircleHR}[\\text{tf$\\_$},\\ell \\_:1,\\kappa \\_,\\text{direcN$\\_$},\\text{r$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}=\\text{CirclePoints3D}[\\text{direcN},\\text{npts}],}\\\\\n\\pmb{\\text{TempSoln},N=\\{\\{1,0,0\\},\\{0,1,0\\},\\{0,0,1\\}\\}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{RelSolnHR}[\\text{tf},\\ell ,r \\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]}\\\\\n\\pmb{\\text{\/.}\\text{Line}\\text{-$>$}\\text{Tube}],\\{i,\\text{Length}@\\text{listpos}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{RGBColor}[0,0.4,0.6],}\\\\\n\\pmb{\\text{Tube}[\\text{circle3D}[\\{0,0,0\\},r,N[[\\text{direcN}]]],0.002]\\}]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosCircleHR2}[\\text{tf$\\_$},\\ell \\_:1,\\kappa \\_,\\text{direcN$\\_$},\\text{r$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}=\\text{CirclePoints3D}[\\text{direcN},\\text{npts}],}\\\\\n\\pmb{\\text{TempSoln},N=\\{\\{1,0,0\\},\\{0,1,0\\},\\{0,0,1\\}\\}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{SolnHR}[\\text{tf},\\ell ,r \\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]}\\\\\n\\pmb{\\text{\/.}\\text{Line}\\text{-$>$}\\text{Tube}],\\{i,\\text{Length}@\\text{listpos}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{RGBColor}[0,0.4,0.6],}\\\\\n\\pmb{\\text{Tube}[\\text{circle3D}[\\{0,0,0\\},r,N[[\\text{direcN}]]],0.002]\\}]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\nStarting at origin with velocities radially outward\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsVelCircle}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\kappa \\_,\\text{direcN$\\_$},\\text{v$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listvel}=\\text{CirclePoints3D}[\\text{direcN},\\text{npts}],\\text{listplots}=\\{\\},\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{RelSoln}[\\text{tf},\\text{config},\\ell ,\\{0,0,0\\},v \\text{listvel}[[i]],\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Hue}[i\/10],\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{Hue}[i\/10],\\text{Arrowheads}[0.03],}\\\\\n\\pmb{\\text{Arrow}[\\text{Tube}[\\{\\{0,0,0\\},v \\text{listvel}[[i]]\\}]]\\}]],\\{i,\\text{Length}@\\text{listvel}\\}];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsVelCircle2}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\kappa \\_,\\text{direcN$\\_$},\\text{v$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listvel}=\\text{CirclePoints3D}[\\text{direcN},\\text{npts}],\\text{listplots}=\\{\\},\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{Soln}[\\text{tf},\\text{config},\\ell ,\\{0,0,0\\},v \\text{listvel}[[i]],\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Hue}[i\/10],\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{Hue}[i\/10],\\text{Arrowheads}[0.03],}\\\\\n\\pmb{\\text{Arrow}[\\text{Tube}[\\{\\{0,0,0\\},v \\text{listvel}[[i]]\\}]]\\}]],\\{i,\\text{Length}@\\text{listvel}\\}];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\nHopf--Ran\\~{a}da\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsVelCircleHR}[\\text{tf$\\_$},\\ell \\_:1,\\kappa \\_,\\text{direcN$\\_$},\\text{v$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listvel}=\\text{CirclePoints3D}[\\text{direcN},\\text{npts}],\\text{listplots}=\\{\\},\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{RelSolnHR}[\\text{tf},\\ell ,\\{0,0,0\\},v \\text{listvel}[[i]],\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Hue}[i\/10],\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{Hue}[i\/10],\\text{Arrowheads}[0.03],}\\\\\n\\pmb{\\text{Arrow}[\\text{Tube}[\\{\\{0,0,0\\},v \\text{listvel}[[i]]\\}]]\\}]],\\{i,\\text{Length}@\\text{listvel}\\}];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsVelCircleHR2}[\\text{tf$\\_$},\\ell \\_:1,\\kappa \\_,\\text{direcN$\\_$},\\text{v$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listvel}=\\text{CirclePoints3D}[\\text{direcN},\\text{npts}],\\text{listplots}=\\{\\},\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{SolnHR}[\\text{tf},\\ell ,\\{0,0,0\\},v \\text{listvel}[[i]],\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Hue}[i\/10],\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{Hue}[i\/10],\\text{Arrowheads}[0.03],}\\\\\n\\pmb{\\text{Arrow}[\\text{Tube}[\\{\\{0,0,0\\},v \\text{listvel}[[i]]\\}]]\\}]],\\{i,\\text{Length}@\\text{listvel}\\}];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\subsection*{Multi-particle (sphere)}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{RandomSpherePoints}[\\text{n$\\_$}]\\text{:=}\\text{Module}[\\{\\text{list}=\\{\\}\\},}\\\\\n\\pmb{\\text{Do}[\\text{AppendTo}[\\text{list},\\text{RandomPoint}[\\text{Sphere}[]]],\\{i,1,n\\}];\\text{list}]}\\)\n\\end{doublespace}\n\nStarting at rest and located on the circle\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosSphereSymmetric}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\kappa \\_,\\text{r$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}= \\text{SpherePoints}[\\text{npts}],\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{RelSoln}[\\text{tf},\\text{config},\\ell ,r \\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},}\\\\\n\\pmb{\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]],\\{i,\\text{Length}@\\text{listpos}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{RGBColor}[0,0.4,0.6],\\text{Sphere}[\\{0,0,0\\},r]\\}]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosSphereSymmetric2}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\kappa \\_,\\text{r$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}= \\text{SpherePoints}[\\text{npts}],\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{Soln}[\\text{tf},\\text{config},\\ell ,r \\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},}\\\\\n\\pmb{\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]],\\{i,\\text{Length}@\\text{listpos}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{RGBColor}[0,0.4,0.6],\\text{Sphere}[\\{0,0,0\\},r]\\}]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosSphereRandom}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\kappa \\_,\\text{r$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}= \\text{RandomSpherePoints}[\\text{npts}],\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{RelSoln}[\\text{tf},\\text{config},\\ell ,r \\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},}\\\\\n\\pmb{\\text{PlotRange}\\to \\text{All}]],\\{i,\\text{Length}@\\text{listpos}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{RGBColor}[0,0.4,0.6],\\text{Sphere}[\\{0,0,0\\},r]\\}]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosSphereRandom2}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\kappa \\_,\\text{r$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}= \\text{RandomSpherePoints}[\\text{npts}],\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{Soln}[\\text{tf},\\text{config},\\ell ,r \\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]],}\\\\\n\\pmb{\\{i,\\text{Length}@\\text{listpos}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{RGBColor}[0,0.4,0.6],\\text{Sphere}[\\{0,0,0\\},r]\\}]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosInhomogeneous}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\kappa \\_,\\text{range$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}= \\text{RandomSpherePoints}[\\text{npts}],\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{RelSoln}[\\text{tf},\\text{config},\\ell ,\\text{RandomReal}[\\{\\text{range}\/10,\\text{range}\\}]}\\\\\n\\pmb{ \\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];\\text{AppendTo}[\\text{listplots},}\\\\\n\\pmb{\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},}\\\\\n\\pmb{\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]],\\{i,\\text{npts}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{RGBColor}[0,0.4,0.6],\\text{Sphere}[\\{0,0,0\\},\\text{range}]\\}]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosInhomogeneous2}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\kappa \\_,\\text{range$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}= \\text{RandomSpherePoints}[\\text{npts}],\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{Soln}[\\text{tf},\\text{config},\\ell ,\\text{RandomReal}[\\{\\text{range}\/10,\\text{range}\\}] }\\\\\n\\pmb{\\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];\\text{AppendTo}[\\text{listplots},}\\\\\n\\pmb{\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},}\\\\\n\\pmb{\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]],\\{i,\\text{npts}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{RGBColor}[0,0.4,0.6],\\text{Sphere}[\\{0,0,0\\},\\text{range}]\\}]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\nHopf--Ran\\~{a}da\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosSphereSymmetricHR}[\\text{tf$\\_$},\\ell \\_:1,\\kappa \\_,\\text{r$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}=\\text{SpherePoints}[\\text{npts}],\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{RelSolnHR}[\\text{tf},\\ell ,r \\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},}\\\\\n\\pmb{\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]],\\{i,\\text{Length}@\\text{listpos}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{RGBColor}[1,0,1],\\text{Sphere}[\\{0,0,0\\},r]\\}]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosSphereSymmetricHR2}[\\text{tf$\\_$},\\ell \\_:1,\\kappa \\_,\\text{r$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}=\\text{SpherePoints}[\\text{npts}],\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{SolnHR}[\\text{tf},\\ell ,r \\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},}\\\\\n\\pmb{\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]],\\{i,\\text{Length}@\\text{listpos}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{RGBColor}[1,0,1],\\text{Sphere}[\\{0,0,0\\},r]\\}]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosSphereRandomHR}[\\text{tf$\\_$},\\ell \\_:1,\\kappa \\_,\\text{r$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}=\\text{RandomSpherePoints}[\\text{npts}],\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{RelSolnHR}[\\text{tf},\\ell ,r \\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},}\\\\\n\\pmb{\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]],\\{i,\\text{Length}@\\text{listpos}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{RGBColor}[1,0,1],\\text{Sphere}[\\{0,0,0\\},r]\\}]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosSphereRandomHR2}[\\text{tf$\\_$},\\ell \\_:1,\\kappa \\_,\\text{r$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}=\\text{RandomSpherePoints}[\\text{npts}],\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{SolnHR}[\\text{tf},\\ell ,r \\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},}\\\\\n\\pmb{\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]],\\{i,\\text{Length}@\\text{listpos}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{RGBColor}[1,0,1],\\text{Sphere}[\\{0,0,0\\},r]\\}]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosInhomogeneousHR}[\\text{tf$\\_$},\\ell \\_:1,\\kappa \\_,\\text{range$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}=\\text{RandomSpherePoints}[\\text{npts}],\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{RelSolnHR}[\\text{tf},\\ell , \\text{RandomReal}[\\{\\text{range}\/10,\\text{range}\\}]}\\\\\n\\pmb{\\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];\\text{AppendTo}[\\text{listplots},}\\\\\n\\pmb{\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},}\\\\\n\\pmb{\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]],\\{i,\\text{npts}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{RGBColor}[0,0.4,0.6],\\text{Sphere}[\\{0,0,0\\},\\text{range}]\\}]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosInhomogeneousHR2}[\\text{tf$\\_$},\\ell \\_:1,\\kappa \\_,\\text{range$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}=\\text{RandomSpherePoints}[\\text{npts}],\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{SolnHR}[\\text{tf},\\ell , \\text{RandomReal}[\\{\\text{range}\/10,\\text{range}\\}]}\\\\\n\\pmb{\\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];\\text{AppendTo}[\\text{listplots},}\\\\\n\\pmb{\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},}\\\\\n\\pmb{\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]],\\{i,\\text{npts}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{RGBColor}[0,0.4,0.6],\\text{Sphere}[\\{0,0,0\\},\\text{range}]\\}]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\nStarting at origin with velocities radially outward\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsVelSphereSymmetric}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\kappa \\_,\\text{v$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listvel}=\\text{SpherePoints}[\\text{npts}],\\text{listplots}=\\{\\},\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{RelSoln}[\\text{tf},\\text{config},\\ell ,\\{0,0,0\\},v \\text{listvel}[[i]],\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Hue}[i\/20],\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{Hue}[i\/20],\\text{Arrowheads}[0.03],}\\\\\n\\pmb{\\text{Arrow}[\\text{Tube}[\\{\\{0,0,0\\},v \\text{listvel}[[i]]\\}]]\\}]],\\{i,\\text{Length}@\\text{listvel}\\}];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsVelSphereRandom}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\kappa \\_,\\text{v$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listvel}=\\text{RandomSpherePoints}[\\text{npts}],\\text{listplots}=\\{\\},\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{RelSoln}[\\text{tf},\\text{config},\\ell ,\\{0,0,0\\},v \\text{listvel}[[i]],\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Hue}[i\/20],\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{Hue}[i\/20],\\text{Arrowheads}[0.03],}\\\\\n\\pmb{\\text{Arrow}[\\text{Tube}[\\{\\{0,0,0\\},v \\text{listvel}[[i]]\\}]]\\}]],\\{i,\\text{Length}@\\text{listvel}\\}];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsVelInhomogeneous}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\kappa \\_,\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listvel}=\\text{RandomSpherePoints}[\\text{npts}],\\text{listplots}=\\{\\},\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{RelSoln}[\\text{tf},\\text{config},\\ell ,\\{0,0,0\\},(i\/100) \\text{listvel}[[i]],\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Hue}[i\/20],\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{Hue}[i\/20],\\text{Arrowheads}[0.03],}\\\\\n\\pmb{\\text{Arrow}[\\text{Tube}[\\{\\{0,0,0\\},(i\/100) \\text{listvel}[[i]]\\}]]\\}]],\\{i,\\text{Length}@\\text{listvel}\\}];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\nHopf--Ran\\~{a}da\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsVelSphereSymmetricHR}[\\text{tf$\\_$},\\ell \\_:1,\\kappa \\_,\\text{v$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listvel}=\\text{SpherePoints}[\\text{npts}],\\text{listplots}=\\{\\},\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{SolnHR}[\\text{tf},\\ell ,\\{0,0,0\\},v \\text{listvel}[[i]],\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Hue}[i\/20],\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{Hue}[i\/20],\\text{Arrowheads}[0.03],}\\\\\n\\pmb{\\text{Arrow}[\\text{Tube}[\\{\\{0,0,0\\},v \\text{listvel}[[i]]\\}]]\\}]],\\{i,\\text{Length}@\\text{listvel}\\}];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsVelSphereRandomHR}[\\text{tf$\\_$},\\ell \\_:1,\\kappa \\_,\\text{v$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listvel}=\\text{RandomSpherePoints}[\\text{npts}],\\text{listplots}=\\{\\},\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{SolnHR}[\\text{tf},\\ell ,\\{0,0,0\\},v \\text{listvel}[[i]],\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Hue}[i\/20],\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{Hue}[i\/20],\\text{Arrowheads}[0.03],}\\\\\n\\pmb{\\text{Arrow}[\\text{Tube}[\\{\\{0,0,0\\},v \\text{listvel}[[i]]\\}]]\\}]],\\{i,\\text{Length}@\\text{listvel}\\}];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsVelInhomogeneousHR}[\\text{tf$\\_$},\\ell \\_:1,\\kappa \\_,\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listvel}=\\text{RandomSpherePoints}[\\text{npts}],\\text{listplots}=\\{\\},\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{SolnHR}[\\text{tf},\\ell ,\\{0,0,0\\},(i\/100) \\text{listvel}[[i]],\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Hue}[i\/20],\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{Hue}[i\/20],\\text{Arrowheads}[0.03],}\\\\\n\\pmb{\\text{Arrow}[\\text{Tube}[\\{\\{0,0,0\\},(i\/100) \\text{listvel}[[i]]\\}]]\\}]],\\{i,\\text{Length}@\\text{listvel}\\}];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\n\n\n\\vspace{12pt}\n\\section{Fluctuation matrix}\n\\vspace{2pt}\n\n\\subsection*{j=0 case}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{U=\\text{Array}\\left[\\Phi _0,3,1\\right];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{V=\\text{Flatten}[\\text{Array}[\\Phi ,\\{3,3\\},\\{1,1\\}]];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{rem3}[\\text{m$\\_$}]\\text{:=}\\text{QuotientRemainder}[m-1,3][[2]]+1}\\\\\n\\pmb{\\text{quo3}[\\text{n$\\_$}]\\text{:=}\\text{Quotient}[n-1,3]+1}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{num}[\\text{p$\\_$},\\text{a$\\_$}]\\text{:=}3\\text{QuotientRemainder}[p-1,3][[2]]}\\\\\n\\pmb{+\\text{QuotientRemainder}[a-1,3][[2]]+4}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $00} =2(1+\\psi [\\tau ])^2\\text{IdentityMatrix}[3];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $01} = \\text{Table}[0,\\{m,3\\},\\{n,9\\}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{Do}\\left[\\text{sq$\\Omega $01}[[m,n]]=\\left(-2 \\psi '[\\tau ]\\right)D[\\text{Sum}[\\text{Sum}[\\text{LeviCivitaTensor}[3][[a,m,k]]\\right.}\\\\\n\\pmb{\\Phi [k,a],\\{k,1,3\\}],\\{a,1,3\\}],V[[n]]],\\{m,1,3\\},\\{n,1,9\\}]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $10}=\\text{Transpose}[\\text{sq$\\Omega $01}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $11d}=(2\\psi [\\tau ]^2+4\\psi [\\tau ]+6) \\text{IdentityMatrix}[9];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $11od}=\\text{Table}[0,\\{m,9\\},\\{n,9\\}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{Do}\\left[\\text{sq$\\Omega $11od}[[m,n]]=2\\left(\\psi [\\tau ]^2-\\psi [\\tau ]-2\\right)\\right.}\\\\\n\\pmb{D[\\text{Sum}[\\text{LeviCivitaTensor}[3][[k,\\text{quo3}[m],p]]}\\\\\n\\pmb{\\text{LeviCivitaTensor}[3][[k,\\text{rem3}[m],a]]}\\\\\n\\pmb{\\Phi [p,a],\\{k,1,3\\},\\{p,1,3\\},\\{a,1,3\\}],}\\\\\n\\pmb{V[[n]]],\\{m,1,9\\},\\{n,1,9\\}]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $11}=\\text{sq$\\Omega $11d}+\\text{sq$\\Omega $11od};}\\)\n\\end{doublespace}\n\n\\subsection*{j=1\/2 case}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{j=\\frac{1}{2};}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{U=\\text{Flatten}\\left[\\text{Array}\\left[\\Phi _0,\\{2,3\\},\\{-1\/2,1\\}\\right]\\right]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{V=\\text{Flatten}[\\text{Array}[\\Phi ,\\{2,3,3\\},\\{-1\/2,1,1\\}]]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{preFac}[\\text{exp$\\_$}]\\text{:=}\\exp \\text{\/.}\\{U[[1]]\\to 1,U[[2]]\\to 1,U[[3]]\\to 1,U[[4]]\\to 1,}\\\\\n\\pmb{U[[5]]\\to 1,U[[6]]\\to 1,V[[1]]\\to 1,V[[2]]\\to 1,V[[3]]\\to 1,V[[4]]\\to 1,}\\\\\n\\pmb{V[[5]]\\to 1,V[[6]]\\to 1,V[[7]]\\to 1,V[[8]]\\to 1,V[[9]]\\to 1,V[[10]]\\to 1,}\\\\\n\\pmb{V[[11]]\\to 1,V[[12]]\\to 1,V[[13]]\\to 1,V[[14]]\\to 1,V[[15]]\\to 1,}\\\\\n\\pmb{V[[16]]\\to 1,V[[17]]\\to 1,V[[18]]\\to 1\\}}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{Pos1}[\\text{exp$\\_$}]\\text{:=}\\text{Extract}\\left[\\text{preFac}[\\exp ]^{-1}\\exp ,1\\right]}\\\\\n\\pmb{\\text{Pos2}[\\text{exp$\\_$}]\\text{:=}\\text{Extract}\\left[\\text{preFac}[\\exp ]^{-1}\\exp ,2\\right]}\\\\\n\\pmb{\\text{Pos3}[\\text{exp$\\_$}]\\text{:=}\\text{Extract}\\left[\\text{preFac}[\\exp ]^{-1}\\exp ,3\\right]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{UmapRp}[\\text{exp$\\_$}]\\text{:=}-2I\\sqrt{\\frac{(j-\\text{Pos1}[\\exp ])(j+\\text{Pos1}[\\exp ]+1)}{2}}\\text{preFac}[\\exp ]}\\\\\n\\pmb{\\Phi _0[\\text{Pos1}[\\exp ]+1,\\text{Pos2}[\\exp ]]}\\\\\n\\pmb{\\text{UmapRm}[\\text{exp$\\_$}]\\text{:=}-2I\\sqrt{\\frac{(j+\\text{Pos1}[\\exp ])(j-\\text{Pos1}[\\exp ]+1)}{2}}\\text{preFac}[\\exp ]}\\\\\n\\pmb{\\Phi _0[\\text{Pos1}[\\exp ]-1,\\text{Pos2}[\\exp ]]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{UmapR}[1][\\text{exp$\\_$}]\\text{:=}\\text{If}\\left[\\text{Pos1}[\\exp ]<100,\\frac{1}{\\sqrt{2}}(\\text{UmapRm}[\\exp ]+\\text{UmapRp}[\\exp\n]),0\\right]}\\\\\n\\pmb{\\text{UmapR}[2][\\text{exp$\\_$}]\\text{:=}\\text{If}\\left[\\text{Pos1}[\\exp ]<100,\\frac{I}{\\sqrt{2}}(\\text{UmapRm}[\\exp ]-\\text{UmapRp}[\\exp ]),0\\right]}\\\\\n\\pmb{\\text{UmapR}[3][\\text{exp$\\_$}]\\text{:=}\\text{If}[\\text{Pos1}[\\exp ]<100,2I \\text{Pos1}[\\exp ]\\exp ,0]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{UmapR}[1][\\text{exp$\\_$}]\\text{:=}0\\text{\/;}\\exp \\text{==}0}\\\\\n\\pmb{\\text{UmapR}[2][\\text{exp$\\_$}]\\text{:=}0\\text{\/;}\\exp \\text{==}0}\\\\\n\\pmb{\\text{UmapR}[3][\\text{exp$\\_$}]\\text{:=}0\\text{\/;}\\exp \\text{==}0}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{VmapRp}[\\text{exp$\\_$}]\\text{:=}-2I\\sqrt{\\frac{(j-\\text{Pos1}[\\exp ])(j+\\text{Pos1}[\\exp ]+1)}{2}}\\text{preFac}[\\exp ]}\\\\\n\\pmb{\\Phi [\\text{Pos1}[\\exp ]+1,\\text{Pos2}[\\exp ],\\text{Pos3}[\\exp ]]}\\\\\n\\pmb{\\text{VmapRm}[\\text{exp$\\_$}]\\text{:=}-2I\\sqrt{\\frac{(j+\\text{Pos1}[\\exp ])(j-\\text{Pos1}[\\exp ]+1)}{2}}\\text{preFac}[\\exp ]}\\\\\n\\pmb{\\Phi [\\text{Pos1}[\\exp ]-1,\\text{Pos2}[\\exp ],\\text{Pos3}[\\exp ]]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{VmapR}[1][\\text{exp$\\_$}]\\text{:=}\\text{If}\\left[\\text{Pos1}[\\exp ]<100,\\frac{1}{\\sqrt{2}}(\\text{VmapRm}[\\exp ]+\\text{VmapRp}[\\exp\n]),0\\right]}\\\\\n\\pmb{\\text{VmapR}[2][\\text{exp$\\_$}]\\text{:=}\\text{If}\\left[\\text{Pos1}[\\exp ]<100,\\frac{I}{\\sqrt{2}}(\\text{VmapRm}[\\exp ]-\\text{VmapRp}[\\exp ]),0\\right]}\\\\\n\\pmb{\\text{VmapR}[3][\\text{exp$\\_$}]\\text{:=}\\text{If}[\\text{Pos1}[\\exp ]<100,2I \\text{Pos1}[\\exp ]\\exp ,0]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{VmapR}[1][\\text{exp$\\_$}]\\text{:=}0\\text{\/;}\\exp \\text{==}0}\\\\\n\\pmb{\\text{VmapR}[2][\\text{exp$\\_$}]\\text{:=}0\\text{\/;}\\exp \\text{==}0}\\\\\n\\pmb{\\text{VmapR}[3][\\text{exp$\\_$}]\\text{:=}0\\text{\/;}\\exp \\text{==}0}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{rem3}[\\text{i$\\_$}]\\text{:=}\\text{QuotientRemainder}[i-1,3][[2]]+1}\\\\\n\\pmb{\\text{quo3}[\\text{i$\\_$}]\\text{:=}\\text{Quotient}[i-1,3]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{j3}[\\text{m$\\_$}]\\text{:=}\\text{Quotient}[m-1,9]-1\/2}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{pNum}[\\text{m$\\_$}]\\text{:=}\\text{QuotientRemainder}[\\text{Quotient}[m-1,3],3][[2]]+1}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{num}[\\text{x$\\_$},\\text{p$\\_$},\\text{a$\\_$}]\\text{:=}3\\text{QuotientRemainder}[p-1,3][[2]]}\\\\\n\\pmb{+\\text{QuotientRemainder}[a-1,3][[2]]+9x+\\frac{23}{2}}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $00d} = \\left(3+2(1+\\psi [\\tau ])^2\\right)\\text{IdentityMatrix}[6];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $00od} =\\text{Table}[0,\\{a,1,6\\},\\{b,1,6\\}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{Do}[\\text{sq$\\Omega $00od}[[m,n]]=2(1+\\psi [\\tau ])D[\\text{Sum}[\\text{UmapR}[a][\\text{Sum}[}\\\\\n\\pmb{\\left.\\left.\\text{LeviCivitaTensor}[3][[a,\\text{rem3}[m],k]]\\Phi _0[\\text{quo3}[m]-1\/2,k],\\{k,1,3\\}\\right]\\right],}\\\\\n\\pmb{\\{a,1,3\\}],U[[n]]],\\{m,1,6\\},\\{n,1,6\\}]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $00}=\\text{sq$\\Omega $00d}+\\text{sq$\\Omega $00od};}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $01} = \\text{Table}[0,\\{i,6\\},\\{j,18\\}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{Do}[\\text{sq$\\Omega $01}[[m,n]]=(-2 \\psi '[\\tau ])D[\\text{Sum}[\\text{Sum}[\\text{LeviCivitaTensor}[3][[a,\\text{rem3}[m],k]]}\\\\\n\\pmb{\\Phi [\\text{quo3}[m]-1\/2,k,a],\\{k,1,3\\}],\\{a,1,3\\}],V[[n]]],\\{m,1,6\\},\\{n,1,18\\}]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $10}=\\text{Transpose}[\\text{sq$\\Omega $01}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $11d}=\\left(7+2(1+\\psi [\\tau ])^2\\right)\\text{IdentityMatrix}[18];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $11od}=\\text{Table}[0,\\{m,18\\},\\{n,18\\}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{Do}[\\text{sq$\\Omega $11od}[[m,n]]=D[\\text{Sum}[2(1+\\psi [\\tau ])\\text{VmapR}[b][\\text{Sum}[}\\\\\n\\pmb{\\text{LeviCivitaTensor}[3][[b,\\text{pNum}[m],p]]\\Phi [\\text{j3}[m],p,\\text{rem3}[m]],\\{p,1,3\\}]]}\\\\\n\\pmb{+2\\text{VmapR}[b][\\text{Sum}[\\text{LeviCivitaTensor}[3][[b,\\text{rem3}[m],a]]\\Phi [\\text{j3}[m],}\\\\\n\\pmb{\\text{pNum}[m],a],\\{a,1,3\\}]]+2\\left(\\psi [\\tau ]^2-\\psi [\\tau ]-2\\right)}\\\\\n\\pmb{\\text{Sum}[\\text{LeviCivitaTensor}[3][[b,\\text{pNum}[m],p]]\\text{LeviCivitaTensor}[3][[b,\\text{rem3}[m],a]]}\\\\\n\\pmb{\\Phi [\\text{j3}[m],p,a],\\{a,1,3\\},\\{p,1,3\\}],\\{b,1,3\\}],V[[n]]],}\\\\\n\\pmb{\\{m,1,18\\},\\{n,1,18\\}]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $11}=\\text{sq$\\Omega $11d}+\\text{sq$\\Omega $11od};}\\)\n\\end{doublespace}\n\n\\subsection*{j=1 case}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{j=1;}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{U=\\text{Flatten}\\left[\\text{Array}\\left[\\Phi _0,\\{3,3\\},\\{-1,1\\}\\right]\\right]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{V=\\text{Flatten}[\\text{Array}[\\Phi ,\\{3,3,3\\},\\{-1,1,1\\}]]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{preFac}[\\text{exp$\\_$}]\\text{:=}\\exp \\text{\/.}\\{U[[1]]\\to 1,U[[2]]\\to 1,U[[3]]\\to 1,U[[4]]\\to 1,}\\\\\n\\pmb{U[[5]]\\to 1,U[[6]]\\to 1,U[[7]]\\to 1,U[[8]]\\to 1,U[[9]]\\to 1,V[[1]]\\to 1,}\\\\\n\\pmb{V[[2]]\\to 1,V[[3]]\\to 1,V[[4]]\\to 1,V[[5]]\\to 1,V[[6]]\\to 1,V[[7]]\\to 1,}\\\\\n\\pmb{V[[8]]\\to 1,V[[9]]\\to 1,V[[10]]\\to 1,V[[11]]\\to 1,V[[12]]\\to 1,V[[13]]\\to 1,}\\\\\n\\pmb{V[[14]]\\to 1,V[[15]]\\to 1,V[[16]]\\to 1,V[[17]]\\to 1,V[[18]]\\to 1,V[[19]]\\to 1,}\\\\\n\\pmb{V[[20]]\\to 1,V[[21]]\\to 1,V[[22]]\\to 1,V[[23]]\\to 1,}\\\\\n\\pmb{V[[24]]\\to 1,V[[25]]\\to 1,V[[26]]\\to 1,V[[27]]\\to 1\\}}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{Pos1}[\\text{exp$\\_$}]\\text{:=}\\text{Extract}\\left[\\text{preFac}[\\exp ]^{-1}\\exp ,1\\right]}\\\\\n\\pmb{\\text{Pos2}[\\text{exp$\\_$}]\\text{:=}\\text{Extract}\\left[\\text{preFac}[\\exp ]^{-1}\\exp ,2\\right]}\\\\\n\\pmb{\\text{Pos3}[\\text{exp$\\_$}]\\text{:=}\\text{Extract}\\left[\\text{preFac}[\\exp ]^{-1}\\exp ,3\\right]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{UmapRp}[\\text{exp$\\_$}]\\text{:=}-2I\\sqrt{\\frac{(j-\\text{Pos1}[\\exp ])(j+\\text{Pos1}[\\exp ]+1)}{2}}\\text{preFac}[\\exp ]}\\\\\n\\pmb{\\Phi _0[\\text{Pos1}[\\exp ]+1,\\text{Pos2}[\\exp ]]}\\\\\n\\pmb{\\text{UmapRm}[\\text{exp$\\_$}]\\text{:=}-2I\\sqrt{\\frac{(j+\\text{Pos1}[\\exp ])(j-\\text{Pos1}[\\exp ]+1)}{2}}\\text{preFac}[\\exp ]}\\\\\n\\pmb{\\Phi _0[\\text{Pos1}[\\exp ]-1,\\text{Pos2}[\\exp ]]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{UmapR}[1][\\text{exp$\\_$}]\\text{:=}\\text{If}\\left[\\text{Pos1}[\\exp ]<100,\\frac{1}{\\sqrt{2}}(\\text{UmapRm}[\\exp ]+\\text{UmapRp}[\\exp\n]),0\\right]}\\\\\n\\pmb{\\text{UmapR}[2][\\text{exp$\\_$}]\\text{:=}\\text{If}\\left[\\text{Pos1}[\\exp ]<100,\\frac{I}{\\sqrt{2}}(\\text{UmapRm}[\\exp ]-\\text{UmapRp}[\\exp ]),0\\right]}\\\\\n\\pmb{\\text{UmapR}[3][\\text{exp$\\_$}]\\text{:=}\\text{If}[\\text{Pos1}[\\exp ]<100,2I \\text{Pos1}[\\exp ]\\exp ,0]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{UmapR}[1][\\text{exp$\\_$}]\\text{:=}0\\text{\/;}\\exp \\text{==}0}\\\\\n\\pmb{\\text{UmapR}[2][\\text{exp$\\_$}]\\text{:=}0\\text{\/;}\\exp \\text{==}0}\\\\\n\\pmb{\\text{UmapR}[3][\\text{exp$\\_$}]\\text{:=}0\\text{\/;}\\exp \\text{==}0}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{VmapRp}[\\text{exp$\\_$}]\\text{:=}-2I\\sqrt{\\frac{(j-\\text{Pos1}[\\exp ])(j+\\text{Pos1}[\\exp ]+1)}{2}}\\text{preFac}[\\exp ]}\\\\\n\\pmb{\\Phi [\\text{Pos1}[\\exp ]+1,\\text{Pos2}[\\exp ],\\text{Pos3}[\\exp ]]}\\\\\n\\pmb{\\text{VmapRm}[\\text{exp$\\_$}]\\text{:=}-2I\\sqrt{\\frac{(j+\\text{Pos1}[\\exp ])(j-\\text{Pos1}[\\exp ]+1)}{2}}\\text{preFac}[\\exp ]}\\\\\n\\pmb{\\Phi [\\text{Pos1}[\\exp ]-1,\\text{Pos2}[\\exp ],\\text{Pos3}[\\exp ]]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{VmapR}[1][\\text{exp$\\_$}]\\text{:=}\\text{If}\\left[\\text{Pos1}[\\exp ]<100,\\frac{1}{\\sqrt{2}}(\\text{VmapRm}[\\exp ]+\\text{VmapRp}[\\exp\n]),0\\right]}\\\\\n\\pmb{\\text{VmapR}[2][\\text{exp$\\_$}]\\text{:=}\\text{If}\\left[\\text{Pos1}[\\exp ]<100,\\frac{I}{\\sqrt{2}}(\\text{VmapRm}[\\exp ]-\\text{VmapRp}[\\exp ]),0\\right]}\\\\\n\\pmb{\\text{VmapR}[3][\\text{exp$\\_$}]\\text{:=}\\text{If}[\\text{Pos1}[\\exp ]<100,2I \\text{Pos1}[\\exp ]\\exp ,0]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{VmapR}[1][\\text{exp$\\_$}]\\text{:=}0\\text{\/;}\\exp \\text{==}0}\\\\\n\\pmb{\\text{VmapR}[2][\\text{exp$\\_$}]\\text{:=}0\\text{\/;}\\exp \\text{==}0}\\\\\n\\pmb{\\text{VmapR}[3][\\text{exp$\\_$}]\\text{:=}0\\text{\/;}\\exp \\text{==}0}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{rem3}[\\text{i$\\_$}]\\text{:=}\\text{QuotientRemainder}[i-1,3][[2]]+1}\\\\\n\\pmb{\\text{quo3}[\\text{i$\\_$}]\\text{:=}\\text{Quotient}[i-1,3]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{j3}[\\text{m$\\_$}]\\text{:=}\\text{Quotient}[m-1,9]-1}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{pNum}[\\text{m$\\_$}]\\text{:=}\\text{QuotientRemainder}[\\text{Quotient}[m-1,3],3][[2]]+1}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $00d} = \\left(8+2(1+\\psi [\\tau ])^2\\right)\\text{IdentityMatrix}[9];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $00od} =\\text{Table}[0,\\{a,1,9\\},\\{b,1,9\\}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{Do}[\\text{sq$\\Omega $00od}[[m,n]]=2(1+\\psi [\\tau ])D[\\text{Sum}[\\text{UmapR}[a][\\text{Sum}[}\\\\\n\\pmb{\\left.\\left.\\text{LeviCivitaTensor}[3][[a,\\text{rem3}[m],k]]\\Phi _0[\\text{quo3}[m]-1,k],\\{k,1,3\\}\\right]\\right],}\\\\\n\\pmb{\\{a,1,3\\}],U[[n]]],\\{m,1,9\\},\\{n,1,9\\}]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $00}=\\text{sq$\\Omega $00d}+\\text{sq$\\Omega $00od};}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $00d} = \\left(8+2(1+\\psi [\\tau ])^2\\right)\\text{IdentityMatrix}[9];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $00od} =\\text{Table}[0,\\{a,1,9\\},\\{b,1,9\\}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{Do}[\\text{sq$\\Omega $00od}[[m,n]]=2(1+\\psi [\\tau ])D[\\text{Sum}[\\text{UmapR}[a][\\text{Sum}[}\\\\\n\\pmb{\\left.\\left.\\text{LeviCivitaTensor}[3][[a,\\text{rem3}[m],k]]\\Phi _0[\\text{quo3}[m]-1,k],\\{k,1,3\\}\\right]\\right],}\\\\\n\\pmb{\\{a,1,3\\}],U[[n]]],\\{m,1,9\\},\\{n,1,9\\}]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $00}=\\text{sq$\\Omega $00d}+\\text{sq$\\Omega $00od};}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $01} = \\text{Table}[0,\\{i,9\\},\\{j,27\\}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{Do}[\\text{sq$\\Omega $01}[[m,n]]=(-2 \\psi '[\\tau ])D[\\text{Sum}[\\text{Sum}[\\text{LeviCivitaTensor}[3][[a,\\text{rem3}[m],k]]}\\\\\n\\pmb{\\Phi [\\text{quo3}[m]-1,k,a],\\{k,1,3\\}],\\{a,1,3\\}],V[[n]]],\\{m,1,9\\},\\{n,1,27\\}]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $10}=\\text{Transpose}[\\text{sq$\\Omega $01}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $11d}=\\left(12+2(1+\\psi [\\tau ])^2\\right)\\text{IdentityMatrix}[27];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $11od}=\\text{Table}[0,\\{m,27\\},\\{n,27\\}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{Do}[\\text{sq$\\Omega $11od}[[m,n]]=D[\\text{Sum}[2(1+\\psi [\\tau ])\\text{VmapR}[b][\\text{Sum}[}\\\\\n\\pmb{\\text{LeviCivitaTensor}[3][[b,\\text{pNum}[m],p]]\\Phi [\\text{j3}[m],p,\\text{rem3}[m]],\\{p,1,3\\}]]}\\\\\n\\pmb{+2\\text{VmapR}[b][\\text{Sum}[\\text{LeviCivitaTensor}[3][[b,\\text{rem3}[m],a]]}\\\\\n\\pmb{\\Phi [\\text{j3}[m],\\text{pNum}[m],a],\\{a,1,3\\}]]}\\\\\n\\pmb{+2\\left(\\psi [\\tau ]^2-\\psi [\\tau ]-2\\right)\\text{Sum}[\\text{LeviCivitaTensor}[3][[b,\\text{pNum}[m],p]]}\\\\\n\\pmb{\\text{LeviCivitaTensor}[3][[b,\\text{rem3}[m],a]]}\\\\\n\\pmb{\\Phi [\\text{j3}[m],p,a],\\{a,1,3\\},\\{p,1,3\\}],\\{b,1,3\\}],V[[n]]],}\\\\\n\\pmb{\\{m,1,27\\},\\{n,1,27\\}]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $11}=\\text{sq$\\Omega $11d}+\\text{sq$\\Omega $11od};}\\)\n\\end{doublespace}\n\n\\subsection*{Matrix eigenvalues}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $} =\\{\\{\\text{sq$\\Omega $00},\\text{sq$\\Omega $01}\\},\\{\\text{sq$\\Omega $10},\\text{sq$\\Omega $11}\\}\\}\\text{\/\/}\\text{ArrayFlatten};}\\) \\\\\n\\(\\pmb{\\text{sq$\\Omega $no$\\psi $dot}=\\text{sq$\\Omega $}\\text{\/.}\\psi '[\\tau ]\\to 0;}\\) \\\\\n\\(\\pmb{\\text{sq$\\Omega $no$\\psi $}=\\text{sq$\\Omega $}\\text{\/.}\\{\\psi [\\tau ]\\to 0,\\psi '[\\tau ]\\to 0\\};}\\) \\\\\n\\(\\pmb{\\text{Eigenvalues}[\\text{sq$\\Omega $no$\\psi $}]}\\) \\\\\n\\(\\pmb{\\text{Eigenvalues}[\\text{sq$\\Omega $no$\\psi $dot}]}\\) \\\\\n\\(\\pmb{\\text{Eigenvalues}[\\text{sq$\\Omega $}]}\\)\n\\end{doublespace}\n\n\\chapter{Polynomials in the characteristic equation}\n\\label{appendPolynomials}\n\n\n\\begin{center}\n\\resizebox{\\linewidth}{10cm}{\n\\begin{tabular}{| P{10mm} | P{25mm} | P{45mm} | P{80mm} |} \n\\hline\n$j\\ ,\\ v $ & P($\\lambda$) & Q($\\lambda$)  & R($\\lambda$) \\\\ [0.5ex] \n \\hline\\hline\n$0\\ ,\\ 0$ \\phantom{\\Big|} & N\/A & $(-2 + 6 \\psi^2) - \\lambda$ & $-(2 - 6 \\psi^2) + \\lambda$ \\\\\n \\hline\n$0\\ ,\\ 1$ \\phantom{\\Big|} & 1 & $(2 + 2 \\psi + 4 \\psi^2) - \\lambda$ & $(4 + 12\\psi + 20 \\psi^2 + 20 \\psi^3 + 8 \\psi^4 - 8 \\dot{\\psi}^2) - (4 + 6 \\psi + 6 \\psi^2) \\lambda + \\lambda^2$ \\\\\n \\hline\n$0\\ ,\\ 2$ \\phantom{\\Big|} & N\/A & $(10 + 6 \\psi) - \\lambda$ & N\/A \\\\\n \\hline\n$\\sfrac12\\ ,\\ \\sfrac12$ \\phantom{\\Big|} & $-(1 + 6 \\psi^2) + \\lambda$ & $-(1 + 2 \\psi^2 + 24 \\psi^4) + (2 + 10 \\psi^2) \\lambda - \\lambda^2$ & $-(1 + 4 \\psi^2 + 28 \\psi^4 + 48 \\psi^6 - 8 \\dot{\\psi}^2 - 48 \\psi^2 \\dot{\\psi}^2) + (3 + 16 \\psi^2 + 44 \\psi^4 - 8 \\dot{\\psi}^2) \\lambda - (3 + 12 \\psi^2) \\lambda^2 + \\lambda^3$ \\\\ \n \\hline\n$\\sfrac12\\ ,\\ \\sfrac32$ \\phantom{\\Big|} & $-(7 + 3 \\psi) + \\lambda$ & $-(49 + 42 \\psi + 32 \\psi^2 + 12 \\psi^3)   + (14 + 6 \\psi + 4 \\psi^2) \\lambda - \\lambda^2$ & $-(343 + 588 \\psi + 574 \\psi^2 + 360 \\psi^3 + 136 \\psi^4 + 24 \\psi^5 - 56 \\dot{\\psi}^2 - 24 \\psi \\dot{\\psi}^2) + (147 + 168 \\psi + 124 \\psi^2 + 48 \\psi^3 + 8 \\psi^4 - 8 \\dot{\\psi}^2) \\lambda - (21 + 12 \\psi + 6 \\psi^2) \\lambda^2 + \\lambda^3$ \\\\ \n \\hline\n$\\sfrac12\\ ,\\ \\sfrac52$ \\phantom{\\Big|} & N\/A & $(17 + 8 \\psi) - \\lambda$ & N\/A \\\\\n \\hline\n$1\\ ,\\ 0$ \\phantom{\\Big|} & $1$ & $(2 - 2 \\psi + 4 \\psi^2) - \\lambda$ & $( 4 - 12 \\psi + 20 \\psi^2 - 20 \\psi^3 + 8 \\psi^4 - 8 \\dot{\\psi}^2) - (4 - 6 \\psi + 6 \\psi^2) \\lambda + \\lambda^2$ \\\\\n \\hline\n$1\\ ,\\ 1$ \\phantom{\\Big|} & $(36 + 36 \\psi^2) - (12 + 6\\psi^2) \\lambda + \\lambda^2$ & $(216 + 192 \\psi^2 + 104 \\psi^4) - (108 + 92 \\psi^2 + 24 \\psi^4) \\lambda + (18 + 10 \\psi^2) \\lambda^2 - \\lambda^3$ & $(1296 + 1584 \\psi^2 + 1008 \\psi^4 + 208 \\psi^6 - 288 \\dot{\\psi}^2 - 288 \\psi^2 \\dot{\\psi}^2) - (864 + 960 \\psi^2 + 432 \\psi^4 + 48 \\psi^6 - 96 \\dot{\\psi}^2 - 48 \\psi^2 \\dot{\\psi}^2) \\lambda + (216 + 188 \\psi^2 + 44 \\psi^4 - 8 \\dot{\\psi}^2) \\lambda^2 - (24 + 12\\psi^2) \\lambda^3 + \\lambda^4$ \\\\\n \\hline\n$1\\ ,\\ 2$ \\phantom{\\Big|} & $-(14 + 4 \\psi) + \\lambda$ & $-(196 + 112 \\psi + 60 \\psi^2 + 16 \\psi^3) + (28 + 8 \\psi + 4 \\psi^2) \\lambda - \\lambda^2$ & $-(2744 + 3136 \\psi + 2128 \\psi^2 + 928 \\psi^3 + 248 \\psi^4 + 32 \\psi^5 - 112 \\dot{\\psi}^2 - 32 \\psi \\dot{\\psi}^2) + (588 + 448 \\psi + 236 \\psi^2 + 64 \\psi^3 + 8 \\psi^4 - 8 \\dot{\\psi}^2) \\lambda - (42 + 16 \\psi + 6 \\psi^2) \\lambda^2 + \\lambda^3$ \\\\\n \\hline\n$1\\ ,\\ 3$ \\phantom{\\Big|} & N\/A & $(26 + 10 \\psi) - \\lambda$ & N\/A \\\\ \n \\hline\n$\\sfrac32\\ ,\\ \\sfrac12$ \\phantom{\\Big|} & $-(7 - 3 \\psi ) + \\lambda$ & $-(49 - 42 \\psi + 32 \\psi^2 - 12 \\psi^3) + (14 - 6 \\psi+ 4 \\psi^2) \\lambda - \\lambda^2$ & $-(343 - 588 \\psi + 574 \\psi^2 - 360 \\psi^3 + 136 \\psi^4 - 24 \\psi^5 - 56 \\dot{\\psi}^2 + 24 \\psi \\dot{\\psi}^2) + (147 - 168 \\psi + 124 \\psi^2 - 48 \\psi^3 + 8 \\psi^4 - 8 \\dot{\\psi}^2) \\lambda - (21 - 12 \\psi + 6 \\psi^2) \\lambda^2 + \\lambda^3$ \\\\ \n \\hline\n$\\sfrac32\\ ,\\ \\sfrac32$ \\phantom{\\Big|} & $(169 + 78 \\psi^2) - (26 + 6 \\psi^2) \\lambda + \\lambda^2$ & $(2197 + 962 \\psi^2 + 216 \\psi^4) - (507 + 204 \\psi^2 + 24 \\psi^4) \\lambda + (39 + 10 \\psi^2) \\lambda^2 - \\lambda^3$ & $(28561 + 16900 \\psi^2 + 4732 \\psi^4 + 432 \\psi^6 - 1352 \\dot{\\psi}^2 - 624 \\psi^2 \\dot{\\psi}^2) + (-8788 - 4628 \\psi^2 - 936 \\psi^4 - 48 \\psi^6 + 208 \\dot{\\psi}^2 + 48 \\psi^2 \\dot{\\psi}^2) \\lambda + (1014 + 412 \\psi^2 + 44 \\psi^4 - 8 \\dot{\\psi}^2) \\lambda^2 - (52 + 12 \\psi^2) \\lambda^3 + \\lambda^4$ \\\\\n \\hline\n$\\sfrac32\\ ,\\ \\sfrac52$ \\phantom{\\Big|} & $-(23 + 5 \\psi) + \\lambda$ & $-(529 + 230 \\psi + 96 \\psi^2 + 20 \\psi^3) + (46 + 10 \\psi + 4 \\psi^2) \\lambda - \\lambda^2$ & $-(12167 + 10580 \\psi + 5566 \\psi^2 + 1880 \\psi^3 + 392 \\psi^4 + 40 \\psi^5 - 184 \\dot{\\psi}^2 + 40 \\psi\\dot{\\psi}^2) + (1587 + 920 \\psi + 380 \\psi^2 + 80 \\psi^3 + 8 \\psi^4 - 8 \\dot{\\psi}^2) \\lambda + (-69 - 20 \\psi - 6 \\psi^2) \\lambda^2 + \\lambda^3$ \\\\\n \\hline\n$\\sfrac32\\ ,\\ \\sfrac72$ \\phantom{\\Big|} & N\/A & $(37 + 12 \\psi) - \\lambda$ & N\/A \\\\ \n \\hline\n$2\\ ,\\ 0$ \\phantom{\\Big|} & N\/A & $(10- 6 \\psi) - \\lambda$ & N\/A \\\\\n \\hline\n$2\\ ,\\ 1$ \\phantom{\\Big|} & $-(14 - 4 \\psi) + \\lambda$ & $-(196 - 112 \\psi + 60 \\psi^2 - 16 \\psi^3) + (28 - 8 \\psi + 4 \\psi^2) \\lambda - \\lambda^2$ & $-(2744 - 3136 \\psi + 2128 \\psi^2 - 928 \\psi^3 + 248 \\psi^4 - 32 \\psi^5 - 112 \\dot{\\psi}^2 + 32 \\psi \\dot{\\psi}^2) + (588 - 448 \\psi + 236 \\psi^2 - 64 \\psi^3 + 8 \\psi^4 - 8 \\dot{\\psi}^2) \\lambda - (42 - 16 \\psi + 6 \\psi^2) \\lambda^2 + \\lambda^3$ \\\\\n \\hline\n$2\\ ,\\ 2$ \\phantom{\\Big|} & $(484 + 132 \\psi^2) - (44 + 6 \\psi^2) \\lambda + \\lambda^2$ & $(10648 + 2816 \\psi^2 + 360 \\psi^4) - (1452 + 348 \\psi^2 + 24 \\psi^4) \\lambda + (66 + 10 \\psi^2) \\lambda^2 - \\lambda^3$ & $(234256 + 83248 \\psi^2 + 13552 \\psi^4 + 720 \\psi^6 - 3872 \\dot{\\psi}^2 - 1056 \\psi^2 \\dot{\\psi}^2) - (42592 + 13376 \\psi^2 + 1584 \\psi^4 + 48 \\psi^6 - 352 \\dot{\\psi}^2 - 48 \\psi^2 \\dot{\\psi}^2) \\lambda + (2904 + 700 \\psi^2 + 44 \\psi^4 - 8 \\dot{\\psi}^2) \\lambda^2 - \n (88 + 12 \\psi^2) \\lambda^3 + \\lambda^4$ \\\\\n \\hline\n$2\\ ,\\ 3$ \\phantom{\\Big|} & $-(34 + 6 \\psi) + \\lambda$ & $-(1156 + 408 \\psi + 140 \\psi^2 + 24 \\psi^3) + (68 + 12 \\psi + 4 \\psi^2) \\lambda - \\lambda^2$ & $-(39304 + 27744 \\psi + 11968 \\psi^2 + 3312 \\psi^3 + 568 \\psi^4 + 48 \\psi^5 - 272 \\dot{\\psi}^2 - 48 \\psi \\dot{\\psi}^2) + (3468 + 1632 \\psi + 556 \\psi^2 + 96 \\psi^3 + 8 \\psi^4 - 8 \\dot{\\psi}^2) \\lambda - (102 + 24 \\psi + 6\\psi^2) \\lambda^2 + \\lambda^3$ \\\\ \n \\hline\n$2\\ ,\\ 4$ \\phantom{\\Big|} & N\/A & $(50 + 14 \\psi) - \\lambda$ & N\/A \\\\ \n \\hline\n\\end{tabular}\n}\n\\end{center}\n\n\\restoregeometry\n\\chapter{Rotation of indices}\n\n\\label{appendRotation}\n\\noindent\nBy construction the gauge potential ${\\cal A}$ is $SO(4)$ invariant, which means it is also invariant under the action of $SO(3)$ generators $\\mathcal{D}_a$ \\eqref{DPdef}. For a complex-valued ${\\cal A} = {\\cal A}_a\\,e^a$ expanded using \\eqref{Asep} (for type I) and \\eqref{XY} at a fixed $j$ this means \n\\begin{equation}\n\\begin{aligned}\n    0 &\\= \\mathcal{D}_a ({\\cal A}) \\\\\n    &\\= \\sum_{m,n,\\tilde{n}} C^{n,\\tilde{n}}_{j,b}\\, \\mathrm{e}^{\\mathrm{i}\\Omega\\tau}\\, \\left( \\mathcal{D}_a(\\Lambda_{m,\\tilde{n}})\\,e^b\\,Y_{j;m,n} + \\Lambda_{m,\\tilde{n}}\\,\\mathcal{D}_a(e^b)\\,Y_{j;m,n} + \\Lambda_{m,\\tilde{n}}\\,e^b\\,\\mathcal{D}_a(Y_{j;m,n})  \\right)\n\\end{aligned}    \n\\end{equation}\nwhere $\\mathcal{D}_a(e^b)$ are determined from \\eqref{LieActionL} and \\eqref{LieActionR} while $\\mathcal{D}_a(Y_{j;m,n})$ are determined from (\\ref{Y-action}-\\ref{JpmAction}). By collecting the coefficients of various linearly independent $e^b$ and $Y_{j;m,n}$ terms in the above expansion for a fixed $\\mathcal{D}_a$ one gets a set of coupled linear equations for $\\mathcal{D}_a(\\Lambda_{m,\\tilde{n}})$, which can be easily solved. The action of the generators $\\mathcal{D}_a$ on $\\Lambda_{m,\\tilde{n}}$ for $j=0,~ 1\/2 ~\\textrm{and}~ 1$ is given in the following table.\n\n\\begin{center}\n\\begin{adjustbox}{width=\\linewidth}\n    \\begin{tabular}{|P{3.25cm}|P{6.3cm}|P{6.25cm}|P{1.8cm}|}\n        \\hline\n      & $\\mathcal{D}_1$ & $\\mathcal{D}_2$ & $\\mathcal{D}_3$ \\\\ [0.5ex] \\hline \\hline\n     $\\left.j=0: \\right.                                               \n     \\begin{aligned}\n        &\\Lambda_{0,-1} \\mapsto \\\\\n        &\\Lambda_{0,0} \\mapsto \\\\\n        &\\Lambda_{0,1} \\mapsto\n     \\end{aligned}$ & \n     $\\begin{aligned}                                              \n         &\\sqrt{2}\\mathrm{i} \\Lambda_{0,0} \\\\\n         &\\sqrt{2}\\mathrm{i} (\\Lambda_{0,1}+\\Lambda_{0,-1}) \\\\\n         &\\sqrt{2}\\mathrm{i} \\Lambda_{0,0}\n     \\end{aligned}$ & \n     $\\begin{aligned}                                              \n         &-\\sqrt{2}  \\Lambda_{0,0} \\\\\n         &\\sqrt{2}  (\\Lambda_{0,-1} - \\Lambda_{0,1}) \\\\\n         &\\sqrt{2}  \\Lambda_{0,0} \n     \\end{aligned}$ &\n     $\\begin{aligned}                                              \n         &-\\sqrt{2}\\mathrm{i}  \\Lambda_{0,-1} \\\\\n         &\\quad 0 \\\\\n         &\\sqrt{2}\\mathrm{i}  \\Lambda_{0,1}\n     \\end{aligned}$ \\\\ \\hline\n     $\\left.\\begin{aligned}\n        &\\quad j=\\sfrac12: \\\\                                               \n        &\\underbrace{\\textrm{Notation}}\\\\\n        &\\pm\\sfrac12\\equiv\\pm \\\\\n        &\\pm\\sfrac32\\equiv\\uparrow\\downarrow\n     \\end{aligned}\\right\\}\n     \\begin{aligned}\n        &\\Lambda_{-,\\downarrow} \\mapsto \\\\[.2em]\n        &\\Lambda_{-,-} \\mapsto \\\\[.2em]\n        &\\Lambda_{-,+} \\mapsto \\\\[.2em]\n        &\\Lambda_{-,\\uparrow} \\mapsto \\\\[.2em]\n        &\\Lambda_{+,\\downarrow} \\mapsto \\\\[.2em]\n        &\\Lambda_{+,-} \\mapsto \\\\[.2em]\n        &\\Lambda_{+,+} \\mapsto \\\\[.2em]\n        &\\Lambda_{+,\\uparrow} \\mapsto\n     \\end{aligned}$ & \n     $\\begin{aligned}                                              \n         &\\mathrm{i}(\\sqrt{3}\\Lambda_{-,-} + \\Lambda_{+,\\downarrow} ) \\\\\n         &\\mathrm{i}(\\sqrt{3}\\Lambda_{-,\\downarrow} + 2\\Lambda_{-,+} + \\Lambda_{+,-} ) \\\\\n         &\\mathrm{i}(2\\Lambda_{-,-} + \\sqrt{3}\\Lambda_{-,\\uparrow} + \\Lambda_{+,+} ) \\\\\n         &\\mathrm{i}(\\sqrt{3}\\Lambda_{-,+} + \\Lambda_{+,\\uparrow} ) \\\\\n         &\\mathrm{i}(\\Lambda_{-,\\downarrow} + \\sqrt{3}\\Lambda_{+,-} ) \\\\\n         &\\mathrm{i}(\\Lambda_{-,-} + \\sqrt{3}\\Lambda_{+,\\downarrow} + 2 \\Lambda_{+,+} ) \\\\\n         &\\mathrm{i}(\\Lambda_{-,+} + 2\\Lambda_{+,-} + \\sqrt{3}\\Lambda_{+,\\uparrow} ) \\\\\n         &\\mathrm{i}(\\Lambda_{-,\\uparrow} + \\sqrt{3}\\Lambda_{+,+} ) \\\\\n     \\end{aligned}$ &\n     $\\begin{aligned}                                              \n         &-\\sqrt{3}\\Lambda_{-,-} - \\Lambda_{+,\\downarrow} \\\\\n         &\\sqrt{3}\\Lambda_{-,\\downarrow} - 2\\Lambda_{-,+} - \\Lambda_{+,-} \\\\\n         &2\\Lambda_{-,-} - \\sqrt{3}\\Lambda_{-,\\uparrow} - \\Lambda_{+,+} \\\\\n         &\\sqrt{3}\\Lambda_{-,+} - \\Lambda_{+,\\uparrow} \\\\\n         &\\Lambda_{-,\\downarrow} - \\sqrt{3}\\Lambda_{+,-} \\\\\n         &\\Lambda_{-,-} + \\sqrt{3}\\Lambda_{+,\\downarrow} -2\\Lambda_{+,+} \\\\\n         &\\Lambda_{-,+} +2\\Lambda_{+,-} -\\sqrt{3}\\Lambda_{+,\\uparrow} \\\\\n         &\\Lambda_{-,\\uparrow} + \\sqrt{3}\\Lambda_{+,+} \\\\\n     \\end{aligned}$ &\n     $\\begin{aligned}                                              \n         &-4\\mathrm{i}\\Lambda_{-,\\downarrow} \\\\[.2em]\n         &-2\\mathrm{i}\\Lambda_{-,-} \\\\[.2em]\n         &0 \\\\[.2em]\n         &2\\mathrm{i}\\Lambda_{-,\\uparrow} \\\\[.2em]\n         &-2\\mathrm{i}\\Lambda_{+,\\downarrow} \\\\[.2em]\n         &0 \\\\[.2em]\n         &2\\mathrm{i}\\Lambda_{+,+} \\\\[.2em]\n         &4\\mathrm{i}\\Lambda_{+,\\uparrow}\n     \\end{aligned}$ \\\\ \\hline\n     $\\left.\\begin{aligned}\n        &\\quad j=1: \\\\                                               \n        &\\underbrace{\\textrm{Notation}}\\\\\n        &\\pm1\\equiv\\pm \\\\\n        &\\pm2\\equiv\\uparrow\\downarrow\n     \\end{aligned}\\right\\}\n     \\begin{aligned}\n        &\\Lambda_{-,\\downarrow} \\mapsto \\\\[.2em]\n        &\\Lambda_{-,-} \\mapsto \\\\[.2em]\n        &\\Lambda_{-,0} \\mapsto \\\\[.2em]\n        &\\Lambda_{-,+} \\mapsto \\\\[.2em]\n        &\\Lambda_{-,\\uparrow} \\mapsto \\\\[.2em]\n        &\\Lambda_{0,\\downarrow} \\mapsto \\\\[.2em]\n        &\\Lambda_{0,-} \\mapsto \\\\[.2em]\n        &\\Lambda_{0,0} \\mapsto \\\\[.2em]\n        &\\Lambda_{0,+} \\mapsto \\\\[.2em]\n        &\\Lambda_{0,\\uparrow} \\mapsto \\\\[.2em]\n        &\\Lambda_{+,\\downarrow} \\mapsto \\\\[.2em]\n        &\\Lambda_{+,-} \\mapsto \\\\[.2em]\n        &\\Lambda_{+,0} \\mapsto \\\\[.2em]\n        &\\Lambda_{+,+} \\mapsto \\\\[.2em]\n        &\\Lambda_{+,\\uparrow} \\mapsto\n     \\end{aligned}$ & \n     $\\begin{aligned}                                              \n         &\\mathrm{i}(2\\Lambda_{-,-} + \\sqrt{2}\\Lambda_{0,\\downarrow} ) \\\\\n         &\\mathrm{i}(2\\Lambda_{-,\\downarrow} + \\sqrt{6}\\Lambda_{-,0} + \\sqrt{2}\\Lambda_{0,-} ) \\\\\n         &\\mathrm{i}(\\sqrt{6}\\Lambda_{-,-} + \\sqrt{6}\\Lambda_{-,+} + \\sqrt{2}\\Lambda_{0,0} ) \\\\\n         &\\mathrm{i}(\\sqrt{6}\\Lambda_{-,0} + 2\\Lambda_{-,\\uparrow} + \\sqrt{2}\\Lambda_{0,+} ) \\\\\n         &\\mathrm{i}(2\\Lambda_{-,+} + \\sqrt{2}\\Lambda_{+,\\uparrow} ) \\\\\n         &\\mathrm{i}(\\sqrt{2}\\Lambda_{-,\\downarrow} + 2\\Lambda_{0,-} + \\sqrt{2}\\Lambda_{+,\\downarrow} ) \\\\\n         &\\mathrm{i}\\sqrt{2}(\\Lambda_{-,-} + \\sqrt{2}\\Lambda_{0,\\downarrow} + \\sqrt{3}\\Lambda_{0,0} + \\Lambda_{+,-} ) \\\\\n         &\\mathrm{i}\\sqrt{2}(\\Lambda_{-,0} + \\sqrt{3}\\Lambda_{0,-} + \\sqrt{3}\\Lambda_{0,+} + \\Lambda_{+,0} ) \\\\\n         &\\mathrm{i}\\sqrt{2}(\\Lambda_{-,+} + \\sqrt{3}\\Lambda_{0,0} + \\sqrt{2}\\Lambda_{0,\\uparrow} + \\Lambda_{+,+} ) \\\\\n         &\\mathrm{i}(\\sqrt{2}\\Lambda_{-,\\uparrow} + 2\\Lambda_{0,+}  + \\sqrt{2}\\Lambda_{+,\\uparrow} ) \\\\\n         &\\mathrm{i}(\\sqrt{2}\\Lambda_{0,\\downarrow} + 2\\Lambda_{+,-} ) \\\\\n         &\\mathrm{i}(\\sqrt{2}\\Lambda_{0,-} + 2\\Lambda_{+,\\downarrow} + \\sqrt{6}\\Lambda_{+,0} ) \\\\\n         &\\mathrm{i}(\\sqrt{2}\\Lambda_{0,0} + \\sqrt{6}\\Lambda_{+,-} + \\sqrt{6}\\Lambda_{+,+} ) \\\\\n         &\\mathrm{i}(\\sqrt{2}\\Lambda_{0,+} + \\sqrt{6}\\Lambda_{+,0} + 2\\Lambda_{+,\\uparrow} ) \\\\\n         &\\mathrm{i}(\\sqrt{2}\\Lambda_{0,\\uparrow} + 2\\Lambda_{+,+} ) \\\\        \n     \\end{aligned}$ &\n     $\\begin{aligned}                                              \n         &-2\\Lambda_{-,-} - \\sqrt{2}\\Lambda_{0,\\downarrow} \\\\\n         &2\\Lambda_{-,\\downarrow} - \\sqrt{6}\\Lambda_{-,0} - \\sqrt{2}\\Lambda_{0,-} \\\\\n         &\\sqrt{6}\\Lambda_{-,-} - \\sqrt{6}\\Lambda_{-,+} - \\sqrt{2}\\Lambda_{0,0} \\\\\n         &\\sqrt{6}\\Lambda_{-,0} - 2\\Lambda_{-,\\uparrow} - \\sqrt{2}\\Lambda_{0,+} \\\\\n         &2\\Lambda_{-,+} - \\sqrt{2}\\Lambda_{0,\\uparrow} \\\\\n         &\\sqrt{2}\\Lambda_{-,\\downarrow} - 2\\Lambda_{0,-} - \\sqrt{2}\\Lambda_{+,\\downarrow} \\\\\n         &\\sqrt{2}(\\Lambda_{-,-} + \\sqrt{2}\\Lambda_{0,\\downarrow} - \\sqrt{3}\\Lambda_{0,0} - \\Lambda_{+,-}) \\\\\n         &\\sqrt{2}(\\Lambda_{-,0} + \\sqrt{3}\\Lambda_{0,-} - \\sqrt{3}\\Lambda_{0,+} - \\Lambda_{+,0}) \\\\\n         &\\sqrt{2}(\\Lambda_{-,+} + \\sqrt{3}\\Lambda_{0,0} - \\sqrt{2}\\Lambda_{0,\\uparrow} - \\Lambda_{+,+}) \\\\\n         &\\sqrt{2}\\Lambda_{-,\\uparrow} + 2\\Lambda_{0,+} - \\sqrt{2}\\Lambda_{+,\\uparrow} \\\\\n         &\\sqrt{2}\\Lambda_{0,\\downarrow} + -2\\Lambda_{+,-} \\\\\n         &\\sqrt{2}\\Lambda_{0,-} + 2\\Lambda_{+,\\downarrow} - \\sqrt{6}\\Lambda_{+,0} \\\\\n         &\\sqrt{2}\\Lambda_{0,0} + \\sqrt{6}\\Lambda_{+,-} - \\sqrt{6}\\Lambda_{+,+} \\\\\n         &\\sqrt{2}\\Lambda_{0,+} + \\sqrt{6}\\Lambda_{+,0} - 2\\Lambda_{+,\\uparrow} \\\\\n         &\\sqrt{2}\\Lambda_{0,\\uparrow} + 2\\Lambda_{+,+} \\\\\n     \\end{aligned}$ &\n     $\\begin{aligned}                                              \n         &-6\\mathrm{i}\\Lambda_{-,\\downarrow} \\\\[.2em]\n         &-4\\mathrm{i}\\Lambda_{-,-} \\\\[.2em]\n         &-2\\mathrm{i}\\Lambda_{-,0} \\\\[.2em]\n         &0 \\\\[.2em]\n         &2\\mathrm{i}\\Lambda_{-,\\uparrow} \\\\[.2em]\n         &-4\\mathrm{i}\\Lambda_{0,\\downarrow} \\\\[.2em]\n         &-2\\mathrm{i}\\Lambda_{0,-} \\\\[.2em]\n         &0 \\\\[.2em]\n         &2\\mathrm{i}\\Lambda_{0,+} \\\\[.2em]\n         &4\\mathrm{i}\\Lambda_{0,\\uparrow} \\\\[.2em]\n         &-2\\mathrm{i}\\Lambda_{+,\\downarrow} \\\\[.2em]\n         &0 \\\\[.2em]\n         &2\\mathrm{i}\\Lambda_{+,0} \\\\[.2em]\n         &4\\mathrm{i}\\Lambda_{+,+} \\\\[.2em]\n         &6\\mathrm{i}\\Lambda_{+,\\uparrow}\n     \\end{aligned}$ \\\\ \\hline\n    \\end{tabular}\n\n\n\\end{adjustbox} \n\\end{center}\n\n\\restoregeometry\n\n\\nopagebreak[0]\n\\chapter{Carter-Penrose transformation}\n\\label{appendCPtransf}\n\\noindent\nThe metric on the Minkowski space $\\mathds R^{1,3}$ in polar coordinates is given by\n\\begin{equation}\n    \\mathrm{d} s_{Mink}^2 \\= -\\mathrm{d} t^2 + \\mathrm{d} r^2 + r^2\\,\\mathrm{d}\\Omega_2^2\\ ;\\qquad \\mathrm{d}\\Omega_2^2 \\= \\mathrm{d}\\theta^2 + \\sin^2\\theta\\,\\mathrm{d}\\phi^2 ,\n\\end{equation}\nwhere $-\\infty < t < \\infty,\\ 0 \\leq r < \\infty,\\ 0 \\leq \\theta \\leq \\pi$, and $0 \\leq \\phi \\leq 2\\pi$. We first employ the light-cone coordinates $(u,v)$ to transform the metric in the following way\n\\begin{equation}\\label{transf1}\n\\begin{aligned}\n    &u\\ :=\\ t-r\\ ,\\quad v\\ :=\\ t+r\\ ;\\quad  -\\infty < u \\leq v < \\infty\\  \\\\[2pt]\n    &\\implies \\mathrm{d} s_{Mink}^2 \\= -\\mathrm{d} v\\, \\mathrm{d} u + \\sfrac14 (v-u)^2\\,\\mathrm{d}\\Omega_2^2\\ .\n\\end{aligned}\n\\end{equation}\nIn the second step we compactify the spacetime with the help of the coordinate (U,V) as follows:\n\\begin{equation}\\label{transf2}\n\\begin{aligned}\n   &U\\ :=\\ \\arctan u\\ ,\\quad V\\ :=\\ \\arctan v\\ ;\\quad  -\\sfrac\\pi2 < U \\leq V < \\sfrac\\pi2\\  \\\\[2pt]\n    &\\implies \\mathrm{d} s_{Mink}^2 \\= \\frac{1}{4\\cos^2V\\cos^2U}\\left[-4\\,\\mathrm{d} V\\, \\mathrm{d} U +  \\sin^2(V-U)\\,\\mathrm{d}\\Omega_2^2\\right]\\ .\n\\end{aligned}\n\\end{equation}\nFinally, we rotate the coordinate system back using $(\\tau,\\chi)$ to obtain the desired form of the metric:\n\\begin{equation}\\label{transf3}\n\\begin{aligned}\n   &\\tau\\ :=\\ V + U\\ ,\\quad \\chi\\ :=\\ \\pi + U - V\\ ;\\quad  0 < \\chi \\leq \\pi\\ , |\\tau| < \\chi\\  \\\\[2pt]\n    &\\implies \\mathrm{d} s_{Mink}^2 \\= \\gamma^{-2}\\left[-\\mathrm{d} \\tau^2 + \\mathrm{d}\\chi^2 +  \\sin^2\\chi\\,\\mathrm{d}\\Omega_2^2\\right]\\ ; \\quad \\gamma \\= \\cos\\tau - \\cos\\chi\\ .\n\\end{aligned}\n\\end{equation}\nWe realize that the Minkowski metric is conformally equivalent to a Lorentzian cylinder $\\mathcal{I}\\times S^3\\ ;\\ \\mathcal{I}=(-\\pi,\\pi)$ with a conformal factor that can be recasted in terms of Minkowski coordinates using the above transformations (\\ref{transf1}-\\ref{transf3}):\n\\begin{equation}\\label{metricCyl}\n      \\mathrm{d} s_{cyl}^2\\ :=\\ -\\mathrm{d}\\tau^2 + \\mathrm{d}\\Omega_3^2 \\= \\gamma^2\\, \\mathrm{d} s_{Mink}^2\\ ;\\quad \\gamma = \\frac{2\\ell^2}{\\sqrt{4\\,t^2\\,\\ell^2 + (r^2-t^2+\\ell^2)^2}}\\ ,\n\\end{equation}\nwhere we have made $\\gamma$ dimensionless using the de Sitter radius $\\ell$. The lightcone structure of the spacetime as presented in the Penrose diagram (see Figure \\ref{CPplot}) is a direct consequence of \\eqref{transf3}.  \n\n\\chapter{Conclusion \\& Outlook}\n\\label{Chapter7} \n\nWe have studied stability behaviour of some well known solutions of $SU(2)$ Yang--Mills fields in $4$-dimensional de Sitter space $\\mathrm{d}{S}_4$ under generic gauge perturbation. These solutions could be of relevance to early time cosmology (before the electro-weak symmetry breaks down) in a scenario recently presented by Friedan \\cite{Friedan}. An $SO(4)$ symmetric sector is analytically solvable and reduces to three coupled anharmonic oscillators (for the metric, an $SU(2)$ Yang\u2013Mills field and the Higgs field, the latter being frozen to its vacuum state). We have presented a complete analysis of the linear gauge-field perturbations of the time-dependent Yang\u2013Mills solution, by diagonalizing the fluctuation operator and studying the long-time behavior of the ensuing Hill's equations using the stroboscopic map and Floquet theory. For parametrically large gauge-field energy (as is required in Friedan's setup) the natural frequencies and monodromies become universal, and some unstable perturbation modes survive even in this limit. This provides strong evidence that such oscillating cosmic Yang\u2013Mills fields are unstable against small perturbations, although we have not yet included metric fluctuations here. Their influence will be analyzed in\nfollow-up work.\n\nWe have also analyzed a family of electromagenetic knot configurations recently developed by making use of a conformal equivalence between $\\mathrm{d}{S}_4$, Minkwoski space $\\mathds R^{1,3}$ and a finite Lorentzian $S^3$-cylinder. These solutions are constructed on the cylinder in an $SO(4)$ covariant way and are then pulled back to the Minkowski space using the conformal map (that leaves the Maxwell's theory invariant). These ``basis-knot\" solutions of Maxwell's equation are labelled with $S^3$ harmonics $Y_{j,m,n}$ and give rise to field configurations of knotted field lines when pulled back to the Minkowski space. We have analyzed the symmetry feature of these basis knotted electromagnetic field configurations with the isometry group $SO(1,4)$ of the de Sitter space. We have further studied, numerically, the effect of these basis configurations on the trajectories of multiple identical charged particles with different initial conditions. Various behaviors were obtained, including a separation of trajectories into different \"solid angle regions\" that converge asymptotically into a beam of charged particles along a few particular regions of space, an ultrarelativistic acceleration of particles and coherent twists\/turns of the trajectories before they go off asymptotically. The results contribute to an effort to better understand the interactions between electromagnetic knots and charged particles \\cite{AT10}. This becomes increasingly relevant as laboratory generation of knotted\nfields progresses \\cite{LSMetal18}. Further work in this direction could be to analyze a single Fourier mode of these\nsolutions to understand its experimental realization via monochromatic laser beams.\n\nFurthermore, we have considered complex linear combinations of these basis-knot field configurations (that can model any finite-energy field configuration) and characterized the corresponding moduli space of null fields. We have also computed Noether charges of such a linearly combined configuration with fixed $j$ for the conformal group $SO(2,4)$, which is the largest symmetry group for the Maxwell theory. Here again the ``de Sitter\" method proved to be advantageous in that the expressions of these charge densities simplifies immensely on the cylinder and, thus, can be computed with ease. We found that many of these charges vanish owing to the orthogonality of the harmonics and that the energy and momentum are the only independent charged in many cases. A nice geometric structure of $1$-forms facilitated the computation of spherical components of vector charges as well. We verified our results against the results for some modified Hopf--Ran{\\~a}da field configurations of \\cite{HSS15}. We would like to further check the validity of our results by comparing them with other solutions presented in \\cite{HSS15}.\n\\chapter{Non-Abelian solution: $SU(2)$}\n\\label{Chapter6} \n\\justifying\n\nHere we present cosmic Yang--Mills solution with $SU(2)$ gauge group and study their stability behaviour. The contents of this chapter, in large parts, are taken from the published work (in Nucl. Phys. B) \\cite{KLP21}. All the graphics in this chapter is due to Gabriel Pican\\c{c}o Costa. I have been thoroughly involved at all stages of this work.\n\n\n\n\\vspace{12pt}\n\\section{SO(4)-symmetric cosmic Yang--Mills solution}\n\\vspace{2pt}\nThe Yang--Mills action on this ansatz simplifies to\n\\begin{equation}\\label{YMaction}\n   S \\= \\frac{-1}{4g^2}\\int_{{\\cal I}\\times S^3} \\!\\!\\! \\mathrm{tr}\\ {\\cal F}\\wedge *{\\cal F} \\=\n   \\frac{6\\pi^2}{g^2} \\int_{\\cal I} \\!\\mathrm{d}\\tau\\ \\bigl[ \\sfrac12\\dot\\psi^2 - V(\\psi) \\bigr]\n   \\quad\\textrm{with}\\quad V(\\psi)\\=\\sfrac12(\\psi^2{-}1)^2\\ ,\n\\end{equation}\nwhere $g$ here denotes the gauge coupling.\nDue to the principle of symmetric criticality~\\cite{Palais}, solutions to the mechanical problem \n\\begin{equation} \\label{Newton}\n   \\ddot{\\psi} + V'(\\psi) \\= 0\n\\end{equation}\nwill, via (\\ref{Aansatz}), provide Yang--Mills configurations which extremize the action.\nConservation of energy implies that\n\\begin{equation} \\label{energylaw}\n   \\sfrac12\\dot{\\psi}^2 +V(\\psi) \\= E \\= \\textrm{constant}\\ ,\n\\end{equation}\nand the generic solution in the double-well potential~$V$ is periodic in~$\\tau$ with a period~$T(E)$.\n\nHence, fixing a value for~$E$ and employing time translation invariance to set $\\dot{\\psi}(0)=0$\nuniquely determines the classical solution~$\\psi(\\tau)$ up to half-period shifts.\nIts explicit form is\n\\begin{equation} \\label{backgrounds}\n   \\psi(\\tau) \\= \n \\begin{cases}\n   \\ \\sfrac{k}{\\epsilon}\\,\\mathrm{cn}\\bigl(\\sfrac{\\tau}{\\epsilon},k\\bigr)  \n   \\qquad\\qquad\\ \\ \\,\\textrm{with}\\quad T=4\\,\\epsilon\\,K(k) \n   & \\textrm{for}\\quad \\sfrac12<E<\\infty \\\\[4pt]\n   \\ 0 \\qquad\\qquad\\qquad\\qquad\\quad\\! \\textrm{with}\\quad T=\\infty \n   & \\textrm{for}\\quad E=\\sfrac12 \\\\[4pt]\n   \\ \\pm\\sqrt{2}\\,\\mathrm{sech}\\bigl(\\sqrt{2}\\,\\tau\\bigr) \n   \\qquad\\ \\textrm{with}\\quad T=\\infty \n   & \\textrm{for}\\quad E=\\sfrac12 \\\\[4pt]\n   \\ \\pm\\sfrac{k}{\\epsilon}\\,\\mathrm{dn}\\bigl(\\sfrac{k\\,\\tau}{\\epsilon},\\sfrac1k\\bigr)  \n   \\qquad\\quad\\ \\textrm{with}\\quad T=2\\,\\sfrac{\\epsilon}{k}\\,K(\\sfrac1k) \\quad\n   & \\textrm{for}\\quad 0<E< \\sfrac12 \\\\[4pt]\n   \\ \\pm 1 \\qquad\\qquad\\qquad\\qquad \\textrm{with}\\quad T=\\pi\n   &  \\textrm{for}\\quad E=0 \n \\end{cases}\\ ,\n\\end{equation}\nwhere cn and dn denote Jacobi elliptic functions, $K$ is the complete elliptic integral of the first kind, and\n\\begin{equation}\n   2\\,\\epsilon^2 \\= 2k^2{-}1 \\= 1\/\\sqrt{2E} \\qquad\\textrm{with}\\quad\n   k=\\sfrac{1}{\\sqrt{2}},1,\\infty \\quad\\Leftrightarrow\\quad E=\\infty,\\sfrac12,0\\ .\n\\end{equation}\nFor $E{\\gg}\\frac12$, we have $k^2{\\to}\\frac12$, and the solution is well approximated by \n$\\frac{2}{\\epsilon}\\cos\\bigl(\\frac{2\\sqrt{\\pi^3}}{\\Gamma(1\/4)^2}\\frac{\\tau}{\\epsilon}\\bigr)$.\nAt the critical value of $E{=}\\sfrac12$ ($k{=}1$), the unstable constant solution coexists \nwith the celebrated bounce solution, and below it the solution bifurcates into oscillations \nin the left or right well of the double-well potential, which halfens the oscillation period. \nThe two constant minima $\\psi=\\pm1$ correspond to the vacua ${\\cal A}=0$ and ${\\cal A}=g^{-1}\\mathrm{d} g$.\nActually, the time translation freedom is broken by the finite range of~${\\cal I}$,\nso that time-shifted solutions differ in their boundary values \nand also in their values for the total energy and action.\n\\begin{figure}[h!]\n\\centering\n\\captionsetup{width=0.9\\linewidth}\n\\includegraphics[width = 0.35\\paperwidth]{psi_large.pdf} \\quad\n\\includegraphics[width = 0.35\\paperwidth]{psi_medium+.pdf} \\\\[12pt]\n\\includegraphics[width = 0.35\\paperwidth]{psi_medium-.pdf} \\quad\n\\includegraphics[width = 0.35\\paperwidth]{psi_small.pdf}\n\\caption{Plots of $\\psi(\\tau)$ over one period, for different values of $k^2$:\n$0.500001$ (top left), $0.9999999$ (top right), $1.0000001$ (bottom left) and $2$ (bottom right).}\n\\end{figure}\n\nThe corresponding color-electric and -magnetic field strengths read\n\\begin{equation}\n   {\\cal E}_a \\= {\\cal F}_{0a} \\= \\sfrac12\\dot\\psi\\,T_a \\quad\\quad\\textrm{and}\\quad\\quad\n   {\\cal B}_a \\= \\sfrac12\\varepsilon_a^{\\ bc}{\\cal F}_{bc} \\= \\sfrac12(\\psi^2{-}1)\\,T_a\\ ,\n\\end{equation}\nwhich yields a finite total energy (on the cylinder) of $6\\pi^2 E\/g^2$ and a finite action~\\cite{ILP17prl,ILP17jhep}\n\\begin{equation}\n   g^2 S[\\psi] \\= 6\\pi^2\\int_{\\cal I} \\!\\mathrm{d}\\tau\\ \\bigl[E-(\\psi^2{-}1)^2\\bigr]\n   \\=6\\pi^2\\int_{\\cal I} \\!\\mathrm{d}\\tau\\ \\bigl[\\dot{\\psi}^2-E\\bigr]\\ \\ge\\ -3\\pi^2|{\\cal I}|\\ .\n\\end{equation}\nThe energy-momentum tensor of our SO(4)-symmetric Yang--Mills solutions is readily found as\n\\begin{equation}\n   T \\= \\frac{3\\,E}{g^2 a^2} \\bigl( \\mathrm{d}\\tau^2 + \\sfrac13\\,\\mathrm{d}\\Omega_3^2 \\bigr)\\ ,\n\\end{equation}\nwhich is traceless as expected.\n\nThe Einstein equations for a closed FLRW universe with cosmological constant~$\\Lambda$\nreduce to two independent relations, which can be taken to be its trace and its time-time component.\nIn conformal time one gets, respectively,\n\\begin{equation} \\label{friedmann}\n   \\left.\\begin{cases}\n   \\quad -R+4\\,\\Lambda \\= 0 \\\\[4pt]\n   \\quad R_{\\tau\\tau}+\\sfrac12R\\,a^2-\\Lambda\\,a^2\\= \\kappa\\,T_{\\tau\\tau}\n   \\end{cases} \\right\\}\n   \\quad\\Leftrightarrow\\quad\n   \\left.\\begin{cases}\n   \\quad \\ddot{a}+W'(a)\\=0 \\\\[4pt]  \\quad \\sfrac12\\dot{a}^2+W(a) \\= \\frac{\\kappa}{2g^2}E \\ =:\\ E'\n   \\end{cases} \\right\\}\n\\end{equation}\nwith a gravitational coupling $\\kappa=8\\pi G$, \na gravitational energy $E'$\nand a cosmological potential \n\\begin{equation} \\label{gravpot}\n   W(a) \\= \\sfrac12 a^2 - \\sfrac{\\Lambda}{6} a^4\\ .\n\\end{equation}\nThe two anharmonic oscillators, with potential~$V$ for the gauge field and potential~$W$ for gravity,\nare coupled only via the balance of their conserved energies,\n\\begin{equation}\n   \\frac1\\kappa \\bigl[ \\sfrac12\\dot{a}^2+W(a)\\bigr] \\= \\frac1{2g^2} \\bigl[ \\sfrac12\\dot{\\psi}^2+V(\\psi)\\bigr]\\ ,\n\\end{equation}\nwhich is nothing but the Wheeler--DeWitt constraint~$H=0$.\n\\begin{figure}[h!]\n\\captionsetup{width=\\linewidth}\n\\centering\n\\includegraphics[width = 0.35\\paperwidth]{Wplot.pdf} \\quad\n\\includegraphics[width = 0.35\\paperwidth]{Vplot.pdf} \n\\caption{Plots of the cosmological potential $W(a)$ for $\\lambda{=}1$ and the double-well potential $V$.}\n\\end{figure}\n\nThe Friedmann equation~(\\ref{friedmann}), being a mechanical system with an inverted anharmonic\npotential~(\\ref{gravpot}), is again easily solved analytically,\n\\begin{equation} \\label{universalscale}\n   a(\\tau) \\= \\begin{cases}\n   \\ \\sqrt{\\sfrac{3}{\\Lambda}}\\,\\sfrac{1}{2\\epsilon'}\\,\n   \\sqrt{\\frac{1-\\mathrm{cn}\\bigl(\\sfrac{\\tau}{\\epsilon'},k'\\bigr)}{1+\\mathrm{cn}\\bigl(\\sfrac{\\tau}{\\epsilon'},k'\\bigr)}}\n   \\qquad\\qquad  \\textrm{with}\\quad T'=2\\,\\epsilon'K(k') \n   & \\textrm{for}\\quad \\sfrac{3}{8\\Lambda}<E'<\\infty \\\\[4pt]\n   \\ \\sqrt{\\sfrac{3}{2\\Lambda}}\\,\\tanh\\bigl(\\tau\/\\sqrt{2}\\bigr) \\qquad\\qquad\\quad\\  \\textrm{with}\\quad T'=\\infty \n   & \\textrm{for}\\quad E'=\\sfrac{3}{8\\Lambda} \\\\[4pt]\n   \\ \\sqrt{\\sfrac{3}{\\Lambda}}\\,\\sfrac{1}{2\\epsilon'}\\,\n   \\sqrt{\\frac{1-\\mathrm{dn}\\bigl(\\sfrac{k'\\tau}{\\epsilon'},\\sfrac{1}{k'}\\bigr)}\n   {1+\\mathrm{dn}\\bigl(\\sfrac{k'\\tau}{\\epsilon'},\\sfrac{1}{k'}\\bigr)}}\n   \\qquad\\quad \\textrm{with}\\quad T'=2\\,\\sfrac{\\epsilon'}{k'}K(\\sfrac{1}{k'}) \\quad\n   & \\textrm{for}\\quad 0<E'< \\sfrac{3}{8\\Lambda} \\\\[10pt]\n   \\ 0 \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\ \\; \\textrm{with}\\quad T'=\\pi\n   &  \\textrm{for}\\quad E'=0 \n   \\end{cases}\\ ,\n\\end{equation}\nwhere we abbreviated~\\footnote{\nOur ${k'}^2$ should not be confused with the dual modulus $1{-}k^2$, which is often denoted this way.}\n\\begin{equation}\n   2\\,{\\epsilon'}^2 \\= 2{k'}^2{-}1 \\= 1\/\\sqrt{\\smash{\\sfrac{8\\Lambda}{3}}\\, E'}\n   \\qquad\\textrm{so that}\\quad\n   k'=\\sfrac{1}{\\sqrt{2}},1,\\infty \\quad\\Leftrightarrow\\quad E'=\\infty,\\sfrac{3}{8\\Lambda},0\\ .\n\\end{equation}\n\\begin{figure}[h!]\n\\centering\n\\captionsetup{width=\\linewidth}\n\\includegraphics[width = 0.35\\paperwidth]{a_large.pdf} \\quad\n\\includegraphics[width = 0.35\\paperwidth]{a_medium+.pdf} \\\\[12pt]\n\\includegraphics[width = 0.35\\paperwidth]{a_medium-.pdf} \\quad\n\\includegraphics[width = 0.35\\paperwidth]{a_small.pdf}\n\\caption{Plots of $a(\\tau)$ over one lifetime, for $\\Lambda{=}1$ and different values of ${k'}^2$:\n$0.505$ (top left), $0.9999999$ (top right), $1.0000001$ (bottom left) and $1.1$ (bottom right).}\n\\end{figure}\nFor $E'{\\gg}\\frac{3}{8\\Lambda}$, we have ${k'}^2{\\to}\\frac12$, and the solution is well approximated by\n$\\sqrt{\\frac{3}{\\Lambda}}\\frac{1}{2\\epsilon'}\\tan\\bigl(\\frac{\\sqrt{\\pi^3}}{\\Gamma(1\/4)^2}\\frac{\\tau}{\\epsilon'}\\bigr)$.\nWe only listed solutions with initial value $a(0)=0$ (big bang). \nThere exist also (for $E'<\\sfrac{3}{8\\Lambda}$) bouncing solutions,\nwhere the universe attains a minimal radius \n$a_{\\textrm{min}}=\\bigl[\\frac{3}{2\\Lambda}(1+\\sqrt{1-\\smash{\\sfrac{8\\Lambda}{3}}E'})\\bigr]^{1\/2}$\nbetween infinite extension in the far past ($t{=}{-}\\infty\\leftrightarrow\\tau{=}0$) and the\nfar future ($t{=}{+}\\infty\\leftrightarrow\\tau{=}T'$). For $E'{>}0$ they are obtained by \nsending $\\mathrm{dn}\\to-\\mathrm{dn}$ in~(\\ref{universalscale}) above.\nThe quantity~$T'$ listed there is the (conformal) lifetime of the universe, \nfrom the big bang until either the big rip (for $E'>\\frac{3}{8\\Lambda}$) \nor the big crunch of an oscillating universe (for $E'<\\frac{3}{8\\Lambda}$).\nThe solution relevant to our Einstein--Yang--Mills system is entirely determined by the Newtonian energy~$E$\ncharacterizing the cosmic Yang--Mills field: above the critical value of\n\\begin{equation}\n   E_{\\textrm{crit}} \\= \\frac{2\\,g^2}{\\kappa}\\,\\frac{3}{8\\,\\Lambda}\n\\end{equation}\nthe universe expands forever (until $t_{\\textrm{max}}{=}\\infty$), \nwhile below this value it recollapses (at $t_{\\textrm{max}}{=}\\int_0^{T'}\\!\\mathrm{d}\\tau\\,a(\\tau)$).\nIt demonstrates the necessity of a cosmological constant (whose role may be played\nby the Higgs expectation value) as well as the nonperturbative nature of the cosmic Yang--Mills field,\nwhose contribution to the energy-momentum tensor is of~$O(g^{-2})$.\n\n\\vspace{12pt}\n\\section{Natural perturbation frequencies}\n\\vspace{2pt}\n\\noindent\nOur main task in this paper is an investigation of the stability of the cosmic Yang--Mills solutions\nreviewed in the previous section. For this, we should distinguish between global and local stability.\nThe former is difficult to assess in a nonlinear dynamics but clear from the outset in case of a compact\nphase space. The latter refers to short-time behavior induced by linear perturbations around\nthe reference configuration. We shall look at this firstly, in the present section and the following one. \nHere, we set out to diagonalize the fluctuation operator for our time-dependent Yang--Mills backgrounds\nand find the natural frequencies. \n\nEven though our cosmic gauge-field configurations are SO(4)-invariant, we must allow for all kinds of\nfluctuations on top of it, SO(4)-symmetric perturbations being a very special subclass of them.\nA generic gauge potential ``nearby'' a classical solution~${\\cal A}$ on ${\\cal I}\\times S^3$ can be expanded as\n\\begin{equation}\n   {\\cal A}+\\Phi \\= {\\cal A}(\\tau,g)\\ +\\ \\sum_{p=1}^3 \\Phi_0^p(\\tau,g)\\,T_p\\,\\mathrm{d}\\tau\\\n   +\\ \\sum_{a=1}^3\\sum_{p=1}^3 \\Phi_a^p(\\tau,g)\\,T_p\\,e^a(g)\n\\end{equation}\nwith, using $(\\mu)=(0,a)$,\n\\begin{equation}\n   \\Phi_\\mu^p(\\tau,g) \\= \\sum_{j,m,n} \\Phi_{\\mu|j;m,n}^p(\\tau)\\,Y_{j;m,n}(g)\\ ,\n\\end{equation}\non which we notice the following actions (suppressing the $\\tau$ and $g$ arguments),\n\\begin{equation}\n\\begin{aligned}\n   &(L_a \\Phi_\\mu^p)_{j;m,n} \\= \\Phi_{\\mu|j;m',n}^p \\bigl(L_a)^{m'}_{\\ m}\\ ,\\qquad\n   (S_a \\Phi)_0^p \\= 0\\ ,\\\\\n   &(S_a \\Phi)_b^p \\= -2\\varepsilon_{abc}\\,\\Phi_c^p \\ ,\\qquad\n   (T_a \\Phi)_\\mu^p \\= -2\\varepsilon_{apq}\\,\\Phi_\\mu^q \\ ,\n\\end{aligned}\n\\end{equation}\nwhere the $L_a$ matrix elements are determined from \\eqref{Y-action} and \\eqref{JpmAction},\nand $S_a$ are the components of the spin operator.\nThe (metric and gauge) background-covariant derivative reads\n\\begin{equation}\n   D_\\tau \\Phi \\= \\partial_\\tau \\Phi \\quad\\textrm{and}\\quad\n   D_a \\Phi \\= L_a \\Phi + [ {\\cal A}_a,\\Phi ] \\quad\\textrm{with}\\quad\n   {\\cal A}_a \\= \\sfrac12\\bigl(1+\\psi(\\tau)\\bigr)\\,T_a\\ ,\n\\end{equation}\nwhich is equivalent to\n\\begin{equation}\n   D_a\\Phi_b^p \\= L_a\\Phi_b^p - \\varepsilon_{abc}\\Phi_c^p + [ {\\cal A}_a,\\Phi_b]^p\n   \\quad\\textrm{since}\\quad \n   D_a\\,e^b \\= L_a\\,e^b -  \\varepsilon_{abc}\\,e^c \\= \\varepsilon_{abc}\\,e^c \\ .\n\\end{equation}\n\nThe background ${\\cal A}$ obeys the Coulomb gauge condition,\n\\begin{equation}\n   {\\cal A}_\\tau =0 \\qquad\\quad\\textrm{and}\\quad\\qquad L_a{\\cal A}_a = 0\\ ,\n\\end{equation}\nbut we cannot enforce these equations on the fluctuation~$\\Phi$. \nHowever, we may impose the Lorenz gauge condition,\n\\begin{equation} \\label{gauge}\n   D^\\mu \\Phi^p_\\mu \\= 0 \\qquad\\Rightarrow\\qquad\n   \\partial_\\tau \\Phi_0^p - L_a\\Phi^p_a - \\sfrac12(1{+}\\psi)(T_a\\Phi_a)^p \\= 0\\ ,\n\\end{equation}\nwhich is seen to couple the temporal and spatial components of~$\\Phi$ in general.\nWe then linearize the Yang--Mills equations around ${\\cal A}$ and obtain\n\\begin{equation}\n   D^\\nu D_\\nu \\Phi_\\mu - R_{\\mu\\nu}\\Phi^\\nu + 2[{\\cal F}_{\\mu\\nu},\\Phi^\\nu] \\= 0\n\\end{equation}\nwith the Ricci tensor\n\\begin{equation}\n   R_{\\mu 0} \\= 0 \\qquad\\quad\\textrm{and}\\quad\\qquad R_{ab} \\= 2\\delta_{ab}\\ .\n\\end{equation}\nAfter a careful evaluation, the $\\mu{=}0$ equation yields\n\\begin{equation} \\label{temporal}\n   \\bigl[\\partial_\\tau^2 - L_b L_b + 2(1{+}\\psi)^2\\bigr] \\Phi_0^p \n   - (1{+}\\psi)L_b(T_b\\Phi_0)^p - \\dot{\\psi}(T_b\\Phi_b)^p \\= 0\\ ,\n\\end{equation}\nwhile the $\\mu{=}a$ equations read\n\\begin{equation} \\label{spatial}\n\\begin{aligned}\n   \\bigl[\\partial_\\tau^2 - L_b L_b + 2(1{+}\\psi)^2{+}4\\bigr]\\Phi_a^p \n   &-(1{+}\\psi)L_b(T_b\\Phi)_a^p - L_b(S_b\\Phi)_a^p \\\\\n   &- \\sfrac12(1{+}\\psi)(2{-}\\psi)(S_bT_b\\Phi)_a^p\n   - \\dot{\\psi}(T_a\\Phi_0)^p \\= 0\\ .\n\\end{aligned}\n\\end{equation}\nIt is convenient to package the orbital, spin, isospin, and fluctuation triplets into formal vectors,\n\\begin{equation}\n   \\vec{L}=(L_a)\\ ,\\qquad \\vec{S}=(S_a)\\ ,\\qquad \\vec{T}=(T_a)\\ ,\\qquad \\vec{\\Phi}=(\\Phi_a)\\ ,\n\\end{equation}\nrespectively, but they act in different spaces, hence on different indices,\nsuch that $\\vec{S}^2=\\vec{T}^2=-8$ on~$\\Phi$. \nIn this notation, (\\ref{gauge}), (\\ref{temporal}) and~(\\ref{spatial}) take the compact form\n(suppressing the color index~$p$)\n\\begin{fleqn}[\\parindent]\n\\begin{equation}\n   \\partial_\\tau\\Phi_0 - \\vec{L}{\\cdot}\\vec{\\Phi} - \\sfrac12(1{+}\\psi)\\vec{T}{\\cdot}\\vec{\\Phi} \\= 0\\ ,\n   \\label{gauge2}\n\\end{equation}\n\\begin{equation}\n   \\bigl[\\partial_\\tau^2{-}\\vec{L}^2+2(1{+}\\psi)^2\\bigr] {\\Phi}_0 \n   - (1{+}\\psi)\\vec{L}{\\cdot}\\vec{T}\\,\\Phi_0 - \\dot{\\psi}\\,\\vec{T}{\\cdot}\\vec{\\Phi} \\= 0 \\ ,\n   \\label{mu=0}\n\\end{equation}\n\\begin{equation}\n\\begin{split}\n   \\bigl[\\partial_\\tau^2{-}\\vec{L}^2{-}\\sfrac12\\vec{S}^2{+}2(1{+}\\psi)^2\\bigr] {\\Phi}_a\n   &- (1{+}\\psi)\\vec{L}{\\cdot}\\vec{T}\\,\\Phi_a - \\vec{L}{\\cdot}(\\vec{S}\\,\\Phi)_a \\\\\n   &- \\sfrac12(1{+}\\psi)(2{-}\\psi)\\vec{T}{\\cdot}(\\vec{S}\\,\\Phi)_a - \\dot{\\psi}\\,T_a\\Phi_0 \\= 0\\ .\n\\end{split}\n\\label{mu=a}\n\\end{equation}\n\\end{fleqn}\n\n\nA few remarks are in order. \nFirst, except for the last term, (\\ref{mu=0}) is obtained from (\\ref{mu=a}) by setting $\\vec{S}=0$,\nsince $\\Phi_0$ carries no spin index. \nSecond, both equations can be recast as\n\\begin{fleqn}\n\\begin{equation} \\label{2spins}\n   \\bigl[ \\partial_\\tau^2 - \\sfrac{1{-}\\psi}{2}\\vec{L}^2 - \\sfrac{1{+}\\psi}{2}(\\vec{L}{+}\\vec{T})^2 \n   - 2(1{+}\\psi)(1{-}\\psi) \\bigr] \\Phi_0 \\= \\dot{\\psi}\\,\\vec{T}\\cdot\\vec{\\Phi}\\ ,  \n\\end{equation}\n\\begin{equation} \\label{3spins}\n\\begin{split}\n   \\bigl[ \\partial_\\tau^2 - \\sfrac{(1{-}\\psi)(2{+}\\psi)}{4}\\vec{L}^2 - \\sfrac{\\psi(1{+}\\psi)}{4}(\\vec{L}{+}\\vec{T})^2\n   + \\sfrac{\\psi(1{-}\\psi)}{4}(\\vec{L}&{+}\\vec{S})^2 - \\sfrac{(1{+}\\psi)(2{-}\\psi)}{4}(\\vec{L}{+}\\vec{T}{+}\\vec{S})^2 \\\\\n   &- 2(1{+}\\psi)(1{-}\\psi) \\bigr] \\vec{\\Phi} = \\dot{\\psi}\\vec{T}\\Phi_0\\ ,\n\\end{split}\n\\end{equation}\n\\end{fleqn}\nwhich reveals a problem of addition of three spins and a corresponding symmetry under\n\\begin{equation} \\label{symmetry}\n   \\psi\\ \\leftrightarrow\\ -\\psi\\ ,\\qquad\n   \\vec{L}\\ \\leftrightarrow\\ \\vec{L}{+}\\vec{T}{+}\\vec{S} \\quad\\quad\\textrm{and}\\quad\\quad\n   \\vec{L}{+}\\vec{S}\\ \\leftrightarrow\\ \\vec{L}{+}\\vec{T}\\ .\n\\end{equation}\nThird, for constant backgrounds $(\\dot{\\psi}{=}0)$ the temporal fluctuation $\\Phi_0$ decouples \nand may be gauged away. \nStill, the fluctuation operator in~(\\ref{3spins}) is easily diagonalized only when \nthe coefficient of one of the first three spin-squares vanishes, i.e.~for $\\vec{L}{=}0$ ($j{=}0$),\nfor the two vacua $\\psi{=}{\\pm}1$, or for the ``meron'' $\\psi{=}0$. \nThe latter case has been analyzed by Hosotani~\\cite{Hosotani}.\n\nLet us decompose the fluctuation problem (\\ref{gauge2})--(\\ref{mu=a}) into finite-dimensional blocks\naccording to a fixed value of the spin~$j\\in\\sfrac12\\mathds N$,\n\\begin{equation}\n   \\vec{L}^2\\, \\Phi^p_{\\mu|j} \\= -4\\,j(j{+}1)\\,\\Phi^p_{\\mu|j}\n\\end{equation}\nand suppress the $j$ subscript. We employ the following coupling scheme,\\footnote{\nAnother (less convenient) scheme couples $\\vec{L}{+}\\vec{S}$, then $(\\vec{L}{+}\\vec{S}){+}\\vec{T}=:\\vec{V}$.}\n\\begin{equation}\n   \\vec{L}{+}\\vec{T}=:\\vec{U} \\qquad\\textrm{then}\\qquad \\vec{U}{+}\\vec{S}=(\\vec{L}{+}\\vec{T}){+}\\vec{S}=:\\vec{V}\\ .\n\\end{equation}\nClearly, $\\vec{U}$ and $\\vec{V}$ act on $\\vec{\\Phi}$ in $su(2)$ representations \n$j\\otimes 1$ and $j\\otimes 1\\otimes 1$, respectively. On $\\Phi_0$, we must put~$\\vec{S}{=}0$\nand have just $\\vec{V}{=}\\vec{U}$ act in a $j\\otimes 1$ representation.\nCombining the coupled equations (\\ref{mu=0}) and (\\ref{mu=a}) to a single linear system for\n$(\\Phi^p_\\mu)=(\\Phi^p_0,\\Phi^p_a)$, we get a $12(2j{+}1)\\times 12(2j{+}1)$ fluctuation matrix~$\\Omega^2_{(j)}$,\n\\begin{equation} \\label{fluctop}\n   \\bigl[ \\delta_{\\mu\\nu}^{pq}\\,\\partial_\\tau^2\\ +\\ (\\Omega^2_{(j)})_{\\mu\\nu}^{pq} \\bigr]\\,\\Phi_\\nu^q \\= 0\\ .\n\\end{equation}\nActually, there is an additional overall $(2j{+}1)$-fold degeneracy present due to the trivial action of\nthe $su(2)_{\\textrm{R}}$ generators~$R_a$, which plays no role here and will be suppressed.\nRoughly speaking, the $3(2j{+}1)$ modes of $\\Phi_0$ are related to gauge modes,\\footnote{\nStrictly, they are gauge modes only when $\\dot\\psi{=}0$. \nOtherwise, the gauge modes are mixtures with the $\\Phi_a$ modes.}\nand we still must impose the gauge condition~(\\ref{gauge2}), which also has $3(2j{+}1)$ components.\nTherefore, a subspace of dimension $6(2j{+}1)$ inside the space of all fluctuations will \nrepresent the physical gauge-equivalence classes in the end.\n\nOur goal is to diagonalize the fluctuation operator~(\\ref{fluctop}) for a given fixed value of~$j$.\nIt has a block structure,\n\\begin{equation} \\label{block}\n   \\Omega^2_{(j)} \\= \\begin{pmatrix} \\bar{N} & -\\dot{\\psi}\\,T^\\top \\\\[6pt] -\\dot{\\psi}\\,T & N \\end{pmatrix}\\ ,\n\\end{equation}\nwhere $\\bar{N}$ and $N$ are given by the left-hand sides of (\\ref{2spins}) and~(\\ref{3spins}), respectively.\nWe introduce a basis where $\\vec{U}^2$, $\\vec{V}^2$ and $V_3$ are diagonal, i.e.\n\\begin{equation}\n   \\vec{U}^2\\,|uvm\\> \\= -4\\,u(u{+}1)\\,|uvm\\> \\quad\\quad\\textrm{and}\\quad\\quad \n   \\vec{V}^2\\,|uvm\\> \\= -4\\,v(v{+}1)\\,|uvm\\>\\ ,\n\\end{equation}\nwith $m=-v,\\ldots,v$ and denote the irreducible $su(2)_v$ representations with those quantum numbers as $\\rep{v}{u}$.\nOn the $\\Phi_0$ subspace, $u$ is redundant since $u{=}v$ as $\\vec{S}{=}0$.\nWorking out the tensor products, we encounter the values\n\\begin{equation} \\label{uvreps}\n\\begin{aligned}\n   \\rep{v}{u} &\\= \\rep{j{-}2}{j{-}1}\\ ;\\!\\!\\!&\\rep{j{-}1}{j{-}1}&\\ ,\\ \\rep{j{-}1}{j}\\ ;\\ \\rep{j}{j{-}1}\\ ,\n   \\!\\!\\!&\\rep{j}{j}&\\ ,\\ \\rep{j}{j{+}1}\\ ;\\ \\rep{j{+}1}{j}\\ ,\\!\\!\\!&\\rep{j{+}1}{j{+}1}&\\ ;\\ \\rep{j{+}2}{j{+}1} \n   &\\quad\\textrm{on}\\ \\vec{\\Phi}\\ ,& \\\\\n   \\rep{v}{u} &\\= &\\rep{j{-}1}{j{-}1}&\\ ; &\\rep{j}{j}&\\ ; &\\rep{j{+}1}{j{+}1}&\n   &\\quad\\textrm{on} \\ \\Phi_0&\\ ,\n\\end{aligned}\n\\end{equation}\nwith some representations obviously missing for $j{<}2$.\n\nLet us treat the $\\dot{\\psi}\\,T$ term in~(\\ref{block}) as a perturbation and momentarily put it to zero,\nso that $\\Omega^2_{(j)}$ is block-diagonal for the time being.\nThen, it is easy to see from (\\ref{2spins}) and (\\ref{3spins}) that $[\\vec{V},\\bar{N}]=[\\vec{U},\\bar{N}]=0$\nand $[\\vec{V},N]=0$, even though $[\\vec{U},N]\\neq0$ because $(\\vec{L}{+}\\vec{S})^2$ \nis not diagonal in our basis. Therefore, we have a degeneracy in~$m$.\nFurthermore, both $\\bar{N}$ and $N$ decompose into at most three respectively five blocks \nwith fixed values of~$v$ ranging from $j{-}2$ to~$j{+}2$ and separated by semicolons in~(\\ref{uvreps}). \nMoreover, the $\\bar{N}$~blocks are irreducible and trivially also carry a value of~$u{=}v$. \nIn contrast, $N$ is not {\\it simply\\\/} reducible; its $\\vec{V}$ representations have multiplicity one, two or three. \nOnly the $N$~blocks with extremal $v$~values in~(\\ref{uvreps}) are irreducible. \nThe other ones are reducible and contain more than one $\\vec{U}$~representation, \nhence the $u$-spin distinguishes between their (two or three) irreducible $v$~subblocks.\nThe only non-diagonal term in~$N$ is the $(\\vec{L}{+}\\vec{S})^2$ contribution, which couples different copies\nof the same $v$-spin to each other, but of course not to any $u{=}v$ block of~$\\bar{N}$, \nand does not lift the $V_3{=}m$~degeneracy. \nAs a consequence, the unperturbed fluctuation equations \nfor $\\Phi_0{=}\\Phi_{(\\bar{v})}$ and $\\vec{\\Phi}{=}\\Phi_{(v,\\alpha)}$ take the form \n(suppressing the $m$ index)\n\\begin{fleqn}\n\\begin{equation}\n\\begin{split}\n   \\mathbbm{1}_{(\\bar{v})}\\bigl[ \\partial_\\tau^2\\ +\\ \\bar\\omega^2_{(\\bar{v})}\\bigr]\\,\\Phi_{(\\bar{v})} \\= 0 \\qquad\\quad\\textrm{and}\\quad\\qquad\n   \\mathbbm{1}_{(v)}\\bigl[ \\partial_\\tau^2\\ +\\ \\omega^2_{(v,\\alpha)}\\bigr]\\,\\Phi_{(v,\\alpha)} \\= 0 \\\\[4pt]\n   \\quad\\textrm{for}\\quad \\dot{\\psi}=0 \\quad\\textrm{with}\\quad \\bar{v} \\in \\{ j{-}1,\\ j,\\ j{+}1\\} \\quad\\textrm{and}\\quad\n   v \\in \\{ j{-}2,\\ j{-}1,\\ j,\\ j{+}1,\\ j{+}2 \\} \\ ,\n\\end{split}\n\\end{equation}\n\\end{fleqn}\nwhere $\\mathbbm{1}_{(v)}$ denotes a unit matrix of size~$2v{+}1$, and $\\alpha$ counts the multiplicity \nof the $v$-spin representation in~$N$ (between one and three). \nAccording to~(\\ref{2spins}) the unperturbed frequency-squares for $\\bar{N}$ are the eigenvalues\n\\begin{equation}\n   \\bar{\\omega}^2_{(\\bar{v})} \\= \n   2(1{-}\\psi)\\,j(j{+}1) + 2(1{+}\\psi)\\,\\bar{v}(\\bar{v}{+}1) - 2(1{+}\\psi)(1{-}\\psi)\n\\end{equation}\nwith multiplicity  $2\\bar{v}{+}1$, hence we get\n\\begin{equation}\n\\begin{aligned}\n   \\bar{\\omega}^2_{(j-1)} &\\= 2\\,\\psi^2-4j\\,\\psi+2(2j^2{-}1)\\ ,\\\\\n   \\bar{\\omega}^2_{(j)} &\\= 2\\,\\psi^2+2(2j^2{+}2j{-}1)\\,\\\\\n   \\bar{\\omega}^2_{(j+1)} &\\= 2\\,\\psi^2+4(j{+}1)\\,\\psi +2(2j^2{+}4j{+}1)\\ .\n\\end{aligned}\n\\end{equation}\nConsidering $N$ in (\\ref{3spins}), we can read off the eigenvalues at $v=j{\\pm}2$\nbecause in these two extremal cases $(\\vec{L}{+}\\vec{S})^2=\\vec{U}^2$ \nis already diagonal in the $\\bigl\\{ |uvm\\>\\bigr\\}$ basis. For the other $v$-values\nwe must diagonalize a $2{\\times}2$ or $3{\\times}3$ matrix to find\n\\begin{equation}\n\\begin{aligned}\n   \\omega^2_{(j-2)} &\\= \\textrm{root of}\\ Q_{j-2}(\\lambda)\n   \\= -2(2j{-}1)\\,\\psi + 2(2j^2{-}2j{+}1)\\ ,\\\\\n   \\omega^2_{(j-1,\\alpha)} &\\= \\textrm{two roots of}\\ Q_{j-1}(\\lambda)\\ ,\\\\\n   \\omega^2_{(j,\\alpha)} &\\= \\textrm{three roots of}\\ Q_j(\\lambda)\\ ,\\\\\n   \\omega^2_{(j+1,\\alpha)} &\\= \\textrm{two roots of}\\ Q_{j+1}(\\lambda)\\ ,\\\\\n   \\omega^2_{(j+2)} &\\= \\textrm{root of}\\ Q_{j+2}(\\lambda)\n   \\= 2(2j{+}3)\\,\\psi + 2(2j^2{+}6j{+}5)\\ ,\n\\end{aligned}\n\\end{equation}\neach with multiplicity $2v{+}1$, where $Q_v$ denotes a linear, quadratic or cubic polynomial.\\footnote{\nFor $j{<}2$ some obvious modifications occur due to the missing of $v{<}0$ representations.}\n\nLet us now turn on the perturbation $\\dot{\\psi}\\,T$, which couples $N$ with~$\\bar{N}$,\nand consider the characteristic polynomial~${\\cal P}_j(\\lambda)$ of our fluctuation problem,\n\\begin{equation}\n\\begin{aligned}\n   {\\cal P}_j(\\lambda) &\\ :=\\\n   \\det\\,\\Bigl(\\begin{smallmatrix} \\bar{N}{-}\\lambda & -\\dot{\\psi}T^\\top \\\\[4pt] \n   -\\dot{\\psi}T & N{-}\\lambda \\end{smallmatrix} \\Bigr)\n   \\= \\det (N{-}\\lambda) \\cdot \\det\\bigl[ (\\bar{N}{-}\\lambda) - \\dot{\\psi}^2\\,T^\\top (N{-}\\lambda)^{-1} T \\bigr] \\\\[4pt]\n   &\\= \\bigl[{\\textstyle\\prod}_v \\det (N_{(v)}{-}\\lambda)\\bigr] \\cdot \n   \\det\\bigl[ (\\bar{N}{-}\\lambda) - \\dot{\\psi}^2\\, T^\\top \\{{\\textstyle\\bigoplus}_{v} (N_{(v)}{-}\\lambda)^{-1}\\}\\,T\\bigr] \\ ,\n\\end{aligned}\n\\end{equation}\nwhere we made use of\n\\begin{equation}\n\\< u\\,v\\,m|\\,N\\,|u'v'm'> \\= \\bigl(N_{(v)}\\bigr)_{uu'}\\,\\delta_{vv'}\\delta_{mm'}\\ .\n\\end{equation}\nSince $T$ furnishes an $su(2)$ representation (and not an intertwiner) \nit must be represented by square matrices and thus cannot connect different $v$ representations.\nHence the perturbation does not couple different $v$ sectors but only links $N$ and $\\bar{N}$\nin a common $\\bar{v}{=}v$~sector. Therefore, it does not affect the extremal sectors $v=j{\\pm}2$.\nMoreover, switching to a diagonal basis $\\{|\\alpha vm\\>\\}$ for $N$ we can simplify to\n\\begin{equation}\n\\begin{aligned}\n   T^\\top \\{{\\textstyle\\bigoplus}_{v} (N_{(v)}{-}\\lambda)^{-1}\\}\\,T\\bigr]\n   &\\= {\\textstyle\\bigoplus}_{\\bar{v}}\\{T^\\top (N{-}\\lambda)^{-1} T\\}_{(\\bar{v})} \\\\\n   &\\={\\textstyle\\bigoplus}_{\\bar{v}}\\Bigl\\{ {\\textstyle\\sum}_\\alpha \n   (\\omega^2_{(\\bar{v},\\alpha)}{-}\\lambda)^{-1} \\bigl(T^\\top|\\alpha\\>\\!\\<\\alpha|\\,T\\bigr)_{(\\bar{v})}\\Bigr\\}\\ .\n\\end{aligned}\n\\end{equation}\nObserving that \n$\\bigl(T^\\top|\\alpha\\>\\!\\<\\alpha|\\,T\\bigr)_{(\\bar{v})}=\n-t_{\\bar{v},\\alpha}\\bigl(\\vec{T}^2\\bigr)_{(\\bar{v})}=8\\,t_{\\bar{v},\\alpha}\\mathbbm{1}_{(\\bar{v})}$\nwith some coefficient functions $t_{\\bar{v},\\alpha}(\\psi)$, with $\\sum_\\alpha t_{\\bar{v},\\alpha}=1$,\nwe learn that the $V_3$ degeneracy remains intact and arrive at ($\\bar{v}\\in\\{j{-}1,j,j{+}1\\}$)\n\\begin{equation} \\label{charP}\n\\begin{aligned}\n   {\\cal P}_j(\\lambda) &\\= \\bigl[{\\textstyle\\prod}_v Q_v(\\lambda)^{2v+1}\\bigr] \\cdot\n   {\\textstyle\\prod}_{\\bar{v}} \\bigl\\{(\\bar{\\omega}^2_{(\\bar{v})}{-}\\lambda) \n   - 8\\dot{\\psi}^2 {\\textstyle\\sum}_\\alpha t_{\\bar{v},\\alpha} \n   (\\omega^2_{(\\bar{v},\\alpha)}{-}\\lambda)^{-1} \\bigr\\}^{2\\bar{v}+1}\\\\[4pt]\n   &\\= (\\omega^2_{(j-2)}{-}\\lambda)^{2j-3}\\cdot(\\omega^2_{(j+2)}{-}\\lambda)^{2j+5}\\cdot\n   {\\textstyle\\prod}_{\\bar{v}} \\bigl\\{ (\\bar{\\omega}^2_{(\\bar{v})}{-}\\lambda)\\,Q_{\\bar{v}}(\\lambda) - 8\\dot{\\psi}^2 P_{\\bar{v}}(\\lambda) \\bigr\\}^{2\\bar{v}+1} \\\\[4pt]\n   &\\= (\\omega^2_{(j-2)}{-}\\lambda)^{2j-3}\\cdot(\\omega^2_{(j+2)}{-}\\lambda)^{2j+5}\\cdot\n   {\\textstyle\\prod}_{\\bar{v}} R_{\\bar{v}}(\\lambda)^{2\\bar{v}+1}\\ ,\n\\end{aligned}\n\\end{equation}\nwhere $P_{\\bar{v}}=Q_{\\bar{v}}\\sum_\\alpha t_{\\bar{v},\\alpha} (\\omega^2_{(\\bar{v},\\alpha)}{-}\\lambda)^{-1}$\nis a polynomial of degree one less than $Q_{\\bar{v}}$ since all poles cancel, and $R_{\\bar{v}}$ is a\npolynomial of one degree more.\nWe list the polynomials $Q_v$, $P_{\\bar{v}}$ and $R_{\\bar{v}}$ for $j{\\le}2$ in the Appendix.\n\nTo summarize, by a successive basis change ($m'=-j,\\ldots,j$ and $m=-v,\\ldots,v$)\n\\begin{equation}\n   \\bigl\\{ |\\mu\\,p\\,m'\\> \\bigr\\} \\quad\\Rightarrow\\quad \n   \\bigl\\{ |\\bar{v} m\\>, |u v m\\> \\bigr\\} \\quad\\Rightarrow\\quad\n   \\bigl\\{ |\\bar{v} m\\>, |\\alpha v m\\> \\bigr\\} \\quad\\Rightarrow\\quad\n   \\bigl\\{ |\\beta v m\\> \\bigr\\}\n\\end{equation}\nwe have diagonalized~(\\ref{fluctop}) to\n\\begin{equation} \\label{diagonalized}\n   \\bigl[ \\partial_\\tau^2 - \\Omega^2_{(j,v,\\beta)} \\bigr] \\, \\Phi_{(v,\\beta)} \\= 0\n   \\qquad\\textrm{with}\\quad v\\in\\{j{-}2,j{-}1,j,j{+}1,j{+}2\\} \\ ,\n\\end{equation}\nwhere $\\Omega^2_{(j,v,\\beta)}$ are the distinct roots of the characteristic polynomial~${\\cal P}_j$ in~(\\ref{charP}),\nand (for $j{\\ge}2$) the multiplicity $\\beta$ takes $1,3,4,3,1$ values, respectively:\n\\begin{equation}\n   \\Omega^2_{(j,j\\pm2)} = \\omega^2_{(j\\pm2)}\\ ,\\quad\n   \\Omega^2_{(j,j\\pm1,\\beta)} = \\textrm{three roots of}\\ R_{j\\pm1}(\\lambda)\\ ,\\quad\n   \\Omega^2_{(j,j,\\beta)} = \\textrm{four roots of}\\ R_{j}(\\lambda)\\ .\n\\end{equation}\nThe reflection symmetry~(\\ref{symmetry}) implies that \n$\\Omega^2_{(v,j,\\cdot)}(\\psi)=\\Omega^2_{(j,v,\\cdot)}(-\\psi)$.\nFor $j{<}2$, obvious modifications occur due to the absence of some $v$~representations.\n\nWe still have to discuss the gauge condition~(\\ref{gauge2}), which can be cast into the form\n\\begin{equation}\n   0 \\= \\mbox{$\\partial$}_\\tau\\Phi_0 - \\bigl[ \\sfrac12(1{-}\\psi)\\vec{L}+\\sfrac12(1{+}\\psi)\\vec{U}\\bigr]\\cdot\\vec\\Phi\n   \\= \\mbox{$\\partial$}_\\tau\\Phi_{(\\bar{v},\\bar{m})} - K_{\\bar{v},\\bar{m}}^{\\ v,m,\\alpha}(\\psi)\\,\\Phi_{(v,m,\\alpha)}\n\\end{equation}\nwith a $3(2j{+}1){\\times}7(2j{+}1)$ linear (in~$\\psi$) matrix function~$K$.\\footnote{\nWe have to bring back the $m$ indices because the gauge condition is not diagonal in them.}\nHere the $v$~sum runs over $(j{-}1,j,j{+}1)$ only,\nsince the gauge condition~(\\ref{gauge2}) has components only in the middle three $v$~sectors,\nlike the gauge-mode equation~(\\ref{mu=0}).\nIt does not restrict the extremal $v$~sectors~$v=j{\\pm}2$, since these fluctuations\ndo not couple to the gauge sector~$\\Phi_0$ and are entirely physical.\nFor the middle three $v$~sectors (labelled by~$\\bar{v}$), the $\\dot{\\psi}\\,T$ perturbation leads to\na mixing of the $N$ modes with the $\\bar{N}$ gauge modes, so their levels will avoid crossing. \nPerforming the corresponding final basis change, the gauge condition takes the form\n\\begin{equation} \\label{gauge4}\n   \\bigl[ L_{\\bar{v},\\bar{m}}^{\\ \\bar{v}'\\!,\\bar{m}'\\!,\\beta}(\\psi)\\,\\mbox{$\\partial$}_\\tau \n   - M_{\\bar{v},\\bar{m}}^{\\ \\bar{v}'\\!,\\bar{m}'\\!,\\beta}(\\psi) \\bigr]\\,\\Phi_{(\\bar{v}'\\!,\\bar{m}'\\!,\\beta)}\\=0\n\\end{equation}\nwith certain $3(2j{+}1){\\times}10(2j{+}1)$ matrix functions $L$ and~$M$.\nThis linear equation represents conditions on the normal mode functions $\\Phi_{(\\bar{v},\\bar{m},\\beta)}$\nand defines a $7(2j{+}1)$-dimensional subspace of physical fluctuations, \nwhich of course still contains a $3(2j{+}1)$-dimensional subspace of gauge modes. \nFor $j{<}1$, these numbers are systematically smaller.\nTogether with the two extremal $v$~sectors, we end up with $(7-3+2)(2j{+}1)=6(2j{+}1)$ \nphysical degrees of freedom for any given value of~$j({\\ge}2)$, as advertized earlier.\n\nWe conclude this section with more details for the simplest examples,\nwhich are constant backgrounds and $j{=}0$ backgrounds.\nFor the vacuum background, say $\\psi=-1$, which is isospin degenerate, one gets\n\\begin{equation}\n   \\bigl( \\partial_\\tau^2 -\\sfrac12\\vec{L}^2 -\\sfrac12(\\vec{L}{+}\\vec{S})^2 \\bigr)\\,\\vec{\\Phi} \\=0, \\qquad\n   \\vec{L}{\\cdot}\\vec{\\Phi} = 0\\ ,\\qquad \\Phi_0 =0\\ .\n\\end{equation}\nIt yields the positive eigenfrequency-squares\n\\begin{equation}\n   \\omega^2_{(j,u')} \\= 2j(j{+}1) + 2u'(u'{+}1) \\=\n   \\begin{cases}\n   \\ 4j^2 \\ \\textrm{at}\\ j{{\\ge}1} & \\quad\\textrm{for}\\quad u'=j{-}1 \\\\ \n   \\ 4j(j{+}1) & \\quad\\textrm{for}\\quad u'=j \\\\\n   \\ 4(j{+}1)^2 & \\quad\\textrm{for}\\quad u'=j{+}1 \n\\end{cases}\n\\end{equation}\nfor $j=0,\\sfrac12,1,\\ldots$,\nbut the $\\vec{L}{\\cdot}\\vec{\\Phi}=0$ constraint removes the $u'{=}j$ modes.\nClearly, all (constant) eigenfrequency-squares are positive, hence the vacuum is stable.\n\nFor the ``meron'' background, $\\psi\\equiv0$, one has\n\\begin{equation}\n   \\bigl( \\partial_\\tau^2 -\\sfrac12\\vec{L}^2 -\\sfrac12(\\vec{L}{+}\\vec{T}{+}\\vec{S})^2 -2 \\bigr)\\,\\vec{\\Phi} \\=0, \\qquad\n   \\bigl(\\vec{L}+\\sfrac12\\vec{T}\\bigr)\\cdot\\vec{\\Phi} = 0\\ ,\\qquad \\Phi_0 =0\\ .\n\\end{equation}\nIn this case, we read off \n\\begin{equation}\n   \\omega^2_{(j,v)} +2 \\= 2j(j{+}1) + 2v(v{+}1) \\=\n   \\begin{cases}\n   \\ 4(j^2{-}j{+}1)  & \\quad\\textrm{for}\\quad v=j{-}2 \\qquad (0\\ \\textrm{to}\\ 1\\times) \\\\ \n   \\ 4j^2 & \\quad\\textrm{for}\\quad v=j{-}1 \\qquad (0\\ \\textrm{to}\\ 2\\times) \\\\\n   \\ 4j(j{+}1) & \\quad\\textrm{for}\\quad v=j \\qquad\\quad\\  (1\\ \\textrm{to}\\ 3\\times) \\\\\n   \\ 4(j{+}1)^2 & \\quad\\textrm{for}\\quad v=j{+}1 \\qquad (1\\ \\textrm{to}\\ 2\\times) \\\\\n   \\ 4(j^2{+}3j{+}3)  & \\quad\\textrm{for}\\quad v=j{+}2 \\qquad (1\\times)\n\\end{cases}\\ ,\n\\end{equation}\nbut the constraint removes one copy from each of the three middle cases (and less when $j{<}1$).\nWe end up with a spectrum $\\{\\omega^2\\}=\\{-2,1,6,7,10,\\ldots\\}$ with certain degeneracies~\\cite{Hosotani}.\nThe single non-degenerate negative mode $\\omega^2_{(0,0)}{=}{-}2$ is a singlet, $\\Phi_a^p=\\delta_a^p\\phi(\\tau)$, \nand it corresponds to rolling down the local maximum of the double-well potential. \nThe meron is stable against all other perturbations.\n\nFor a time-varying background, the natural frequencies $\\Omega_{(j,v,\\beta)}$\ninherit a $\\tau$ dependence from the background~$\\psi(\\tau)$.\nDirect diagonalization is still possible for $j{=}0$, where we should solve\n\\begin{equation}\n\\begin{aligned}\n   & \\partial_\\tau\\Phi_0 - \\sfrac12(1{+}\\psi)\\vec{T}{\\cdot}\\vec{\\Phi} \\= 0 \\ ,\\\\[4pt]\n   & \\bigl[\\partial_\\tau^2+2(1{+}\\psi)^2\\bigr] {\\Phi}_0 - \\dot{\\psi}\\,\\vec{T}{\\cdot}\\vec{\\Phi} \\= 0\\ ,\\\\[4pt]\n   & \\bigl[ \\partial_\\tau^2 +2(3\\psi^2{-}1) -\\sfrac14(1{+}\\psi)(2{-}\\psi)(\\vec{S}{+}\\vec{T})^2 \\bigr]\\ \n   \\vec\\Phi - \\dot{\\psi}\\,\\vec{T}\\,\\Phi_0 \\= 0\\ ,\n\\end{aligned}\n\\end{equation}\nwith\n\\begin{equation}\n   (\\vec{S}{+}\\vec{T})^2\\=\\vec{V}^2\\=-4\\,v(v{+}1)\\=0,-8,-24 \\qquad\\textrm{for}\\quad v=0,1,2\\ .\n\\end{equation}\nIt implies the unperturbed frequencies (suppressing the $j$ index)\n\\begin{equation} \\label{unperturbed}\n\\begin{aligned}\n   \\bar\\omega_{(1)}^2 &= 2(\\psi{+}1)^2\\ \\ (3\\times)\\ ,\\quad\n   \\omega_{(0)}^2 = 2(3\\psi^2{-}1)\\ \\ (1\\times)\\ ,\\\\\n   \\omega_{(1)}^2 &= 2(2\\psi^2{+}\\psi{+}1)\\ \\ (3\\times) ,\\quad\n   \\omega_{(2)}^2 = 2(3\\psi{+}5)\\ \\ (5\\times)\n\\end{aligned}\n\\end{equation}\nfor\n\\begin{equation}\n\\begin{aligned}\n   (\\Phi_0)^p &\\equiv \\bigl(\\Phi_{(\\bar{v}=1)}\\bigr)^p\\ =:\\ \\delta^{pb}\\bar\\phi_b \\ ,\\\\\n   (\\vec\\Phi)_a^p \\ &\\equiv \\bigl(\\Phi_{(0)}+\\Phi_{(1)}+\\Phi_{(2)}\\bigr)^p_a\\ =:\\\n   \\phi\\,\\delta_a^p + \\epsilon^p_{\\ ab}\\,\\phi_b + (\\phi_{(ab)}{-}\\delta_{ab}\\phi)\\delta^{bp}\\ ,\n\\end{aligned}\n\\end{equation}\nas long as $\\dot{\\psi}$ is ignored.\nThere are no $v$-spin multiplicities (larger than one) here.\nTurning on $\\dot{\\psi}$ and observing that $(\\vec{T}{\\cdot}\\vec\\Phi)^p\\sim\\delta^{pb}\\phi_b$, \nthe characteristic polynomial of the coupled $12{\\times}12$ system in the $|uvm\\>$ basis reads\n\\begin{equation}\n   {\\cal P}_0(\\lambda) \\= \\det \\begin{pmatrix}\n   (\\bar{\\omega}_{(1)}^2{-}\\lambda)\\mathbbm{1}_3 & \n   0 & -\\dot{\\psi}\\,T_{(1)}^\\top & 0 \\\\[4pt]\n   0 & (\\omega_{(0)}^2{-}\\lambda)\\mathbbm{1}_1 & 0 & 0 \\\\[4pt]\n   -\\dot{\\psi}\\,T_{(1)} & 0 & (\\omega_{(1)}^2{-}\\lambda)\\mathbbm{1}_3 & 0 \\\\[4pt]\n   0 & 0 & 0 & (\\omega_{(2)}^2{-}\\lambda)\\mathbbm{1}_5 \n\\end{pmatrix}\\ .\n\\end{equation}\nSpecializing the general discussion above to $j{=}0$, we find just $t_1{=}1$ so that $P_1{=}1$ and arrive at\n\\begin{equation}\n\\begin{aligned}\n   {\\cal P}_0(\\lambda) &\\= (\\omega_{(0)}^2{-}\\lambda)^1\n   (\\omega_{(1)}^2{-}\\lambda)^3 (\\omega_{(2)}^2{-}\\lambda)^5\n   \\bigl[(\\bar{\\omega}_{(1)}^2{-}\\lambda) - 8\\dot{\\psi}^2(\\omega_{(1)}^2{-}\\lambda)^{-1} \\bigr]^3 \\\\\n   &\\= (\\omega_{(0)}^2{-}\\lambda) (\\omega_{(2)}^2{-}\\lambda)^5\n   \\bigl\\{ (\\bar{\\omega}_{(\\bar{1})}^2{-}\\lambda)(\\omega_{(1)}^2{-}\\lambda) - 8\\dot{\\psi}^2 \\bigr\\}^3 \\ .\n\\end{aligned}\n\\end{equation}\nWe see that the frequencies $\\Omega_{(0)}^2{=}\\omega_{(0)}^2$ and $\\Omega_{(2)}^2{=}\\omega_{(2)}^2$ \nare unchanged and given by~(\\ref{unperturbed}), while the gauge mode $\\bar{\\omega}_{(\\bar{1})}^2$ gets\nentangled with the (unphysical) $v{=}1$ mode to produce the pair\n\\begin{equation}\n\\begin{aligned}\n   \\Omega^2_{(1,\\pm)} &\\= \\sfrac12(\\bar\\omega_{(\\bar{1})}^2{+}\\omega_{(1)}^2)\n   \\,\\pm\\sqrt{\\sfrac14(\\bar\\omega_{(\\bar{1})}^2{+}\\omega_{(1)}^2)^2-\\bar\\omega_{(\\bar{1})}^2\\omega_{(1)}^2+8\\dot{\\psi}^2}  \\\\\n   &\\= 3\\psi^2{+}3\\psi{+}2\\, \\pm\\sqrt{\\psi^2(\\psi{-}1)^2+8\\dot{\\psi}^2}\n\\end{aligned}\n\\end{equation}\nwith a triple degeneracy. There are avoided crossings at $\\psi{=}0$ and $\\psi{=}1$.\nRemoving the unphysical and gauge modes in pairs, we remain with the singlet mode\n$\\Omega_{(0,0)}^2$ and the fivefold-degenerate $\\Omega_{(0,2)}^2$. \nFor all higher spins $j{>}0$, analytic expressions for the natural frequencies $\\Omega_{(j,v,\\beta)}$\nnow require merely solving a few polynomial equations of order four at worst.\nWe have done so up to $j{=}2$ and list them in the Appendix \nbut refrain from giving further explicit examples here.\n\\begin{figure}[h!]\n\\centering\n\\captionsetup{width=0.9\\linewidth}\n\\includegraphics[width = 0.35\\paperwidth]{om2zero_51.pdf} \\quad\n\\includegraphics[width = 0.35\\paperwidth]{om2zero_99.pdf} \\\\[12pt]\n\\includegraphics[width = 0.35\\paperwidth]{om2zero_01.pdf} \\quad\n\\includegraphics[width = 0.35\\paperwidth]{om2zero_50.pdf}\n\\caption{Plots of $\\Omega^2_{(0,v,\\beta)}(\\tau)$ over one period, for different values of $k^2$: \n$0.51$ (top left), $0.99$ (top right), $1.01$ (bottom left) and $5$ (bottom right).}\n\\end{figure}\n\\begin{figure}[h!]\n\\centering\n\\captionsetup{width=0.9\\linewidth}\n\\includegraphics[width = 0.35\\paperwidth]{om2two_505.pdf} \\quad\n\\includegraphics[width = 0.35\\paperwidth]{om2two_550.pdf} \\\\[12pt]\n\\includegraphics[width = 0.35\\paperwidth]{om2two_999.pdf} \\quad\n\\includegraphics[width = 0.35\\paperwidth]{om2two_001.pdf}\n\\caption{Plots of $\\Omega^2_{(2,v,\\beta)}(\\tau)$ over one period, for different values of $k^2$: \n$0.505$ (top left), $0.550$ (top right), $0.999$ (bottom left) and $1.001$ (bottom right).}\n\\end{figure}\nFrom the cases of $j{=}0$ and $j{=}2$ displayed below one can see that some of the normal modes\ndip into the negative regime, i.e.~their frequency-squares become negative, for a certain fraction of the time~$\\tau$.\nBecause of this and, quite generally, due to the $\\tau$ variability of the natural frequencies,\nit is not easy to predict the long-term evolution of the fluctuation modes.\nClearly, the stability of the zero solution $\\Phi{\\equiv}0$, \nequivalent to the linear stability of the background Yang--Mills configuration, \nis not simply decided by the sign of the $\\tau$-average of the corresponding frequency-square.\n\n\\vspace{12pt}\n\\section{Stability analysis: stroboscopic map and Floquet theory}\n\\vspace{2pt}\n\\noindent\nThe diagonalized linear fluctuation equation~(\\ref{diagonalized}) represents\na bunch of Hill's equations, where the frequency-squared is a root of a polynomial of order up to four\nwith coefficients given by a polynomial of twice that order in Jacobi elliptic functions.\nA unique solution requires fixing two initial conditions, and so\nfor each fluctuation~$\\Phi_{(j,v,\\beta)}$ there is a two-dimensional solution space.\nIt is well known that Hill's equation, e.g.~in the limit of Mathieu's equation, displays parametric resonance phenomena,\nwhich can stabilize otherwise unstable systems or destabilize otherwise stable ones.\n\nFor oscillating dynamical systems with periodically varying frequency, \nthere exist some general tools to analyze linear stability.\nSwitching to a Hamiltonian picture and to phase space, \nit is convenient to transform the second-order differential equation into\na system of two coupled first-order equations (suppressing all quantum numbers),\n\\begin{equation} \\label{first-order}\n   \\bigl[\\partial^2_\\tau-\\Omega^2(\\tau)\\bigr] \\Phi(\\tau) \\= 0 \\quad\\Leftrightarrow\\quad\n   \\partial_\\tau \\begin{pmatrix} \\Phi \\\\[4pt] \\dot\\Phi \\end{pmatrix} \\=\n   \\begin{pmatrix} 0 & 1 \\\\[4pt] -\\Omega^2 & 0 \\  \\end{pmatrix}\n   \\begin{pmatrix} \\Phi \\\\[4pt] \\dot\\Phi \\end{pmatrix} \\ =:\\\n   \\mathrm{i}\\,\\widehat\\Omega(\\tau)\\,\\begin{pmatrix} \\Phi \\\\[4pt] \\dot\\Phi \\end{pmatrix}\\ ,\n\\end{equation}\nwhere the frequency~$\\Omega(\\tau)$ is $T$-periodic (sometimes $\\frac{T}{2}$-periodic) in~$\\tau$.\nThe solution to this first-order system is formally given by\n\\begin{equation}\n   \\begin{pmatrix} \\Phi \\\\[4pt] \\dot\\Phi \\end{pmatrix}(\\tau) \\=\n   {\\cal T} \\exp\\,\\Bigl\\{ \\int_0^\\tau \\!\\mathrm{d}\\tau'\\ \\mathrm{i}\\,\\widehat\\Omega(\\tau') \\Bigr\\} \\,\n   \\begin{pmatrix} \\Phi \\\\[4pt] \\dot\\Phi \\end{pmatrix}(0) \\ ,\n\\end{equation}\nwhere ${\\cal T}$ denotes time ordering.\nBecause of the time dependence of $\\Omega$, the time evolution operator above\nis not homogeneous thus does not constitute a one-parameter group, except when\nthe propagation interval is an integer multiple of the period~$T$. For $\\tau{=}T$,\none speaks of the stroboscopic map \\cite{Arnold}\n\\begin{equation}\n   M\\ :=\\ {\\cal T} \\exp\\,\\Bigl\\{ \\int_0^T \\!\\mathrm{d}\\tau\\ \\mathrm{i}\\,\\widehat\\Omega(\\tau) \\Bigr\\} \n   \\qquad\\Rightarrow\\qquad\n   \\begin{pmatrix} \\Phi \\\\[4pt] \\dot\\Phi \\end{pmatrix}(nT) \\= M^n\n   \\begin{pmatrix} \\Phi \\\\[4pt] \\dot\\Phi \\end{pmatrix}(0)\\ .\n\\end{equation}\nThe linear map~$M$ is a functional of the chosen background solution~$\\psi$\nand hence depends on its parameter~$E$ or~$k$.\nThis background is Lyapunov stable if the trivial solution $\\Phi{\\equiv}0$ is,\nwhich is decided by the two eigenvalues $\\mu_1$ and $\\mu_2$ of~$M$. \nSince the system is Hamiltonian, $\\det M{=}1$, we have three cases:\n\\begin{equation}\n\\begin{aligned}\n   |\\mathrm{tr}\\,M| > 2 &\\quad\\Leftrightarrow\\quad \\mu_i\\in\\mathds R  \n   &\\quad\\;\\Leftrightarrow\\quad& \\textrm{hyperbolic\/boost} \n   &\\quad\\Leftrightarrow\\quad& \\textrm{strongly unstable} \\ ,\\\\\n   |\\mathrm{tr}\\,M| = 2  &\\quad\\Leftrightarrow\\quad \\mu_i=\\pm1  \n   &\\quad\\Leftrightarrow\\quad& \\textrm{parabolic\/translation} \n   &\\quad\\Leftrightarrow\\quad& \\textrm{marginally stable}\\ , \\\\\n   |\\mathrm{tr}\\,M| < 2  &\\quad\\Leftrightarrow\\quad \\mu_i\\in\\textrm{U}(1)  \n   &\\quad\\Leftrightarrow\\quad& \\textrm{elliptic\/rotation} \n   &\\quad\\Leftrightarrow\\quad&\\textrm{strongly stable}\\ .\n\\end{aligned}\n\\end{equation}\nClearly, $|\\mathrm{tr}\\,M|$ determines the linear stability of our classical solution.\n\nLet us thus try to evaluate the trace of the stroboscopic map~$M$,\nmaking use of the special form of the matrix~$\\widehat\\Omega$,\n\\begin{equation}\n\\begin{aligned}\n   \\mathrm{tr}\\,M &\\= \\sum_{n=0}^\\infty \\mathrm{i}^n \n   \\int_0^T\\!\\!\\mathrm{d}\\tau_1\\int_0^{\\tau_1}\\!\\!\\!\\mathrm{d}\\tau_2\\ \\ldots\\int_0^{\\tau_{n-1}}\\!\\!\\!\\!\\mathrm{d}\\tau_n\\\n   \\mathrm{tr}\\,\\bigl[ \\widehat\\Omega(\\tau_1)\\,\\widehat\\Omega(\\tau_2)\\cdots\\widehat\\Omega(\\tau_n)\\bigr] \\\\[4pt]\n   &\\= 2 +  \\sum_{n=1}^\\infty(-1)^n \n   \\int_0^T\\!\\!\\mathrm{d}\\tau_1\\int_0^{\\tau_1}\\!\\!\\!\\mathrm{d}\\tau_2\\ \\ldots\\int_0^{\\tau_{n-1}}\\!\\!\\!\\!\\mathrm{d}\\tau_n\\\n   H_n(\\tau_1,\\tau_2,\\ldots,\\tau_n)\\Omega^2(\\tau_1)\\,\\Omega^2(\\tau_2)\\,\\cdots\\Omega^2(\\tau_n) \\\\[4pt]\n   \\textrm{with}& \\quad H_n(\\tau_1,\\tau_2,\\ldots,\\tau_n) \\= \n   (\\tau_1{-}\\tau_2)(\\tau_2{-}\\tau_3)\\cdots(\\tau_{n-1}{-}\\tau_n)(\\tau_n{-}\\tau_1{+}1)\n   \\quad\\textrm{and}\\quad H_1(\\tau_1)=1\\ .\n\\end{aligned}\n\\end{equation}\nIt is convenient to scale the time variable such as to normalize the period to unity,\n\\begin{equation}\n   \\tau = T\\, x  \\qquad\\quad\\textrm{and}\\quad\\qquad \\Omega^2(Tx) =: \\omega^2(x)\\ ,\\quad H(\\{Tx\\})=:h(\\{x\\}) \\ ,\n\\end{equation}\nhence\n\\begin{equation} \\label{Ansum}\n\\begin{aligned}\n   \\mathrm{tr}\\,M &\\= 2 +  \\sum_{n=1}^\\infty \\bigl(-T^2\\bigr)^n \n   \\int_0^1\\!\\!\\!\\mathrm{d} x_1\\int_0^{x_1}\\!\\!\\!\\mathrm{d} x_2 \\ldots\\int_0^{x_{n-1}}\\!\\!\\!\\!\\mathrm{d} x_n\\,\n   h_n(x_1,x_2,\\ldots,x_n)\\omega^2(x_1)\\omega^2(x_2)\\cdots\\omega^2(x_n) \\\\[4pt]\n   &\\= \\sum_{n=0}^\\infty \\frac{2}{(2n)!}\\,M_n\\,\\bigl(-T^2\\bigr)^n \n   \\ =:\\  2 - M_1 T^2 + \\sfrac{1}{12}M_2 T^4 - \\sfrac{1}{360}M_3 T^6 + \\sfrac{1}{20160}M_4 T^8 - \\ldots\\ .\n\\end{aligned}\n\\end{equation}\n\nIt is impossible to evaluate the integrals $M_n$ without explicit knowledge of~$\\omega^2(x)$. \nAs a crude guess, we replace the weight function by its (constant) average value\n\\begin{equation}\n   \\<h_n\\> \\ :=\\ \\frac{1}{n!} \\int_0^1\\!\\!\\mathrm{d} x_1\\int_0^{x_1}\\!\\!\\!\\mathrm{d} x_2\\ \\ldots\\int_0^{x_{n-1}}\\!\\!\\!\\!\\mathrm{d} x_n\\ \n   h_n(x_1,x_2,\\ldots,x_n) \\= \\frac{2\\,n!}{(2n)!}\n\\end{equation}\nand obtain\n\\begin{equation}\n   M_n \\= \\frac{(2n)!}{2}\\,\\<h_n\\> \n   \\int_0^1\\!\\!\\mathrm{d} x_1\\int_0^{x_1}\\!\\!\\!\\mathrm{d} x_2\\ \\ldots\\int_0^{x_{n-1}}\\!\\!\\!\\!\\mathrm{d} x_n\\ \\prod_{i=1}^n \\omega^2(x_i)\n   \\= \\Bigl( \\int_0^1\\!\\!\\mathrm{d} x\\ \\omega^2(x) \\Bigr)^n \\ =:\\ \\< \\,\\omega^2\\>^n\\ ,\n\\end{equation}\nwhich yields\n\\begin{equation}\n   \\mathrm{tr}\\,M \\= 2\\,\\sum_{n=0}^\\infty \\frac{(-1)^n}{(2n)!}\\,\\<\\,\\omega^2\\>^n\\,T^{2n} \n   \\= 2\\,\\cos\\bigl(\\sqrt{\\<\\,\\omega^2\\>}\\,T\\bigr)\\ .\n\\end{equation}\nThis expression indicates stability as long as $\\<\\omega^2\\>>0$.\nHowever, the result for the $j{=}0$ singlet mode $\\omega^2=\\Omega^2_{(0,0)}$ in (\\ref{j0average}) \nalready showed that the averaged frequency-squared may turn negative in certain domains\nthus changing the cos into a cosh there.\n\nTo do better, let us look at the individual terms~$M_n$ in~(\\ref{Ansum}) for the simplest case\nof the SO(4) singlet fluctuation, i.e.~$\\Omega^2_{(0,0)}=6\\psi^2{-}2$ in~(\\ref{unperturbed}). \nIts average frequency-square is easily computed to be\n\\begin{equation} \\label{j0average}\n   \\<\\,\\Omega_{(0,0)}^2 \\> \\= \\frac{1}{\\epsilon^2}\\Bigl(6 \\frac{E(k)}{K(k)}+4k^2-5\\Bigr)\\ , \n\\end{equation}\nwhere $E(k)$ and $K(k)$ denote the second and first complete elliptic integrals, respectively.\nPlotting this expression as a function of the modulus~$k$, we see that it\nbecomes negative only in a very narrow range around $k{=}1$, namely for $|k{-}1|\\lesssim0.00005$.\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width = 0.35\\paperwidth]{om2zeroavg1.pdf} \\quad\n\\includegraphics[width = 0.35\\paperwidth]{om2zeroavg2.pdf}\n\\caption{Plot of $\\<\\Omega^2_{(0,0)}\\>$ as a function of~$k$, with detail on the right.}\n\\end{figure}\nWe have only been able to analytically evaluate (with $k{<}1$ for simplicity)\n\\begin{equation}\n   M_1 \\= \\<\\,\\Omega_{(0,0)}^2 \\>  \\quad\\textrm{and}\\quad\n   M_2 \\= \\<\\,\\Omega_{(0,0)}^2 \\> ^2\\ -\\ \\frac{1}{\\epsilon^4}\\Bigl(\n   9\\frac{2{-}k^2}{K(k)^2} - 27\\frac{E(k)}{K(k)^3} + \\frac{9\\,\\pi^2}{4\\,K(k)^4} \\Bigr)\\ ,\n\\end{equation}\nwhich does not suffice to rule out instability.\nIndeed, numerical studies show that $M_n$ as a function of~$k$ looses its positivity in a range\naround $k{=}1$ which increases with~$n$, where the series~(\\ref{Ansum}) ceases to be alternating.\nMoreover, even in the limit of a very large background amplitude, $k^2\\to\\frac12$, we find that\n\\begin{equation}\n   \\<\\,\\Omega_{(0,0)}^2 \\>\\ \\to\\ \\frac{24\\,\\pi^2}{\\epsilon^2\\,\\Gamma(\\frac14)^4}\\ \\approx\\ \\frac{1.37}{\\epsilon^2}\n   \\qquad\\Rightarrow\\qquad \\sqrt{\\<\\,\\Omega^2\\>}\\,T\\ \\to\\ \\sqrt{24\\,\\pi}\\ \\approx\\ 8.68\\ ,\n\\end{equation}\nimplying that we must push the series in~(\\ref{Ansum}) at least to $O(M_{10}T^{20})$,\neven though it turns out that $M_n<\\<\\,\\Omega^2_{(0,0)}\\>^n$ at $k^2=\\frac12$ for $n>1$.\n\nFor a more complete analysis of linear stability in an oscillating system with time-dependent frequency\nwe can take recourse to Floquet theory.\nIt tells us that a general fundamental matrix solution\n\\begin{equation}\n   \\widehat\\Phi(\\tau) \\= \\begin{pmatrix} \\Phi_1 & \\Phi_2 \\\\[4pt] \\dot\\Phi_1 & \\dot\\Phi_2 \\end{pmatrix}(\\tau)\n   \\qquad\\Rightarrow\\qquad \\partial_\\tau \\widehat\\Phi(\\tau) \\= \\mathrm{i}\\,\\widehat\\Omega(\\tau)\\,\\widehat\\Phi(\\tau)\n\\end{equation}\nof our system~(\\ref{first-order}) with some initial condition $\\widehat\\Phi(0)=\\widehat\\Phi_0$ \ncan be expressed in so-called Floquet normal form as\n\\begin{equation}\n   \\widehat\\Phi(\\tau) \\= Q(\\tau)\\;\\mathrm{e}^{\\tau R} \n   \\qquad\\textrm{with}\\qquad Q(\\tau{+}2T) \\= Q(\\tau)\\ ,\n\\end{equation}\nwhere $Q(\\tau)$ and $R$ are real $2{\\times}2$ matrices,\nso that the time dependence of the frequency can be transformed away by a change of coordinates,\n\\begin{equation}\n   \\Psi(\\tau) \\ :=\\ Q(\\tau)^{-1} \\widehat\\Phi(\\tau) \\qquad\\Rightarrow\\qquad\n   \\partial_\\tau \\Psi(\\tau) \\= R\\,\\Psi(\\tau)\\ .\n\\end{equation}\nDue to the identity\n\\begin{equation}\n   \\widehat\\Phi(\\tau{+}T) \\= \\widehat\\Phi(\\tau)\\,\\widehat\\Phi(0)^{-1}\\,\\widehat\\Phi(T) \n   \\= \\widehat\\Phi(T)\\,\\widehat\\Phi(0)^{-1}\\,\\widehat\\Phi(\\tau) \\= M\\,\\widehat\\Phi(\\tau)\n\\end{equation}\nwe see that our stroboscopic map~$M$ is nothing but the monodromy, and\n\\begin{equation}\n   M^2 \\= \\widehat\\Phi(2T)\\,\\widehat\\Phi(0)^{-1} \n   \\= Q(0)\\,\\widehat\\Phi(0)^{-1}\\,\\widehat\\Phi(2T)\\,Q(0)^{-1}\n   \\= Q(0)\\,\\mathrm{e}^{2 R T}\\,Q(0)^{-1}\\ ,\n\\end{equation}\nso that its eigenvalues (or characteristic multipliers)\n\\begin{equation}\n   \\mu_i = \\mathrm{e}^{\\rho_i T} \\qquad\\textrm{for}\\quad i=1,2\n\\end{equation}\ndefine a pair of (complex) Floquet exponents $\\rho_i$ \nwhose real parts are the Lyapunov exponents.\nSince $\\mu_1\\mu_2=1$ implies that $\\rho_1{+}\\rho_2=0$,\nour system is linearly stable if and only if both eigenvalues~$\\rho_i$ of~$R$\nare purely imaginary (or zero).\n\nGenerally it is impossible to find analytically the monodromy pertaining to\na normal mode~$\\Phi_{(j,v,\\beta)}$.\\footnote{\nAn exception is the SO(4) singlet perturbation~$\\Phi_{(0,0)}$, to be treated in the following section.}\nHowever, we can get a qualitative understanding by looking numerically at some examples. \nBefore numerically integrating Hill's equation, however, let us estimate at which energies~$E$\nor, rather, moduli~$k$, possible resonance frequencies might occur.\nTo this end, we determine the period-average of the natural frequency $\\Omega_{(j,v,\\beta)}$ \nand compare it to its modulation frequency~$\\sfrac{2\\pi}{T}$. If we model \n\\begin{equation}\n   \\Omega^2(\\tau) \\= \\<\\,\\Omega^2\\> \\,\\bigl(1+h(\\tau)\\bigr)  \\quad\\textrm{with}\\quad\n   \\<\\,\\Omega^2\\> = \\sfrac{1}{T}\\smallint_0^T \\!\\mathrm{d}\\tau\\ \\Omega^2(\\tau) \\quad\\textrm{and}\\quad\n   h(\\tau) \\ \\propto\\ \\cos(2\\pi\\tau\/T) \\ ,\n\\end{equation}\nwhere $T=4\\,\\epsilon\\,K(k)$, then the resonance condition is met for\n\\begin{equation}\n   \\sqrt{\\<\\Omega^2\\>} \\= \\ell\\,\\frac{\\pi}{T} \\qquad\\Rightarrow\\qquad\n   k=k_\\ell(j,v,\\beta)\\qquad\\textrm{for}\\quad \\ell=1,2,3,\\ldots\\ .\n\\end{equation}\nSince this model reproduces only the rough features of $\\Omega^2(\\tau)$, we expect potential instability\ndue to parametric resonance effects in a band around or near the values~$k_\\ell$.\n\nBelow we display, together with the would-be resonant values~$k_\\ell$,\nthe function $\\mathrm{tr} M(k)$ for the sample cases of $(j,v)=(2,0)$ and $(2,2)$.\n\\begin{figure}[h!]\n\\centering\n\\captionsetup{width=\\linewidth}\n\\includegraphics[width = 0.35\\paperwidth]{trM20a.pdf} \\quad\n\\includegraphics[width = 0.35\\paperwidth]{trM20b.pdf} \n\\caption{Plot of $\\mathrm{tr}\\,M(k)$ for $(j,v)=(2,0)$, with detail on the right. Would-be resonances marked in red.}\n\\end{figure}\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width = 0.35\\paperwidth]{trM22_1.pdf} \\quad\n\\includegraphics[width = 0.35\\paperwidth]{trM22_2.pdf} \\\\[8pt]\n\\includegraphics[width = 0.35\\paperwidth]{trM22_3.pdf} \\quad\n\\includegraphics[width = 0.35\\paperwidth]{trM22_4.pdf} \n\\caption{Plots of $\\mathrm{tr}\\,M(k)$ for $(j,v)=(2,2)$ and $\\beta=1,2,3,4$. Would-be resonances marked in red.}\n\\end{figure}\nOne sees that, on both sides of the critical value of $E{=}\\sfrac12$ (or $k{=}1$), \ncorresponding to the double-well local maximum, the $k_\\ell$ values accumulate at the critical point.\nBut while for $k{>}1$ (energy below the critical point) $\\mathrm{tr} M(k)$ oscillates between values close to $2$ \nin magnitude and thus exponential growth is rare and mild, for $k{<}1$ (energy above the critical point)\nthe oscillatory behavior of $\\mathrm{tr} M(k)$ comes with an amplitude exceeding~2 and growing with energy.\nHence, in this latter regime stable and unstable bands alternate.\nThis is supported by long-term numerical integration, as we demonstrate by plotting\n$\\Phi(\\tau)$ for $(j,v,\\beta)=(2,2,1)$ with initial values $\\Phi(0){=}1$ and $\\dot\\Phi(0){=}0$ \non both sides very close to the end of the first instability \n(at the highest value of~$E$ or the lowest value of~$k$).\n\\begin{figure}[h!]\n\\centering\n\\captionsetup{width=\\linewidth}\n\\includegraphics[width = 0.35\\paperwidth]{phi_22_k073198.pdf} \\quad\n\\includegraphics[width = 0.35\\paperwidth]{phi_22_k073199.pdf} \n\\caption{Plot of $\\Phi(\\tau)$ for $(j,v,\\beta)=(2,2,1)$ and $k{=}0.73198$ (left) and $k{=}0.73199$ (right).}\n\\end{figure}\n\nMost relevant for the cosmological application is the regime of very large energies, $E\\to\\infty$ (or $k\\to 1\/\\sqrt{2}$).\nIn this limit, we observe the following universal behavior. \nBecause the period~$T$ collapses with $\\epsilon{=}\\sqrt{k^2{-}1\/2}$, we rescale\n\\begin{equation}\n   \\sfrac{\\tau}{\\epsilon} = z \\in [0,4K(\\sfrac12)]\\ ,\\quad\n   \\epsilon\\,\\psi = \\tilde\\psi\\ ,\\quad \n   \\epsilon^2\\dot\\psi = \\partial_z\\tilde\\psi\\ ,\\quad \n   \\epsilon^2\\Omega^2 = \\tilde\\Omega^2\\ ,\\quad \n   \\epsilon^2\\lambda = \\tilde\\lambda\n\\end{equation}\nso that the tilded quantities remain finite in the limit,\nand find, with $\\bar\\omega^2_{(\\bar{v})}\\to 2\\psi^2$,\\footnote{\nFor the cases $(j,v)=(0,1)$ and $(1,0)$, the factor $\\tilde\\lambda$ is missing; \nfor $(j,v)=(0,0)$, one only has $R=Q\\sim(\\tilde\\lambda{-}6\\tilde\\psi^2)$.}\n\\begin{equation}\n\\begin{aligned}\n   &Q_{(v\\pm2)}\\ \\sim\\ \\tilde\\lambda\\ , \\\\\n   &Q_{(v\\pm1)}\\ \\sim\\ \\tilde\\lambda\\,(\\tilde\\lambda-4\\tilde\\psi^2)\\ ,\\qquad\\qquad R_{(v\\pm1)}\\ \\sim\\ \\tilde\\lambda\\,\\bigl[(\\tilde\\lambda-2\\tilde\\psi^2)(\\tilde\\lambda-4\\tilde\\psi^2)-8(\\mbox{$\\partial$}_z\\tilde\\psi)^2\\bigr]\\ ,\\\\\n   &Q_{(v)}\\ \\sim\\ \\tilde\\lambda\\,(\\tilde\\lambda-4\\tilde\\psi^2)(\\tilde\\lambda-6\\tilde\\psi^2)\\ ,\\quad R_{(v)}\\ \\sim\\ \\tilde\\lambda\\,\\bigl[(\\tilde\\lambda-2\\tilde\\psi^2)(\\tilde\\lambda-4\\tilde\\psi^2)-8(\\mbox{$\\partial$}_z\\tilde\\psi)^2\\bigr](\\tilde\\lambda-6\\tilde\\psi^2)\\ ,\n\\end{aligned}\n\\end{equation}\nbecause all $j$-dependent terms in the polynomials are subleading and drop out in the limit.\nFactorizing the $R$~polynomials, we find the four universal natural frequency-squares\n\\begin{equation}\n   \\tilde\\Omega^2_1 = 0\\ ,\\quad\n   \\tilde\\Omega^2_2 = 3\\,\\tilde\\psi^2-\\sqrt{\\tilde\\psi^4+8(\\mbox{$\\partial$}_z\\tilde\\psi)^2}\\ ,\\quad\n   \\tilde\\Omega^2_3 = 3\\,\\tilde\\psi^2+\\sqrt{\\tilde\\psi^4+8(\\mbox{$\\partial$}_z\\tilde\\psi)^2}\\ ,\\quad\n   \\tilde\\Omega^2_4 = 6\\,\\tilde\\psi^2\\ .\n\\end{equation}\nOne must pay attention, however, to the fact that the avoided crossings disappear in the $\\epsilon\\to0$ limit.\nTherefore, the correct limiting frequencies to input into\n\\begin{equation}\n   \\bigl[ \\mbox{$\\partial$}_z - \\tilde\\Omega^2_{(j,v,\\beta)} \\bigr]\\,\\tilde\\Phi_{(j,v,\\beta)} \\= 0\n\\end{equation}\nare\n\\begin{equation}\n\\begin{aligned}\n   &\\tilde\\Omega^2_{(j,j\\pm2)}\\ \\ = 0\\ ,\\\\\n   &\\tilde\\Omega^2_{(j,j\\pm1,\\beta)} \\in \\bigl\\{ \n   \\textrm{min}(\\tilde\\Omega^2_1,\\tilde\\Omega^2_2),\\ \\textrm{max}(\\tilde\\Omega^2_1,\\tilde\\Omega^2_2),\\\n   \\tilde\\Omega^2_3 \\bigr\\}\\ ,\\\\\n   &\\tilde\\Omega^2_{(j,j,\\beta)} \\ \\ \\ \\in \\bigl\\{ \n   \\textrm{min}(\\tilde\\Omega^2_1,\\tilde\\Omega^2_2),\\ \\textrm{max}(\\tilde\\Omega^2_1,\\tilde\\Omega^2_2),\\ \n   \\textrm{min}(\\tilde\\Omega^2_3,\\tilde\\Omega^2_4),\\ \\textrm{max}(\\tilde\\Omega^2_3,\\tilde\\Omega^2_4) \\bigr\\}\\ ,\n\\end{aligned}\n\\end{equation}\nof which we show below the last list as a function of~$z$. \n\\begin{figure}[h!]\n\\centering\n\\captionsetup{width=0.9\\linewidth}\n\\includegraphics[width = 0.5\\paperwidth]{om2universal.pdf} \n\\caption{Plot of the universal limiting natural frequency-squares \n$\\tilde\\Omega^2_{(j,v,\\beta)}$ for $v{=}j$ and $\\beta=1,2,3,4$.}\n\\end{figure}\nThe monodromies are easily computed numerically,\\footnote{\nFor the cases $(j,v)=(0,1)$ and $(1,0)$ one gets $\\{56.769,\\ -1.659\\}$;\nfor $(j,v)=(0,0)$ we have $\\mathrm{tr}\\,M=2$.}\n\\begin{equation}\n\\begin{aligned}\n   &\\mathrm{tr}\\,M_{(j,j\\pm2)}(E{\\to}\\infty)\\ \\ \\ = 2\\ ,\\\\\n   &\\mathrm{tr}\\,M_{(j,j\\pm1,\\beta)}(E{\\to}\\infty)\\ \\in \\bigl\\{ 306.704,\\ -1.842,\\ -1.659 \\bigr\\}\\ ,\\\\\n   &\\mathrm{tr}\\,M_{(j,j,\\beta)}(E{\\to}\\infty) \\ \\ \\ \\ \\in \\bigl\\{ 306.704,\\ -1.842,\\ 2.462,\\ -1.067 \\bigr\\}\\ ,\n\\end{aligned}\n\\end{equation}\nin agreement with the figures above. In particular, the extremal $v$-values become marginally stable,\nwhile part of the non-extremal cases are unstable for high energies. \n\nOf course, for each non-extremal value of~$v$ we still have to project out unphysical modes\nby imposing the gauge condition~(\\ref{gauge4}). However, in the $12(2j{+}1)$-dimensional fluctuation space\nthe gauge condition has rank~$3(2j{+}1)$ while we see that (for $j{\\ge}2$) in total $4(2j{+}1)$ normal modes \nare unstable at high energy. Therefore, the projection to physical modes cannot remove all instabilities. \nWe must conclude that, for sufficiently high energy~$E$, some fluctuations grow exponentially,\nimplying that the solution $\\Phi{\\equiv}0$ is linearly unstable, and thus is the Yang--Mills background.\n\n\n\\vspace{12pt}\n\\section{Singlet perturbation: exact treatment}\n\\vspace{2pt}\n\\noindent\nEven though the Floquet representation helped to reduce the long-time behavior of the perturbations\nto the analysis of a single period~$T$, it normally does not give us an exact solution to Hill's equation.\nHowever, for the SO(4) singlet fluctuation around $\\psi(\\tau)$, we can employ the fact that \n$\\dot{\\psi}$ trivially solves the fluctuation equation,\n\\begin{equation} \\label{timeshift}\n   (\\dot\\psi)^{\\cdot\\cdot} = (\\ddot\\psi)^{\\cdot} = -\\bigl(V'(\\psi)\\bigr)^{\\cdot} \n   = -V''(\\psi)\\,\\dot{\\psi} \\= -(6\\psi^2{-}2)\\,\\dot\\psi \\= -\\Omega^2_{(0,0)}(\\tau)\\,\\dot\\psi\\ ,\n\\end{equation}\nwith a frequency function which is $\\frac{T}{2}$-periodic.\nThis implies that all fluctuation modes are $T$-periodic.\nWith the knowledge of an explicit solution to the fluctuation equation \nwe can reduce the latter to a first-order equation and solve that one to find a second solution.\nThe normalizations are arbitrary, so we choose\n\\begin{equation}\n   \\Phi_1(\\tau)\\=-\\sfrac{\\epsilon^3}{k}\\,\\dot{\\psi}(\\tau) \\quad\\textrm{and}\\quad\n   \\Phi_2(\\tau) \\= \\Phi_1(\\tau)\\,\\int^\\tau\\frac{\\mathrm{d}\\sigma}{\\Phi_1(\\sigma)^2}\n   \\= -\\sfrac{k}{\\epsilon^3}\\,\\dot{\\psi}(\\tau)\\,\\int^\\tau \\frac{\\mathrm{d}\\sigma}{\\dot{\\psi}^2(\\sigma)}\\ ,\n\\end{equation}\nwhich are linearly independent since\n\\begin{equation}\n   W(\\Phi_1,\\Phi_2)\\ \\equiv\\ \\Phi_1\\dot\\Phi_2-\\Phi_2\\dot\\Phi_1 \\= 1\\ .\n\\end{equation}\n\nFor simplicity, we restrict ourselves to the energy range \n$\\sfrac12{<}E{<}\\infty$, i.e.~$1{>}k^2{>}\\sfrac12$. \nExplicitly, we have\n\\begin{equation}\n\\begin{aligned}\n   \\!\\!\\!\\Phi_1(\\tau) &\\= \\epsilon\\,{\\mathrm{sn}\\bigl(\\sfrac{\\tau}{\\epsilon},k\\bigr)}\\,{\\mathrm{dn}\\bigl(\\sfrac{\\tau}{\\epsilon},k\\bigr)} \\ , \\\\[4pt]\n   \\!\\!\\!\\Phi_2(\\tau) &\\= \\sfrac{1}{1-k^2}\\,{\\mathrm{cn}\\bigl(\\sfrac{\\tau}{\\epsilon},k\\bigr)}\n   \\bigl[ (2k^2{-}1)\\,\\textrm{dn}^2\\bigl(\\sfrac{\\tau}{\\epsilon},k\\bigr) -k^2 \\bigr] \\\\\n   &\\qquad + {\\mathrm{sn}\\bigl(\\sfrac{\\tau}{\\epsilon},k\\bigr)}\\,{\\mathrm{dn}\\bigl(\\sfrac{\\tau}{\\epsilon},k\\bigr)} \\bigl[ \\sfrac{\\tau}{\\epsilon} +\n   \\sfrac{2k^2{-}1}{1{-}k^2}\\,E\\bigl(\\textrm{am}(\\sfrac{\\tau}{\\epsilon},k),k\\bigr)\\bigr]\\;, \n\\end{aligned}\n\\end{equation}\nwhere $\\textrm{am}(z,k)$ denotes the Jacobi amplitude and $E(z,k)$ is the elliptic integral of the second kind.\n\\begin{figure}[h!]\n\\centering\n\\captionsetup{width=\\linewidth}\n\\includegraphics[width = 0.35\\paperwidth]{phi1.pdf} \\quad\n\\includegraphics[width = 0.35\\paperwidth]{phi2.pdf} \n\\caption{Plot of the SO(4) singlet fluctuation modes $\\Phi_1$ and $\\Phi_2$ over eight periods for $k^2{=}0.81$.}\n\\end{figure}\nAs can be checked, the initial conditions are\n\\begin{equation}\n   \\Phi_1(0)=0\\ ,\\quad \\dot\\Phi_1(0)=1 \\qquad\\quad\\textrm{and}\\quad\\qquad\n   \\Phi_2(0)=-1\\ ,\\quad \\dot\\Phi_2(0)=0\\ ,\n\\end{equation}\nwhich fixes the ambiguity of adding to $\\Phi_2$ a piece proportional to~$\\Phi_1$. Hence,\n\\begin{equation}\n   \\widehat\\Phi(0) \\= \\Bigl(\\begin{smallmatrix} 0 & \\!{-}1 \\\\[4pt] 1 & 0 \\end{smallmatrix} \\Bigr)\n   \\qquad\\Rightarrow\\qquad\n   M\\= \\widehat\\Phi(T)\\, \\Bigl(\\begin{smallmatrix} 0 & 1 \\\\[4pt] {-}1 & 0 \\end{smallmatrix} \\Bigr)\\ .\n\\end{equation}\nWe know that $\\Phi_1\\sim\\dot\\psi$ is $T$-periodic, and so is $\\dot\\Phi_1$, \nbut not the second solution,\n\\begin{equation}\n   \\Phi_2(\\tau{+}T) \\= \\Phi_2(\\tau) + \\gamma\\,T\\,\\Phi_1(\\tau) \\quad\\textrm{with}\\quad \\gamma \\= \\frac{1}{T}\\int_0^T\\frac{\\mathrm{d}\\sigma}{\\Phi_1(\\sigma)^2}\\bigg|_{\\textrm{reg}}\n   \\ =:\\ \\sfrac{k^2}{\\epsilon^6}\\, \\bigl\\langle \\dot\\psi^{-2}\\bigr\\rangle_{\\textrm{reg}}\\ ,\n\\end{equation}\nwhere the integral diverges at the turning points and must be regularized by subtracting\nthe Weierstra\\ss\\ $\\wp$~function with the appropriate half-periods.\nSince $\\Phi_1$ has periodic zeros, $\\Phi_2$ does return to~${-}1$ at integer multiples of~$T$.\nIt follows that the $\\Phi_2$ oscillation linearly grows in amplitude with a rate (per period) of\n\\begin{equation}\n   \\gamma \\= \\frac{1}{\\epsilon^2}\\,\\Bigl[\\,1 + \\frac{2k^2{-}1}{1{-}k^2}\\,\\frac{E(k)}{K(k)} \\,\\Bigr]\\ ,\n\\end{equation}\nwhich is always larger than 7.629, attained at $k\\approx0.882$.\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width = 0.5\\paperwidth]{gamma.pdf} \n\\caption{Plot of the linear growth rate~$\\gamma$ as a function of~$k$.}\n\\end{figure}\n\nIn essence, we have managed to compute the monodromy\n\\begin{equation}\n   M \\= \\begin{pmatrix} {-}\\Phi_2(T) & \\Phi_1(T) \\\\[4pt] {-}\\dot\\Phi_2(T) & \\dot\\Phi_1(T) \\ \\end{pmatrix} \n   \\= \\begin{pmatrix} \\ 1 & 0 \\\\[4pt] \\ \\!\\!{-}\\gamma\\,T & 1 \\ \\end{pmatrix} \n   \\= \\exp \\Bigl\\{ {-}\\gamma\\,T\\,\\bigl(\\begin{smallmatrix} 0 & 0 \\\\[4pt] 1 & 0 \\end{smallmatrix}\\bigr) \\Bigr\\}\n\\end{equation}\nand thus easily obtain the Floquet representation,\n\\begin{equation}\n   R \\= \\begin{pmatrix} 0 & \\gamma \\\\[4pt] 0 & 0 \\end{pmatrix} \\quad\\Rightarrow\\quad\n   \\mathrm{e}^{\\tau R} \\= \\begin{pmatrix} 1 & \\gamma\\,\\tau \\\\[4pt] 0 & 1 \\end{pmatrix}  \\quad\\textrm{and}\\quad\n   Q(\\tau)\\ = \\begin{pmatrix} \\Phi_1 & \\Phi_2{-}\\Phi_1\\gamma\\,\\tau \\\\[4pt] \n   \\dot\\Phi_1 & \\dot\\Phi_2{-}\\dot\\Phi_1\\gamma\\,\\tau \\end{pmatrix}\\ .\n\\end{equation}\nObviously, we have encountered a marginally stable situation, since $M$ is of parabolic type. \nThere is no exponential growth, and $\\Phi_1$ is periodic thus bounded, but $\\Phi_2$ \ngrows without bound as long as one stays in the linear regime.\nNote that we never made use of the form of our Newtonian potential.\nIn fact, this behavior is typical for a conservative mechanical system with oscillatory motion.\n\nWhat to make of this linear growth? It can be (and actually is) easily overturned by nonlinear effects.\nGoing beyond the linear regime, though, requires expanding the Yang--Mills equation to higher orders\nabout our classical Yang--Mills solution~(\\ref{Aansatz}).\nWhile this is a formidable task in general, it can actually be done to all orders for the singlet perturbation!\nThe reason is that a singlet perturbation leaves us in the SO(4)-symmetric subsector,\nthus connecting only to a neighboring ``cosmic background'', $\\psi\\to\\tilde\\psi$.\nSince (\\ref{backgrounds}) gives us analytic control over all solutions~$\\psi(\\tau)$,\nthe full effect of such a shift can be computed exactly. \nSplitting an exact solution~$\\tilde{\\psi}$ into a background part and its (full) deviation,\n\\begin{equation}\n   \\tilde{\\psi}(\\tau) \\= \\psi(\\tau)\\ +\\ \\eta(\\tau)\\ ,\n\\end{equation}\nand inserting $\\tilde{\\psi}$ into the equation of motion~(\\ref{Newton}), we obtain\n\\begin{equation} \\label{nonlinear}\n   0 \\= \\ddot\\eta + V''(\\psi)\\,\\eta + \\sfrac12 V'''(\\psi)\\,\\eta^2 + \\sfrac16 V''''(\\psi)\\,\\eta^3\n   \\= \\ddot\\eta + (6\\psi^2{-}2)\\,\\eta + 6\\psi\\,\\eta^2 + 2\\,\\eta^3\\ ,\n\\end{equation}\nextending the linear equation~(\\ref{timeshift}) by two nonlinear contributions.\nPerturbation theory introduces a small parameter~$\\epsilon$ and formally expands\n\\begin{equation}\n   \\eta \\= \\epsilon\\eta_{(1)} + \\epsilon^2\\eta_{(2)} + \\epsilon^3\\eta_{(3)} + \\ldots\\ ,\n\\end{equation}\nwhich yields the infinite coupled system\n\\begin{equation}\n\\begin{aligned}\n   & \\bigl[\\mbox{$\\partial$}_\\tau^2+(6\\psi^2{-}2)\\bigr]\\,\\eta_{(1)} \\= 0 \\ ,\\\\\n   & \\bigl[\\mbox{$\\partial$}_\\tau^2+(6\\psi^2{-}2)\\bigr]\\,\\eta_{(2)} \\= -6\\psi\\,\\eta_{(1)}^2 \\ ,\\\\\n   & \\bigl[\\mbox{$\\partial$}_\\tau^2+(6\\psi^2{-}2)\\bigr]\\,\\eta_{(3)} \\= -12\\psi\\,\\eta_{(1)}\\eta_{(2)} -2\\,\\eta_{(1)}^3\\ ,\\\\\n   & \\ldots \\ ,\n\\end{aligned}\n\\end{equation}\nwhich could be iterated with a seed solution~$\\eta_{(1)}$ of the linear system.\n\nHowever, we know that the exact solutions to the full nonlinear equation~(\\ref{nonlinear}) \nis simply given by the difference\n\\begin{equation}\n   \\eta(\\tau) \\= \\tilde\\psi(\\tau) - \\psi(\\tau)\n\\end{equation}\nof two analytically known backgrounds. The SO(4)-singlet background moduli space is parametrized by\ntwo coordinates, e.g.~the energy~$E$ (or elliptic modulus~$k$) and the choice of an initial condition\nwhich fixes the origin~$\\tau{=}0$ of the time variable. In~(\\ref{backgrounds}), we selected $\\dot\\psi(0)=0$,\nbut relaxing this we can reintroduce this collective coordinate by allowing shifts in~$\\tau$. \nWe may then parametrize the SO(4)-invariant Yang--Mills solutions as\n\\begin{equation}\n   \\psi_{k,\\ell}(\\tau) \\= \\psi(\\tau{-}\\ell) \n   \\qquad\\textrm{with}\\qquad 2E=1\/(2k^2{-}1)^2 \\quad\\quad\\textrm{and}\\quad\\quad \\ell\\in\\mathds R\n\\end{equation}\nwhere $\\psi$ is taken from~(\\ref{backgrounds}). \nNote that $\\dot\\psi_{k,\\ell}$ solves the background equation~(\\ref{timeshift}) with a frequency-squared\n$\\omega_{k,\\ell}^2=6\\psi_{k,\\ell}^2{-}2$.\nWithout loss of generality we assign $\\psi=\\psi_{k,0}$ and $\\tilde\\psi=\\psi_{k{+}\\delta k,\\delta\\ell}$,\nhence\n\\begin{equation}\n\\begin{aligned}\n   \\eta(\\tau) &\\= \\delta k\\,\\mbox{$\\partial$}_k\\psi(\\tau) - \\delta\\ell\\,\\dot\\psi(\\tau) \n   + \\sfrac12(\\delta k)^2\\,\\mbox{$\\partial$}_k^2\\psi(\\tau) - \\delta k \\delta\\ell\\,\\mbox{$\\partial$}_k\\dot\\psi(\\tau) \n   + \\sfrac12(\\delta\\ell)^2\\,\\ddot\\psi(\\tau) + \\ldots \\\\[4pt]\n   &\\=  \\delta k\\,\\mbox{$\\partial$}_k\\psi(\\tau{-}\\delta\\ell) + \\sfrac12(\\delta k)^2\\,\\mbox{$\\partial$}_k^2\\psi(\\tau{-}\\delta\\ell)\n   + \\sfrac16(\\delta k)^3\\,\\mbox{$\\partial$}_k^3\\psi(\\tau{-}\\delta\\ell) + \\ldots\\ ,\n\\end{aligned}\n\\end{equation}\nbecause $\\mbox{$\\partial$}_\\ell\\psi=-\\dot\\psi$. \nClearly, a shift in~$\\ell$ only shifts the time dependence of the frequency and does not alter the energy~$E$,\nwhich is not very interesting. Its linear part corresponds to the mode $\\Phi_1\\sim\\dot\\psi$ of the previous section.\nA change in~$k$, in contract, will lead to a solution with an altered frequency and energy.\nIts linear part is given by $\\Phi_2$, which grows linearly in time. \nHowever, due to the boundedness of the full motion, the nonlinear corrections have to limit this growth and\nultimately must bring the fluctuation back close to zero. This is the familiar wave beat phenomenon:\nthe difference of two oscillating functions, $\\tilde\\psi$ and $\\psi$, with slightly different frequencies,\nwill display an amplitude oscillation with a beat frequency given by the difference.\nThis is borne out in the following plots.\n\\begin{figure}[h!]\n\\centering\n\\captionsetup{width=\\linewidth}\n\\includegraphics[width = 0.35\\paperwidth]{eta_linear.pdf} \\quad\n\\includegraphics[width = 0.35\\paperwidth]{eta_nonlinear.pdf}\n\\caption{Plots of the full perturbation $\\eta$ at $k{=}0.95$ for $\\eta(0){=}0.02$, $\\dot{\\eta}(0){=}0$, giving a beat ratio of ${\\sim}19$.}\n\\end{figure}\nAs a result, we can assert a long-term stability of the cosmic Yang--Mills fields \nagainst the SO(4) singlet perturbation, even though on shorter time scales an excursion \nto a nearby solution is not met with a linear backreaction.\n\n\n\n\n\\chapter{Yang-Mills fields on $\\mathrm{d}{S}_4$}\n\\label{Chapter4}\n\\justifying\n\nHere we review the conformal relation of the $4$-dimensional de Sitter space with a finite Lorentzian cylinder with $S^3$-slicing, study calculus on $S^3$ and present the Yang--Mills field equations on the cylinder. The content of this chapter is partially taken from \\cite{KL20,KPH21,KLP21}\\footnote{The role of $L_a$ and $R_a$ in \\cite{KL20} is interchanged as compared to this thesis.}.\n\n\n\\vspace{12pt}\n\\vspace{2pt}\n\\section{The de Sitter-Minkowski correspondence via $S^3$-cylinder}\n\\noindent\nThe de Sitter space $\\mathrm{d} S_4$ in four dimensions has a natural embedding as a single-sheeted hyperboloid in five-dimensional Minkowski space $\\mathds R^{1,4}$ with coordinates $(q_0, q_{_A}); A=1,2,3,4$ and global length scale $\\ell$ and is given by\n\\begin{equation} \\label{dS4}\n   -q_0^2\\, + q_1^2\\, + q_2^2\\, + q_3^2\\, + q_4^2\\, \\= \\ell^2 .\n\\end{equation}\nOne can use the flat metric on $\\mathds R^{1,4}$\n\\begin{equation} \\label{flat_metric}\n   \\mathrm{d} s_{(1,4)}^2 \\= -\\mathrm{d} q_0^2\\, + \\mathrm{d} q_1^2\\, + \\mathrm{d} q_2^2\\, + \\mathrm{d} q_3^2\\, + \\mathrm{d} q_4^2\n\\end{equation}\nto construct the corresponding metric on $\\mathrm{d} S_4$ using the coordinate constraint \\eqref{dS4}. The metric so obtained is conformally equivalent to the one on a finite Lorentzian cylinder $\\mathcal{I}\\times S^3$ with $\\mathcal{I} = \\left(0, \\pi \\right)$ over the $3$-sphere $S^3$. To see this, we employ the following coordinates\n\\begin{equation}\n   q_{_A} \\= \\ell\\, \\omega_{_A}\\, \\csc\\tau \\quad\\textrm{and}\\quad q_{_5} \\= -\\ell\\, \\cot\\tau \\quad\\quad\\textrm{with}\\quad\\quad \\tau\\, \\in\\, (0,\\pi)\\ ,\n\\end{equation}\nwhere $\\omega_{_A}$ are the natural embedding coordinates of $S^3$ in $\\mathds R^4$: $\\omega_{_A}\\omega_{_A} \\!=\\! 1$\\footnote{Repeated indices are summed over.}. A natural hyperspherical parametrisation of $\\omega_{_A}$ is given by\n\\begin{equation}\\label{omegas}\n   \\omega_1 \\= \\sin\\chi\\,\\sin\\theta\\,\\cos\\phi\\ ,\\quad \\omega_2 \\= \\sin\\chi\\,\\sin\\theta\\,\\sin\\phi\\ ,\\quad \\omega_3 \\= \\sin\\chi\\,\\cos\\theta\\ ,\\quad \\omega_4 \\= \\cos\\chi\\ ,\n\\end{equation}\nwith $0\\leq \\chi,\\theta \\leq \\pi$ and $0\\leq\\phi\\leq2\\pi$. The modified metric has the following form:\n\\begin{equation}\\label{metricdS}\n   \\mathrm{d} s^2 \\= \\mathrm{d} s_{(1,4)}^2|_{\\mathrm{d} S_4} \\= \\frac{\\ell^2}{\\sin^2\\tau}\\left( -\\mathrm{d}\\tau^2 + \\mathrm{d}\\Omega_3^2 \\right); \\quad \\mathrm{d}\\Omega_3^{\\ 2}\\=\\mathrm{d}\\omega_{A}\\, \\mathrm{d}\\omega_{A}|_{\\mathrm{d} S_4}\\ .\n\\end{equation}\nBy gluing two copies of these Lorentzian cylinders at $\\tau\\!=0$, such that now $\\tau \\in \\widetilde{\\cal I} = \\left(-\\pi, \\pi \\right)$, one finds that half of the resultant cylinder $\\widetilde{\\cal I}\\times S^3$ is conformally equivalent to the 4 dimensional Minkowski space. To see this, consider the following parametrization of $(t,x,y,z)\\in \\mathds R^{1,3}$\n\\begin{equation} \\label{S3toMink}\n   \\cot\\tau \\= \\frac{r^2 {-} t^2 {+} \\ell^2}{2\\,\\ell\\,t}\\ ,\\\n   \\omega_1 \\= \\gamma\\,\\frac{x}{\\ell}\\ ,\\\n   \\omega_2 \\= \\gamma\\,\\frac{y}{\\ell}\\ ,\\\n   \\omega_3 \\= \\gamma\\,\\frac{z}{\\ell}\\ ,\\\n   \\omega_4 \\= \\gamma \\frac{r^2 {-} t^2 {-} \\ell^2}{2\\,\\ell^2}\\ ,\n\\end{equation}\nwhere we have abbreviated\n\\begin{figure}[!htbp]\n\\centering\n\\captionsetup{width=\\linewidth}\n\\includegraphics[width = 0.65\\paperwidth]{Figures\/halfcyl.pdf}\n\\caption{An illustration of the map between a cylinder $2\\mathcal{I}\\times S^3$ and Minkowski space $R^{1,3}$.\nThe Minkowski coordinates cover the shaded area. The boundary of this area is given by the curve $\\omega_4{=}\\cos\\chi{=}\\cos\\tau$. \nEach point is a two-sphere spanned by $\\{\\hat{\\omega}_1,\\hat{\\omega}_2,\\hat{\\omega}_3\\}$, which is mapped to a sphere of constant $r$ and $t$. }\n\\label{patching}\n\\end{figure}\n\\begin{equation}\\label{gamma}\n   r^2 = x^2+y^2+z^2\\ \\ \\&\\ \\\n   \\gamma \\= \\frac{2\\,\\ell^2}{\\sqrt{4\\,\\ell^2 t^2 + (r^2-t^2+\\ell^2)^2}}\n   =  \\frac{2\\,\\ell^2}{\\sqrt{4\\,\\ell^2 r^2 + (t^2-r^2+\\ell^2)^2}}\\ .\n\\end{equation}\nA straightforward computation yields\n\\begin{equation}\n   \\cos\\tau-\\cos\\chi\\= \\gamma\\ >0\\ ,\n\\end{equation}\nwhich shows that only half of the cylinder, constrained by $\\cos\\tau\\ <\\cos\\chi$, is allowed by the map \\eqref{S3toMink}. Plugging this map back into the de Sitter metric \\eqref{metricdS} we obtain\n\\begin{equation}\\label{metricMink}\n   \\mathrm{d} s^2 \\=\n   \\frac{\\ell^2}{t^2}\\,\\bigl(-\\mathrm{d}t^2 +\\mathrm{d} x^2 +\\mathrm{d} y^2 +\\mathrm{d} z^2\\bigr) \n   \\quad\\textrm{with}\\quad (x,y,z)\\equiv(x^1,x^2,x^3)\\in\\mathds R^3\\ \\&\\ t\\in\\mathds R \\ ,\n\\end{equation}\nwhich is the Minkowski metric up to a conformal factor\\footnote{In fact, the map \\eqref{S3toMink} covers only the positive half of the Minkowski space i.e. $t\\in\\mathds R_+$ for the original cylinder ${\\cal I}\\times S^3$.}. A smooth gluing of the two cylinders across the time slice $t\\!=\\tau\\!=0$ is nicely depicted in (Fig. \\ref{patching}). Now, looking at the maps \\eqref{omegas} and \\eqref{S3toMink} we see a SO(3)-symmetry which we can exploit by writing\n\\begin{equation}\\label{OmegaToHatOmega}\n   \\omega_a \\= \\sin\\chi\\; \\hat{\\omega}_a\\ ~\\textrm{and}~\\ x^a \\= r\\,\\hat{\\omega}_a\\ ~\\textrm{for}~\\ a\\=1,2,3\\ ~\\textrm{and}~ r\\= \\sfrac\\ell\\gamma\\,\\sin\\chi\\ ,\n\\end{equation}\nwhere the unit $S^2$ coordinates $\\hat{\\omega}_a$ are given by\n\\begin{equation}\\label{S2coord}\n   \\hat{\\omega}_1 \\= \\sin\\theta\\,\\cos\\phi\\ , \\quad \\hat{\\omega}_2 \\= \\sin\\theta\\,\\sin\\phi \\quad\\textrm{and}\\quad \\hat{\\omega}_3 \\= \\cos\\theta\\ .\n\\end{equation}\n\\begin{figure}[!ht]\n\\centering\n\\captionsetup{width=\\linewidth}\n\\includegraphics[width = 0.4\\linewidth]{Figures\/penrose.pdf}\n\\caption{\nPenrose diagram of Minkowski space~$\\mathds R^{1,3}$. Each point hides a two-sphere $S^2{\\ni}\\{\\theta,\\phi\\}$.\nBlue curves indicate $t{=}\\textrm{const}$ slices while brown curves depict the world volumes of $r{=}\\textrm{const}$ spheres.\nThe lightcone of the Minkowski-space origin is drawn in red.\n}\n\\label{CPplot}\n\\end{figure}\nWe can identify the unit $S^2$ between the Minkowski space and the de Sitter space using  (\\ref{S3toMink}) and (\\ref{OmegaToHatOmega}), so that we have an effective map between coordinates $(t,r)$ and $(\\tau,\\chi)$:\n\\begin{equation} \\label{tr2TauChi}\n   \\frac{t}{\\ell} \\= \\frac{\\sin\\tau}{\\cos\\tau\\,-\\,\\cos\\chi} \\quad\\quad\\textrm{and}\\quad\\quad\n   \\frac{r}{\\ell} \\= \\frac{\\sin\\chi}{\\cos\\tau\\,-\\,\\cos\\chi} \\qquad\\quad\\textrm{for}\\quad\\quad\n   \\chi>|\\tau|\\ ,\n\\end{equation}\nwhich reveals that the triangular $(\\tau,\\chi)$ domain (where the points represent unit $S^2$) is nothing but the Penrose diagram of Minkowski space (See Fig. \\ref{CPplot}). The special lines and points in Fig. \\ref{CPplot} are given by\n\\begin{center}\n\\resizebox{\\columnwidth}{!}{\n \\begin{tabular}{|c|ccc|ccc|c|}\n  \\hline\n   & \\quad$\\tau{=}0$ & south pole & boundary & \\quad --- \\quad{} & \\!\\!\\!north pole\\!\\!\\! & $\\quad\\tau{=}{\\pm}\\pi\\quad$ & $\\chi{\\pm}\\tau{=}\\pi$ \\\\[2pt]\n   $(\\tau,\\chi)$ & \\quad$(0,\\chi)$ & $(\\tau,\\pi)$ & $(\\pm\\chi,\\chi)$ & $(0,\\pi)$ & $(0,0)$ & $(\\pm\\pi,\\pi)$ & $(\\pm\\pi{\\mp}\\chi,\\chi)$ \\\\[4pt]\n   $(t,r)$ & \\quad$(0,r)$ & $(t,0)$ & $(\\pm\\infty,\\infty)$ & $(0,0)$ & $(t,\\infty)$ & $(\\pm\\infty,r)$ & $(\\pm r,r)$ \\\\[2pt]\n   & \\quad$t{=}0$ & $r{=}0$ & $\\mathscr{I}^\\pm$ & \\ \\ origin \\ {} & $i^0$ & $i^\\pm$ & \\ \\  lightcone \\ \\ {} \\\\\n  \\hline\n \\end{tabular}\n }\n\\end{center}\nwith the Minkowski spatial and temporal infinity $i^0$ and $i^\\pm$ corresponding to the edges while the Minkowski null infinity~$\\mathscr{I}^\\pm$ corresponding to the conformal boundary $\\chi{=}|\\tau|$ of the Penrose diagram.\n\nEquation \\eqref{tr2TauChi} can be used to obtain the Jacobian for the transformation between the coordinates $y^m\\in\\{\\tau,\\chi,\\theta,\\phi\\}$ and $x^{\\mu}\\in\\{t,r,\\theta,\\phi\\}$:\n\\begin{equation} \\label{Jacobian}\n   \\bigl(J^m_{\\ \\ \\mu}\\bigr) \\ :=\\ \\frac{\\mbox{$\\partial$}(\\tau,\\chi,\\theta,\\phi)}{\\mbox{$\\partial$}(t,r,\\theta,\\phi)}\n   \\= \\frac1\\ell\\biggl(\\begin{matrix} p & -q \\\\[4pt] q & -p \\end{matrix} \\biggr) \\oplus\\mathds{1}_2\\ ,\n\\end{equation}\nwhere the polynomials $p,q$ are given by\n\\begin{equation}\n    p\\= \\sfrac{\\gamma^2}{\\ell^2}\\,(r^2{+}t^2{+}\\ell^2)\/2 \\= 1{-}\\cos\\tau\\cos\\chi \\quad\\textrm{and}\\quad q\\= \\sfrac{\\gamma^2}{\\ell^2}\\,t\\,r \\= \\sin\\tau\\sin\\chi\\ .\n\\end{equation}\n\nA more direct way of seeing the conformal correspondence between Minkwoski space and the $S^3$-cylinder is via Carter-Penrose transformation that readily produces the lightcone picture (see Appendix \\ref{appendCPtransf}). \n\n\\subsection{Structure of 3-sphere and its harmonics}\n\\noindent\nThe presence of $S^3$ in the metric \\eqref{metricdS} is an added advantage, which can be exploited to write a $SO(4)$-invariant gauge connection and obtain the corresponding fields by solving Yang--Mills equation on the Lorentzian cylinder. These quantities can be later exported to the Minkowski spacetime owing to the conformal invariance of the vacuum Yang--Mills equation in four dimensions. To this end, we start with the group $SO(4)$, which is isomorphic to two copies of $SU(2)$ (up to a $\\mathds{Z}_2$ grading). Each of these $SU(2)$ has a group action that generates a left (right) multiplication (a.k.a. translation) on $S^3$. This can easily be checked using the map\n\\begin{equation}\\label{map}\n   g:\\; S^3 \\rightarrow SU(2);~~ (\\omega_1,\\ \\omega_2,\\ \\omega_3,\\ \\omega_4)\\, \\mapsto\\, -i\n     \\begin{pmatrix}\n     \\beta & \\alpha^* \\\\ \\alpha & -\\beta^*\n     \\end{pmatrix}\\ ,\n\\end{equation} \nwith $\\alpha := \\omega_1 + \\mathrm{i}\\omega_2\\ ~\\textrm{and}~\\ \\beta := \\omega_3 +\\mathrm{i}\\omega_4 $. This parameterization of $g$ ensures that the identity element $e = \\mathds{1}_2$ of the group $SU(2)$ can be obtained from $(0,0,0,1)$ i.e. the North pole of $S^3$. It is well known that $S^3$ is the group manifold of $SU(2)$. Keeping this in mind, we consider the Cartan one-form\n\\begin{equation} \\label{Cartan1}\n   \\Omega_l(g)\\, :=\\, g^{-1}\\,\\mathrm{d} g \\= e^a\\,T_a,~~\\textrm{with}~~ T_a = -i\\sigma_a\n\\end{equation} \nbeing the $SU(2)$ generators. The left-invariant one-form\\footnote{Named so because they remain invariant under the dragging induced by the left SU(2) multiplication.} $e^a$ can alternatively be expressed using the so called self-dual 't Hooft symbol $\\eta^a_{_{BC}}$:\n\\begin{equation}\\label{one-form}\n   e^a \\= -\\eta^a_{_{\\ BC}}\\, \\omega_{_B}\\, \\mathrm{d} \\omega_{_C} \\quad\\textrm{with}\\quad \\eta^a_{\\ bc} \\= \\varepsilon_{abc} \\quad\\textrm{and}\\quad \\eta^a_{\\ b4} \\= -\\eta^a_{\\ 4b} \\= \\delta^a_b\\ .\n\\end{equation}\nThey satisfy the following useful identities\n\\begin{equation}\\label{MaurerCartan}\n\\delta_{ab}\\, e^a\\, e^b \\= \\mathrm{d}\\Omega_3^2 \\quad\\quad\\textrm{and}\\quad\\quad \\mathrm{d} e^a + \\varepsilon_{abc}\\, e^b\\wedge e^c \\= 0.\n\\end{equation} \nThe left-invariant vector fields $L_a$ generating the right translations are dual to $e^a$ and are given by\n\\begin{equation}\\label{leftVF}\n   L_a \\= -\\eta^a_{_{\\ BC}}\\,\\omega_{_B}\\,\\sfrac{\\partial}{\\partial \\omega_{_C}} \\qquad\\Rightarrow\\qquad \\left[ L_a , L_b \\right] \\= 2\\,\\varepsilon_{abc}\\,L_c\\ .\n\\end{equation}\nIn a similar way, the right-invariant vector fields $R_a$ generating the left translations are given by\n\\begin{equation}\\label{rightVF}\n   R_a \\= -\\tilde{\\eta}^a_{_{\\ BC}}\\,\\omega_{_B}\\,\\sfrac{\\partial}{\\partial\\omega_{_C}} \\qquad\\Rightarrow\\qquad\n   \\left[ R_a , R_b \\right] = 2\\,\\varepsilon_{abc}\\,R_c\\ ,\n\\end{equation}\nwhere the anti self--dual 't Hooft symbols $\\tilde{\\eta}^a_{_{\\ BC}}$ are obtained from \\eqref{one-form} by flipping the $B,C\\!=4$ sign. Furthermore, the vector fields $L_a$ and $R_a$ act on the one-forms $e^a$ via their Lie derivative, which can be performed using Cartan formula (see Section \\ref{covectorFields}):\n\\begin{equation}\\label{LieActionL}\n\\begin{aligned}\n    L_a\\,e^b\\ :=\\ \\mathcal{L}_{L_a}\\,e^b &\\= \\mathrm{d}\\circ\\iota_{L_a} e^b + \\iota_{L_a}\\circ\\mathrm{d}{e^b} \\\\\n    &\\= \\mathrm{d}{\\left(e^b(L_a)\\right)} - \\varepsilon^b_{\\ ij}\\,\\iota_{L_a}\\left(e^i\\wedge e^j\\right) \\\\\n    &\\= \\mathrm{d}{\\left(\\delta^b_a\\right)} - \\varepsilon^b_{\\ ij}\\left( e^i(L_a) e^j - e^j(L_a) e^i \\right) \\\\\n    &\\= 2\\,\\varepsilon_{abc}\\,e^c\\ ,\n\\end{aligned}\n\\end{equation}\nwhere in the second line we have used \\eqref{MaurerCartan}. A similar calculation for the action of $R_a$ yields\n\\begin{equation}\\label{LieActionR}\n    R_a\\, e^b \\= 0\\ .\n\\end{equation}\nWe can now write the differential $\\mathrm{d}$ of the functions $f\\in C^{\\infty}\\left(\\mathcal{I}\\times S^3\\right)$ using $L_a$\\footnote{One can use $R_a$ and its dual $1$-form as well.} as\n\\begin{equation}\n   \\mathrm{d} f \\= \\mathrm{d}\\tau\\; \\partial_{\\tau} f\\ +\\ e^a\\,L_a f\\ .\n\\end{equation}\n\nFunctions on $S^3$ can be expanded in a basis of harmonics~$Y_j(\\chi,\\theta,\\phi)$ with $2j\\in\\mathds N_0$, \nwhich are eigenfunctions of the scalar Laplacian,\\footnote{\nThe SO(4) spin of these functions is actually $2j$, but we label them with half their spin, for reasons to be clear below.}\n\\begin{equation}\n-\\mathop{}\\!\\mathbin\\bigtriangleup_3 Y_j \\= 2j(2j{+}2)\\,Y_j \\= 4j(j{+}1)\\,Y_j \\=\n-\\sfrac12(L^2+R^2)\\,Y_j \\= -\\sfrac14({\\cal D}^2+{\\cal P}^2)\\,Y_j\\ ,\n\\end{equation}\nwhere $L^2=L_a L_a$ and $R^2=R_a R_a$ are (minus four times) the Casimirs of $su(2)_L$ and $su(2)_R$, respectively,\n\\begin{equation}\n-\\sfrac14 L^2\\,Y_j \\= -\\sfrac14 R^2\\,Y_j \\= -\\sfrac14\\mathop{}\\!\\mathbin\\bigtriangleup_3 Y_j \\= j(j{+}1)\\,Y_j\\ .\n\\end{equation}\nWe have also introduced ${\\cal D}^2={\\cal D}_a {\\cal D}_a$ and ${\\cal P}^2={\\cal P}_a {\\cal P}_a$ with\n\\begin{equation} \\label{DPdef}\n{\\cal D}_a \\= R_a + L_a \\= -2\\,\\varepsilon_a^{\\ bc}\\,\\omega_b\\,\\mbox{$\\partial$}_c \\quad\\textrm{and}\\quad \n{\\cal P}_a \\= R_a - L_a \\= 2\\,\\omega_{[a}\\,\\mbox{$\\partial$}_{4]}\n\\end{equation}\nwith $\\mbox{$\\partial$}_{_A}\\equiv\\sfrac{\\mbox{$\\partial$}}{\\mbox{$\\partial$}\\omega_A}$ so that\n\\begin{equation}\n\\left[ {\\cal D}_a , {\\cal D}_b \\right] \\= 2\\,\\varepsilon_{ab}^{\\ \\ c}\\,{\\cal D}_c \\ ,\\qquad\n\\left[ {\\cal D}_a , {\\cal P}_b \\right] \\= 2\\,\\varepsilon_{ab}^{\\ \\ c}\\,{\\cal P}_c \\ ,\\qquad\n\\left[ {\\cal P}_a , {\\cal P}_b \\right] \\= 2\\,\\varepsilon_{ab}^{\\ \\ c}\\,{\\cal D}_c \\ .\n\\end{equation}\nHence, $\\{{\\cal D}_a\\}$ spans the diagonal subalgebra~$su(2)_D\\subset so(4)$, which generates the stabilizer subgroup~SO(3) \nin the coset representation~$S^3\\cong\\textrm{SO}(4)\/\\textrm{SO}(3)$.\nTherefore, ${\\cal D}^2$ is (minus four times) the Casimir of~$su(2)_D$, with eigenvalues $l(l{+}1)$ for $l=0,1,\\ldots,2j$,\nand $\\sfrac14 {\\cal D}^2=\\mathop{}\\!\\mathbin\\bigtriangleup_2$ is the scalar Laplacian on the $S^2$ slices traced out in~$S^3$ by the SO(3)$_D$ action.\n\nTo further characterize a complete basis of $S^3$ harmonics, there are two natural options,\ncorresponding to two different complete choices of mutually commuting operators to be diagonalized.\nFirst, the left-right (or toroidal) harmonics~$Y_{j;m,n}$ are eigenfunctions of $L^2=R^2$, $L_3$ and~$R_3$,\n\\begin{equation} \\label{Y-action}\n\\sfrac{\\mathrm{i}}{2}\\,L_3\\,Y_{j;m,n} \\= n\\,Y_{j;m,n} \\quad\\quad\\textrm{and}\\quad\\quad\n\\sfrac{\\mathrm{i}}{2}\\,R_3\\,Y_{j;m,n} \\= m\\,Y_{j;m,n} \\ ,\n\\end{equation}\nand hence the corresponding ladder operators \n\\begin{equation}\nL_\\pm \\= (L_1\\pm\\mathrm{i} L_2)\/\\sqrt{2} \\quad\\quad\\textrm{and}\\quad\\quad R_\\pm \\= (R_1\\pm\\mathrm{i} R_2)\/\\sqrt{2}\n\\end{equation}\nact as\n\\begin{equation}\\label{JpmAction}\n\\begin{aligned}\n  \\sfrac{\\mathrm{i}}{2}\\,L_\\pm\\,Y_{j;m,n} &\\= \\sqrt{(j{\\mp}n)(j{\\pm}n{+}1)\/2}\\,Y_{j;m,n\\pm1}\\ ,\\\\[4pt]\n  \\sfrac{\\mathrm{i}}{2}\\,R_\\pm\\,Y_{j;m,n} &\\= \\sqrt{(j{\\mp}m)(j{\\pm}m{+}1)\/2}\\,Y_{j;m\\pm1,n}\\ .\n\\end{aligned}\n\\end{equation}\nThe normalized harmonics $Y_{j;m,n}$ can be expanded in terms of functions $\\alpha,\\beta$ and their complex conjugates as\n\\begin{equation}\\label{S3Harmonics}\n\\begin{aligned}\n    Y_{j;m,n}(\\omega) &\\= \\sum_{k=0}^{2j}\\,K_{j,m,n}(k)\\,\\alpha^{n+m+k}\\, \\bar{\\alpha}^{k}\\, \\beta^{j-m-k}\\, \\bar{\\beta}^{j-n-k} \\qquad\\quad\\textrm{with}\\quad \\\\\n    K_{j,m,n}(k) &\\= (-1)^{m+n+k}\\,\\sqrt{\\frac{2j+1}{2\\pi^2}}\\,\\frac{\\sqrt{(j+m)!(j-m)!(j+n)!(j-n)!}}{(n+m+k)!(j-n-k)!(j-m-k)!k!}\\ .\n\\end{aligned}    \n\\end{equation}\nThey satisfy the orthonormality condition\n\\begin{equation}\\label{orthonormality}\n    \\int \\mathrm{d}^3\\Omega_3\\, Y_{j;m,n}\\, \\bar{Y}_{j';m',n'} \\= \\delta_{j,j'}\\, \\delta_{m,m'}\\, \\delta_{n,n'} \\quad\\textrm{with}\\quad \\mathrm{d}^3\\Omega_3 \\= \\sin^2\\chi\\sin\\theta\\, \\mathrm{d}\\chi\\, \\mathrm{d}\\theta\\, \\mathrm{d}\\phi\\ .\n\\end{equation}\nSecond, the adjoint (or hyperspherical) harmonics~${\\widetilde{Y}}_{j;l,M}$ are eigenfunctions of $L^2=R^2$, ${\\cal D}^2$ and~${\\cal D}_3$,\n\\begin{equation} \n-\\sfrac{1}{4}\\,{\\cal D}^2\\,{\\widetilde{Y}}_{j;l,M} \\= l(l{+}1)\\,{\\widetilde{Y}}_{j;l,M} \\quad\\quad\\textrm{and}\\quad\\quad\n\\sfrac{\\mathrm{i}}{2}\\,{\\cal D}_3\\,{\\widetilde{Y}}_{j;l,M} \\= M\\,{\\widetilde{Y}}_{j;l,M} \\ ,\n\\end{equation}\nwith the ladder-operator actions~\\cite{wybourne}\n\\begin{equation}\n\\begin{aligned}\n\\sfrac{\\mathrm{i}}{2}\\,{\\cal D}_\\pm\\,{\\widetilde{Y}}_{j;l,M} &\\= \\sqrt{(l{\\mp}M)(l{\\pm}M{+}1)\/2}\\,{\\widetilde{Y}}_{j;l,M\\pm1} \\ ,\\\\[4pt]\n\\sfrac{\\mathrm{i}}{2}\\,{\\cal P}_\\pm\\,{\\widetilde{Y}}_{j;l,M} &\\= \n\\mp\\sqrt{(l{\\mp}M{-}1)(l{\\mp}M)\/2}\\,c_{j,l}\\,{\\widetilde{Y}}_{j;l-1,M\\pm1} \\\\[2pt] \n&\\qquad\\pm\\ \\sqrt{(l{\\pm}M{+}1)(l{\\pm}M{+}2)\/2}\\,c_{j,l+1}\\,{\\widetilde{Y}}_{j;l+1,M\\pm1} \\ ,\\\\[4pt]\n\\sfrac{\\mathrm{i}}{2}\\,{\\cal P}_3\\,{\\widetilde{Y}}_{j;l,M} &\\= \n\\sqrt{l^2{-}M^2}\\,c_{j,l}\\,{\\widetilde{Y}}_{j;l-1,M} \\ +\\ \n\\sqrt{(l{+}1)^2{-}M^2}\\,c_{j,l+1}\\,{\\widetilde{Y}}_{j;l+1,M} \\ ,\n\\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\nc_{j,l} \\= \\sqrt{\\bigl((2j{+}1)^2-l^2\\bigr)\/\\bigl((2l{-}1)(2l{+}1)\\bigr)}\\ .\n\\end{equation}\nIn this case, there exists a recursive construction for harmonics on $S^{k+1}$ from those on $S^k$,\n\\begin{equation}\\label{split-Y}\n{\\widetilde{Y}}_{j;l,M}(\\chi,\\theta,\\phi) \\= R_{j,l}(\\chi)\\,Y_{l,M}(\\theta,\\phi)\\ ;\\\nR_{j,l}(\\chi) \\= \\mathrm{i}^{2j+l}\\,\\sqrt{\\sfrac{2j+1}{\\sin\\chi}\\sfrac{(2j+l+1)!}{(2j-l)!}}\\, P_{2j+\\frac12}^{-l-\\frac12}(\\cos\\chi)\\ ,\n\\end{equation}\nwhere $Y_{l,M}$ are the standard $S^2$ spherical harmonics and $P_a^b$ denote the associated Legendre polynomials of the first kind.\\footnote{\nWith fractional indices, it is rather a Gegenbauer polynomial, but also a hypergeometric function (see eq.~(2.8) of~\\cite{Higuchi,HiguchiErratum}).} \nThe two bases of harmonics are related by the standard Clebsch-Gordan series for the angular momentum addition\n$j\\otimes j = 0\\oplus 1\\oplus\\ldots\\oplus 2j$,\n\\begin{equation}\\label{Y-new}\nY_{j;m,n} \\= \\sum\\limits_{l=0}^{2j}\\sum\\limits_{M=-l}^{l}\\,C_{m,n}^{l,M}\\,{\\widetilde{Y}}_{j;l,M}\\ ,\n\\qquad\\textrm{with}\\qquad C_{m,n}^{l,M} = \\<2j;l,M|j,m;j,n\\>\n\\end{equation}\nbeing the Clebsch-Gordan coefficients enforcing $m{+}n{=}M$ and $l\\in\\{0,1,\\ldots,2j\\}$.\n\nA somewhat cumbersome calculation involving \\eqref{S3toMink}, \\eqref{gamma} and \\eqref{one-form} gives us the cylinder one--forms in Minkowski coordinates:\n\\begin{equation}\\label{1formsExpanded}\n\\begin{aligned}\ne^0\\ &:=\\ \\mathrm{d}{\\tau} \\= \\sfrac{\\gamma^2}{\\ell^3}\\bigl(\n\\sfrac12(t^2{+}r^2{+}\\ell^2)\\,\\mathrm{d} t - t\\,x^k \\mathrm{d} x^k \\bigr) \\\\[2pt]\n&\\= \\sfrac{\\gamma^2}{\\ell^3}\\bigl(\n\\sfrac12(t^2{+}r^2{+}\\ell^2)\\,\\mathrm{d} t - t\\,r\\,\\mathrm{d} r \\bigr)\n\\qquad\\quad\\textrm{and}\\quad \\\\[4pt]\ne^a &\\= \\sfrac{\\gamma^2}{\\ell^3}\\bigl(\nt\\,x^a \\mathrm{d} t - \\bigl[\\sfrac12(t^2{-}r^2{+}\\ell^2)\\,\\delta^{ak} +\nx^a x^k + \\ell\\,\\varepsilon^{ajk}x^j \\bigr]\\,\\mathrm{d} x^k \\bigr) \\\\[2pt]\n&\\= \\sfrac{\\gamma^2}{\\ell^3}\\bigl(\n\\hat{x}^a \\bigl[ r\\,t\\,\\mathrm{d} t -\\sfrac12(t^2{+}r^2{+}\\ell^2)\\,\\mathrm{d} r\\bigr]\n- \\sfrac12(t^2{-}r^2{+}\\ell^2)\\,r\\,\\mathrm{d}\\hat{x}^a -\n\\ell\\,r^2\\varepsilon^{ajk}\\hat{x}^j\\mathrm{d}\\hat{x}^k \\bigr)\\ .\n\\end{aligned}\n\\end{equation}\nWe further note down the expressions for the vector fields $L_a$ in terms of $S^3$ angles by using \\eqref{omegas} in \\eqref{leftVF} for later purposes:\n\\begin{equation}\\label{Lfields}\n \\begin{aligned}\n   L_1 &\\= \\sin\\theta\\cos\\phi\\, \\mbox{$\\partial$}_\\chi\\ +\\ (\\cot\\chi\\cos\\theta\\cos\\phi + \\sin\\phi)\\,\\mbox{$\\partial$}_\\theta\\ -\\ (\\cot\\chi\\csc\\theta\\sin\\phi - \\cot\\theta\\cos\\phi)\\, \\mbox{$\\partial$}_\\phi\\ , \\\\\n   L_2 &\\= \\sin\\theta\\sin\\phi\\, \\mbox{$\\partial$}_\\chi\\ +\\ (\\cot\\chi\\cos\\theta\\sin\\phi - \\cos\\phi)\\,\\mbox{$\\partial$}_\\theta\\ +\\ (\\cot\\chi\\csc\\theta\\cos\\phi + \\cot\\theta\\sin\\phi)\\, \\mbox{$\\partial$}_\\phi\\ , \\\\\n   L_3 &\\= \\cos\\theta\\, \\mbox{$\\partial$}_\\chi\\ -\\ \\cot\\chi\\sin\\theta\\, \\mbox{$\\partial$}_\\theta\\ -\\ \\mbox{$\\partial$}_\\phi\\ .\n \\end{aligned}\n\\end{equation}\n\n\n\n\\section{Yang-Mills field equations}\nOne can make use of these left-invariant one forms $e^a$ to expand a general Yang--Mills gauge potential $\\mathcal{A}$ (in the temporal gauge $\\mathcal{A}_\\tau {=}0$) by invoking the so called SU(2)-equivariant ansatz\n\\begin{equation}\\label{one-formAnsatz}\n\\mathcal{A} \\= X_a(\\tau,g)\\,e^a,\n\\end{equation}\nwhere the three $SU(2)$ matrices $X_a$ depends, in general, on both the cylinder parameter $\\tau$ and internal $S^3$ coordinates $\\omega$ via the map $g$ \\eqref{map}. The corresponding field strength $\\mathcal{F}$, with this choice of the gauge potential $\\mathcal{A}$, looks like:\n\\begin{equation} \\label{FcalExpanded}\n\\mathcal{F} \\= \\mathrm{d}{{\\cal A}} + \\mathcal{A}\\wedge\\mathcal{A} \\= \\dot{X}_a\\,e^0\\wedge e^a + \\frac{1}{2} \\left( L_{[b}\\, X_{c]} - 2\\,\\epsilon_{abc}\\, X_a + \\left[ X_b, X_c \\right] \\right) e^b\\wedge e^c\\ ,\n\\end{equation}\nwhere $e^0 := \\mathrm{d}{\\tau}$ and the dot on $X_a$ refers to its derivative w.r.t. $\\tau$. The Yang--Mills equation for this gauge field ${\\cal A}$ i.e.\n\\begin{equation}\n    \\mathrm{d}*\\mathcal{F} + {\\cal A}\\wedge *\\mathcal{F} - *\\mathcal{F}\\wedge\\mathcal{A} \\= 0\\ ,\n\\end{equation}\nafter a straightforward calculation, yields the constraint condition\n\\begin{equation}\\label{constraint}\n    -2\\,\\mathrm{i}\\,J_a\\,X_a + [X_a,\\dot{X}_a] \\= 0\\ .\n\\end{equation}\nand the field equation\n\\begin{align} \\label{ge_eq}\n\\begin{split}\n\\ddot{X}_a =& -4\\left(J^2{+}1\\right)X_a - 4\\,\\mathrm{i}\\,\\epsilon_{abc}\\,J_b\\,X_c + 4\\,J_a\\,J_b\\,X_b + 3\\,\\epsilon_{abc}\\,[X_b,X_c]\\\\\n&  + 2\\,\\mathrm{i}\\,[X_a,J_b\\,X_b] + 2\\,\\mathrm{i}\\,[X_b,J_a\\,X_b] + 4\\,\\mathrm{i}\\,[J_b\\,X_a,X_b] - [X_b,[X_a,X_b]]\\ ,\n\\end{split}\n\\end{align}\nwhere the operators $J_a$ and $J^2$ are given by\n\\begin{equation}\\label{opJ}\n    J_a\\ :=\\ \\frac{\\mathrm{i}}{2}\\,L_a \\quad\\textrm{and}\\quad J^2\\ :=\\ J_a\\,J_a\\ .\n\\end{equation}\nFinding a general solution for this equation is a daunting task, but progress can be made in the two limiting cases below. \n\nOne of the limiting case of \\eqref{ge_eq} is that of the Abelian one, where all the commutators vanish to yield\n\\begin{equation}\\label{eq_abelian}\n\\ddot{X}_a = -4\\left( J^2 {+} 1 \\right) X_a + 2\\,\\mathrm{i}\\,\\epsilon_{abc}\\,J_b\\,X_c\\ .\n\\end{equation}\nFurthermore the constraint \\eqref{constraint} in this case can be brought to the following form after integrating w.r.t to time and choosing the constant to be zero:\n\\begin{equation}\\label{constraintAbelian}\n    J_a\\,X_a \\= 0\\ .\n\\end{equation}\nOne thing to note here is that the above constraint along with the temporal gauge $A_\\tau\\!=0$ is not the usual Coulomb gauge on Minkowski space. In fact, we can make use of the inverse Jacobian \\eqref{Jacobian} while promoting the gauge potential to the Minkowski space ${\\cal A} \\!= {\\cal A}_a\\, e^a \\!= A_\\mu\\,dx^\\mu \\!=A $ to get\n\\begin{equation}\\label{gaugeCond2}\n  0 = {\\cal A}_\\tau = {\\cal A} \\left(\\partial_\\tau \\right) = A \\left( \\sfrac{\\ell^2}{\\gamma\n  ^2}\\left(p\\,\\partial_t + q\\,\\partial_r \\right) \\right) \\implies \\left(r^2+t^2+\\ell^2\\right)A_t + 2\\,r\\,t\\,A_r = 0\\ .\n\\end{equation}\nSolution to \\eqref{eq_abelian} was obtained in \\cite{LZ17} using hyperspherical harmonics $Y_{j;m,n}$ as we will see in the next chapter.\n\nAnother limiting case of \\eqref{ge_eq} is when a more symmetric $SO(4)$-equivariant condition is imposed yielding \\cite{Friedan,Luescher77} the following form of $X_a$:\n\\begin{equation}\\label{Aansatz}\nX_a \\= \\frac{1}{2}\\left( 1 + \\psi(\\tau) \\right)T_a\n\\end{equation}\nwith some function $\\psi : \\mathcal{I} \\rightarrow \\mathds R$ and $SU(2)$ generators $T_a$ satisfying the Lie algebra\n\\begin{equation}\n   [T_a,T_b] = 2\\,\\epsilon_{abc}\\,T_c\\ .\n\\end{equation}\nMoreover, $T_a$ act in adjoint representation where $\\mathrm{tr}{(T_a\\,T_b)} = -8\\,\\delta_{ab}$. The constraint \\eqref{constraint} for this symmetric ansatz i.e.\n\\begin{equation}\n    [X_a,\\dot{X}_a] \\= 0\n\\end{equation}\nis automatically satisfied and the equation \\eqref{ge_eq} becomes\n\\begin{equation}\n\\ddot{X}_a = -4\\,X_a + 3\\,\\epsilon_{abc}\\,[X_b,X_c] - [X_b,[X_a,X_b]].\n\\end{equation}\nSolution to this equation was obtained in \\cite{ILP17jhep} as we will see in chapter 6.\n\n\n\n\n\\chapter{Abelian solution: $U(1)$}\n\\label{Chapter5} \n\\justifying\n\nIn this chapter we present the knotted electromagnetic fields arising via the ``de Sitter\" method and study its various properties. The content of this chapter has generated one published work (in J. Phys. A) \\cite{KL20} and two pre-print articles \\cite{KPH21,KLP22}. Section \\ref{nullFields} on null solutions is due to Olaf Lechtenfeld with verification by Colin Becker and some clarifications from Harald Skarke. I have been involved at all stages of the following works for the rest of the sections in this chapter.\n\n\\vspace{12pt}\n\\vspace{2pt}\n\\section{Family of ``knot\" solutions}\nIt was shown in \\cite{LZ17}, that the general solution of \\eqref{eq_abelian} decomposes into spin-$j$ representations of~$so(4)$ and are labelled with hyperspherical harmonics of $S^3$. We review the construction of these solutions below in two steps: first we solve \\eqref{eq_abelian} on the $S^3$-cylinder, and then we pull these solution back to Minkowski space using $(\\tau,\\chi)\\rightarrow (t,r)$ coordinate transformation. \n\n\\subsection{Generic solution on the $S^3$-cylinder}\n\\noindent\nTo solve \\eqref{eq_abelian} we first write it down in terms of the ladder operators $J_{\\pm}$ and $J_3$ \\eqref{opJ} together with the redefined functions\n\\begin{equation}\n    X_{\\pm}\\ :=\\ \\frac{1}{2}\\left(X_1 \\pm \\mathrm{i} X_2 \\right) \n\\end{equation}\nas follows\n\\begin{equation} \\label{knotEquations}\n \\begin{aligned}\n    \\mbox{$\\partial$}_{\\tau}^2\\,X_+ &\\= -4\\,(J^2-J_3+1)X_+ - 4\\,J_+\\,X_3\\ ,\\\\[4pt]\n    \\mbox{$\\partial$}_{\\tau}^2\\,X_3 &\\= -4\\,(J^2+1)X_3 + 4\\,J_+\\,X_- - 4\\,J_-\\,X_+\\ ,\\\\[4pt]\n    \\mbox{$\\partial$}_{\\tau}^2\\,X_- &\\= -4\\,(J^2+J_3+1)X_- + 4\\, J_-\\,X_3\\ .\n \\end{aligned}\n\\end{equation}\nSimilarly, the constraint condition \\eqref{constraintAbelian} takes the following form\n\\begin{equation} \\label{constraintAbelian2}\n    0 \\= J_3\\,X_3 + J_+\\,X_- +J_-\\,X_+\\ .\n\\end{equation}\nWe can solve \\eqref{knotEquations} by the following ansatz\n\\begin{equation}\n\\begin{aligned}\n    X_+ &\\= \\sum\\limits_{j,m,n} Z^{j;m,n}_+\\,e^{i\\Omega_{j,m,n}\\tau}\\ ;\\quad Z^{j;m,n}_+ \\= c_+\\,Y_{j;m,n+1}\\ ,\\\\\n    X_3 &\\= \\sum\\limits_{j,m,n} Z^{j;m,n}_3\\,e^{i\\Omega_{j,m,n}\\tau}\\ ; \\quad Z^{j;m,n}_3 \\= c_3\\,Y_{j;m,n}\\ , \\quad\\textrm{and}\\quad \\\\\n    X_- &\\= \\sum\\limits_{j,m,n} Z^{j;m,n}_-\\,e^{i\\Omega_{j,m,n}\\tau}\\ ;\\quad Z^{j;m,n}_- \\= c_-\\,Y_{j;m,n-1}\\ ,\n\\end{aligned}\n\\end{equation}\nwhere $X_*(j,m,n)$ with $*\\in \\{ +,3,- \\}$ has been expanded in terms of $S^3$ harmonics. Plugging this ansatz in \\eqref{knotEquations} and using \\eqref{JpmAction} we find, for every mode $(j,m,n)$, an eigenvalue equation for the vector $(c_+\\ c_3\\ c_-)^T$:\n\\begin{equation}\n\\begin{aligned}\n    &\\qquad\\qquad M\\begin{pmatrix}\n    c_+ \\\\ c_3 \\\\ c_-\n    \\end{pmatrix} \\= \\Omega(j,m,n)^2\\begin{pmatrix}\n    c_+ \\\\ c_3 \\\\ c_-\n    \\end{pmatrix}\\ ,\\ \\quad\\textrm{with}\\quad \\\\\n    &M \\= \\begin{psmallmatrix}\n    4(j^2+j-n) & 2\\sqrt{2(j-n)(j+n+1)} & 0 \\\\\n    2\\sqrt{2(j-n)(j+n+1)} & 4(j^2+j+1) & -2\\sqrt{2(j+n)(j-n+1)} \\\\\n    0 & -2\\sqrt{2(j+n)(j-n+1)} & 4(j^2+j+n)\n    \\end{psmallmatrix}\\ ,\n\\end{aligned}\n\\end{equation}\nwhich admits an eigensystem with $3$ distinct eigenvalues $\\Omega_j^2$ (that turns out to be independent of $m$ and $n$) and their corresponding eigenvectors. One of these eigenvectors does not satisfy the constraint \\eqref{constraintAbelian2} and is, therefore, discarded. We label the remaining two eigensystems as type I, with eigenfrequency $\\Omega_j^2=4(j{+}1)^2$, and type II, with eigenfrequency $\\Omega_j^2=4\\,j^2$, as follows\n\\begin{itemize}\n\\addtolength{\\itemsep}{-4pt}\n\\item type I : \\quad\n$j{\\geq}0\\ ,\\quad m = -j,\\ldots,+j\\ ,\\quad n = -j{-}1,\\ldots,j{+}1\\ ,\\ \\Omega_j=\\pm 2\\,(j{+}1)\\ ,$\n\\\\\n\\begin{equation} \\label{type1}\n\\begin{aligned}\nZ_{+\\ \\textrm{I}}^{(j;m,n)}(\\omega) &\\= \\sqrt{(j{-}n)(j{-}n{+}1)\/2} \\ \\ Y_{j;m,n+1}(\\omega) \\ ,\\\\\nZ_{3\\ \\textrm{I}}^{(j;m,n)}(\\omega)\\,&\\= \\sqrt{(j{+}1)^2-n^2} \\ \\ Y_{j;m,n}(\\omega) \\ ,\\\\\nZ_{-\\ \\textrm{I}}^{(j;m,n)}(\\omega)   &\\= -\\sqrt{(j{+}n)(j{+}n{+}1)\/2} \\ \\ Y_{j;m,n-1}(\\omega) \\ .\n\\end{aligned}\n\\end{equation}\n\\item type II :\\quad\n$j{\\geq}1\\ ,\\quad m = -j,\\ldots,+j\\ ,\\quad n = -j{+}1,\\ldots,j{-}1\\ ,\\ \\Omega_j=\\pm2\\,j\\ ,$\n\\\\\n\\begin{equation} \\label{type2}\n\\begin{aligned}\nZ_{+\\ \\textrm{II}}^{(j;m,n)}(\\omega)  &\\= -\\sqrt{(j{+}n)(j{+}n{+}1)\/2} \\ \\ Y_{j;m,n+1}(\\omega)\\ ,\\\\\nZ_{3\\ \\textrm{II}}^{(j;m,n)}(\\omega)\\,&\\= \\sqrt{j^2-n^2} \\ \\ Y_{j;m,n}(\\omega) \\ ,\\\\\nZ_{-\\ \\textrm{II}}^{(j;m,n)}(\\omega)   &\\= \\sqrt{(j{-}n)(j{-}n{+}1)\/2} \\ \\ Y_{j;m,n-1}(\\omega) \\ .\n\\end{aligned}\n\\end{equation}\n\\end{itemize}\n\nWe take a linear combination of these basis solutions and write down the (real-valued) gauge potential as\n\\begin{equation}\\label{Asep}\n{\\cal A} \\=  \n\\Bigl\\{ \\sum_{2j=0}^\\infty X_{a\\ \\textrm{I}}^j(\\omega) \\ \\mathrm{e}^{2(j+1)\\mathrm{i}\\tau} \\ +\\ \\textrm{c.c.} \\Bigr\\}e^a \\ +\\ \n\\Bigl\\{ \\sum_{2j=2}^\\infty X_{a\\ \\textrm{II}}^j(\\omega) \\ \\mathrm{e}^{2j\\,\\mathrm{i}\\tau} \\ +\\ \\textrm{c.c.} \\Bigr\\}e^a\\ ,\n\\end{equation}\nwhere we have reorganized the complex angular functions~$X_a^j$ as\n\\begin{equation}\\label{XZ}\nX_1^j=\\sfrac{1}{\\sqrt{2}} \\bigl(Z_+^j + Z_-^j \\bigr)\\ ,\\qquad \nX_2^j=\\sfrac{\\mathrm{i}}{\\sqrt{2}} \\bigl( Z_-^j - Z_+^j \\bigr)\\ ,\\qquad \nX_3^j= Z_3^j\n\\end{equation}\nfor both types and expanded the functions~$Z^j_\\pm$ and $Z^j_3$ into the above spin-$j$ basis solutions of type I \\eqref{type1} and type II \\eqref{type2} (for $*\\in\\{+,3,-\\}$),\n\\begin{equation}\\label{basisZ}\n\\begin{aligned}\nZ_{*\\ \\textrm{I}}^j(\\omega) &\\= \\sum_{m=-j}^j \\sum_{n=-j-1}^{j+1} \\lambda_{j;m,n}^{\\textrm{I}}\\ Z_{*\\ \\textrm{I}}^{(j;m,n)}(\\omega)  \\quad\\textrm{and}\\quad \\\\\nZ_{*\\ \\textrm{II}}^j(\\omega) &\\= \\sum_{m=-j}^j \\sum_{n=-j+1}^{j-1} \\lambda_{j;m,n}^{\\textrm{II}}\\ Z_{*\\ \\textrm{II}}^{(j;m,n)}(\\omega)\\ ,\n\\end{aligned}\n\\end{equation}\nwith $(2j{+}1)(2j{+}3)$ arbitrary complex coefficients $\\lambda^{\\textrm{I}}_{j;m,n}$\nand $(2j{+}1)(2j{-}1)$ coefficients $\\lambda^{\\textrm{II}}_{j;m,n}$.\n(Note that type-II solutions are absent for $j{=}0$ and $j{=}\\sfrac12$.)\n\n\nInserting \\eqref{type1} and \\eqref{type2} into \\eqref{basisZ} and the result into~\\eqref{XZ} provides a harmonic expansion\n\\begin{equation}\\label{XY}\nX_a^j(\\omega) \\= \\sum_{m=-j}^j \\sum_{n=-j}^j X_a^{j;m,n}\\ Y_{j;m,n}(\\omega)\n\\end{equation}\nfor both types of angular functions in~\\eqref{Asep}. \n(Note the different range of~$n$ for $X_a^{j;m,n}$ and $Z_*^{j;m,n}$; they are not easily related as $X_a^j$ and $Z_*^j$ are in~\\eqref{XZ}.)\n\nIt is useful for later purposes to introduce here the ``sphere-frame\" electric and magnetic fields,\n\\begin{equation}\\label{2FormEM}\n    {\\cal F} \\= {\\cal E}_a\\,e^a {\\wedge} e^\\tau + \\sfrac12\\,{\\cal B}_a\\,\\varepsilon^a_{\\ bc}\\,e^b{\\wedge} e^c\\ .\n\\end{equation}\nFor a fixed type (I or II) and spin~$j$, we may eliminate $L_{[b}A_{c]}$ in \\eqref{FcalExpanded} (without the commutator term) by using~\\eqref{eq_abelian} and employ\n\\begin{equation}\n\\mbox{$\\partial$}_\\tau^2 {\\cal A}^{(j)} = -\\Omega_j^2\\,{\\cal A}^{(j)} \\quad\\quad\\textrm{and}\\quad\\quad\nL^2\\,{\\cal A}^{(j)} \\= -4j(j{+}1)\\,{\\cal A}^{(j)}\n\\end{equation}\nto obtain\n\\begin{equation}\\label{cur-field}\n{\\cal E}_a^{(j)} \\= -\\mbox{$\\partial$}_\\tau X_a^{(j)} \\quad\\quad\\textrm{and}\\quad\\quad {\\cal B}_a^{(j)} \\= \\mp\\Omega_j\\,X_a^{(j)}\\ ,\n\\end{equation}\nwhere the upper sign pertains to type~I and the lower one to type~II.\nWe note in passing that, due to the compactness of the Lorentzian cylinder, the sphere-frame energy and action are always finite.\n\nDue to the linearity of Maxwell theory, the overall scale of any solution is arbitrary.\nFurthermore, the parity transformation $L\\leftrightarrow R$ and $m\\leftrightarrow n$ \ninterchanges a spin-$j$ solution of type~I with a spin-$(j{+}1)$ solution of type~II.\nFinally, electromagnetic duality at fixed $j$ is realized by shifting $\\Omega_j\\tau$ by $\\sfrac{\\pi}{2}$ for type~I\nor by $-\\sfrac{\\pi}{2}$ for type~II, which maps ${\\cal A}$ to a dual configuration~${\\cal A}_\\textrm{D}$\nand likewise ${\\cal F}$ to ${\\cal F}_\\textrm{D}$.\n\n\\subsection{Pulling the solution back to Minkowski space}\n\n\\noindent\nWe have completely solved the vacuum Maxwell equations on the Lorentzian cylinder ${\\cal I}\\times S^3$ and, hence, on \nde Sitter space~dS$_4$. By conformal invariance, this solution carries over to any conformally equivalent spacetime. We translate our Maxwell solutions from $\\widetilde{{\\cal I}}\\times S^3$ to $\\mathds R^{1,3}$\nsimply by the coordinate change\n\\begin{equation}\n\\begin{aligned}\n\\tau=\\tau(t,x,y,z) &\\quad\\textrm{and}\\quad \\omega_{_A}=\\omega_{_A}(t,x,y,z) \\\\[2pt]\n\\textrm{or}\\qquad\n\\tau=\\tau(t,r) &\\quad\\textrm{and}\\quad \\chi=\\chi(t,r)\\ .\n\\end{aligned}\n\\end{equation}\nIn other words, abbreviating $x\\equiv\\{x^\\mu\\}$ and $y\\equiv\\{y^\\rho\\}$ and expanding\n\\begin{equation}\n\\begin{aligned}\n{\\cal A} &\\= X_a\\bigl(\\tau(x),g(x)\\bigr)\\,e^a(x) \n\\= A_\\mu(x)\\,\\mathrm{d} x^\\mu\n\\= A_\\rho(y)\\,\\mathrm{d} y^\\rho \\quad\\quad\\textrm{and}\\quad \\\\[4pt]\n\\mathrm{d}{\\cal A} &\\= \\mbox{$\\partial$}_\\tau X_a\\,e^0{\\wedge} e^a + \\bigl( L_b\\,X_c - X_a\\,\\varepsilon^a_{\\ bc}\\bigr)\\,e^b {\\wedge} e^c \\\\\n&\\= \\sfrac12 F_{\\mu \\nu}(x)\\,\\mathrm{d} x^\\mu {\\wedge} \\mathrm{d} x^\\nu \n\\= \\sfrac12 F_{\\rho\\lambda}(y)\\,\\mathrm{d} y^\\rho{\\wedge}\\mathrm{d} y^\\lambda\n\\end{aligned}\n\\end{equation}\nusing \\eqref{1formsExpanded} we may read off $A_\\mu$ (note that $A_t\\neq0$, as discussed before) and \n$F_{\\mu\\nu}$ and thus the electric and magnetic fields\n\\begin{equation}\\label{EMCartesian}\n{\\cal F} \\= F \\= E_a\\,\\mathrm{d}{x}^a\\wedge\\mathrm{d}{t} + \\frac{1}{2}B_a\\,\\varepsilon^a_{\\ bc}\\,\\mathrm{d}{x}^b{\\wedge} \\mathrm{d}{x}^c\n\\end{equation}\nin Cartesian or in spherical coordinates. In general, the knotted electromagnetic fields arising from the basis configurations \\eqref{type1} are complex, the basis configurations on Minkowski space will also be complex. Hence, they combine two physical solutions, namely the real and imaginary parts\\footnote{The notation should not be confused with the type I configuration \\eqref{type1}.}, which we denote as\n\\begin{equation*}\n    (j;m,n)_R~ \\textrm{configuration} \\quad\\textrm{and}\\quad (j;m,n)_I ~\\textrm{configuration}\\ ,\n\\end{equation*}\nrespectively. These knotted electromagnetic fields \\eqref{EMCartesian} for the basis configurations \\eqref{type1} and \\eqref{type2} increase in complexity with increasing $j$, as shown in Figure \\ref{fieldLines}.\n\nFurthermore, it comes in handy that ${\\cal A}$ finally contains only even powers of~$\\gamma$\nand depends on~$\\tau$ only through integral powers of\n\\begin{equation}\n\\exp (2\\mathrm{i}\\,\\tau) \\= \\frac{[(\\ell+\\mathrm{i}t)^2+r^2]^2}{4\\,\\ell^2 t^2 + (r^2-t^2+\\ell^2)^2}\\ .\n\\end{equation}\nTherefore, our Minkowski solutions have the remarkable property of being rational functions of~$(t,x,y,z)$.\nMore precisely, their electric and magnetic fields are of the form\n\\begin{equation}\n\\textrm{type I:} \\quad \\frac{P_{2(2j+1)}(x)}{Q_{2(2j+3)}(x)} \\ ,\\qquad\n\\textrm{type II:}\\quad \\frac{P_{2(2j-1)}(x)}{Q_{2(2j+1)}(x)} \n\\end{equation}\nwhere $P_r$ and $Q_r$ denote polynomials of degree~$r$.\nThus, as expected, their energy and action are finite.\nIndeed, the fields fall off like $r^{-4}$ at spatial infinity for fixed time, but they decay\nmerely like $(t{\\pm}r)^{-1}$ along the light-cone.\nHence, the asymptotic energy flow is concentrated on past and future null infinity~$\\mathscr{I}^\\pm$,\nas it should be, but peaks on the light-cone of the spacetime origin.\nSince on de Sitter space our basis solutions (\\ref{type1}) and~(\\ref{type2}) form a complete set, \ntheir Minkowski relatives are also complete in the space of finite-action configurations. \n\n\\begin{figure}[h!]\n\\captionsetup{width=\\linewidth}\n\\centering\n   \\includegraphics[width = 5cm, height = 5cm]{fieldLines_0,0,1.pdf}\n   \\includegraphics[width = 5cm, height = 5cm]{fieldLines_h,-h,3h.pdf}\n   \\includegraphics[width = 5cm, height = 5cm]{fieldLines_1,0,2.pdf}\n \\caption{\n Electric (red) and magnetic (green) field lines at $t{=}0$ with 4 fixed seed points.\n Left: $(j;m,n)=(0;0,1)_R$ configuration, Center: $(j;m,n)=(\\sfrac12;-\\sfrac12,\\sfrac32)_R$ configuration, Right: $(j;m,n)=(1;0,2)_R$ configuration. More self-knotted field lines start appearing with additional seed points in the simulation.}\n\\label{fieldLines}\n\\end{figure}\n\nFor illustration, we display a type I basis solution with $(j;m,n)=(1;0,0)$\nobtained from \n\\begin{equation}\\label{j1Prototype}\nX_\\pm \\ \\propto\\ \\sfrac{1}{\\sqrt{2}}\\,(\\omega_1{\\pm}\\mathrm{i}\\omega_2)(\\omega_3{\\pm}\\mathrm{i}\\omega_4)\\,\\cos 4\\tau \\quad\\textrm{and}\\quad\nX_3\\ \\propto\\  (\\omega_1^2{+}\\omega_2^2{-}\\omega_3^2{-}\\omega_4^2)\\,\\cos 4\\tau\\ .\n\\end{equation}\nIt bodes well to combine these field configurations into the Riemann--Silberstein vector \n\\begin{equation}\\label{RSvector}\n    {\\bf S} \\= {\\bf E} + \\mathrm{i} {\\bf B}\n\\end{equation}\nwhose components, for \\eqref{j1Prototype}, are (up to overall scale)\n{\\small\n\\begin{equation}\n\\begin{aligned}\nS_x &\\= -\\frac{2\\mathrm{i}}{N}\n\\Bigl\\{ 2y +3 \\mathrm{i} t y -x z +2 t^2 y + 2\\mathrm{i} t x z-8x^2 y-8y^3+4yz^2 + 4\\mathrm{i} t^3 y \\\\%[4pt]\n&\\qquad\\qquad  -6t^2 xz - 8\\mathrm{i} t x^2 y-8\\mathrm{i} t y^3+4\\mathrm{i} t yz^2 +10x^3 z+10xy^2 z -2xz^3 \\\\%[4pt]\n&\\qquad\\qquad\\qquad\\qquad + 2(\\mathrm{i} t x z + yr^2)(-t^2{+}r^2)+(\\mathrm{i} t y- x z) (-t^2{+}r^2)^2\\Bigr\\}\n\\ ,\\quad{}\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\n\\begin{aligned}\nS_y &\\= \\frac{2\\mathrm{i}}{N}\n\\Bigl\\{ 2x +3 \\mathrm{i} t x + y z+2 t^2 x-2\\mathrm{i} t y z-8x^3-8xy^2 +4 xz^2 + 4 \\mathrm{i} t^3 x \\\\%[4pt]\n&\\qquad\\qquad + 6 t^2y z -8\\mathrm{i} t x^3-8\\mathrm{i} txy^2 +4 \\mathrm{i} t x z^2-10 x^2 yz-10y^3 z +2y z^3 \\\\%[4pt]\n&\\qquad\\qquad\\qquad\\qquad + 2 (-\\mathrm{i} t y z {+} x r^2)(-t^2{+} r^2) +(\\mathrm{i} t x+ y z)(-t^2{+}r^2)^2\\Bigr\\}\n\\ ,\\qquad\\&\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\n\\begin{aligned}\nS_z &\\= \\frac{\\mathrm{i}}{N}\n\\Bigl\\{1+2\\mathrm{i} t+t^2-11 x^2- 11 y^2 +3 z^2 +4 \\mathrm{i} t^3-16\\mathrm{i} t x^2-16\\mathrm{i} t y^2+4\\mathrm{i} t z^2 - t^4 \\\\%[4pt]\n&\\qquad\\qquad -2 t^2 x^2 -2t^2 y^2-2t^2 z^2+11 x^4+22x^2y^2-10 x^2 z^2+11y^4-10 y^2 z^2 \\\\%[4pt]\n&\\qquad\\qquad\\qquad\\qquad\\qquad +3z^4 + 2 \\mathrm{i} t (t^2 {-} 3 r^2{+}2 z^2) (t^2{-}r^2)-(t^2{+}r^2)(-t^2{+}r^2)^2\\Bigr\\}\n\\ ,\\quad{}\n\\end{aligned}\n\\end{equation}\n}\nwith $N=\\bigl((t{-}\\mathrm{i})^2-r^2\\bigr)^5$. We can also obtain electromagnetic field configurations corresponding to the gauge field \\eqref{Asep}, that consists of complex linear combinations of these basis solutions. We demonstrate this for the famous Hopf--Ran\\~{a}da field configuration, which was first discovered by Ran\\~{a}da in 1989 \\cite{Ranada89} using the Hopf fibration. The Riemann--Silberstein vector ${\\bf S}$ for this field configuration is given by \\cite{LZ17}\n\\begin{equation} \\label{HRfields}\n    {\\bf S} \\= \\frac{1}{((t{-}\\mathrm{i})^2{-}r^2)^3} \\begin{pmatrix}\n        (x{-}\\mathrm{i}\\,y)^2 {-} (t{-}\\mathrm{i}{-}z)^2 \\\\\n        \\mathrm{i}(x{-}\\mathrm{i}\\,y)^2 {+} \\mathrm{i}(t{-}\\mathrm{i}{-}z)^2 \\\\\n        -2(x{-}\\mathrm{i}\\,y)(t{-}\\mathrm{i}{-}z)\n    \\end{pmatrix}\\ .\n\\end{equation}\nWe find that this solution, in our construction, is obtained by taking linear combination of $j{=}0$, and hence type I, basis configurations \\eqref{type1} with the following coefficients in \\eqref{basisZ}\n\\begin{equation}\n    \\lambda^I_{0;0,+1} \\= 0\\ ,\\quad \\lambda^I_{0;0,0} \\= 0\\ ,\\quad\\textrm{and}\\quad \\lambda^I_{0;0,-1} \\= -\\mathrm{i}\\frac{\\pi}{8}\\ .\n\\end{equation}\n\n\n\n\\vspace{12pt}\n\\section{Symmetry analysis}\n\\vspace{2pt}\n\\noindent\nThe main advantage of constructing Minkowski-space electromagnetic field configurations \nvia the detour over de Sitter space is the enhanced manifest symmetry of our construction. \nThe isometry group SO(1,4) of dS$_4$ is generated by \n($A,B=1,2,3,4$ and $a,b,c=1,2,3$, abbreviate $\\frac{\\partial}{\\partial q_{_B}}\\equiv\\partial_{_B}$)\n\\begin{equation}\n\\{{\\cal M}_{_{AB}}{\\equiv}\\,{-}q_{[{\\scriptscriptstyle A}}\\partial_{{\\scriptscriptstyle B}]}\\,,\\ \n{\\cal M}_{0{\\scriptscriptstyle B}}{\\equiv}\\,q_{({\\scriptstyle 0}}\\partial_{{\\scriptscriptstyle B})} \\} \\= \n\\{ {\\cal M}_{ab}{=}\\varepsilon_{abc}{\\cal D}_c\\,,\\ {\\cal M}_{4a}{=}{\\cal P}_a\\,,\\ {\\cal M}_{04}{=}{\\cal P}_0\\,,\\ {\\cal M}_{0b}{=}{\\cal K}_b\\}\\ ,\n\\end{equation}\nwhich can be contracted (with $\\ell{\\to}\\infty$) to the isometry group ISO(1,3) of $\\mathds R^{1,3}$ (the Poincar\\'e group)\ngenerated by ($\\mu,\\nu=0,1,2,3$ and $i,j,k=1,2,3$)\n\\begin{equation}\n\\{M_{\\mu\\nu}\\,,\\ P_\\mu \\} \\=\n\\{ M_{ij}{=}\\varepsilon_{ijk}D_k\\,,\\ P_i\\,,\\ P_0\\,,\\ M_{0j}{=}K_j\\}\\ ,\n\\end{equation}\nwhere the two sets are ordered likewise, \nand we employ (as aleady earlier) calligraphic symbols for de Sitter quantities and straight symbols for Minkowskian ones.\nHere, $D$ denotes spatial rotations, $P$ are translations, and $K$ stand for boosts in Minkowski space.\n\nSince the two spaces are conformally equivalent already at $\\ell{<}\\infty$ via~\\eqref{S3toMink},\nthe corresponding generators should be related. Indeed, the common SO(3) subgroup in\n\\begin{equation}\nSO(1,4) \\supset SO(4) \\supset SO(3) \\quad\\quad\\textrm{and}\\quad\\quad ISO(1,3) \\supset SO(1,3) \\supset SO(3)\n\\end{equation}\nis identified, ${\\cal D}_i{=}D_i{=}{-}2\\varepsilon_{ij}^{\\ k}x^j\\partial_k$. \nHowever, any other generator becomes nonlinearly realized when \nmapped to the other space via \\eqref{tr2TauChi}.\nFor example, the would-be translation ${\\cal P}_3$ defined in~\\eqref{DPdef} reads\n\\begin{equation}\n\\begin{aligned}\n{\\cal P}_3 = L_3-R_3 &\\= -2\\,\\cos\\theta\\,\\mbox{$\\partial$}_\\chi + 2\\,\\cot\\chi\\sin\\theta\\,\\mbox{$\\partial$}_\\theta \\\\[2pt]\n&\\= \\sfrac1{\\ell}\\,\\cos\\theta\\,\\bigl( 2\\,r\\,t\\,\\mbox{$\\partial$}_t + (t^2{+}r^2{+}\\ell^2)\\,\\mbox{$\\partial$}_r \\bigr) \n- \\sfrac1{\\ell\\,r} (t^2{-}r^2{+}\\ell^2)\\,\\sin\\theta\\,\\mbox{$\\partial$}_\\theta \\\\[2pt]\n&\\ \\to\\ 2\\ell\\,\\bigl( \\cos\\theta\\,\\mbox{$\\partial$}_r - \\sfrac1r\\,\\sin\\theta\\,\\mbox{$\\partial$}_\\theta \\bigr)\n\\= 2\\ell\\,\\mbox{$\\partial$}_z \\= \\ell\\,P_z \\quad\\quad\\textrm{for}\\quad\\ell\\to\\infty\n\\end{aligned}\n\\end{equation}\nas it should be. Similarly, ${\\cal P}_0\\to \\ell\\,P_0$ and ${\\cal K}_b\\to K_j$ for $\\ell{\\to}\\infty$ \nwhen expanded around $(t,r)=(\\ell,0)$ corresponding to the $S^3$ south pole at $q_0{=}0$.\nNevertheless, the de Sitter construction enjoys an SO(4) covariance (generated by ${\\cal D}_a$ and ${\\cal P}_a$)\nwhich extends the obvious SO(3) covariance in Minkowski space.\nIt allows us to connect all solutions of a given type (I or II) with a fixed value of the spin~$j$\nby the action of SO(4) ladder operators $L_\\pm$ and $R_\\pm$ or ${\\cal D}_\\pm$ and ${\\cal P}_\\pm$, \nwhich is non-obvious on the Minkowski side. \nOn the other hand, Minkowski boosts and translations have no simple realization on de Sitter space.\n\nActually, Maxwell theory on either space is also invariant under conformal transformations.\nThese may be generated by the isometry group together with a conformal inversion.\nOn the Minkowski side, the latter is\n\\begin{equation}\nJ : \\quad x^\\mu \\ \\mapsto\\ \\frac{x^\\mu}{x\\cdot x} \\quad\\quad\\textrm{with}\\quad x\\cdot x=r^2-t^2\\ .\n\\end{equation}\nWe have to distinguish two cases:\n\\begin{equation}\n\\begin{aligned}\n\\textrm{spacelike:}\\quad & t^2<r^2 \\quad\\Rightarrow\\quad\nJ_> : \\quad\\bigl(t,r,\\theta,\\phi\\bigr) \\ \\mapsto\\ \\bigl(\\sfrac{t}{r^2-t^2}, \\sfrac{r}{r^2-t^2},\\theta,\\phi\\bigr)\\ ,\\\\[2pt]\n\\textrm{timelike:} \\quad & t^2>r^2 \\quad\\Rightarrow\\quad\nJ_< : \\quad\\bigl(t,r,\\theta,\\phi\\bigr) \\ \\mapsto\\ \\bigl(\\sfrac{-t}{t^2-r^2}, \\sfrac{r}{t^2-r^2},\\pi{-}\\theta,\\phi{+}\\pi\\bigr)\\ .\n\\end{aligned}\n\\end{equation}\nOn the de Sitter side, this is either (spacelike) a reflection on the $S^3$ equator~$\\chi{=}\\frac{\\pi}{2}$\nor (timelike) a $\\pi$-shift in cylinder time~$\\tau$ plus an $S^2$ antipodal flip,\n\\begin{equation}\n\\begin{aligned}\n\\textrm{spacelike:}\\quad & |\\tau|{+}\\chi<\\pi \\quad\\Rightarrow\\quad\n{\\cal J}_> :  \\quad \\bigl(\\tau,\\chi,\\theta,\\phi\\bigr)\\ \\mapsto\\ \\bigl(\\tau,\\pi{-}\\chi,\\theta,\\phi\\bigr)\\ , \\\\[2pt]\n\\textrm{timelike:} \\quad & |\\tau|{+}\\chi>\\pi \\quad\\Rightarrow\\quad\n{\\cal J}_< :  \\quad \\bigl(\\tau,\\chi,\\theta,\\phi\\bigr)\\ \\mapsto\\ \\bigl(\\tau{\\pm}\\pi,\\chi,\\pi{-}\\theta,\\phi{+}\\pi\\bigr)\\ .\n\\end{aligned}\n\\end{equation}\nIn the spacelike case, merely the sign of $\\omega_4{\\equiv}\\cos\\chi$ gets flipped, \nwhich amounts to a parity flip $L\\leftrightarrow R$. \nIn the timelike case, both $\\cos\\tau$ and $\\sin\\tau$ change sign, \nwhich combines a time reversal with a reflection at $\\tau{=}\\frac{\\pi}{2}$ or $\\tau{=}{-}\\frac{\\pi}{2}$.\nNote that it is different from the $S^3$ antipodal map, which\nis not a reflection but a proper rotation, $\\omega_{_A}\\mapsto-\\omega_{_A}$ or \n$(\\chi,\\theta,\\phi)\\mapsto(\\pi{-}\\chi,\\pi{-}\\theta,\\phi{+}\\pi)$.\nThe lightcone is singular under the inversion; it is mapped to the conformal boundary\n$r{=}{\\pm}t{=}\\infty$ or $\\chi{=}{\\pm}\\tau$.\nWe infer that the conformal inversion allows us to relate type-I and type-II solutions of the same spin.\nIt is easily checked that the spatial fall-off behavior of our rational solutions is not modified by the inversion.\n\nFinally, one may consider dilatations in Minkowski space,\n\\begin{equation}\nx^\\mu \\ \\mapsto\\ \\lambda\\,x^\\mu \\quad\\quad\\textrm{for}\\quad \\lambda\\in\\mathds R_+\\ .\n\\end{equation}\nHowever, this amounts to a trivial rescaling also achieved by changing the de Sitter radius,\n$\\ell\\mapsto\\lambda\\,\\ell$, as the scale~$\\ell$ was removed on the Lorentzian cylinder. \n\n\\vspace{12pt}\n\\section{Conformal group and Noether charges}\n\\vspace{2pt}\n\\noindent\nIt is well known \\cite{BR16} that free Maxwell theory on $\\mathds R^{1,3}$ arising from the action\n\\begin{equation}\n    S\\left[A_\\mu\\right] \\= \\int d^4x\\, \\mathcal{L}\\ ;\\qquad \\mathcal{L} \\= -\\sfrac14 F^{\\mu\\nu}F_{\\mu\\nu}\n\\end{equation}\nis invariant under the conformal group $SO(2,4)$. Furthermore, the above action is also invariant under the gauge transformations: $A_\\mu(x)\\rightarrow A_\\mu(x)+\\mbox{$\\partial$}_\\mu\\lambda(x)$. The conformal group is generated by transformations $x^\\mu \\rightarrow x^\\mu + \\zeta^\\mu(x)$, where the vector fields $\\zeta^\\mu$ obey the conformal Killing equations:\n\\begin{equation}\\label{Killing}\n    \\zeta_{\\mu,\\nu}\\, +\\, \\zeta_{\\nu,\\mu} \\= \\sfrac12\\, \\eta_{\\mu\\nu}\\, \\zeta^\\alpha_{~,\\alpha} \\quad\\textrm{with}\\quad \\{\\eta_{\\mu\\nu}\\} \\= \\textrm{diag}(-1,1,1,1)\\ .\n\\end{equation}\nThe conserved Noether current $J^\\mu$ is obtained by equating the ``on-shell variation\" of the action where the variations $\\delta A_\\mu$ are arbitrary and the fields $A_\\mu$ satisfies the Euler-Langrange equations with its ``symmetry variation\" where the fields $A_\\mu$ are arbitrary but variations $\\delta A_\\mu$ satisfy the symmetry condition. The correct variation $\\delta A_\\mu$ is obtained by imposing the gauge invariance on the Lie derivative of $A_\\mu$ w.r.t. the vector field $\\zeta^\\mu$:\n\\begin{equation}\n    \\mathcal{L}_{\\zeta^\\alpha}A_\\mu := A'_\\mu (x) - A_\\mu (x) = -\\zeta^\\alpha\\, \\mbox{$\\partial$}_\\alpha A_\\mu - A_\\alpha\\, \\mbox{$\\partial$}_\\mu \\zeta^\\alpha ~\\longrightarrow ~ \\delta A_\\mu \\= F_{\\mu\\nu}\\zeta^\\nu\\ .\n\\end{equation}\nFinally, the conserved current is obtained as\n\\begin{equation}\\label{currentJ}\n    J^\\mu \\= \\frac{\\mbox{$\\partial$} \\mathcal{L}}{\\mbox{$\\partial$} A_{\\rho,\\mu}} \\delta A_\\rho + \\zeta^\\mu \\mathcal{L} \\= \\zeta^\\rho \\left( F^{\\mu\\alpha}F_{\\rho\\alpha} - \\sfrac14\\delta^\\mu_\\rho F^2   \\right) \\quad\\textrm{with}\\quad F^2 = F_{\\beta \\gamma}F^{\\beta \\gamma}\\ ,\n\\end{equation}\nwhich satisfies the continuity equation and gives the conserved (in time) charge $Q$\\footnote{For source-free fields the current $\\textbf{J}$ is assumed to vanish when the surface $\\mbox{$\\partial$} V$ is taken to infinity.}:\n\\begin{equation}\n    \\mbox{$\\partial$}_\\mu\\, J^\\mu = 0 \\quad\\implies\\quad \\frac{\\mathrm{d} Q}{\\mathrm{d} t} \\= -\\int_{\\mbox{$\\partial$} V} \\mathrm{d}^2{\\bf s}\\cdot{\\bf J} \\= 0\\ ;\\quad Q \\= \\int_{V} \\mathrm{d}^3x\\, J^0\\ .\n\\end{equation}\nBefore proceeding further, a couple of remarks pertaining to the subsequent calculations are in order:\n\\begin{itemize}\n   \\item All the charges $Q$ are computed at $t{=}0$ owing to the simple $J^0$ expressions on this time-slice. To that end, we record the following useful identities at $t{=}\\tau{=}0$:\n   \\begin{equation}\\label{t0Results}\n      \\begin{aligned}\n       e^a_i &\\= \\sfrac1\\ell \\left( \\gamma\\, \\omega_4\\, \\delta^a_i - \\omega_a\\, \\omega_i + \\epsilon_{aic}\\, \\gamma\\, \\omega_c \\right)\\ ,\\quad  e^a_i\\, e^b_i \\= \\sfrac{\\gamma^2}{\\ell^2} \\delta^{ab} \\\\\n       \\gamma &\\= 1-\\omega_4\\ ,~ \\mathrm{d}^3x \\= \\sfrac{\\ell^3}{\\gamma^3}\\mathrm{d}^3\\Omega_3\\ ~;~ \\mathrm{d}^3\\Omega_3 := e^1\\wedge e^2\\wedge e^3\\ .\n      \\end{aligned}\n   \\end{equation}\n   Moreover, the electromagnetic fields at $t{=}0$ are given in terms of tetrads $e^\\tau {=} e^\\tau_\\mu\\, \\mathrm{d} x^\\mu $ and $ e^a {=} e^a_\\mu\\, \\mathrm{d} x^\\mu$:\n   \\begin{equation}\\label{EMfields}\n    E_i \\= e^\\tau_0\\, e^a_i\\, \\mathcal{E}_a \\quad\\quad\\textrm{and}\\quad\\quad  B_i \\= \\sfrac12\\, \\epsilon_{ijk}\\, \\epsilon_{abc}\\, e^b_j\\, e^c_k\\, \\mathcal{B}_a\\ \n   \\end{equation}\n   using the $2$-form \\eqref{2FormEM} and its Minkowski counterpart \\eqref{EMCartesian}.\n   \n   \\item The charges are computed for type I \\eqref{type1} solutions only (except for the energy and the related helicity) and we relabel the complex coefficients \\eqref{basisZ} $\\lambda^I_{j;m,n}$ as $\\Lambda_{j;m,n}$ for convenience.\n   \n   \\item  Furthermore, these charges $Q$ are computed for a fixed spin-$j$ and, thus, we will suppress the index $j$ from now onward, unless necessary. Note that the sphere-frame EM fields for fixed $j$ can be obtained by using the expansion \\eqref{Asep} (for type I only) in \\eqref{cur-field} as\n   \\begin{equation} \\label{EBj}\n     {\\cal A}_a \\= X_a(\\omega)\\,\\mathrm{e}^{\\Omega\\,\\mathrm{i}\\tau} + \\bar{X}_a(\\omega)\\,\\mathrm{e}^{-\\Omega\\,\\mathrm{i}\\tau}\n     \\Rightarrow \\biggl\\{ \\begin{array}{l}\n     {\\cal E}_a \\= -\\mathrm{i}\\,\\Omega\\,X_a\\,\\mathrm{e}^{\\Omega\\,\\mathrm{i}\\tau} + \\mathrm{i}\\,\\Omega\\,\\bar{X}_a\\,\\mathrm{e}^{-\\Omega\\,\\mathrm{i}\\tau} \\\\[2pt]\n     {\\cal B}_a \\= \\,-\\,\\Omega\\,X_a\\,\\mathrm{e}^{\\Omega\\,\\mathrm{i}\\tau}\\,-\\,\\Omega\\,\\bar{X}_a\\,\\mathrm{e}^{-\\Omega\\,\\mathrm{i}\\tau} \\end{array} \\biggr\\}\n   \\end{equation}\n   with $\\bar{X}_a$ denoting the complex conjugate of $X_a$. Notice that for type II the overall sign in ${\\cal B}_a$ flips.\n \n   \\item  The simplifications below for the charge density $J^0$ are carried out using the harmonic expansion \\eqref{XY} for the type I solution \\eqref{type1}.\n   \n   \\item We frequently use below the well known fact that an odd $\\{\\omega_{_A}\\}$ integral over $S^3$ vanishes because of the opposite contributions coming from the antipodal points on the sphere. In particular, it can be checked that the following integral vanishes\\footnote{Note that the power of $\\omega_{_A}$ in $Y_{j;m,n}\\bar{Y}_{j;m',n'}$ is always even. One way to check this is by employing the toroidal coordinates: $\\omega_1 = \\cos\\eta\\cos\\kappa_1,\\omega_2 = \\cos\\eta\\sin\\kappa_1, \\omega_3 = \\sin\\eta\\cos\\kappa_2,\\omega_4 = \\sin\\eta\\sin\\kappa_2$ with $\\eta\\in(0,\\sfrac\\pi2)$ and $\\kappa_1,\\kappa_2\\in(0,2\\pi)$ in \\eqref{S3Harmonics}. The resultant selection rules coming from $\\kappa_1,\\kappa_2$ integral would yield $m-m'\\in \\sfrac{2k+1}{2}$ with $k\\in\\mathds N_0$, which is not feasible for fixed $j$.}:\n   \\begin{equation}\n      \\int \\mathrm{d}^3\\Omega_3\\, (\\omega_1)^a\\, (\\omega_2)^b\\, (\\omega_3)^c\\, (\\omega_4)^d\\, Y_{j;m,n}\\, \\overline{Y}_{j;m',n'} \\= 0 \\quad\\textrm{for}\\quad a+b+c+d\\ \\in\\ 2\\mathds N_0 + 1\\ .\n   \\end{equation}\n\\end{itemize}\nHaving made these remarks, we now proceed to compute the charges $Q$ for various conformal transformations $\\zeta^\\mu$ obeying \\eqref{Killing} in following four categories.\n\n\n\\subsection{Translations}\n\\noindent\nAn easily seen solution to \\eqref{Killing} is the set of four constant translations\n\\begin{equation}\n    \\zeta^\\mu = \\epsilon^\\mu\\ ,\n\\end{equation}\nwhich also partly generates the Poincar\\'e group and give rise to the usual stress-energy tensor of electrodynamics\n\\begin{equation}\\label{stress-energy}\n    J^\\mu \\= T^\\mu_{\\ \\; \\nu} \\=  F^{\\mu\\alpha}F_{\\nu\\alpha} - \\sfrac14\\delta^\\mu_\\nu F^2\\ ,\n\\end{equation}\ncorresponding to the $\\mu$-component of the translation for an arbitrary $\\epsilon^\\nu$. The corresponding charges are the energy $E$ and the momentum $\\textbf{P}$.\n\n\\noindent\n{\\bf Energy.} The expression of the energy density $e:=T^{00}$ simplifies to  \n\\begin{equation}\n    e\\, :=\\, \\frac{1}{2} \\left( E_i^2 + B_i^2 \\right) \\= \\frac{\\gamma}{\\ell}^4\\, \\rho \\quad\\quad\\textrm{with}\\quad\\quad \\rho \\= \\sfrac12 \\left( {\\cal E}_a^2 + {\\cal B}_a^2 \\right)\\ ,\n\\end{equation}\nwhich, in turn, simplifies the expression for the energy $E$ to\n\\begin{equation}\n    E \\= \\int_V \\mathrm{d}^3x\\;e \\= \\sfrac{1}{2\\ell}\\int_{S^3} \\mathrm{d}^3\\Omega_3 \\  (1-\\cos\\chi)\\,\\bigl({\\cal E}_a^2 + {\\cal B}_a^2\\bigr)\\ .\n\\end{equation}\nNotice here that the orientation of the $S^3$ volume measure~$\\mathrm{d}^3\\Omega_3$ is chosen to provide a positive result. From \\eqref{EBj} we find that the ``sphere-frame'' energy density is time-independent and has similar expression for both solution types (with appropriate eigenfrequency $\\Omega$):\n\\begin{equation}\n\\sfrac12 \\bigl( {\\cal E}_a^2 + {\\cal B}_a^2 \\bigr) \\= 2\\,\\Omega^2\\,X_a\\bar{X}_a(\\omega)\\ .\n\\end{equation}\n\nThe resultant expression for $E$ in terms of complex parameters $\\lambda_{m,n}$ (for both solution types) is given by\n\\begin{equation}\n    E \\= \\frac{1}{\\ell}\\, (2j+1)\\,\\Omega^3 \\sum_{m,n}\\,|\\lambda_{m,n}|^2\\ .\n\\end{equation}\nNotice again here that for type I solutions we would have $\\Lambda_{m,n}$ in the above expression.\n\n{\\bf Helicity.} Although helicity is not a Noether charge of the conformal group, it is nevertheless a conserved quantity for the Maxwell system and turns out to be related to the energy. The expression for the helicity is metric-free and can thus be evaluated over any spatial slice. \nChoosing again $t{=}\\tau{=}0$,\n\\begin{equation} \\label{helicity}\nh = h_{\\textrm{mag}}+h_{\\textrm{el}} \\= \\sfrac12 \\int_{\\mathds R^3} \\ \\bigl( A\\wedge F + A_D \\wedge F_D \\bigr)\n\\= -\\sfrac12 \\int_{S^3} \\mathrm{d}^3\\Omega_3 \\  (1-\\cos\\chi)\\,\\bigl({\\cal A}_a{\\cal B}_a + {\\cal A}^D_a{\\cal E}_a\\bigr)\\ ,\n\\end{equation}\nwhere the subscript\/superscript $`D'$ refers to the dual fields. Once again, taking type~I (upper sign) or type~II (lower sign) and fixing the spin~$j$ we obtain\n\\begin{equation}\n{\\cal A}^D_a \\= \\pm\\mathrm{i}\\,X_a(\\omega)\\,\\mathrm{e}^{\\Omega\\,\\mathrm{i}\\tau} \\mp\\mathrm{i}\\,\\bar{X}_a(\\omega)\\,\\mathrm{e}^{-\\Omega\\,\\mathrm{i}\\tau}\\ ,\n\\end{equation}\nwhich yields a constant ``sphere-frame'' helicity density\n\\begin{equation}\n-\\sfrac12 \\bigl( {\\cal A}_a{\\cal B}_a + {\\cal A}^D_a{\\cal E}_a \\bigr) \\= \\pm 2\\,\\Omega\\,X_a\\bar{X}_a(\\omega)\\ .\n\\end{equation}\nAs a result, even before performing the $S^3$ integration, we find a linear helicity-energy relation \n\\begin{equation}\n\\Omega\\, h \\= \\pm \\ell\\, E \\qquad\\textrm{for fixed spin and type}\\ .\n\\end{equation}\n\nSince the helicity measure an average of the linking numbers of any two electric or magnetic field lines~\\cite{moffatt17, moffatt69, berger},\nthe latter must be related to the value~$j$ of the spin. The individual linking number of two field lines, however, appears neither to be independent of the lines chosen nor constant in time, as our observations indicate.\nAn exception are the Ra\\~nada--Hopf knots discussed before, \nwhich display a conserved linking number of unity between any pair of electric or magnetic field lines.\n\n\n\\noindent\n{\\bf Momentum.} For the momentum densities $p_i = T^{0i} = (\\textbf{E}\\times\\textbf{B})_i$ we obtain an interesting correspondence relating the one-form $p := p_i\\, \\mathrm{d} x^i$ on Minkowski space with a similar one on de Sitter space:\n\\begin{equation}\\label{1formP}\n    p \\= (\\sfrac\\gamma\\ell)^3\\, \\mathcal{P}_a\\, e^a\\ =:\\ (\\sfrac\\gamma\\ell)^3\\, \\mathcal{P} \\quad\\quad\\textrm{with}\\quad\\quad \\mathcal{P}_a\\ :=\\ \\varepsilon_{abc}\\, \\mathcal{E}_b\\, \\mathcal{B}_c\\ .\n\\end{equation}\nA straightforward calculation then yields the expression of momenta $P_i$:\n\\begin{equation}\n    P_i \\= \\int_V \\mathrm{d}^3x\\, p_i \\= \\int_{S^3} \\mathrm{d}^3\\Omega_3\\, \\mathcal{P}_a\\, e^a_i \\= 2\\mathrm{i}\\ \\Omega^2\\ \\varepsilon_{abc}\\ \\int_{S^3} \\mathrm{d}^3\\Omega_3\\, e^a_i\\, Z_b\\, \\bar{Z}_c\n\\end{equation}\nwith $e^a_i$ given by \\eqref{t0Results}. The results for $j=0$ are\n\\begin{equation}\n  \\begin{aligned}\n    P_1^{(j=0)} &\\= -\\sfrac{\\sqrt{2}}{\\ell}\\left(\\left(\\bar{\\Lambda}_{0,-1} + \\bar{\\Lambda}_{0,1}\\right) \\Lambda_{0,0} + \\bar{\\Lambda}_{0,0}\\left(\\Lambda_{0,-1}+\\Lambda_{0,1}\\right)\\right)\\ , \\\\\n    P_2^{(j=0)} &\\=  \\sfrac{\\mathrm{i}\\sqrt{2}}{\\ell} \\left(\\left(-\\bar{\\Lambda}_{0,-1} + \\bar{\\Lambda}_{0,1}\\right) \\Lambda_{0,0} + \\bar{\\Lambda}_{0,0}\\left(\\Lambda_{0,-1}-\\Lambda_{0,1}\\right)\\right)\\ , \\\\\n    P_3^{(j=0)} &\\= \\sfrac2\\ell \\left( |\\Lambda_{0,-1}|^2 - |\\Lambda_{0,1}|^2 \\right)\\ .\n  \\end{aligned}\n\\end{equation}\nAs a consistency requirement, we check that the vector charges $P_i$ are rotated according to the algebra of $\\mathcal{D}_a$ \\eqref{DPdef} (See appendix \\ref{appendRotation}):\n\\begin{equation}\\label{rotP}\n    \\mathcal{D}_a\\, P_b \\= 2\\,\\varepsilon_{abc}\\, P_c\\ .\n\\end{equation}\nWe also note down $P_3$ for $j=1\/2\\ ~\\textrm{and}~ 1$ in table \\ref{table1}.\n\\begin{table}[]\n    \\centering\\setcellgapes{4pt}\\makegapedcells \\renewcommand\\theadfont{\\normalsize\\bfseries}\n \\resizebox{\\linewidth}{!}{\n    \\begin{tabular}{|c|P{6cm}|P{8cm}|}\n        \\hline\n      & $j=1\/2$ & $j=1$ \\\\ [0.5ex] \\hline \\hline\n     $P_3$ & \n     $\\begin{aligned}\n         \\sfrac{9}{\\ell} \\Big( &|\\Lambda_{-1\/2,-3\/2}|^{2} - |\\Lambda_{-1\/2, 1\/2}|^{2} \\\\\n         &- 2 |\\Lambda_{-1\/2, 3\/2}|^{2} + 2 |\\Lambda_{1\/2,-3\/2}|^{2} \\\\\n         &+ |\\Lambda_{1\/2,-1\/2}|^{2} - |\\Lambda_{1\/2, 3\/2}|^{2}\\Big)\n     \\end{aligned}$ & \n     $\\begin{aligned}\n         \\sfrac{24}{\\ell}\\Big( &|\\Lambda_{-1,-2}|^{2} - |\\Lambda_{-1,0}|^{2} - 2|\\Lambda_{-1,1}|^{2} - 3|\\Lambda_{-1,2}|^{2} \\\\\n         &+ 2|\\Lambda_{0,-2}|^{2} + |\\Lambda_{0,-1}|^{2} - |\\Lambda_{0, 1}|^{2} - 2|\\Lambda_{0,2}|^{2} \\\\\n         &+ 3|\\Lambda_{1,-2}|^{2} + 2|\\Lambda_{1,-1}|^{2} + |\\Lambda_{1,0}|^{2} - |\\Lambda_{1,2}|^{2} \\Big)\n     \\end{aligned}$  \\\\ \\hline\n     $P_\\phi$ & \n     $\\begin{aligned}\n         9 \\Big( &2|\\Lambda_{-1\/2,-3\/2}|^{2} + |\\Lambda_{-1\/2,-1\/2}|^{2} \\\\\n         &-|\\Lambda_{-1\/2,3\/2}|^{2} + |\\Lambda_{1\/2,-3\/2}|^{2} \\\\\n         &- |\\Lambda_{1\/2,1\/2}|^{2} - 2|\\Lambda_{1\/2, 3\/2}|^{2}\\Big)\n     \\end{aligned}$ & \n     $\\begin{aligned}\n         24 \\Big( &3|\\Lambda_{-1,-2}|^{2} + 2|\\Lambda_{-1,-1}|^{2} + |\\Lambda_{-1,0}|^{2} - |\\Lambda_{-1,2}|^{2} \\\\\n         &+ 2|\\Lambda_{0,-2}|^{2} + |\\Lambda_{0,-1}|^{2} - |\\Lambda_{0,1}|^{2} - 2|\\Lambda_{0,2}|^{2} \\\\\n         &+ |\\Lambda_{1,-2}|^{2} - |\\Lambda_{1,0}|^{2} - 2|\\Lambda_{1,1}|^{2} - 3|\\Lambda_{1,2}|^{2} \\Big)\n     \\end{aligned}$  \\\\ \\hline\n     $L_3$ & \n     $\\begin{aligned}\n         9\\ell\\, \\Big( &-2|\\Lambda_{-1\/2,-3\/2}|^{2} - |\\Lambda_{-1\/2,-1\/2}|^{2} \\\\\n         &+|\\Lambda_{-1\/2, 3\/2}|^{2} - |\\Lambda_{1\/2,-3\/2}|^{2} \\\\\n         &+ |\\Lambda_{1\/2,1\/2}|^{2} + 2|\\Lambda_{1\/2, 3\/2}|^{2}\\Big)\n     \\end{aligned}$ & \n     $\\begin{aligned}\n         -24\\ell\\, \\Big( &3|\\Lambda_{-1,-2}|^{2} + 2|\\Lambda_{-1,-1}|^{2} + |\\Lambda_{-1,0}|^{2} \\\\\n         & - |\\Lambda_{-1,2}|^{2} + 2|\\Lambda_{0,-2}|^{2} + |\\Lambda_{0,-1}|^{2} \\\\\n         & - |\\Lambda_{0, 1}|^{2} - 2|\\Lambda_{0,2}|^{2} + |\\Lambda_{1,-2}|^{2} \\\\\n         &- |\\Lambda_{1,0}|^{2} - 2|\\Lambda_{1,1}|^{2} - 3|\\Lambda_{1,2}|^{2} \\Big)\n     \\end{aligned}$  \\\\ \\hline\n    \\end{tabular}\n\n    \\caption{Expressions of $P_3$, $P_\n    \\phi$ and $L_3$ for $j=1\/2 ~\\textrm{and}~ 1$.}\n    \\label{table1}\n\\end{table}\nOne can compute the corresponding $P_1$ and $P_2$ for $j=1\/2$ and $j=1$ by employing the action of an appropriate $\\mathcal{D}_a$ of the table in Appendix \\ref{appendRotation}.\n\nWe can additionally compute the spherical components of the momentum $(P_r,P_\\theta,P_\\phi)$ by letting the one-form $e^a$ in \\eqref{1formP} act on the vector fields $(\\mbox{$\\partial$}_r,\\mbox{$\\partial$}_\\theta, \\mbox{$\\partial$}_\\phi)$. In practice, we first write $\\mbox{$\\partial$}_r = -\\sfrac\\gamma\\ell \\mbox{$\\partial$}_\\chi$ using \\eqref{Jacobian} at $t{=}0$ and then invert the vector fields $(\\mbox{$\\partial$}_r,\\mbox{$\\partial$}_\\theta, \\mbox{$\\partial$}_\\phi)$ in terms of the left invariant vector fields $(L_1,L_2,L_3)$ using \\eqref{Lfields}. Finally, using the duality relation $e^a(L_b)=\\delta^a_b$ we obtain\n\\begin{equation}\\label{sphericalP}\n \\begin{aligned}\n   P_r &\\= -\\frac1\\ell\\int_{S^3} \\mathrm{d}^3\\Omega_3\\,(1-\\cos\\chi)\\, \\left( \\sin\\theta\\cos\\phi\\, \\mathcal{P}_1 + \\sin\\theta\\sin\\phi\\, \\mathcal{P}_2 + \\cos\\theta\\, \\mathcal{P}_3 \\right)\\ , \\\\\n   P_\\theta &\\= \\int_{S^3} \\mathrm{d}^3\\Omega_3\\,\\sin\\chi\\cos\\chi \\Big( (\\cos\\theta\\cos\\phi + \\tan\\chi\\sin\\phi)\\, \\mathcal{P}_1 \\\\\n   &\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad + (\\cos\\theta\\sin\\phi - \\tan\\chi\\cos\\phi)\\mathcal{P}_2 - \\sin\\theta\\, \\mathcal{P}_3 \\Big) ,\\\\\n   P_\\phi &\\= \\int_{S^3} \\mathrm{d}^3\\Omega_3\\, \\sin^2\\chi\\sin\\theta \\Big( (\\cos\\theta\\cos\\phi - \\cot\\chi\\sin\\phi)\\, \\mathcal{P}_1 \\\\\n   &\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad + (\\cos\\theta\\sin\\phi + \\cot\\chi\\cos\\phi)\\, \\mathcal{P}_2 - \\sin\\theta\\, \\mathcal{P}_3 \\Big) \\ .\n \\end{aligned}\n\\end{equation}\nFrom the expression of $P_r$ we see that the integrand over $S^2$ i.e. $\\hat{\\omega}_a\\mathcal{P}_a$ \\eqref{OmegaToHatOmega} is an odd function\\footnote{The terms of $\\mathcal{P}_a$ using \\eqref{split-Y} are proportional to  $Y_{l,M}Y_{l',M'}$, which is even in $\\hat{\\omega}_a$ for a fixed $j$.}, which makes $P_r$ vanish. We also find with explicit calculations (verified for up to $j=1$) that $P_\\theta$ vanishes. For $j=0$ we find that $P_\\phi$ is proportional to $P_3$:\n\\begin{equation}\n    P_\\phi^{(j=0)} \\= \\ell\\, P_3^{(j=0)} \\=  2\\left(|\\Lambda_{0,-1}|^{2}-|\\Lambda_{0,1}|^{2}\\right)\\ .\n\\end{equation}\nThe expressions of $P_\\phi$ for $j=1\/2$ and $1$ has been recorded in the table \\ref{table1}.\n\\subsection{Lorentz transformations}\n\\noindent\nAnother solution of \\eqref{Killing} is given by\n\\begin{equation}\n    \\zeta^\\mu(x) \\= \\epsilon^{\\mu}_{\\ \\; \\nu}\\, x^\\nu \\quad\\textrm{where}\\quad \\epsilon_{\\mu\\nu} \\= -\\epsilon_{\\nu\\mu}\\ ,\n\\end{equation}\nwhich correspond to the six generators of the Lorentz group $SO(1,3)$. These six together with the above four translations generates the full Poincar\\'e group. The corresponding six charges are grouped into the boost ${\\bf K}$ and the angular momentum ${\\bf L}$.\n\n\\noindent\n{\\bf Boost.} The conserved charge densities arising from $J^0$ \\eqref{currentJ} corresponding to $\\epsilon_{0i}$ are the boost densities ${\\bf k} = \\rho\\ {\\bf x} - {\\bf p}\\ t$, which simplify for $t{=}0$ to\n\\begin{equation}\n    k_i \\= (\\sfrac\\gamma\\ell)^3\\, \\mathcal{K}_i \\quad\\quad\\textrm{with}\\quad\\quad \\mathcal{K}_i \\= \\rho\\, \\omega_i\\ .\n\\end{equation}\nThe corresponding charges $K_i$ vanishes because of the odd integrand as discussed in an earlier remark:\n\\begin{equation}\nK_i \\= \\int_V \\mathrm{d}^3x\\, k_i \\= \\int_{S^3} \\mathrm{d}^3\\Omega_3\\, \\mathcal{K}_i \\= 2 \\int_{S^3} \\mathrm{d}^3\\Omega_3\\, \\omega_i\\, X_a^j\\, \\bar{X}_a^j \\= 0\\ .\n\\end{equation}\n\n\\noindent\n{\\bf Angular momentum.} The other three conserved charge densities $J^0$ corresponding to $\\epsilon_{ij}$ in \\eqref{currentJ} are the angular momentum densities ${\\bf l} = {\\bf p}\\times{\\bf x}$, which takes a simple form just like in momentum \\eqref{1formP}:\n\\begin{equation}\n    l_i\\, \\mathrm{d} x^i \\= (\\sfrac\\gamma\\ell)^2\\, \\mathcal{L}_a\\, e^a \\quad\\quad\\textrm{with}\\quad\\quad \\mathcal{L}_a \\= \\varepsilon_{abc}\\, \\mathcal{P}_b\\, \\omega_c\\ .\n\\end{equation}\nThe expressions for the charges $L_i$ simplify to\n\\begin{equation}\n    L_i \\= \\int_V \\mathrm{d}^3x\\, l_i \\= \\int_{S^3} \\mathrm{d}^3\\Omega_3\\, (\\sfrac\\ell\\gamma)\\, \\mathcal{L}_a\\, e^a_i \\= 2\\mathrm{i}\\ell\\ \\Omega^2\\ \\int_{S^3} \\mathrm{d}^3\\Omega_3\\, \\sfrac{1}{\\gamma}\\, e^a_i \\left( \\bar{X}_a\\, \\omega_b X_b - X_a\\, \\omega_b\\bar{X}_b \\right)\\ .\n\\end{equation}\nExplicit calculation show that for $j{=}0$ the angular momenta $L_i$ are proportional to the momenta $P_i$:\n\\begin{equation}\n    L_i^{(j=0)} \\= -\\ell^2\\, P_i^{(j=0)}\\ .\n\\end{equation}\nThis is, however, not true for higher spin $j$. The angular momenta $L_i$ has the same rotation behaviour as for the momenta $P_i$ \\eqref{rotP}. We therefore note down the results of $L_3$ for $j=1\/2$ and $1$ in table \\ref{table1}, from which the corresponding expressions of $L_1$ and $L_2$ can be obtained using the table in Appendix \\ref{appendRotation}.\n\nWe can again compute the spherical components of the angular momentum $(L_r,L_\\theta,L_\\phi)$ using the relations \\eqref{sphericalP} by replacing $\\mathcal{P}$ with $\\mathcal{L}$ in it. We realize that the $S^2$ integrand for $L_r$ would only have terms like $\\hat{\\omega}_a\\hat{\\omega}_b\\mathcal{P}_c$, which are all odd functions\\footnote{Here again the terms of $\\mathcal{P}_c$, which goes like $Y_{l,M}Y_{l',M'}$ \\eqref{split-Y}, are all even functions of $\\phi$ for a fixed $j$.} of $\\phi$ and, therefore, the $\\phi$ integral over the domain $(0,2\\pi)$ would make $L_r$ vanish. We also find, with explicit computations, that the charges $L_\\theta$ and $L_\\phi$ for $j{=}0$ are proportional to $P_3$:\n\\begin{equation}\n    L_\\theta^{(j=0)} \\= \\sfrac43\\, \\ell^2\\, P_3^{(j=0)} \\quad\\quad\\textrm{and}\\quad\\quad L_\\phi^{(j=0)} \\= -\\sfrac13\\, \\ell^2\\, P_3^{(j=0)}\\ .\n\\end{equation}\nAs a non trivial example, we collect these charges for $j=1\/2$ below:\n\\begin{equation}\n  \\begin{aligned}\n     L_\\theta^{\\left(j=1\/2\\right)} \\= \\sfrac{12}{5}\\ell\\, \\Big( &9|\\Lambda_{-1\/2,-3\/2}|^{2} + 6|\\Lambda_{-1\/2,-1\/2}|^{2} +|\\Lambda_{-1\/2, 1\/2}|^{2}\n     - 6|\\Lambda_{-1\/2,3\/2}|^{2} \\\\ \n     & + 6|\\Lambda_{1\/2,-3\/2}|^{2} - |\\Lambda_{1\/2, -1\/2}|^{2} - 6|\\Lambda_{1\/2,1\/2}|^{2} - 9|\\Lambda_{1\/2, 3\/2}|^{2} \\Big)\\ ,\n  \\end{aligned}\n\\end{equation}\n\\begin{equation}\n  \\begin{aligned}\n      L_\\phi^{\\left(j=1\/2\\right)} \\= -\\sfrac35 \\ell\\, \\Big(  &6|\\Lambda_{-1\/2,-3\/2}|^{2} - |\\Lambda_{-1\/2,-1\/2}|^{2} -6|\\Lambda_{-1\/2, 1\/2}|^{2} - 9|\\Lambda_{-1\/2,3\/2}|^{2} \\\\\n      &+ 9|\\Lambda_{1\/2,-3\/2}|^{2} + 6|\\Lambda_{1\/2,-1\/2}|^{2} + |\\Lambda_{1\/2,1\/2}|^{2} - 6|\\Lambda_{1\/2, 3\/2}|^{2}\\\\ \n      &+ \\sqrt{3}\\big( \\bar{\\Lambda}_{1\/2,-3\/2}\\Lambda_{-1\/2,-1\/2} -\\bar{\\Lambda}_{1\/2,1\/2}\\Lambda_{-1\/2,3\/2} \\\\\n      &+ \\bar{\\Lambda}_{-1\/2,-1\/2}\\Lambda_{1\/2,-3\/2} - \\bar{\\Lambda}_{-1\/2,3\/2}\\Lambda_{1\/2,1\/2}\\big) \\Big)\\ .\n  \\end{aligned}\n\\end{equation}\n\n\\subsection{Dilatation} \nIt is easy to verify that a constant rescaling by $\\lambda$:\n\\begin{equation}\n    \\zeta^\\mu \\= \\lambda\\, x^\\mu\n\\end{equation}\nis also a solution of \\eqref{Killing}. The charge density corresponding to this single generator of the conformal group is ${\\bf p}\\cdot{\\bf x}- e\\,t$, which for $t{=}0$ simplifies to \n\\begin{equation}\n    p_i\\, x_i \\= (\\sfrac\\gamma\\ell)^3\\, \\mathcal{P}_a\\,\\omega_a\\ .\n\\end{equation}\nThe corresponding charge $D$ vanishes because of the odd integrand:\n\\begin{equation}\n    D \\= \\int_V \\mathrm{d}^3x\\, p_i\\, x_i \\= 2\\mathrm{i}\\,\\Omega^2\\, \\varepsilon_{abc} \\int_{S^3} \\mathrm{d}^3\\Omega_3\\, \\omega_a\\, X_b\\, \\bar{X}_c \\= 0\\ .\n\\end{equation}\n\\subsection{Special conformal transformations}\nA fairly straightforward calculation shows that the following not so obvious transformation\n\\begin{equation}\n    \\zeta^\\mu \\= 2\\, x^\\mu\\,b_\\nu x^\\nu\\ -\\ b^\\mu\\, x^\\nu x_\\nu \n\\end{equation}\nalso satisfies \\eqref{Killing}. The four generators corresponding to $b_\\mu$ give rise to four different charges $V_0$ and ${\\bf V}$.\n\n\\noindent\n{\\bf Scalar SCT.} The charge density $J^0$ corresponding to $b_0$ is \\begin{equation}\n    v_0 \\= ({\\bf x}^2+t^2)\\, e - 2t\\, {\\bf p}\\cdot{\\bf x}\\ ,\n\\end{equation}\nwhich for $t{=}0$ simplifies to\n\\begin{equation}\n    v_0 \\= {\\bf x}^2\\, e \\= (\\sfrac\\gamma\\ell)^2 \\rho\\, \\omega_a^2\\ .\n\\end{equation}\nThe expression for the corresponding charge $V_0$ takes the following simple form:\n\\begin{equation}\n    V_0 \\= \\int_V \\mathrm{d}^3x\\, v_0 \\= \\int_{S^3} \\mathrm{d}^3\\Omega_3\\, (\\sfrac\\ell\\gamma)\\, (1-\\omega_4^2)\\, \\rho \\= 2\\ell\\, \\Omega^2 \\int_{S^3} \\mathrm{d}^3\\Omega_3\\, (1+\\omega_4)\\, X_a\\, \\bar{X}_a\\ .\n\\end{equation}\nHere again the $\\omega_4$ term of the integral, being odd, vanishes and yields\n\\begin{equation}\n    V_0 \\= \\ell^2\\, E \\= 8\\,\\ell\\, (j+1)^3(2j+1)\\, \\sum_{m,n}\\,|\\Lambda_{m,n}|^2\\ .\n\\end{equation}\n\n\\noindent\n{\\bf Vector SCT.} The charge densities $J^0$ that correspond to $b_i$ read:\n\\begin{equation}\n    {\\bf v} \\= 2{\\bf x}({\\bf x}\\cdot{\\bf p}) - 2t\\,{\\bf x}\\,e - ({\\bf x}^2-t^2){\\bf p}\\ .\n\\end{equation}\nThis simplify at $t{=}0$ and takes a structure similar to the momentum densities $p_i$ \\eqref{1formP}:\n\\begin{equation}\\label{vectorSCT}\n    v_i\\, dx^i \\= (\\sfrac\\gamma\\ell)\\, \\mathcal{V}_a\\, e^a \\quad\\textrm{with}\\quad \\mathcal{V}_a \\= 2\\omega_a\\, (\\mathcal{P}_b\\,\\omega_b) - \\omega_b^2\\,\\mathcal{P}_a\\ . \n\\end{equation}\nThe expressions for the charges $V_i$ then simplify to\n\\begin{equation}\n    V_i \\= \\int_V \\mathrm{d}^3x\\, v_i \\= 2\\mathrm{i}\\,\\ell^2\\,\\Omega^2 \\int_{S^3} \\mathrm{d}^3\\Omega_3\\, \\gamma^{-2} e^a_i \\left( 2\\varepsilon_{bcd}\\,\\omega_a\\,\\omega_b\\,X_c\\,\\bar{X}_d - \\varepsilon_{abc}\\,(1-\\omega_4^2)\\,X_b\\,\\bar{X}_c \\right)\\ .\n\\end{equation}\nWith explicit computation we observe that the charges $V_i$ are proportional to the momenta $P_i$ (verified explicitly for up to $j{=}1$)\n\\begin{equation}\n    V_i \\= \\ell^2\\, P_i\\ .\n\\end{equation}\n\nAs before, we can compute the spherical components $(V_r,V_\\theta,V_\\phi)$ by using the expressions \\eqref{sphericalP} and replacing $\\mathcal{P}$ with $\\mathcal{V}$ in it. We notice that the charge $V_r$ vanishes owing to the odd $S^2$ integrand\\footnote{Observe that the terms in $\\mathcal{V}_a$ \\eqref{vectorSCT} are all even in $\\hat{\\omega}_a$.} just like in the case of $P_r$. However, unlike $P_\\theta$ here the charge $V_\\theta$ is non-vanishing. Explicit calculations show that for $j=0$ the charges $V_\\theta$ and $V_\\phi$ are proportional to the momentum $P_3$:\n\\begin{equation}\n    V_\\theta^{(j=0)} \\= -\\sfrac43 \\ell^3\\, P_3^{(j=0)} \\quad\\quad\\textrm{and}\\quad\\quad V_\\phi^{(j=0)} \\= -\\sfrac53 \\ell^3\\, P_3^{(j=0)}\\ .\n\\end{equation}\nAdditionally, we record below the charges $V_\\theta$ and $V_\\phi$ for the non-trivial case of $j{=}1\/2$\n\\begin{equation}\n  \\begin{aligned}\n     V_\\theta^{\\left(j=1\/2\\right)} \\= -\\sfrac{6}{5}\\ell^2\\, \\Big( &9|\\Lambda_{-1\/2,-3\/2}|^{2} + |\\Lambda_{-1\/2,-1\/2}|^{2} - 9|\\Lambda_{-1\/2,1\/2}|^{2} - 21|\\Lambda_{-1\/2,3\/2}|^{2} \\\\ \n     & + 21|\\Lambda_{1\/2,-3\/2}|^{2} +\n     9|\\Lambda_{1\/2,-1\/2}|^{2} - |\\Lambda_{1\/2,1\/2}|^{2} - 9|\\Lambda_{1\/2, 3\/2}|^{2} \\Big)\\ ,\n  \\end{aligned}\n\\end{equation}\n\\begin{equation}\n  \\begin{aligned}\n      V_\\phi^{\\left(j=1\/2\\right)} \\= -\\sfrac35 \\ell^2\\, \\Big(  &42|\\Lambda_{-1\/2,-3\/2}|^{2} + 33|\\Lambda_{-1\/2,-1\/2}|^{2} + 8|\\Lambda_{-1\/2,1\/2}|^{2} - 33|\\Lambda_{-1\/2,3\/2}|^{2} \\\\\n      &+ 33|\\Lambda_{1\/2,-3\/2}|^{2} - 8|\\Lambda_{1\/2,-1\/2}|^{2} - 33|\\Lambda_{1\/2,1\/2}|^{2} - 42|\\Lambda_{1\/2,3\/2}|^{2}\\\\ \n      &\\qquad + 8\\sqrt{3}\\big( -\\bar{\\Lambda}_{1\/2,-3\/2}\\Lambda_{-1\/2,-1\/2} +\\bar{\\Lambda}_{1\/2,1\/2}\\Lambda_{-1\/2,3\/2} \\\\\n      &\\qquad\\qquad\\qquad - \\bar{\\Lambda}_{-1\/2,-1\/2}\\Lambda_{1\/2,-3\/2} + \\bar{\\Lambda}_{-1\/2,3\/2}\\Lambda_{1\/2,1\/2}\\big) \\Big)\\ .\n  \\end{aligned}\n\\end{equation}\nOne can compute these charges for higher spin-$j$ by following the same strategy.\n\n\\subsection{Applications}\n\\label{sec4}\n\\noindent\nThe method of constructing rational electromagnetic fields presented in this paper has the added advantage that it produces a complete set labelled by $(j,m,n)$; any electromagnetic field configuration having finite energy can, in principle, be obtained from an expansion like in (\\ref{Asep}-\\ref{basisZ}), albeit with a varying $j$. The operational difficulty involved in this procedure has to do with the fact that this set is infinite as $j\\in \\sfrac{\\mathds N}{2}$. There are, however, many important cases where only a finite number of knot-basis solutions (sometimes only with a fixed $j$) need to be combined to get the desired EM field configuration. One such very important case is that of the Hopfian solution discussed before. Below we analyse two very interesting generalisations of the Hopfian solution presented in \\cite{HSS15} in the context of present construction. It is imperative to note here that while the scope of construction of a new solution from the known ones as presented in \\cite{HSS15} is limited, the same is not true for the method presented in this paper, which by design can produce arbitrary number of new field configurations. Some of these possible new field configurations obtained from the $j{=}0$ sector (possibly from $j{=}1\/2~ \\textrm{or}~ 1$ as well) could find experimental application with improved experimental techniques like in \\cite{LSMetal18}. \n\n\\begin{figure}[h!]\n\\captionsetup{width=\\linewidth}\n\\centering\n   \\includegraphics[width = 5cm, height = 5cm]{Figures\/HopfianTT.pdf}\n   \\hspace{2cm}\n   \\includegraphics[width = 5cm, height = 5cm, trim = {2cm 3cm 1.5cm 2cm},clip]{Figures\/HopfianR.pdf}\n \\caption{\n Sample electric (red) and magnetic (green) field lines at $t{=}0$.\n Left: time-translated Hopfian with $c=60$, Right: Rotated Hopfian with $\\theta=1$.} \n\\label{fig3}\n\\end{figure}\nBateman's construction, employed in \\cite{HSS15}, hinges on a judicious ansatz for the Riemann-Silberstein vector \\eqref{RSvector} satisfying Maxwell's equations:\n\\begin{equation}\n    {\\bf S} \\= \\nabla\\alpha \\times \\nabla\\beta \\quad\\implies\\quad \\nabla\\cdot{\\bf S}\\=0 ~\\ \\& ~\\ \\mathrm{i}\\, \\partial_t{\\bf S} \\= \\nabla\\times{\\bf S}\\ ,\n\\end{equation}\nusing a pair of complex functions $(\\alpha,\\beta)$. An interesting generalization of the Hopfian solution is obtained in equations (3.16-3.17) of \\cite{HSS15} using (complex) time-translation (TT) to obtain the following $(\\alpha,\\beta)$ pair (up to a normalization)\n\\begin{equation}\n    (\\alpha,\\beta)_{TT} \\= \\left(\\frac{A-1-\\mathrm{i} z}{A+\\mathrm{i}(t+\\mathrm{i} c)}, \\frac{x-\\mathrm{i} y}{A+\\mathrm{i}(t+\\mathrm{i} c)} \\right)\\ ;\\quad A\\= \\sfrac12 \\left(x^2+y^2+z^2-(t+\\mathrm{i} c)^2+1\\right)\\ ,\n\\end{equation}\nwhere $c$ is a constant real parameter. The corresponding  EM field configuration is obtained, in our case, by choosing $\\ell {=} 1{-}c$ and only the following $j{=}0$, and hence type I, complex coefficients in \\eqref{basisZ}\n\\begin{equation}\\label{j0TT}\n    \\left(\\Lambda_{0;0,1}\\ ,~\\Lambda_{0;0,0}\\ ,~\\Lambda_{0;0,-1}\\right)_{TT} \\= \\left( 0\\ ,~0\\ ,~-\\mathrm{i}\\frac{\\pi}{2\\ell^2} \\right)\\ .\n\\end{equation}\nA sample electromagnetic knot configuration of this modified Hopfian is illustated in figure \\ref{fig3}. Another interesting generalization of the Hopfian is constructed in equations (3.20-3.21) of \\cite{HSS15} using a (complex) rotation (R) to get the following $(\\alpha,\\beta)$ pair (again, up to a normalization)\n\\begin{equation}\n    (\\alpha,\\beta)_{R} \\= \\left(\\frac{A-1+\\mathrm{i}(z \\cos{\\mathrm{i}\\theta}+x\\sin{\\mathrm{i}\\theta})}{A+\\mathrm{i} t}, \\frac{x \\cos{\\mathrm{i}\\theta}-z\\sin{\\mathrm{i}\\theta}-\\mathrm{i} y}{A+\\mathrm{i} t} \\right) \\ ,\n\\end{equation}\nwhere $A = \\sfrac12 \\left(x^2+y^2+z^2-t^2+1\\right)$. To get this particular EM field configuration we need to set $\\ell{=}1$ and use the following combination of only $j{=}0$ type I coefficients in \\eqref{basisZ}\n\\begin{equation}\\label{j0R}\n    \\left(\\Lambda_{0;0,1}\\ ,~\\Lambda_{0;0,0}\\ ,~\\Lambda_{0;0,-1}\\right)_{R} \\= \\left( \\mathrm{i}\\frac{\\pi}{4}(\\cosh{\\theta}-1)\\ ,~-\\frac{\\pi}{2\\sqrt{2}}\\sinh{\\theta}\\ ,~-\\mathrm{i}\\frac{\\pi}{4}(\\cosh\\theta+1) \\right)\\ .\n\\end{equation}\nWe illustrate the EM field lines for a sample $\\theta$ value of this modified Hopfian in figure \\ref{fig3}. We note down the conformal charges corresponding to these solutions in table \\ref{table3} by plugging the $j{=}0$ coefficients \\eqref{j0TT} and \\eqref{j0R} in the appropriate formulae of the previous section. The results matches with the ones given in \\cite{HSS15} up to a rescaling of the energy, which can be achieved by an appropriate choice of normalization.\n\\begin{table}[]\n    \\centering\\setcellgapes{4pt}\\makegapedcells \\renewcommand\\theadfont{\\normalsize\\bfseries}\n    \\begin{tabular}{|c|P{5cm}|P{5cm}|}\n        \\hline\n      & Time-translated Hopfian & Rotated Hopfian \\\\ [0.5ex] \\hline \\hline\n     Energy (E) & \n     $\\frac{2\\pi^2}{(1-c)^5}=:E_{TT}$ & \n     $2\\pi^2\\cosh^2\\theta =: E_{R}$  \\\\ \\hline\n     Momentum ($\\textbf{P}$) & \n     $\\left(0\\ ,~0\\ ,~\\frac{1}{4}\\right)E_{TT}$ & \n     $\\left(0\\ ,~-\\frac{1}{4}\\tanh{\\theta}\\ ,~\\frac{1}{4}\\sech\\theta\\right)E_{R}$ \\\\ \\hline\n     Boost ($\\textbf{K}$) & \n     $\\left(0\\ ,~0\\ ,~0\\right)$ & \n     $\\left(0\\ ,~0\\ ,~0\\right)$ \\\\ \\hline\n     Ang. momentum ($\\textbf{L}$) & \n     $\\left(0\\ ,~0\\ ,~-\\frac{1}{4}(1-c)^2\\right)E_{TT}$ & \n     $\\left(0\\ ,~\\frac{1}{4}\\tanh{\\theta}\\ ,~-\\frac{1}{4}\\sech\\theta\\right)E_{R}$ \\\\ \\hline\n      Dilatation (D) & \n     $0$ & \n     $0$ \\\\ \\hline\n     Scalar SCT ($V_0$) & \n     $(1-c)^2E_{TT}$ & \n     $E_{R}$  \\\\ \\hline\n     Vector SCT ($\\textbf{V}$) & \n     $\\left(0\\ ,~0\\ ,~\\frac{1}{4}(1-c)^2\\right)E_{TT}$ & \n     $\\left(0\\ ,~-\\frac{1}{4}\\tanh{\\theta}\\ ,~\\frac{1}{4}\\sech\\theta\\right)E_{R}$ \\\\ \\hline\n    \\end{tabular}\n    \\caption{Conformal charges for the time-translated and rotated Hopfian configurations.}\n    \\label{table3}\n\\end{table}\n\n\\vspace{12pt}\n\\section{Null fields}\n\\label{nullFields}\n\\vspace{2pt}\n\\noindent\nAn interesting subset of vacuum electromagnetic fields are those with vanishing Lorentz invariants,\n\\begin{equation}\n\\vec{E}^2-\\vec{B}^2 =0 \\quad\\quad\\textrm{and}\\quad\\quad \\vec{E}\\cdot\\vec{B} =0 \\qquad\\Longleftrightarrow\\qquad\n\\bigl(\\vec{E}\\pm\\mathrm{i}\\vec{B}\\bigr)^2 =\\ 0\\ .\n\\end{equation}\nAs a scalar equation it must equally hold on the de Sitter side, and so\nwe can try to characterize such configurations with our SO(4) basis above. For a given type and spin, \nthe expressions in~\\eqref{EBj} immediately give the Riemann-Silberstein vector on the $S^3$~cylinder,\n\\begin{equation}\n{\\cal E}_a \\pm\\mathrm{i}\\,{\\cal B}_a \\= -2\\mathrm{i}\\,\\Omega\\,X_a(\\omega)\\,\\mathrm{e}^{\\Omega\\,\\mathrm{i}\\tau}\\ ,\n\\end{equation}\nwhere the upper (lower) sign pertains to type~I (II).\nNote that the negative-frequency part of this field has cancelled.\nThe vanishing of $({\\cal E}_a{\\pm}\\mathrm{i}{\\cal B}_a)({\\cal E}_a{\\pm}\\mathrm{i}{\\cal B}_a)$ is then equivalent to a condition on the angular functions,\n\\begin{equation} \\label{nullcondition}\n0 \\= X_1(\\omega)^2 + X_2(\\omega)^2 + X_3(\\omega)^2 \\= 2\\,Z_+(\\omega) Z_-(\\omega) + Z_3(\\omega)^2\\ .\n\\end{equation}\nWhen expanding the angular functions~$Z^j_{*\\ \\textrm{I}}$ or $Z^j_{*\\ \\textrm{II}}$ into basis solutions with ~\\eqref{basisZ},\none arrives at a system of homogeneous quadratic equations for the free coefficients~$\\lambda_{j;m,n}^{\\textrm{I\/II}}$.\n\nLet us analyze the situation for type~I and spin~$j$. \nThe functions~$Z^j_*(\\omega)$ transform under a $(j,j)$ representation of $su(2)_L\\oplus su(2)_R$. \nThe null condition~\\eqref{nullcondition} then yields a representation content of $(0,0)\\oplus(1,1)\\oplus\\ldots\\oplus(2j,2j)$\nand may thus be expanded into the corresponding harmonics. Furthermore, The independent vanishing of all coefficients produces\n$\\frac16(4j{+}1)(4j{+}2)(4j{+}3)$ equations for the $(2j{+}1)(2j{+}3)$ parameters~$\\lambda_{j;m,n}$ (note the ranges of $m$ and~$n$ for type~I).\nClearly, this system is vastly overdetermined. However, it turns out that only $4j^2{+}6j{+}1$ equations are independent,\nstill leaving $2j{+}2$ free complex parameters for the solution space. The independent equations can be organized as (suppressing~$j$)\n\\begin{equation}\n\\begin{aligned}\n\\lambda_{m,n}^2 &\\ \\sim\\ \\lambda_{m,n-1}\\,\\lambda_{m,n+1} \\qquad\\ \\quad\\textrm{for}\\quad m,n=-j\\,\\ldots,j \\ ,\\\\\n\\lambda_{m,j+1}\\,\\lambda_{m+1,-j-1} &\\= \\lambda_{m+1,j+1}\\,\\lambda_{m,-j-1} \\quad\\quad\\textrm{for}\\quad m=-j,\\ldots,j{-}1\\ .\n\\end{aligned}\n\\end{equation}\nWe have checked for $j{\\le}5$ that the upper equations are solved by~\\footnote{These are the generic solutions. There exist also special solutions given by \n\\eqref{extweights} and $\\lambda_{m,n}=0$ for $|n|\\neq j{+}1$, for arbitrarily selected choices of~$m\\in\\{-j,\\ldots,j\\}$.}\n\\begin{equation}\n\\lambda_{m,n}^{2j+2} \\= {\\textstyle\\sqrt{\\binom{2j+2}{j+1-n}}}\\ \\lambda_{m,-j-1}^{j+1-n}\\,\\lambda_{m,j+1}^{j+1+n}\n\\quad\\quad\\textrm{for}\\quad m=-j,\\ldots,j \\quad\\textrm{and}\\quad n=-j{-}1,\\ldots,j{+}1\\ ,\n\\end{equation}\nwhile the lower ones imply that the highest weights $n{=}j{+}1$ are proportional to the lowest weights $n{=}{-}j{-}1$ (independent of~$m$), \n\\begin{equation} \\label{extweights}\n\\lambda_{m,-j-1} \\= w\\,\\lambda_{m,j+1} \\quad\\quad\\textrm{for}\\quad w\\in\\mathds{C}^*\\ .\n\\end{equation}\nTherefore, the full (generic) solution reads\n\\begin{equation}\n\\lambda_{m,n} \\= {\\textstyle\\sqrt{\\binom{2j+2}{j+1-n}}}\\ w^{\\frac{j+1-n}{2j+2}}\\ \\mathrm{e}^{2\\pi\\mathrm{i} k_m\\frac{j+1-n}{2j+2}}\\ z_m \n\\qquad\\quad\\textrm{with}\\quad z_m\\in\\mathds{C} \\quad\\textrm{and}\\quad k_m\\in\\{0,1,\\ldots,2j{+}1\\}\\ ,\n\\end{equation}\ncontaining $2j{+}2$ complex parameters $z_m$ and~$q$ as well as $2j$ discrete choices~$\\{k_m\\}$ \n(one of them can be absorbed into~$z_m$). This completely specifies the type-I null fields for a given spin.\nType-II null fields are easily obtained by applying electromagnetic duality to type-I null fields.\n\nIn the simplest case of $j{=}0$, the single equation $\\lambda_{0,0}^2=2\\lambda_{0,-1}\\lambda_{0,1}$ describes \na generic rank-3 quadric in $\\mathds{C} P^2$, or a cone over a sphere $\\mathds{C} P^1$ inside the parameter space~$\\mathds{C}^3$. For higher spin,\nthe moduli space of type-I null fields remains a complete-intersection projective variety of complex dimension~$2j{+}1$.\n\nWe conclude the Section with a display of typical field lines (see Figure \\ref{nullFieldLines}) for a type I $j{=}\\sfrac12$ and $j{=}1$ null field at $t{=}0$. For $t{\\neq}0$ the pictures get smoothly distorted.\n\\begin{figure}[h!]\n\\captionsetup{width=\\linewidth}\n\\centering\n\\includegraphics[width = 0.35\\paperwidth]{j=half_field_lines.pdf}\n\\includegraphics[width = 0.35\\paperwidth]{j=one_field_lines.pdf}\n\\caption{\nSample electric (red) and magnetic (green) field lines at $t{=}0$.\nLeft: a pair of electric and a pair of magnetic field lines for the $(j;m,n) = \\left(\\sfrac12;\\sfrac12,\\sfrac32 \\right)_R$ field configuration. Right: a pair of electric field lines, a magnetic field line\nof self-linking one and a magnetic field line of self-linking seven for the $(j;m,n) = (1;0,2)_R$ field configuration.\n}  \n\\label{nullFieldLines}\n\\end{figure}\n\n\\vspace{12pt}\n\\section{Flux transport}\n\\vspace{2pt}\n\\noindent\nWe have seen that electromagnetic energy is radiated away along the light-cones. \nLet us try to quantify its amount over future null infinity~$\\mathscr{I}^+$. Before proceeding further we note down the determinant of the Jacobian $J$ \\eqref{Jacobian}\n\\begin{equation}\n|\\textrm{det}J| \\= \\frac{p^2{-}q^2}{\\ell^2} \\= \\frac{\\gamma^2}{\\ell^2} \n\\= \\frac{\\sin^2\\!\\tau}{t^2} \\= \\frac{\\sin^2\\!\\chi}{r^2}\n\\quad\\quad\\textrm{with}\\quad \\gamma^2=p^2{-}q^2\n\\end{equation}\nand the spherical Minkowski components\n\\begin{equation}\nA_t \\= {\\cal A}_\\tau J^\\tau_{\\ t} + {\\cal A}_\\chi J^\\chi_{\\ t}  \\ ,\\qquad\nA_r \\= {\\cal A}_\\tau J^\\tau_{\\ r} + {\\cal A}_\\chi J^\\chi_{\\ r}  \\ ,\\qquad\nA_\\theta = {\\cal A}_\\theta \\ ,\\qquad A_\\phi = {\\cal A}_\\phi\\ ,\n\\end{equation}\nor any other such tensor component arising due to \\eqref{Jacobian}.\nFor later use, we also note here the transformation of the volume form\n\\begin{equation} \\label{measure}\n\\mathrm{d}^4x \\= \\mathrm{d} t\\,r^2\\mathrm{d} r\\,\\mathrm{d}^2\\Omega_2 \\= r^2\\,|\\textrm{det}J|^{-1}\\mathrm{d}\\tau\\,\\mathrm{d}\\chi\\,\\mathrm{d}^2\\Omega_2\n\\= \\sin^2\\!\\chi\\,|\\textrm{det}J|^{-2}\\mathrm{d}\\tau\\,\\mathrm{d}\\chi\\,\\mathrm{d}^2\\Omega_2\n\\=\\frac{\\ell^4}{\\gamma^4}\\,\\mathrm{d}\\tau\\,\\mathrm{d}^3\\Omega_3\\ .\n\\end{equation}\n\n\n\n\nThe energy flux at time~$t_0$ passing through a two-sphere of radius~$r_0$ centered at the spatial origin is given by\n\\begin{equation}\n\\Phi(t_0,r_0) \\= \\int_{S^2(r_0)}\\!\\!\\mathrm{d}^2\\vec{\\sigma}\\cdot \\bigl(\\vec{E}\\times\\vec{B}\\bigr)(t_0,r_0,\\theta,\\phi) \n\\= \\int_{S^2}r_0^2\\,\\mathrm{d}^2\\Omega_2\\ T_{t\\,r}^{\\mathrm{(M)}}(t_0,r_0,\\theta,\\phi)\\ ,\n\\end{equation}\nwhere $\\mathrm{d}^2\\Omega_2=\\sin\\theta\\,\\mathrm{d}\\theta\\,\\mathrm{d}\\phi$, \nand $T_{t\\,r}^{\\mathrm{(M)}}$ is the $(t,r)$ component of the Minkowski-space stress-energy tensor\n\\begin{equation}\\label{stress-energyMink}\nT_{\\mu\\,\\nu}^{\\mathrm{(M)}} \\= F_{\\mu\\rho}F_{\\nu\\lambda}\\,g^{\\rho\\lambda} - \\sfrac14 g_{\\mu\\nu}F^2 \n\\quad\\quad\\textrm{with}\\quad (g_{\\mu\\nu}) = \\textrm{diag}(-1,1,r^2,r^2\\sin^2\\!\\theta)\\ \n\\end{equation}\nfor $\\mu,\\nu,\\ldots\\in\\{t,r,\\theta,\\phi\\}$. We carry out this computation in the $S^3$-cylinder frame by using the conformal relations\n\\begin{equation}\n\\ell^2\\,T_{\\mu\\,\\nu}^{\\mathrm{(dS)}} \\= t^2\\,T_{\\mu\\,\\nu}^{\\mathrm{(M)}} \\= \\sin^2\\!\\tau\\,T_{\\mu\\,\\nu}^{\\mathrm{(cyl)}} \n\\= \\sin^2\\!\\tau\\,T_{m\\,n}^{\\mathrm{(cyl)}}\\, J^m_{\\ \\ \\mu}\\,J^n_{\\ \\ \\nu} \n\\quad\\textrm{for}\\quad m,n\\in\\{\\tau,\\chi,\\theta,\\phi\\}\n\\end{equation}\nwith the Jacobian~\\eqref{Jacobian} and the fact that \\ $r\\sin\\tau=t\\sin\\chi$ \\ so that \n\\begin{equation}\n\\Phi(\\tau_0,\\chi_0) \\= \\int_{S^2}\\sin^2\\!\\chi\\,\\mathrm{d}^2\\Omega_2\\ T_{t\\,r}^{\\mathrm{(cyl)}}(\\tau_0,\\chi_0,\\theta,\\phi)\n\\=  \\int_{S^2}\\sin^2\\!\\chi\\,\\mathrm{d}^2\\Omega_2\\ T_{m\\,n}^{\\mathrm{(cyl)}}\\, J^m_{\\ \\ t}\\,J^n_{\\ \\ r} \\ .\n\\end{equation}\nA straightforward computation using $(g_{mn})=\\textrm{diag}(-1,1,\\sin^2\\!\\chi,\\sin^2\\!\\chi\\sin^2\\!\\theta)$ then yields\n\\begin{equation}\n\\begin{aligned}\n\\Phi(\\tau_0,\\chi_0) &\\= \\frac{p\\,q}{\\ell^2}\\int \\mathrm{d}^2\\Omega_2\\,\\Bigl(({\\cal F}_{\\tau\\theta})^2 + ({\\cal F}_{\\chi\\theta})^2 \n+ \\sfrac{1}{\\sin^2\\!\\theta}\\bigl[({\\cal F}_{\\tau\\phi})^2 + ({\\cal F}_{\\chi\\phi})^2\\bigr]\\Bigr) \\\\\n&\\quad +\\ \\frac{p^2{+}q^2}{\\ell^2}\\int \\mathrm{d}^2\\Omega_2\\,\\Bigl( {\\cal F}_{\\tau\\theta}\\,{\\cal F}_{\\chi\\theta} \n+ \\sfrac{1}{\\sin^2\\!\\theta}{\\cal F}_{\\tau\\phi}\\,{\\cal F}_{\\chi\\phi}\\Bigr)\\ .\n\\end{aligned}\n\\end{equation}\nThe sphere-frame components ${\\cal F}_{mn}$ can be computed by expanding\n$e^a=e^a_{\\ m}\\,\\mathrm{d}\\xi^m$ in\n\\begin{equation}\n{\\cal F} \\=  {\\cal E}_a\\,e^a {\\wedge} e^\\tau + \\sfrac12\\,{\\cal B}_a\\,\\varepsilon^a_{\\ bc}\\,e^b{\\wedge} e^c \\= {\\cal F}_{mn}\\,\\mathrm{d}\\xi^m{\\wedge}\\mathrm{d}\\xi^n\n\\quad\\quad\\textrm{with}\\quad\\xi^n\\in\\{\\tau,\\chi,\\theta,\\phi\\}\\ .\n\\end{equation}\nThe expression for the flux in sphere-frame fields then becomes\n\\begin{equation}\\label{flux}\n\\begin{aligned}\n\\ell^2\\,\\Phi &\\= p\\,q\\,\\sin^2\\!\\chi \\int_{S^2}\\mathrm{d}^2\\Omega_2\\ \\Bigl[\n(\\sin\\phi\\,{\\cal E}_1-\\cos\\phi\\,{\\cal E}_2)^2 + (\\cos\\theta\\cos\\phi\\,{\\cal E}_1+\\cos\\theta\\sin\\phi\\,{\\cal E}_2 \\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad-\\sin\\theta\\,{\\cal E}_3)^2 + (\\sin\\phi\\,{\\cal B}_1-\\cos\\phi\\,{\\cal B}_2)^2 \\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad+ (\\cos\\theta\\cos\\phi\\,{\\cal B}_1+\\cos\\theta\\sin\\phi\\,{\\cal B}_2-\\sin\\theta\\,{\\cal B}_3)^2 \\Bigr] \\\\\n&\\qquad +\\ (p^2{+}q^2)\\,\\sin^2\\!\\chi \\int_{S^2}\\mathrm{d}^2\\Omega_2\\ \\Bigl[\n(\\sin\\phi\\,{\\cal B}_1-\\cos\\phi\\,{\\cal B}_2) (\\cos\\theta\\cos\\phi\\,{\\cal E}_1 \\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\quad +\\cos\\theta\\sin\\phi\\,{\\cal E}_2-\\sin\\theta\\,{\\cal E}_3) - (\\sin\\phi\\,{\\cal E}_1-\\cos\\phi\\,{\\cal E}_2) \\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\quad \\cdot(\\cos\\theta\\cos\\phi\\,{\\cal B}_1+\\cos\\theta\\sin\\phi\\,{\\cal B}_2-\\sin\\theta\\,{\\cal B}_3) \\Bigr]\\ .\n\\end{aligned}\n\\end{equation}\n\nThe total energy flux across future null infinity is obtained by evaluating this expression on $\\mathscr{I}^+$ and integrating over it. Introducing cylinder light-cone coordinates\n\\begin{equation}\nu=\\tau{+}\\chi \\quad\\textrm{and}\\quad v=\\tau{-}\\chi \\qquad\\textrm{so that}\\qquad\nt{+}r=-\\ell\\,\\cot\\sfrac{v}{2} \\quad\\textrm{and}\\quad t{-}r=-\\ell\\,\\cot\\sfrac{u}{2}\n\\end{equation}\nwe characterize $\\mathscr{I}^+$ as\n\\begin{equation}\n\\biggl\\{ \\begin{array}{l} t{+}r\\to\\infty \\\\[4pt]  t{-}r \\in\\mathds R \\end{array} \\biggr\\} \\quad\\Leftrightarrow\\quad\n\\biggl\\{ \\begin{array}{l} u\\in(0,2\\pi) \\\\[4pt]  v=0 \\end{array} \\biggr\\} \\qquad\\Rightarrow\\qquad\np=q=\\sin^2\\!\\chi \\quad\\textrm{and}\\quad \\gamma=0\\ .\n\\end{equation}\nFurther noticing that\n\\begin{equation}\n\\mathrm{d}(t{-}r) \\= \\frac{\\ell\\ \\mathrm{d} u}{p{+}q} \\= \\frac{\\ell\\ \\mathrm{d} u}{1-\\cos u} \\quad\\quad\\textrm{and}\\quad\\quad\n\\sin^2\\!\\chi \\= \\sin^2\\!\\sfrac{u-v}{2} \\= \\sfrac12\\bigl(1-\\cos(u{-}v)\\bigr)\\ ,\n\\end{equation}\nwe may express this total flux as\n\\begin{equation}\n\\Phi_+ \\= \\int_{-\\infty}^\\infty \\mathrm{d}(t{-}r)\\ \\Phi\\big|_{\\mathscr{I}^+}\n\\= \\int_0^{2\\pi} \\frac{\\ell\\ \\mathrm{d} u}{1-\\cos u}\\ \\Phi(\\sfrac{u}{2},\\sfrac{u}{2})\n\\end{equation}\nto obtain\n\\begin{equation} \\label{totalflux}\n\\begin{aligned}\n\\Phi_+ &\\=\\frac{1}{8\\ell} \\int\\!\\mathrm{d} u\\,(1{-}\\cos u)^2 \\int\\!\\mathrm{d}^2\\Omega_2\\ \\Bigl[\n\\bigl\\{ \\cos\\theta\\cos\\phi\\,{\\cal E}_1+\\cos\\theta\\sin\\phi\\,{\\cal E}_2-\\sin\\theta\\,{\\cal E}_3+\\sin\\phi\\,{\\cal B}_1 \\\\\n&-\\cos\\phi\\,{\\cal B}_2 \\bigr\\}^2 +\\ \\bigl\\{ \\cos\\theta\\cos\\phi\\,{\\cal B}_1+\\cos\\theta\\sin\\phi\\,{\\cal B}_2-\\sin\\theta\\,{\\cal B}_3-\\sin\\phi\\,{\\cal E}_1+\\cos\\phi\\,{\\cal E}_2 \\bigr\\}^2 \\Bigr]\n\\end{aligned}\n\\end{equation}\nThe square bracket expression above can be further simplified for a fixed spin and type \nby employing~\\eqref{EBj} along with \\eqref{XZ}, \\eqref{basisZ}, \\eqref{type1} and~\\eqref{type2} to get\n\\begin{equation}\n\\Phi_+^{(j)} = \\frac{\\Omega^2}{4\\ell}\\int\\!\\mathrm{d} u\\,(1{-}\\cos u)^2 \\int\\!\\mathrm{d}^2\\Omega_2\\,\n \\big| \\pm Z_+^j e^{-\\mathrm{i}\\phi}(1\\pm \\cos\\theta)\\mp Z_-^j e^{\\mathrm{i}\\phi}(1\\mp \\cos\\theta)-\\sqrt{2}Z_3^j\\sin\\theta\\, \\big|^2\\ ,\n\\end{equation}\nwhere the upper (lower) sign corresponds to a type-I (type-II) solution. \nIn the special case of $j=0$ $(\\Omega{=}2)$ the contribution to the two-sphere integral only comes \nfrom the part which is independent of $(\\theta,\\phi)$, i.e.~$\\frac{4}{3}\\left(|Z^0_+|^2+|Z^0_-|^2+|Z^0_3|^2\\right)$, \nso that the integration can easily be performed by passing to the adjoint harmonics \n$\\tilde{Y}_{j;l,M}$ \\eqref{Y-new} and using \\eqref{split-Y}  to get\n\\begin{equation}\n\\Phi_+^{(0)} \\= \\frac{16}{3\\,\\ell}\\int\\limits_0^{2\\pi}\\!\\mathrm{d} u\\ \\sin^4\\!\\sfrac{u}{2}\\,\n\\bigl| R_{0,0}(\\sfrac{u}{2})\\bigr|^2\\sum\\limits_{n=-1}^{1}|\\lambda_{0,n}|^2 \n\\= \\frac{8}{\\ell}\\sum\\limits_{n=-1}^{1}|\\lambda_{0,n}|^2 \\= E^{(0)}\\ .\n\\end{equation}\n{\\setstretch{2.0} The same equality $\\Phi_+=E$ continues to hold true as we go up in spin $j$ \n(we verified it for $j{=}\\sfrac12$ and $j{=}1$), \nthus validating the  energy conservation $\\mbox{$\\partial$}^\\mu T_{\\mu 0}=0$.}\n\n\\vspace{12pt}\n\\section{Trajectories}\n\\vspace{2pt}\n\\noindent\nGiven a knotted electromagnetic field configuration, a natural issue that arises is the behavior of charged particles in the background of such a field. We proceed to address this issue by analyzing, with numerical simulations, the trajectories of several (identical) charged point particles for the family of knotted field configurations \\eqref{EMCartesian} that we encountered before. We will consider type I \\eqref{type1} basis field configurations (up to $j{=}1$ for simplicity) and the Hopf--Ra\u00f1ada field configuration \\eqref{HRfields}. \n\nIn the simulations we employ the maximum of the energy density at time $t$, i.e. $E_{max}(t)$ (that occurs at several points ${\\bf x}_{max}$ that are located symmetrically with respect to the origin), for different initial conditions and field configurations, and in each case we have employed a parameter $R_{max}(t)$ of ``maximal'' radius defined via\n\\begin{equation}\\label{Rmax}\n    E\\left( t,{\\bf x}_{max}(t) \\right) \\= E_{max}(t)\\quad \\implies \\quad R_{max}(t)\\ :=\\ |{\\bf x}_{max}(t)|\\ .\n\\end{equation}\nA characteristic feature of these basis knot electromagnetic fields is that they have a preferred $z$-axis direction which is clearly exemplified in Figure \\ref{EnDen}, where the energy density $E := \\sfrac12 ({\\bf E}^2 + {\\bf B}^2)$ decreases along the $z$-axis. As a result, the basis fields along the $z$-axis (i.e. ${\\bf E}(t,x{=}0,y{=}0,z)$ and ${\\bf B}(t,x{=}0,y{=}0,z)$) are either directed in the $xy$-plane or along the $z$-axis. In fact, for extreme field configurations $(j;\\pm j,\\pm (j{+}1))$, for any $j{>}0$, the fields along the $z$-axis vanish for all times.\n\\begin{figure}[h!]\n\\captionsetup{width=\\linewidth}\n\\centering\n   \\includegraphics[width = 5cm, height = 5cm]{EnDenHR.pdf}\n   \\includegraphics[width = 5cm, height = 5cm]{EnDen_h,-h,-3h.pdf}\n   \\includegraphics[width = 5cm, height = 5cm]{EnDen_1,-1,1.pdf}\n \\caption{\n Contour plots for energy densities at $t{=}0$ (yellow), $t{=}1$ (cyan) and $t{=}1.5$ (purple) with contour value $0.9E_{max}(t{=}1.5)$.\n Left: Hopf--Ran\\~{a}da configuration, Center: $(j;m,n)=(\\sfrac12;-\\sfrac12,-\\sfrac32)_R$ configuration, Right: $(j;m,n)=(1;-1,1)_R$ configuration.}\n\\label{EnDen}\n\\end{figure}\n\nThe trajectories of these particles are governed by the relativistic Lorentz equation\n\\begin{equation}\\label{eq:force1}\n\t\\frac{\\mathrm{d} {\\bf p}}{\\mathrm{d} t} = q({\\bf E}_{\\ell}+{\\bf v} \\times {\\bf B}_{\\ell})\\ ,\n\\end{equation}\nwhere $q$ is the charge of the particle, ${\\bf p} = \\widetilde{\\gamma} m {\\bf v}$ is the relativistic three-momentum, ${\\bf v}$ is the usual three-velocity of the particle, $m$ is its mass, $\\widetilde{\\gamma} = (1-{\\bf v}^{\\,2})^{-1\/2}$ is the Lorentz factor, and ${\\bf E}_{\\ell}$ and ${\\bf B}_{\\ell}$ are dimensionful electric and magnetic fields respectively. With the energy of the particle $E_{p} = \\widetilde{\\gamma} m$ and $\\mathrm{d} E_p\/ \\mathrm{d}{t} = q\\, {\\bf v} \\cdot {\\bf E}$, one can rewrite (\\ref{eq:force1}) in terms of the derivative of ${\\bf v}$ \\cite{landau2} as\n\\begin{equation}\\label{eq:force2}\n\t\\frac{\\mathrm{d} {\\bf v}}{\\mathrm{d} t} = \\frac{q}{\\gamma m} \\left( {\\bf E}_{\\ell} + {\\bf v} \\times {\\bf B}_{\\ell} - ({\\bf v} \\cdot {\\bf E}_{\\ell})\\,{\\bf v} \\right)\\ .\n\\end{equation}\nEquations (\\ref{eq:force1}) and (\\ref{eq:force2}) are equivalent, and either one can be used for a simulation purpose; they only differ by the position of the nonlinearity in ${\\bf v}$. In natural units $\\hbar {=} c {=} \\epsilon_0 {=} 1$, every dimensionful quantity can be written in terms of a length scale. We relate all dimensionful quantities to the de Sitter radius $\\ell$ from equation \\eqref{dS4} and work with the corresponding dimensionless ones as follows:\n\\begin{equation}\n    T := \\frac{t}{\\ell}\\ ,\\quad {\\bf X} := \\frac{{\\bf x}}{\\ell}\\ ,\\quad {\\bf E} := \\ell^2 {\\bf E}_{\\ell}, \\quad\\textrm{and}\\quad {\\bf B} := \\ell^2 {\\bf B}_{\\ell}\\ .\n\\end{equation}\nMoreover, the fields are solutions of the homogeneous (source-free) Maxwell equations, so they can be freely rescaled by any dimensionless constant factor $\\lambda$. Combining the above considerations, one can rewrite (\\ref{eq:force1}) (or analogously (\\ref{eq:force2})) fully in terms of dimensionless quantities as\n\\begin{equation}\n\t\t\\frac{\\mathrm{d} (\\widetilde{\\gamma}{\\bf V})}{\\mathrm{d} T} = \\kappa({\\bf E} + {\\bf V} \\times {\\bf B})\\ ,\n\\end{equation}\nwhere $\\kappa = \\sfrac{q\\ell^3\\lambda}{m}$ is a dimensionless parameter. As for the initial conditions, we mostly work in the following two scenarios:\n\\begin{enumerate}[label=(\\alph*)]\n\\item ${\\bf V}_0 \\equiv {\\bf V}(T{=}0) = 0$ with particles symmetrically (with respect to the origin) positioned along a line, a circle, or a sphere around the origin at $T{=}0$, and\n\\item ${\\bf X}_0 \\equiv {\\bf X}(T{=}0) = 0$ with particle velocities directed radially outward in a symmetric (with respect to the origin) fashion along a line, in a plane or in space at $T{=}0$.\n\\end{enumerate} \n\nWe vary several parameters including the initial conditions (a) and (b) with different directions of lines and planes for each configuration, the value of $\\kappa$, and the simulation time in order to study the behavior of the trajectory. In general, we used the value of $R_{max}(0)$ for each field configuration to have particles with initial conditions positioned near the region of maximum energy of the field. In this scenario, the effect of the field on the trajectories of the particles is more prominent, as expected. For particles positioned along a sphere centered around the origin we use (i) the radius of the sphere to be $R_{max}$, and (ii) the radius of the sphere to be very small. The latter scenario helps us understand small perturbations of the trajectories as compared to a particle starting at rest from the origin. Moreover, for the initial condition of kind (b) we use the particle initial speeds in the range where it is (i) non-relativistic, (ii) relativistic (usually between $0.1$ and $0.9$), and (iii) ultrarelativistic (here, $0.99$ or higher).\n\nWe observe a variety of different behaviors for these trajectories, some of which we summarize below with the aid of figures. Firstly, it is worth noticing that, even with all fields decreasing as powers of both space and time coordinates, in most field configurations we observe particles getting accelerated from rest up to ultrarelativistic speeds. The limit of these ultrarelativistic speeds for higher times depend on the magnitude of the fields (see, for example, Figure \\ref{singleTrajectory}).\n\\begin{figure}[H]\n\\captionsetup{width=\\linewidth}\n\\centering\n   \\includegraphics[width = 5cm, height = 5cm]{Trajectory_h,-h,-3h.pdf}\n   \\includegraphics[width = 5cm, height = 5cm]{Speed_h,-h,-3h_10k.pdf}\n   \\includegraphics[width = 5cm, height = 5cm]{Speed_h,-h,-3h_100k.pdf}\n \\caption{\n Trajectory of a charged particle for $(j;m,n)=(\\sfrac12;-\\sfrac12,-\\sfrac32)_R$ configuration with initial condition ${\\bf X}_0 = (0.01,0.01,0.01)$ and ${\\bf V}_0 = 0$ simulated for $t \\in [-1,1]$.\n Left: Particle trajectory, Center: absolute velocity profile for $\n\\kappa {=} 10$, Right: absolute velocity profile for $\\kappa {=} 100$.}\n\\label{singleTrajectory}\n\\end{figure}\n\nWith fixed initial conditions (of kind (a) or (b)) and for higher values of $\\kappa$ one can expect, in general, that the initial conditions may become increasingly less relevant. To this end, we find that the particles get focused and accumulate like a beam of charged particles (the beam is quite narrow for some configurations and higher $\\kappa$) along some specific region of space and move asymptotically for higher simulation times. This is exemplified below with two $j{=}0$ configurations: the $(0,0,1)_I$ configuration in Figure \\ref{j001plots}, and the Hopf--Ra\u00f1ada configuration in Figure \\ref{HRplots}. We have verified this feature not just with symmetric initial conditions of particles like that with initial conditions (a) and (b) (as in Figure \\ref{j001plots}), but also in several initial conditions asymmetric with respect to the origin, like particles located randomly inside a sphere of fixed radius about the origin with zero initial velocity, and particles located at the origin but with different magnitudes of velocities. Figure \\ref{HRplots} is an illustrative example for both of these latter scenarios of asymmetric initial conditions.\n\n\\begin{figure}[H]\n\\captionsetup{width=\\linewidth}\n\\centering\n   \\includegraphics[width = 7cm, height = 5cm]{PosSymm_0,0,1_100k.pdf}\\hspace{1cm}\n   \\includegraphics[width = 7cm, height = 5cm]{VelSymm_0,0,1_100k.pdf}\n \\caption{\n Simulation of trajectories of $18$ identical charged particles for $(j;m,n)=(0,0,1)_I$ field configuration with $\\kappa {=} 100$ for $t\\in[0,1]$. Left: particles starting from rest and located symmetrically on a sphere of radius $|{\\bf X}_0|{=}R_{max}(0)$. Right: particles located at origin and directed symmetrically in space (shown with colored arrows) with $|{\\bf V}_0|=0.75$.}\n\\label{j001plots}\n\\end{figure}\n\n\\begin{figure}[H]\n\\captionsetup{width=\\linewidth}\n\\centering\n   \\includegraphics[width = 7cm, height = 5cm]{HRposInhomo1000k.pdf}\\hspace{1cm}\n   \\includegraphics[width = 7cm, height = 5cm]{HRvelRandom1000k.pdf}\n \\caption{\n Simulation of trajectories of $20$ identical charged particles for Hopf--Ran\\~{a}da field configuration with $\\kappa {=} 1000$ for $t\\in[0,1]$. Left: particles starting from rest and located randomly inside a solid ball of radius $r{=}0.01$. Right: particles located at origin and directed randomly (shown with colored arrows) with $|{\\bf V}_0|{=}0.45$.}\n\\label{HRplots}\n\\end{figure}\n\nThis is not always the case though. For some $j{=}\\tfrac12$ and $j{=}1$ configurations, and with initial particle positions in a sphere of very small radius about the origin, we are able to observe the splitting of particle trajectories (starting in some specific solid angle regions around the origin) into two, three or even four such asymptotic beams that converge along some particular regions of space (depending on the initial location of these particles in one of these solid angle regions). Trajectories generated by two such $j{=}1$ configurations have been illustrated in Figure \\ref{3-4furcation}. \n\\begin{figure}[H]\n\\captionsetup{width=\\linewidth}\n\\centering\n   \\includegraphics[width = 7cm, height = 5cm]{Trifurcation_1,-1,-2.pdf}\\hspace{1cm}\n   \\includegraphics[width = 7cm, height = 5cm]{Quadrifurcation_1,-1,-1.pdf}\n \\caption{\n Simulation of trajectories of $18$ identical charged particles located symmetrically on a sphere with $|{\\bf X}_0|=0.01$ and ${\\bf V}_0=0$ for $t \\in [0,3]$. Left: $(j;m,n)=(1,-1,-2)_R$ configuration with $\\kappa {=} 500$, Right: $(j;m,n)=(1,-1,-1)_R$ configuration with $\\kappa {=} 10$.}\n\\label{3-4furcation}\n\\end{figure}\n\nNaturally, there are also regions of unstable trajectories for particles starting between these solid angle regions (see Figure \\ref{Bifurcation}), which generally include the preferred $z$-axis, since in some cases trajectories that start at rest in the $z$-axis never leave it.\n\\begin{figure}[H]\n\\captionsetup{width=\\linewidth}\n\\centering\n   \\includegraphics[width = 7cm, height = 5cm]{Bifurcation_h,-h,-3h.pdf}\\hspace{1cm}\n   \\includegraphics[width = 7cm, height = 5cm]{Bifurcation_1,0,-2.pdf}\n \\caption{\n Simulation of trajectories of $18$ identical charged particles located symmetrically on a sphere with $|{\\bf X}_0|=0.01$, ${\\bf V}_0=0$ and $\\kappa {=} 10$ for $t \\in [0,3]$. Left: $(j;m,n)=(\\sfrac12;-\\sfrac12,-\\sfrac32)_R$ configuration, Right: $(j;m,n)=(1,0,-2)_I$ configuration.}\n\\label{Bifurcation}\n\\end{figure}\n\nFor particles with initial condition (a), we find that their trajectories are symmetric as far as time reversal around $T{=}0$ is concerned. We illustrate this feature with Figure \\ref{1Dpos} where particles with initial position along a line has been considered for better visualization. Of course, choosing a different initial condition would violate this bevaviour as is clear from Figure \\ref{singleTrajectory}. \n\nWe employ the paramter $R_{max}$ \\eqref{Rmax} in the the following Figures \\ref{1Dpos}, \\ref{1Dvel}, \\ref{2Dvel}, \\ref{3Dvel}, \\ref{2Dpos}, and \\ref{3Dpos} for both kinds of initial conditions viz. (a) and (b) (it is especially relevant for the former) to understand the effect of field intensity on particle trajectories.\n\\begin{figure}[H]\n\\captionsetup{width=\\linewidth}\n\\centering\n   \\includegraphics[width = 7cm, height = 5cm]{PosLine_h,h,h_10kZ.pdf}\\hspace{1cm}\n   \\includegraphics[width = 7cm, height = 5cm]{PosLine_h,h,h_10kXY.pdf}\n \\caption{\n Trajectories of $11$ identical charged particles with ${\\bf V}_0=0$ and ${\\bf X}_0 \\propto \\pm\\sfrac14 R_{max}(0)$ (including one at the origin) for $(j;m,n)=(\\sfrac12;\\sfrac12,\\sfrac12)_R$ configuration with $\\kappa {=} 10$ simulated for $t \\in [-1,1]$.\n Left: particles located along $z$-axis (blue line), Right: particles located along some (blue) line in $xy$-plane.}\n\\label{1Dpos}\n\\end{figure}\n\nOne very interesting feature of trajectories for some of these field configurations is that they twist and turn in a coherent fashion owing to the symmetry of the background field. For particles with initial condition of kind (b), we see that their trajectories take sharp turns, up to two times, with mild twists before going off asymptotically. This has to do with the presence of strong background electromagnetic fields with knotted field lines. This is clearly demonstrated below in Figures \\ref{1Dvel}, \\ref{2Dvel}, and \\ref{3Dvel}. It is worthwhile to notice in Figure \\ref{1Dvel} that the particle which was initially at rest moves unperturbed along the $z$-axis; again, this has to do with the fact that these fields have preferred $z$-direction.\n\\begin{figure}[H]\n\\captionsetup{width=\\linewidth}\n\\centering\n   \\includegraphics[width = 7cm, height = 5cm]{VelLine_1,0,0_1tf.pdf}\\hspace{1cm}\n   \\includegraphics[width = 7cm, height = 5cm]{VelLine_1,0,0_3tf.pdf}\n \\caption{\n Trajectories of $11$ identical charged point particles (shown with colored arrows) with ${\\bf X}_0=0$ and ${\\bf V}_0 \\propto \\pm\\sfrac14 R_{max}(0,1,0)$ (including one at rest) for $(j;m,n)=(1,0,0)_I$ configuration with $\\kappa {=} 10$.\n Left: particles simulated for $t \\in [0,1]$, Right: particles simulated for $t \\in [0,3]$.}\n\\label{1Dvel}\n\\end{figure}\n\nThis feature is even more pronounced in Figure \\ref{2Dvel} and (the right subfigure of) Figure \\ref{3Dvel} where we see that particles with ultrarelativistic initial speeds are forced to turn (almost vertically upwards) due to the strong electromagnetic field. These particles later take very interesting twists in a coherent manner. This twisting feature is much more refined for the case where initial particle velocities were directed along the $xy$-plane. Here also, we can safely attribute this behavior of the particle trajectories to the special field configurations, with preferred $z$-direction, that we are working with.\n\\begin{figure}[H]\n\\captionsetup{width=\\linewidth}\n\\centering\n   \\includegraphics[width = 7cm, height = 5cm]{VelCirc_1,-1,1_Ynormal.pdf}\\hspace{1cm}\n   \\includegraphics[width = 7cm, height = 5cm]{VelCirc_1,-1,1_Znormal.pdf}\n \\caption{\n Trajectories of $10$ identical charged particles with ${\\bf X}_0=0$ and $|{\\bf V}_0|=0.99$ for $(j;m,n)=(1,-1,1)_R$ configuration with $\\kappa {=} 100$ simulated for $t \\in [0,3]$.\n Left: particles directed symmetrically along the $xz$-plane at $T{=}0$ (shown with colored arrows), Right: particles directed symmetrically along the $xy$-plane at $T{=}0$ (shown with colored arrows).}\n\\label{2Dvel}\n\\end{figure}\n\\begin{figure}[H]\n\\captionsetup{width=\\linewidth}\n\\centering\n   \\includegraphics[width = 5cm, height = 7cm]{VelSphere_1,0,0_10k.pdf}\\hspace{2cm}\n   \\includegraphics[width = 7cm, height = 6.5cm]{VelSphere_1,0,0_100k.pdf}\n \\caption{\n Trajectories of $10$ identical charged particles directed symmetrically away from the origin at $T{=}0$ (shown with colored arrows) with ${\\bf X}_0=0$ for $(j;m,n)=(1,0,0)_I$ configuration simulated for $t \\in [0,2]$.\n Left: $|{\\bf V}_0|=R_{max}$ and $\\kappa {=} 10$, Right: $|{\\bf V}_0|=0.99$ and $\\kappa {=} 100$.}\n\\label{3Dvel}\n\\end{figure}\n\n We see in Figure \\ref{2Dpos} that the trajectories of particles that were initially located on a circle whose normal is along the $z$-axis flow quite smoothly along the $z$-axis with mild twists for some time before they all turn symmetrically in a coherent way and went off asymptotically. Comparing this with the other case in Figure \\ref{2Dpos}, where particles split into two asymptotic beams, we realize that this is yet another instance of the preferred choice of direction for the electromagnetic fields influencing the trajectories of particles.\n\n\\begin{figure}[H]\n\\captionsetup{width=\\linewidth}\n\\centering\n   \\includegraphics[width = 7cm, height = 5cm]{PosCirc_1,0,0_Xnormal.pdf}\\hspace{1cm}\n   \\includegraphics[width = 7cm, height = 5cm]{PosCirc_1,0,0_Znormal.pdf}\n \\caption{\n Trajectories of $10$ identical charged particles located on a circle with $|{\\bf X}_0|=R_{max}$ and ${\\bf V}_0=0$ for $(j;m,n)=(1,0,0)_I$ configuration with $\\kappa {=} 10$ simulated for $t \\in [0,2]$.\n Left: the normal of the circle is along the $x$-axis, Right: the normal of the circle is along the $z$-axis.}\n\\label{2Dpos}\n\\end{figure}\n\nIn Figures \\ref{3Dpos} and \\ref{2Dvel} we find examples of kind (a) and (b) respectively where both twisting as well as turning of trajectories is prominent. We see in Figure \\ref{3Dpos} that the particles that start very close to the origin takes a longer time to show twists as compared to the ones that start off on a sphere of radius $R_{max}$. This is due to the fact that the field is maximal at $R_{max}$ and hence its effect on particles is prominent, as discussed before. We also notice here that the particles sitting along the $z$-axis at $T{=}0$ (either on the north pole or on the south pole of this sphere) keep moving along the $z$-axis without any twists or turns. This exemplifies the fact that these background fields have a preferred direction.\n\\begin{figure}[H]\n\\captionsetup{width=\\linewidth}\n\\centering\n   \\includegraphics[width = 3cm, height = 7cm]{PosSphere_1,-1,1_10k.pdf}\\hspace{2cm}\n   \\includegraphics[width = 8cm, height = 7cm]{PosSphere_1,-1,1_100k.pdf}\n \\caption{\n Trajectories of $18$ identical charged particles for $(j;m,n)=(1,-1,1)_R$ configuration with ${\\bf V}_0=0$ simulated for $t \\in [0,1]$.\n Left: particles located symmetrically on a sphere with $|{\\bf X}_0|=0.01$ and $\\kappa {=} 10$, Right: particles located symmetrically on a sphere with $|{\\bf X}_0|=R_{max}$ and $\\kappa {=} 100$.}\n\\label{3Dpos}\n\\end{figure}\n\n\nFor field configurations with $j {=} \\tfrac32$ and $2$ we found in some simulations (with the same parameters i.e. $\\kappa$, initial conditions, and simulation time) that their effect on particle trajectories is much less prominent as compared to the $j\\leq 1$ field configurations. This indicates that the field intensity may decrease quickly for higher $j$ configurations. This prompts the speculation that, depending on the desired degree of precision, for practical purposes it may be enough to expand any finite energy field configuration in terms of up to $j{=}1$ modes of these basis configurations.\n\n\n\n\\chapter{Introduction} \n\\label{Chapter1} \n\\def\\={\\ =\\ }\n\\justifying\n\nThe $4$-dimensional de Sitter space $\\mathrm{d}{S}_4$ plays an important role in gravity. It is one of the three (topological) types\\footnote{There are $3$ types of FLRW spacetime viz. Minkowski space ($\\kappa{=}0$), de Sitter space ($\\kappa{=}1$), and Anti-de Sitter space ($\\kappa{=}-1$) according to the global topology of the backgroud $3$-space (labelled by $\\kappa$).} of the Friedmann--Lema\u00eetre--Robertson--Walker (FLRW) spacetime that can model a homogeneous and isotropic universe like ours (at distance scales of 100 Mpsec). This has a positive global scalar curvature of the underlying $3$-space that is consistent with the observed positive cosmological constant $\\Lambda$ aka {\\it dark energy} that is fueling the accelerated expansion of our universe. It is believed that our universe is asymptotically de Sitter which means in the future, when the dark energy dominates, our universe would become de Sitter.\n\nGauge theory, in particular Yang--Mills theory, is of central importance in the classical description of fundamental forces of nature like electromagnetism, weak and strong nuclear forces. Classical Yang--Mills theory also has applications in other physics areas such as QCD confinement of high energy physics and spin-orbit interaction in condensed matter physics. Even gravity can be understood as a gauge theory. It is, therefore, natural to seek solutions of Yang--Mills theory on four-dimensional de Sitter space. Furthermore, owing to the conformal relation of $\\mathrm{d}{S}_4$ with Minkowski space $\\mathds R^{1,3}$, it turns out that even Abelian Yang--Mills theory aka electromagnetism studied on the former has nice application since the solutions can be pulled back to Minkowski space (of our laboratory) owing to the conformal invariance of the Yang--Mills theory in $4$-dimensions.\n\n\\vspace{12pt}\n\\section{Electromagnetic knots}\n\\vspace{2pt}\nTheoretical discovery of electromagnetic knots dates back to 1989 when Ra\u00f1ada \\cite{Ranada89} constructed them using the Hopf map. These finite-energy finite-action vacuum solutions of Maxwell's equations are constructed from a pair of complex scalar fields $\\phi$ and $\\theta$ on the $4$-dimensional spacetime where the $3$-space is compatified to $S^3$ with the addition of a point at infinity. These solutions are thus characterised by a topological quantity called the Hopf index of the following Hopf map ($A=1,2,3,4$):\n\\begin{equation}\n    h:\\ S^3 \\to S^2\\ ,\\quad \\{ \\omega_{_A} \\} \\mapsto \\{x_1,x_2,x_3\\}\n\\end{equation}\nwhere $S^2$, arising from the compactification of the complex plane $\\mathds{C}$, has coordinates $x_i$, satisfying $x_1^2 + x_2^2 + x_3^2 = 1$, that are constructed from the $S^3$ coordinates $\\omega_{_A}$, satisfying $\\omega_1^2 + \\omega_2^2 + \\omega_3^2 + \\omega_4^2 = 1$, as follows\n\\begin{equation}\n    x_1\\ :=\\ 2(\\omega_1\\omega_2 + \\omega_3\\omega_4)\\ ,\\ x_2\\ :=\\ 2(\\omega_1\\omega_4 - \\omega_2\\omega_3)\\ ,\\ x_3\\ :=\\ (\\omega_1^2+\\omega_3^2) - (\\omega_2^2 + \\omega_4^2)\\ .\n\\end{equation}\nThe level curves of these complex-valued functions $\\theta$ and $\\phi$ are identified with electric and magnetic field lines. Several other approaches to construct such electromagnetic knots have been developed since then such as  Bateman's complex Euler potentials, conformal inversion and Penrose twistors (see \\cite{ABT17} for a full review).\n\nApart from these there exists another way of constructing these knotted electromagnetic fields via the conformal correspondence between de Sitter space $\\mathrm{d}{S}_4$ and Minkowski space $\\mathds R^{1,3}$, while passing through a finite Lorentzian $S^3$-cylinder, as was shown in \\cite{LZ17}. In this method, one obtains a complete family\\footnote{In the sense that any given finite-energy rational Maxwell solution can be expanded in terms of these basis configurations.} of such electromagnetic knotted configurations that are labelled with hyperspherical harmonics $Y_{j;m,n}$ of the $3$-sphere. The de Sitter space enjoys a larger symmetry group in $SO(1,4)$ whose subgroup $SO(4)\\cong (SU(2)\\times SU(2))\/\\mathds{Z}_2$ is made use of in this construction by working in the $S^3$-cylinder. Here one employs the right-action of $SU(2)$, which is the group manifold of $SO(4)$, on the $3$-sphere to write down the gauge field in terms of the left-invariant one-forms of $S^3$. The resulting Maxwell's equation can be solved analytically and are then pulled back to the Minkowski space using the conformal map. This ``de Sitter\" method of construction has advantage over the others because of the $SO(4)$ covariant treatment of Maxwell theory. \n\nKnotted electromagnetic fields might become important for future applications because of their unique topological properties. It is, therefore, important to seek experimental settings to generate those fields and to study scenarios with them. Irvine and Bouwmeester \\cite{Irvine08} discuss the generation of knotted fields using Laguerre--Gaussian beams and predict potential applications in atomic particle trapping, the manipulation of cold atomic ensembles, helicity injection for plasma confinement, and in the generation of soliton-like solutions in a nonlinear medium. Moreover, laser beams with knotted polarization singularities were recently employed to produce some simple knotted field configurations, including the one with figure-$8$ topology in the lab \\cite{LSMetal18}.\n\n\\vspace{12pt}\n\\section{Cosmic $SU(2)$ Yang--Mills fields}\n\\vspace{2pt}\n\nFinding analytic solution to the Einstein--Yang--Mills system of equations arising from the following action\n\\begin{equation}\n    S \\= S_{YM} + S_{EH} \\= \\int_M \\mathrm{d}^4x \\sqrt{-g}\\,R + \\int_M \\mathrm{Tr}(F{\\wedge} *F)\n\\end{equation}\nis not possible in general. There is, however, one scenario where such a solution can be obtained and that is FLRW cosmology. Here the Yang--Mills equation decouple from Einstein equation due to the conformal invariance of the former in $4$-dimensional spacetime $M$. This means that given a solution of Yang--Mills equation one of these FLRW spacetime, the corresponding scale factor can be obtained via the Friedmann equation. Such a solution with finite energy and action do exist for the de Sitter case together with the $SU(2)$ gauge group \\cite{AFF76,Luescher77,Schechter77}. \n\nA {\\it crucial} ingredient of the Standard Model of cosmology called inflation can also be tackled with homogeneous and isotropic non-Abelian Yang--Mills fields in the Minkowski type (spatially flat) FLRW background in theories of gauge-flation or chromo-natural inflation (see \\cite{MSS13} for a review). Another more minimalistic approach towards tackling this issue was recently put forth by Daniel Friedan \\cite{Friedan}. He considers a coupled Einstein--Yang--Mills--Higgs system where a rapidly oscillating isotropic $SU(2)$ gauge field stabilizes the symmetric Higgs vacuum in a de Sitter type (spatially closed) FLRW spacetime. \n\nBased on these, it is only natural to analyze the stability behaviour of such ``cosmic Yang--Mills fields\" under generic linear perturbation of the Yang--Mills field equation. For the gauge-flation scenario such an analysis has been done before, but for the later scenario there only exits result for spin-$j{=}0$ case of these $SU(2)$ gauge fields \\cite{Hosotani}.\n\nIn light of this, we present here a complete stability analysis of these $SU(2)$ solutions in a closed FLRW universe. This analysis is in contrast with that of the guage-flation one where conformal invariance is broken; our homogeneous and isotropic gauge field would give rise to inhomogeneous Yang--Mills fields on flat FLRW spacetime. For the sake of simplicity we keep the background metric fixed in this perturbation analysis. While this does mean that our analysis is still partial, we can argue for the relevance of our analysis on at least two grounds:\n\\begin{enumerate}[(a)]\n    \\item Because of the decoupling of the gauge fields with the metric in $4$-dimensional spacetime, the fluctuation of the latter does not affect the gauge field.\n    \\item The fluctuation of the gauge fields is too rapid for the evolution of the background metric and thus any non-conformal metric fluctuation would not have much of an effect on the gauge fields.\n\\end{enumerate}\n\nThe only known family of finite-energy $SU(2)$ Yang-Mills field configuration on FLRW spacetime are obtained, in an efficient manner, by employing the following conformal correspondence between de Sitter space and the cylinder ${\\cal I}\\times S^3$ for ${\\cal I} := (0,\\pi)$. This conformal map arises via a temporal reparametrization\nand Weyl rescaling~\\cite{Hosotani,Volkov,ILP17prl,Friedan},\n\\begin{equation} \\label{deSittermetric}\n\\begin{aligned}\n   \\mathrm{d} s_{\\textrm{dS}_4}^2 &\\= -\\mathrm{d} t^2 + \\ell^2\\cosh^2\\!\\sfrac{t}{\\ell}\\,\\mathrm{d}\\Omega_3^2\n   \\= \\sfrac{\\ell^2}{\\sin^2\\!\\tau} \\bigl(-\\mathrm{d}\\tau^2 + \\mathrm{d}\\Omega_3^2\\bigr) \\\\\n   &\\qquad \\quad\\textrm{for}\\quad t\\in(-\\infty,+\\infty)\\ \\Leftrightarrow\\ \\tau \\in (0,\\pi)\\ ,\n\\end{aligned}\n\\end{equation}\nwhere $\\mathrm{d}\\Omega_3^{\\ 2}$ is the round metric on $S^3$, and $\\ell$ is the de Sitter radius. We observe that the relation between the conformal time $\\tau$ and co-moving time~$t$ in \\eqref{deSittermetric} fixes the cosmological constant to~$\\Lambda=3\/\\ell^2$. At this point, we can employ an $S^3$-symmetric ansatz for the gauge field by noting that $SU(2)$ is the group manifold of $S^3$. This yields an ODE for some scalar function $\\psi(\\tau)$ parametrized by the conformal time, that is nothing but a Newton's equation for a classical point particle under the influence of a double-well potential\n\\begin{equation} \\label{Vpot}\nV(\\psi) \\= \\sfrac12 (\\psi^2-1)^2\\ .\n\\end{equation}\nThe solution for these anharmonic oscillators are well known in terms of Jacobi elliptic functions that depend on time. Although these solutions exists for all three FLRW metrics, only spatially closed one admits an isotropic solution under the above conformal transformation. For de Sitter type FLRW spacetime we have\n\\begin{equation} \\label{FLRW}\n\\begin{aligned}\n  \\mathrm{d} s^2 &\\= -\\mathrm{d} t^2 + a(t)^2\\,\\mathrm{d}\\Omega_3^{\\ 2}\n  \\= a(\\tau)^2\\bigl(-\\mathrm{d}\\tau^2 + \\mathrm{d}\\Omega_3^{\\ 2}\\bigr) \\\\\n  &\\qquad\\quad\\textrm{for}\\quad t\\in(0,t_{\\textrm{max}})\\ \\Leftrightarrow\\ \\tau \\in {\\cal I} \\equiv (0,T')\\ ,\n\\end{aligned}\n\\end{equation}\nwhere we impose a big-bang initial condition~$a(0){=}0$, so that\n\\begin{equation} \n\\mathrm{d}\\tau \\= \\frac{\\mathrm{d} t}{a(t)} \\qquad\\textrm{with}\\qquad \n\\tau(t{=}0) = 0 \\quad\\quad\\textrm{and}\\quad\\quad \\tau(t{=}t_{\\textrm{max}}) =: T'<\\infty\\ .\n\\end{equation}\nThe lifetime $t_{\\textrm{max}}$ of the universe \ncan be infinite (big rip, $a(t_{\\textrm{max}}){=}\\infty$) \nor finite (big crunch, $a(t_{\\textrm{max}}){=}0$).\nMoreover, bouncing cosmologies as in (\\ref{deSittermetric}) are also allowed but will not be pursued here. \n\nWe notice that the contribution of these $SO(4)$-symmetric Yang--Mills fields, and their stress-energy tensor, in the one-way coupling with the background de Sitter FLRW metric via Friedmann equation (as discussed before) is such that it only modifies the scale factor $a(\\tau)$. It is well known that the equation of motion governing this scale factor arises as a Newton's equation with the following (cosmological) potential \n\\begin{equation} \\label{Wpot}\nW(a) \\= \\sfrac12 a^2 - \\sfrac{\\Lambda}{6} a^4\\ ,\n\\end{equation}\nwhich is another anharmonic oscillator (although inverted). \n\nThe pair of solutions $(\\psi,a)$ corresponding to \\eqref{Vpot} and \\eqref{Wpot} yields an exact classical Einstein--Yang--Mills configuration. The conserved mechanical energy $E$ for $\\psi$ fixes the same i.e. $\\tilde{E}$ for $a$ via the Wheeler--DeWitt constraint\n\\begin{equation}\n    E + \\epsilon \\tilde{E} \\= 0\\ ;\\quad E\\ :=\\ \\frac{1}{2}\\dot{\\psi}^2 + V(\\psi) \\quad\\textrm{and}\\quad \\tilde{E}\\ :=\\ \\frac{1}{2}\\dot{a}^2 + W(a)\\ ,\n\\end{equation}\nwhere the overdot denotes a derivative with respect to conformal time and $\\epsilon$ depends on coupling constants. \n\nOne can also introduce a complex scaler Higgs field $\\phi$ in the fundamental SU(2) representation to the Standard Model of cosmology with Higgs potential\n\\begin{equation}\nU(\\phi) \\= \\sfrac12\\,\\lambda^2\\,\\bigl(\\phi^\\+\\phi-\\sfrac12 v^2\\bigr)^2\\ ,\n\\end{equation}\nwhere $v\/\\sqrt{2}$ is the Higgs vev and $\\lambda v$ is the Higgs mass. It turns out that imposing an $SO(4)$-invariance makes the Higgs field $\\phi\\equiv0$, which provides us with a definite positive cosmological constant of\n\\begin{equation}\n\\Lambda \\= \\kappa\\,U(0) \\= \\sfrac18\\,\\kappa\\,\\lambda^2 v^4\\ ,\n\\end{equation}\nwhere $\\kappa$ is the gravitational coupling.\nThe full Einstein--Yang--Mills--Higgs action (in standard notation),\n\\begin{equation}\nS \\= \\int\\!\\mathrm{d}^4 x\\ \\sqrt{-g}\\ \\Bigl\\{ \\sfrac{1}{2\\kappa} R + \\sfrac{1}{8g^2} \\mathrm{tr} F_{\\mu\\nu}F^{\\mu\\nu}\n-D_\\mu\\phi^\\+ D^\\mu\\phi - U(\\phi) \\Bigr\\}\\ ,\n\\end{equation}\nreduces in the SO(4)-invariant sector to\n\\begin{equation}\nS[a,\\psi,\\Lambda] \\= 12\\pi^2 \\int_0^{T'}\\!\\!\\mathrm{d}\\tau\\ \\Bigl\\{\n\\sfrac{1}{\\kappa} \\bigl(-\\sfrac12\\dot{a}^2+W(a)\\bigr) + \n\\sfrac{1}{2g^2} \\bigl(\\sfrac12\\dot{\\psi}^2-V(\\psi)\\bigr) \\Bigr\\}\\ ,\n\\end{equation}\nwhere $g$ is the gauge coupling.\n\nIf the Yang--Mills energy $E$ is large enough it could propel an eternal expansion of the universe that is accompanied by rapid fluctuations of the gauge field. The coupling of this Yang--Mills field with the Higgs field stabilizes the symmetric vacuum $\\phi\\equiv0$ at the local maximum of~$U$ through a parametric resonance effect, as long as $a$ is not too large. Eventually, when $a$ exceeds a critical value of electro-weak symmetry breaking scale $a_{\\textrm{EW}}$, the Higgs field will begin to roll down towards a minimum of~$U$, thus breaking the SO(4) symmetry. The corresponding time $t_{\\textrm{EW}}$ signifies the electroweak phase transition in the early universe. This is a rather unconventional scenario put forward recently by Friedan~\\cite{Friedan}. \n\n\\vspace{12pt}\n\\section{Outline and summary of results}\n\\vspace{2pt}\n\nIn the next two chapters we review the mathematical preliminaries that builds up gauge theory. In chapter \\ref{Chapter2} we present a brief but thorough review of the mathematics of the background geometry such as manifold, fibre bundles, etc. and the symmetry in physics such as Lie groups, Lie algebra and their representations. Next in chapter \\ref{Chapter3} we review the construction of gauge theory via principal bundle formalism.\n\nIn chapter \\ref{Chapter4} we discuss the geometry of and calculus on the $3$-sphere apart from demonstrating the conformal equivalence of the $S^3$-cylinder with the full Minkowski space. We then present Yang--Mills equation for a $SO(4)$-asymmetric $SU(2)$ Yang--Mills theory on the $S^3$-cylinder and discuss its two limiting cases where analytic solutions can be obtained. \n\nWe present the construction of electromagnetic knotted field configurations in chapter \\ref{Chapter5} and study their symmetry feature and other properties. We analyze the effect of the de Sitter group $SO(1,4)$, i.e. the isometry group of $\\mathrm{d}{S}_4$, on these solutions. To that end, we demonstrate the emergence of the Poincare group $ISO(4)$ of $\\mathds R^{1,3}$ from $SO(1,4)$ in the limit $\\ell \\to \\infty$. We observe that only the subgroup $SO(3)$ is common in the two cases. \n\nWe then proceed, also in \\ref{Chapter5}, to compute all the Noether charges associated with the conformal group $SO(2,4)$ viz. energy, momentum, angular momentum, boost, dilatation and special conformal transformations (SCT) for a complex linear combination, in terms of $\\Lambda_{j,m,n}$, of these basis-knot configurations. These conserved charge densities are evaluated on de Sitter space at $\\tau{=}0$ where considerable simplifications occur demonstrating the usefulness of this ``de Sitter method\". We find that the dilatation vanishes while the scalar SCT charge $V_0$ is proportional to the energy $E$. Furthermore, the boosts $K_i$ vanish and the vector SCT charges $V_i$ are proportional to the momenta $P_i$. Interestingly, for the vector charge densities viz. momenta $p_i$, angular momenta $l_i$ and vector SCT $v_i$ we find that the one-form $p_i\\mathrm{d} x^i$ constructed on the spatial slice $\\mathds R^3 \\hookrightarrow \\mathds R^{1,3}$ is proportional to a similar one on de Sitter space. This correspondence allows us to compute additional charges $(p_r,p_\\theta,p_\\phi)$ by the action of such one-forms on spherical vector fields $(\\mbox{$\\partial$}_r,\\mbox{$\\partial$}_\\theta,\\mbox{$\\partial$}_\\phi)$. At $j{=}0$ it turns out that there are only four independent non-zero charges: the energy $E$ and the momenta $P_i$. The situation for higher spin $j$ is more complicated, but some of the components of the charges in spherical coordinates are found to vanish for arbitrary $j$. The action of the $so(3)$ generators $\\mathcal{D}_a$ on the indices of $\\Lambda_{j,m,n}$ can easily be obtained for a fixed $j$ owing to the $SO(4)$ isometry. This allows for an action of these generators on the charges. For (the Cartesian components of) the vector charges this action is found to inherit the original $so(3)$ Lie algebraic structure, as expected. We also compute the correct coefficients $\\Lambda_{0;0,n}$ corresponding to two interesting generalisations of the Hopfian solution obtained via Bateman's construction in \\cite{HSS15}, which allow us to validate our generic formulae of these charges. We also demonstrate the relationship of energy with the conserved helicity.\n\nFurthermore, in chapter \\ref{Chapter5} we characterize the moduli space of null solutions which turns out to be a complete-intersection projective complex variety of complex dimension $2j{+}1$. We also demonstrate how the energy flux is radiated to infinity with an energy profile that is concentrated along the lightcone situated at the origin (selected by these solutions). Finally, we study the trajectories of multiple identical charged particles in the background of these basis knot configurations. We employ several initial conditions for these charged particles and find interesting features of the trajectories like coherent twisting, ultrarelativistic acceleration of particles starting from rest and a quick convergence of their trajectories into a few narrow cones asymptotically for sufficiently high value of the coupling. \n\nIn chapter \\ref{Chapter6} we first review the classical configurations $(A_\\mu,g_{\\mu\\nu})$ \nin terms of Newtonian solutions $(\\psi,a)$ for the anharmonic oscillator pair~$(V,W)$. We then investigate arbitrary small perturbations of the gauge field departing from the time-dependent background~$A_\\mu$ parametrized by the ``gauge energy'' $E$. Later on we linearize the Yang--Mills equation around it and diagonalize the fluctuation operator to obtain a spectrum of time-dependent natural frequencies. To decide about the linear stability of the cosmic Yang--Mills configurations we have to analyze the long-time behavior of the solutions to Hill's equation for all these normal modes. To that end, we employ Floquet theory to learn that their growth rate is determined \nby the stroboscopic map or monodromy, which is easily computed numerically for any given mode. \nWe do so for a number of low-frequency normal modes and find, when varying~$E$, an alternating sequence of stable (bounded) and unstable (exponentially growing) fluctuations. The unstable bands roughly correspond to the parametric resonance frequencies.\nWith growing ``gauge energy'' the runaway perturbation modes become more prominent,\nand some of them persist in the infinite-energy limit, where we detect universal natural frequencies and monodromies. A special role is played by the SO(4)-invariant fluctuation of $j{=}0$ mode, \nwhich merely shifts the parameter~$E$ of the background. We treat it exactly and beyond the linear regime. This ``singlet'' mode turns out to be marginally stable, i.e. it has a vanishing Lyapunov exponent. Its linear growth, however, gets limited by nonlinear effects of the full fluctuation equation, whose analytic solutions exhibit wave beat behavior.\n\nFinally we present a brief summary of our work in chapter \\ref{Chapter7} and present the future outlook. We also collect several relevant data in various appendices and present a direct map between the cylinder and Minkowski space via Carter--Penrose transformation avoiding the de Sitter detour in appendix \\ref{appendCPtransf}.\n\n\n\n\\chapter{Geometry \\& Symmetry}\n\\label{Chapter2}\n\n\n\\newcommand{\\ex}[1]{{\\bf Example #1}}\n\\newcommand{\\defn}[1]{\\noindent{\\bf Definition #1}}\n\\newcommand{\\vecF}[1]{\\mathfrak{X}(#1)}\n\\newcommand{\\rem}[1]{\\noindent{\\bf Remark #1}}\n\\newcommand{\\thm}[1]{\\noindent{\\bf Theorem #1}}\n\n\\justifying\nTwo of the most important concepts that play a fundamental role in physics are the geometry of background space and the presence of symmetries. They have well understood mathematical foundation in differential geometry and Lie groups\/algebras respectively. Here we present a short exposition of these mathematical topics that is essential towards building the subsequent Yang--Mills theory. The following contents are built upon some basic mathematical structures like vector spaces, groups and topological spaces all of which can be found in \\cite{Nakahara}. We refrain from presenting proofs of the statements in this chapter and suggest the references \\cite{Baez&Muniain} and \\cite{Bleecker} which are the source for most of the contents in this chapter. For details on Lie groups\/algebras and their representations we refer to classic texts \\cite{wybourne} and \\cite{Hall}.\n\n\n\\vspace{12pt}\n\\section{Manifolds}\n\\vspace{2pt}\n\\rem{2.1.1} The idea of a Manifold generalizes the notion of differentiation, or more precisely calculus on $\\mathds R^n$, in the same way as a Topological space allows one to study the notion of continuity in a more abstract way. In this thesis, we will only be concerned with (finite-dimensional) smooth manifolds, and subsequently, smooth structures on it.\n\n\\defn{2.1.1} An n-dimensional {\\it manifold} $M$ is a topological space\\footnote{Some required technicalities like paracompactness and Haussdorffness have been assumed.} equipped with an atlas consisting of charts $(U_i,\\phi_i)$ such that\n\\begin{itemize}\n    \\item $U_i$ are open sets and the maps $\\phi_i$ are homeomorphism between $U_i$ and some open ball in $\\mathds R^n$, and\n    \\item the {\\it transition maps} $\\phi_2\\circ \\phi_1^{-1}:\\ \\phi_1(U_1\\cap U_2) \\rightarrow \\phi_2(U_1\\cap U_2)$ is infinitely differentiable for any two charts $(U_1,\\phi_1)$ and $(U_2,\\phi_2)$.\n\\end{itemize}\n\n\\ex{2.1.1} A nice example of a smooth manifold is the round $n$-sphere\n\\begin{equation*}\n    S^n := \\lbrace {\\bf x} \\in \\mathds R^{n+1} ~|~ {\\bf x\\cdot x} = 1 \\rbrace\n\\end{equation*}\nembedded in $\\mathds R^{n+1}$ that requires two charts: $U_N = S^n - \\{(0,0,\\ldots,1)\\}$ and $U_S := S^n - \\{(0,0,\\ldots,-1)\\}$, along-with their respective stereographic projections:\n\\begin{equation*}\n \\begin{aligned}\n    \\phi_N(\\bf{x})\\ &:=\\ \\left( \\sfrac{x_1}{1-x^{n+1}},\\sfrac{x_2}{1-x_{n+1}},\\ldots,\\sfrac{x_n}{1-x_{n+1}} \\right)\\quad\\textrm{and}\\quad \\\\\n    \\phi_S(\\bf{x})\\ &:=\\ \\left( \\sfrac{x_1}{1+x_{n+1}},\\sfrac{x_2}{1+x_{n+1}},\\ldots,\\sfrac{x_n}{1+x_{n+1}} \\right)\n \\end{aligned}\n\\end{equation*}\nwith $x_{n+1} = 1 - x_1^2 - x_2^2 - \\ldots - x_n^2$.\n\n\\defn{2.1.2} Given a manifold $M$, a map $f: M \\rightarrow \\mathds R$ is called {\\it smooth} or $C^\\infty$ if it is infinitely differentiable. The set of all such smooth maps are denoted as $C^\\infty(M)$.\n\n\\defn{2.1.3} {\\it Diffeomorphism} $f:M \\rightarrow N$ between any two manifolds $M$ and $N$ is a bijection such that both $f$ and $f^{-1}$ are smooth. The set of all diffeomorphisms from $M$ to itself forms a group called $\\mathrm{Diff}(M)$.\n\n\\rem{2.1.2} Notice here that the differentiability of such a map $f$ is decided using local charts: e.g. given a chart $(U,\\phi)$ in $M$ containing $p\\in U$ and another chart $(V,\\psi)$ in $N$ containing $f(p)\\in V$, one can differentiate the map $\\psi\\circ f\\circ\\phi^{\\text -1}$ using standard calculus. A crucial notion in differential geometry is that of a tangent vector. It can intrinsically be defined in terms of a {\\it curve} on a manifold.\n\n\\defn{2.1.4} A {\\it curve} $\\sigma :\\ (-\\epsilon,\\epsilon) \\subset \\mathds R \\to M$ on a manifold $M$ is defined as a smooth map from some open interval of the real line to $M$.\n\n\\defn{2.1.5} The {\\it pull-back of a map $f$} by another map $\\phi$ is defined as\n\\begin{equation*}\n    \\phi^*f\\ :=\\ f\\circ\\phi\\ .\n\\end{equation*}\n\n\\defn{2.1.6} A tangent to a curve $\\sigma$ at a point $p=\\sigma(0)$ on a manifold $M$ is defined as the map\n\\begin{equation*}\n    \\sigma'(t):\\ C^\\infty(M) \\rightarrow \\mathds R\\ ,\\quad \\sigma'(t{=}0)[f]\\ :=\\ \\sfrac{\\mbox{$\\partial$}}{\\mbox{$\\partial$} t}(\\sigma^*f)\\big|_{t=0}\n\\end{equation*}\nThis idea can be generalized as follows.\n\n\\defn{2.1.7} A {\\it tangent vector} at a point $m\\in M$ is a map $v_m:\\ C^\\infty(M) \\rightarrow \\mathds R$ that satisfies the following properties \\begin{itemize}\n    \\item $v_m[f+g] \\= v_m[f] + v_m[g]\\quad  \\forall\\quad f,g \\in C^\\infty(M)$\\ ,\n    \\item $v_m[a\\,f] \\= a\\,v_m[f]\\quad \\forall\\quad a\\in \\mathds R\\ \\&\\ f\\in C^\\infty(M)$\\ , and\n    \\item $v_m[f\\,g] \\= f(p)\\,v_m[g] + g(p)\\,v_m[f] \\quad\\forall\\quad f,g\\in C^\\infty(M)$.\n\\end{itemize}\n\n\\defn{2.1.8} The set of all such tangent vectors at $m\\in M$ is known as the {\\it tangent space} at $m$ and is denoted by $T_mM$, which forms a vector space over $\\mathds R$. \n\n\\rem{2.1.3} This vector space is spanned by vectors $\\big\\{\\sfrac{\\scriptsize\\mbox{$\\partial$}}{\\scriptsize\\mbox{$\\partial$} x^i}\\big|_m \\equiv \\mbox{$\\partial$}_i|_m \\big\\}$ where $x^i$ are the coordinate functions in some chart $(U, \\phi)$ containing $m$ i.e. $\\phi(m) = (x^1,x^2,\\ldots,x^n)$ and whose actions is defined by \n\\begin{equation*}\n    \\sfrac{\\scriptsize\\mbox{$\\partial$}}{\\scriptsize\\mbox{$\\partial$} x^i}\\big|_m[f] \\ :=\\ \\sfrac{\\scriptsize\\mbox{$\\partial$} \\phi^* f}{\\scriptsize\\mbox{$\\partial$} x^i}\\quad \\forall \\quad f\\in C^\\infty(N)\\ .\n\\end{equation*}\nThe dimension of $T_mM$ is, therefore, equal to the dimension of the manifold $M$.\n\n\\defn{2.1.9} Given a map $\\varphi: M \\rightarrow N$ between two manifolds $M$ and $N$ and a vector $v\\in T_mM$, the {\\it push-forward} of $v$ by $\\varphi$ is defined by\n\\begin{equation*}\n    \\varphi_*v[f]\\ :=\\ v[\\phi^*f]\\quad \\forall \\quad f\\in C^\\infty(N)\\ .\n\\end{equation*}\n\n\\rem{2.1.4} Note that the push-forward induces a map between following tangent spaces:\n\\begin{equation*}\n    \\phi_*:\\ T_mM \\rightarrow T_{h(m)}N\n\\end{equation*}\nfor manifolds $M$ and $N$.\n\n\\defn{2.1.10} The vector space dual to $T_mM$ is called {\\it cotangent space} and is denoted by $T^*_mM$.\n\n\\rem{2.1.5} The vector space $T^*_mM$ is spanned by covectors $\\mathrm{d}{x}^i|_m$ defined as \n\\begin{equation*}\n    \\langle \\mathrm{d}{x}^i|_m\\, ,\\, v\\rangle\\ :=\\ v[x^i]\\quad \\forall \\quad v \\in T_mM\n\\end{equation*}\nand has the same dimension as $T_mM$.\n\n\\defn{2.1.11} The {\\it pull-back} of a covector $\\omega \\in T^*_nN$ by a map $\\varphi: M \\rightarrow N$ between two manifolds $M$ and $N$ is defined by\n\\begin{equation*}\n    \\langle\\varphi^*\\omega\\, ,\\, v \\rangle\\ :=\\ \\langle \\omega\\, , \\, \\varphi_*v \\rangle\n\\end{equation*}\nwith $n = \\varphi(m)$.\n\n\\vspace{12pt}\n\\section{Fibre bundles}\n\\vspace{2pt}\n\\rem{2.2.1} The theory of bundles has proved to be the correct way of studying classical gauge theories like general relativity and Yang-Mills theory. Here we shall confine ourselves to bundles constructed on\/with manifolds.\n\n\\defn{2.2.1} A {\\it bundle} is a triple $(E,M,\\pi)$ consisting of a manifold $E$, {\\it aka} the target space, a manifold $M$, {\\it aka} the base space, and a continuous surjection $\\pi: E \\rightarrow M$, {\\it aka} the projection map. It is denoted diagrammatically as \n\\begin{tikzcd}\n   &E \\arrow[d, \"\\pi\"] \\\\\n   &M\n\\end{tikzcd}\n\n\\defn{2.2.2} A bundle $(E,M,\\pi)$ is called a {\\it fibre bundle} with typical fibre $F$ if the inverse image of all $p \\in M$ under $\\pi$ is isomorphic to some space $F$ i.e. $\\pi^{-1}(p) \\cong F$. If $F$ is a vector space then the bundle $(E,M,\\pi)$ becomes a {\\it vector bundle}.\n\n\\defn{2.2.3} A pair of fibre bundles $(E,M,\\pi)$ and $(\\widetilde{E},\\widetilde{M},\\widetilde{\\pi})$ are called {\\it isomorphic} (as bundles) if there exist a pair of diffeomorphisms $\\varphi: E \\rightarrow \\widetilde{E}$ and $\\psi: M \\rightarrow \\widetilde{M}$ such that $\\widetilde{\\pi}\\circ \\varphi = \\psi\\circ\\pi$ and $\\pi\\circ\\varphi^{-1} = \\psi^{-1}\\circ\\widetilde{\\pi}$. Diagrammatically this means that the following diagram, and its inverse, commutes \n\\begin{tikzcd}\n   E \\arrow[r, \"\\varphi\"] \\arrow[d, \"\\pi\"] &\\widetilde{E} \\arrow[d, \"\\widetilde{\\pi}\"] \\\\\n   M \\arrow[r, \"\\psi\"] &\\widetilde{M}\n\\end{tikzcd}\n\n\\defn{2.2.4} A vector bundle $(E,M,\\pi)$ with typical fibre $F$ is called {\\it locally trivial} if for any $U \\subset M$ the induced bundle $(\\pi^{-1}(U),U,\\pi|_{\\pi^{-1}(U)})$ is isomorphic to the product bundle $(U\\times F,U,\\pi_1)$, where $\\pi_1$ is the projection in the first slot. The set $(U,\\varphi)$\\footnote{Notice, here, that $\\psi=Id_U$.} is known as a {\\it local trivialization} of the vector bundle. A {\\it trivial} bundle is one where $E=M\\times F$ and $\\pi = \\pi_1$.\n\n\\rem{2.2.2} We shall only deal with locally trivial bundles here. A vector bundle $(E,M,\\pi)$ will, sometimes, be simply denoted $E$. A classic example of a bundle that is not globally trivial is the following.\n\n\\ex{2.2.1} A M\\\"{o}bius strip $(E,S^1,\\pi)$ with fibre $[0,1]$ is locally isomorphic\\footnote{Meaning that they share local trivialization $(U,\\varphi)$ for any $x\\in U$.} to the trivial bundle $(S^1\\times [0,1],S^1,\\pi_1)$. The former, however, is not trivial as transporting any vector across a loop yields the corresponding vector inverted.\n\n\\defn{2.2.5} Given a fibre bundle $(E,M,\\pi)$ with typical fibre $F$ and a pair of local trivializations $(U_i,\\varphi_i)$ and $(U_j,\\varphi_j)$ with $U_i\\cup U_j \\neq 0$ on it, we obtain {\\it transition functions} $g_{ij}(x)$ with $x\\in U_i\\cap U_j$ of the vector bundle $U_i\\cap U_j\\times F$ by realizing that $\\varphi_j\\circ \\varphi_i(x,v) = (x,g_{ij}(x)v)$ for any vector $v\\in F$. The set of such transition functions $g_{ij}(x)$ forms a group $G \\subset End(F)$ called the {\\it structure group}.\n\n\\defn{2.2.6} A (local) {\\it section} $\\sigma$ (for some local trivialization $(U,\\varphi)$) of a fibre bundle $(E,M,\\pi)$ is a smooth map $\\sigma: (U) M \\rightarrow E$ such that $\\pi\\circ\\sigma = (Id_U) Id_M$. The set of all such (local) global sections are denoted $(\\Gamma^\\infty(U,E))$ $\\Gamma^\\infty(M,E)$. The global section is also denoted more simply as $\\Gamma^\\infty(E)$. \n\n\\vspace{12pt}\n\\section{Lie groups}\n\\vspace{2pt}\n\\rem{2.3.1} The notion of symmetry in modern physics is analyzed with the tools from the theory of Lie groups and Lie algebras. We will only consider finite, matrix Lie groups in what follows.\n\n\\defn{2.3.1} A {\\it Lie group} ({\\it aka} continuous group) $G$ is both a group and a smooth, finite-dimensional manifold where the group operation, $\\cdot: G\\times G \\rightarrow G$, as well as the inversion map, $^{-1}: G \\rightarrow G$ satisfying $g^{-1}\\cdot g = g\\cdot g^{-1}=e~\\forall~ g\\in G$ with identity element $e$, are smooth.\n\n\\rem{2.3.2} We will omit the group multiplication symbol $\\cdot$ and use $\\mathds{1}$ interchangeably with $e$ for the matrix Lie groups (to be considered below) from now onward.\n\n\\defn{2.3.2} A Lie group homomorphism $\\varphi: G \\rightarrow H$ between Lie groups $G$ and $H$ is a smooth map that is also a group homomorphism. The map $\\varphi$ becomes an isomorphism if it is bijective and its inverse $\\varphi^{-1}$ is smooth; the Lie groups $G$ and $H$ then becomes isomorphic.\n\n\\defn{2.3.3} A {\\it positive-definite inner product} is a bilinear map $\\<\\cdot,\\cdot\\>: V\\times V \\xrightarrow{\\sim} \\mathds R$ for a vector space $V\\ni x,y$ that is\n\\begin{enumerate}[(a)]\n    \\item symmetric: $\\<x,y\\>=\\<y,x\\>$\\ ,\n    \\item positive-definite: $\\<x,x\\>>0$\\ , and\n    \\item nondegenerate: i.e. if $\\<x,y\\>=0$ for all $y \\implies x=0$\\ .\n\\end{enumerate}\nIf $\\<x,x\\>\\not>0$ then the inner product is called {\\it indefinite}.\n\n\\rem{2.3.3} Most of the important Lie groups in physics emerges as matrix Lie groups by considering invertible linear maps $V\\xrightarrow{\\sim}V$ on some finite-dimensional vector space $V$ that preserves a given inner product on $V$. Such {\\it general linear groups} are denoted $GL(V)$. Furthermore, these are Lie subgroups\\footnote{A Lie subgroup $H\\subset G$ is a subgroup as well as a submanifold of $G$. It turns out to be a Lie group in itself.} of the general Linear groups $GL(n,\\mathds R)$ or $GL(n,{\\mathds C})$ consisting of invertible linear maps on vector field $\\mathds R$ or $\\mathds{C}$ respectively. This becomes evident when we consider Riemannian manifolds which contains a metric (see Section \\ref{covectorFields}).\n\n\\ex{2.3.1} The special linear groups $SL(n,\\mathds R)$ (over $\\mathds R$) and $SL(n,{\\mathds C})$ (over ${\\mathds C}$) are $n\\times n$ invertible matrices with unit determinant i.e.\n\\begin{equation*}\n    SL(n,\\mathds R\/{\\mathds C}) \\= \\{A \\in GL(n,\\mathds R\/{\\mathds C})~|~ detA = 1 \\}\\ .\n\\end{equation*}\n\n\\ex{2.3.2} The Unitary group $U(n)$ are the $n\\times n$ complex-valued matrices whose inverse is the same as its conjugate transpose i.e.\n\\begin{equation*}\n    U(n) \\= \\{A \\in GL(n,{\\mathds C})~ |~ A^{-1} = \\bar{A}^T\\equiv A^\\+ \\}\\ .\n\\end{equation*}\nThe special unitary group is defined as\n\\begin{equation*}\n    SU(n) \\= \\{ A \\in U(n)~ |~ detA = 1 \\}\\ .\n\\end{equation*}\n\n\\rem{2.3.4} The special unitary groups $SU(n)$ preserves the standard inner product on ${\\mathds C}^n \\ni {\\bf x},{\\bf y}$ given by \n\\begin{equation*}\n    \\langle {\\bf x}, {\\bf y} \\rangle_{{\\mathds C}} \\= \\sum\\limits_{i=1}^n\\bar{x}_iy_i\n\\end{equation*}\nand also the norm of a given vector (induced from this inner product).\n\n\\ex{2.3.3} The orthogonal group $O(k,n)$ are the $(k+n)\\times (k+n)$ matrices that preserve the following inner product (${\\bf x},{\\bf y} \\in \\mathds R^{k+n} $):\n\\begin{equation*}\n    \\langle {\\bf x}, {\\bf y} \\rangle_{k,n}\\ :=\\ -x_1y_1 - \\ldots - x_ky_k + x_{k+1}y_{k+1} +\\ldots + x_{k+n}y_{k+n}\\ .\n\\end{equation*}\nOne can show that $A$ is in $O(k,n)$ if and only if\n\\begin{equation*}\n    A^T\\eta^{(k,n)} A \\= \\eta^{(k,n)}\\ ,\n\\end{equation*}\nwhere $\\eta^{(k,n)}$ is just the diagonal matrix $diag(1,\\ldots1,-1,\\ldots,-1)$. The special orthogonal group is defined as\n\\begin{equation*}\n    SO(k,n)\\ :=\\ \\{ A \\in O(k,n)~|~ detA = 1\\}\\ .\n\\end{equation*}\nThe orthogonal group $O(0,n)$ is denoted $O(n)$ and is more famously defined as\n\\begin{equation*}\n    O(n) := \\{ A \\in GL(n,\\mathds R) ~|~ A^{-1} = A^T \\}\\ .\n\\end{equation*}\nMoreover, the special orthogonal group $SO(0,n)$ is denoted as $SO(n)$.\n\n\\rem{2.3.5} The group $SO(n)$ consists of rotations in $n$-dimensions and is thus knows as isometry group of the $n$-sphere $S^n$. The group $O(n)$ consists of rotations as well as reflections. The groups $SO(1,3)$ and $SO(2,4)$ known, respectively, as the {\\it Lorentz group} and the {\\it Conformal group} are of special significance in physics. Another important group for us is the {\\it de Sitter group} $SO(1,4)$.\n\n\\ex{2.3.4} The {\\it Poincar\\'{e} group} or the {\\it inhomogeneous Lorentz group} $ISO(1,3)$ consists of Lorentz transformation together with translations and is defined as\n\\begin{equation*}\n    ISO(1,3) := \\{ I_{{\\bf x}} A ~|~ A \\in SO(1,3) \\}\\ ,\n\\end{equation*}\nwhere the translations $I_{{\\bf x}}$ acts on a given vector ${\\bf y} \\in \\mathds R^{1,3}$ as\n\\begin{equation*}\n    I_{{\\bf x}}{\\bf y} = {\\bf x} + {\\bf y}\\ .\n\\end{equation*}\n\n\\rem{2.3.6} It can be shown that the group $SO(4)$ decomposes into two copies of $SU(2)$ and that there exist a $2{-}1$ homomorphism\n\\begin{equation*}\n    SU(2)\\times SU(2) \\xrightarrow{2{-}1} SO(4)\\ .\n\\end{equation*}\nThis is a rather generic fact that arises when one considers spin-groups (e.g. $Spin(4)$ here), which provide universal cover\\footnote{This is a topological term that refers to a connected, simply connected group that projects down to the given group $G$ via a smooth surjection $\\pi$ such that any open $U \\in G$ lifts to a disjoint union $\\pi^{-1}(U)$ whose members are isomorphic to $U$.} to some group (e.g. $SO(4)$ here). To this end, we note down the following relevant facts here:\n\\begin{equation*}\n    Spin(4)\\cong SU(2)\\times SU(2)\\ ,\\quad Spin(3)\\cong SU(2)\\ ,\\ \\mathrm{and}\\quad Spin(1,3)\\cong SL(2,{\\mathds C})\\ .\n\\end{equation*}\n\n\\defn{2.3.4} The {\\it left action} of a Lie group $G$ (aka left $G$-action) on a set $M$ is defined by the map\n\\begin{equation*}\n    \\varphi_l :\\ G\\times M \\rightarrow M\\ ,\\quad (g,m) \\mapsto \\varphi_l(g,m)\\ \\equiv\\ gm\n\\end{equation*}\nthat satisfies the following properties:\n\\begin{itemize}\n    \\item $em \\= m$ for the identity element $e \\in G$ and every $m\\in M$, and\n    \\item $g_1(g_2m) \\= (g_1g_2)m$ for all $g_1,g_2 \\in G$ and $m \\in M$.\n\\end{itemize}\nThe set $M$ is known as a {\\it homogeneous space}, that splits into an {\\it orbit space} $M\/G$ of equivalence classes of orbits\n\\begin{equation*}\n    O_m\\ :=\\ \\{ n \\in M ~|~ \\exists\\ g \\in G : n = \\varphi_l(g,m) \\}\\ .\n\\end{equation*}\n\n\\defn{2.3.5} The {\\it left translations} or {\\it left multiplications} of a Lie group $G$ is its diffeomorphism i.e. $l_g\\in Diff(G)$ and is defined by\n\\begin{equation*}\n    l_g :\\ G \\rightarrow G\\ ,\\quad h \\mapsto gh\\ .\n\\end{equation*}\n\n\\rem{2.3.7} An equivalent notion of {\\it right action} defined by $\\varphi_r(g,m) := mg$ also exits and is of prime importance for the principle $G$-bundles that we will discuss in the next chapter. It is important to note here that one can induce a right-action $\\varphi_r$ from a given left-action $\\varphi_l$ as follows:\n\\begin{equation*}\n    \\varphi_r(g,m)\\ :=\\ \\varphi_l(g^{-1},m) \\quad \\forall\\quad m \\in M\\ . \n\\end{equation*}\nSimilarly, we have the notion of right translations $r_g$ for the Lie group $G$ but we will only deal with left translations here.\n\n\\defn{2.3.6} The {\\it coset space} $G\/H$ for a Lie group $G$ and its subgroup $H\\subset G$ is defined as\n\\begin{equation*}\n    G\/H\\ :=\\ \\{ gH ~|~ g \\in G \\}\\quad \\mathrm{where}\\quad gH\\ :=\\ \\{ gh ~|~ h \\in H \\}\\ .\n\\end{equation*}\n\n\\rem{2.3.8} There exist a natural left $G$-action (and hence, a left $H$-action) on $G\/H$ defined by\n\\begin{equation*}\n    \\varphi_l(g',gH) = g'gH\\quad \\forall\\quad g,g' \\in G\\ .\n\\end{equation*}\n\n\\defn{2.3.7} A left $G$-action on $M$ is called {\\it free} if, for all $m\\in M$, $gm=m$ implies that $g=e$. The action would be {\\it transitive} if for all $m,m'\\in M$ there exists $g\\in G$ such that $m=gm'$.\n\n\\rem{2.3.9} If the left action of $G$ on $M$ is free then every orbit $O_m$ is diffeomorphic to the Lie group $G$.\n\n\\defn{2.3.8} The {\\it stability\/isotropy subgroup} $G_m$ of a left $G$-action on $M\\ni m$ for a Lie group $G$ is its closed subgroup defined by\n\\begin{equation*}\n    G_m\\ :=\\ \\{ g \\in G ~|~ gm = m \\}\\ .\n\\end{equation*}\n\n\\thm{2.3.1} For a transitive left $G$-action there exist an isomorphism\\footnote{Some technicalities like $G$ be locally compact and $M$ be locally compact and connected are required.} $G\/G_m \\cong M$ between the coset space $G\/G_m$ and the homogeneous space $M$ for any $m\\in M$ given by\n\\begin{equation*}\n    j_p :\\ G\/G_p \\rightarrow M\\ ,\\quad gG_p \\mapsto gp\\ .\n\\end{equation*}\n\n\\rem{2.3.10} For a Lie group $G$ and its closed subgroup $H$ (e.g. $G_m$) the homogeneous space $G\/H$ can be canonically endowed with the structure of a smooth manifold. Moreover, there exist a canonical projection from $G$ to $G\/H$ given by\n\\begin{equation*}\n    \\pi_0 :\\ G \\rightarrow G\/H\\ ,\\qquad g \\mapsto gH\\ .\n\\end{equation*}\n\n\\ex{2.3.5} The round $n$-sphere $S^n$ is diffeomorphic to the following homogenoeus space:\n\\begin{equation*}\n    S^n\\ \\cong\\ SO(n{+}1)\/SO(n)\\ .\n\\end{equation*}\nIn particular, we have that $S^3 \\cong SO(4)\/SO(3)$.\n\n\\ex{2.3.6} A $(2n{+}1)$-sphere can be realized as a homogeneous space:\n\\begin{equation*}\n    S^{2n{+}1}\\ \\cong\\ SU(n{+}1)\/SU(n)\\ .\n\\end{equation*}\nA special case is $S^3\\cong SU(2)$, where the group $SU(1)$ is trivial.\n\n\\vspace{12pt}\n\\section{Vector fields}\n\\vspace{2pt}\n\\defn{2.4.1} The {\\it tangent bundle} $(TM,M,\\pi)$ over a manifold $M$ is nothing but a union of tangent spaces at all point of the manifold i.e.\n\\begin{equation*}\n    TM \\= \\bigcup\\limits_{p\\in M} T_p M\\ ,\n\\end{equation*}\nwhere the projection $\\pi$ just picks out the base point of a given vector in $TM$. \n\n\\rem{2.4.1} The dimension of the tangent bundle for an $n$-dimensional manifold is $2n$, as its members are ($n$-dimensional) vectors of some $T_pM$ labelled by the coordinate ($n$-tuple) of the base point $p\\in M$.\n\n\\defn{2.4.2} A {\\it vector field} is a smooth section of the tangent bundle $(TM,M,\\pi)$. The set of all such vector fields are denoted as $\\vecF{M} := \\Gamma^\\infty(TM)$.\n\n\\rem{2.4.2} Given a vector field $X\\in \\vecF{M}$ and a smooth function $f\\in C^\\infty(M)$, one can show that $Xf$ defined by\n\\begin{equation*}\n    Xf(p)\\ :=\\ X_p[f]\\ ,\n\\end{equation*}\nwhere $X_p \\in T_pM$, is also smooth i.e. $Xf \\in C^\\infty(M)$. The set of vector fields $\\vecF{M}$ do not form a vector space rather a module over the algebra of smooth functions $C^\\infty(M)$\\footnote{An algebra is just a vector space $V$ equipped with a multiplication operation $V\\times V \\rightarrow V$, which for functions is just composition. A module over an algebra is the same thing as a vector space over a field.}. Nevertheless, one can choose a basis of vector fields $\\left\\{ \\sfrac{\\mbox{$\\partial$}}{\\mbox{$\\partial$} x^i} \\equiv \\mbox{$\\partial$}_i \\right\\}$ on some local chart $(U,\\varphi)$ with coordinate functions $x^i$ (see Remark 2.1.3) to write $X \\in\\vecF{M}$ as\n\\begin{equation*}\n    X \\= \\sum\\limits_{i=1}^n Xx^i\\, \\mbox{$\\partial$}_i\\ .\n\\end{equation*}\n\n\\defn{2.4.3} There is a well defined notion of {\\it Lie bracket} associated with vector fields defined as\n\\begin{equation*}\n    [X,Y]f\\ :=\\ X(Yf) - Y(Xf)\\ ,\n\\end{equation*}\nwhich satisfy the following properties  \n\\begin{itemize}\n    \\item bilinearity: $[\\cdot,\\cdot] :\\ \\vecF{M}\\times \\vecF{M} \\xrightarrow{\\sim} \\vecF{M}$,\n    \\item anti-symmetry: $[X,Y] = -[Y,X]$, and\n    \\item Jacobi identity: $[X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]] = 0$,\n\\end{itemize} \nfor all $X,Y,Z \\in \\vecF{M}$.\n\n\\rem{2.4.3} In general, it is not possible to induce a push-forward (see Definition 2.1.9) between vector fields $\\vecF{M} \\ni X$ and $\\vecF{N} \\ni Y$ of two manifolds $M$ and $N$ with a given map $\\varphi: M\\rightarrow N$ (it works if $\\varphi$ is a diffeomorphism). Nevertheless, it is useful to define the following relation.\n\n\\defn{2.4.4} A vector field $X \\in \\vecF{M}$ is called {\\it h-related} to another vector field $Y \\in \\vecF{M}$ i.e. $Y = \\varphi_*X$ if for all $p\\in M$ we have that\n\\begin{equation*}\n    \\varphi_*X_p \\= Y_{\\varphi(p)}\\ .\n\\end{equation*}\n\n\\rem{2.4.4} If $X_1$ and $X_2$ is h-related to $Y_1$ and $Y_2$ respectively then $[X_1,X_2]$ is h-related to $[Y_1,Y_2]$ for all $X_1,X_2,Y_1,Y_2 \\in \\vecF{M}$ i.e.\n\\begin{equation*}\n    \\varphi_*[X_1,X_2] \\= [Y_1,Y_2]\\ .\n\\end{equation*}\n\n\\defn{2.4.5} Given a vector field $X\\in \\vecF{M}$ its {\\it integral curve} through $p\\in M$ is a curve \n\\begin{equation*}\n    \\sigma_X :\\ (-\\epsilon,\\epsilon) \\rightarrow M\\ ,\n\\end{equation*}\nsuch that \n\\begin{equation*}\n    \\sigma_X(0) \\= p \\quad\\textrm{and}\\quad \\sigma_X'(t) \\= X_{\\sigma(t)}\n\\end{equation*}\nfor all $t\\in (-\\epsilon,\\epsilon)$.\n\n\\defn{2.4.6} If the interval $(-\\epsilon,\\epsilon)$ can be extended to whole of $\\mathds R$ for any $p \\in M$ then the vector field, and also the underlying manifold, is called {\\it complete}.\n\n\\rem{2.4.5} Presence of singularities (e.g. a black hole) on a manifold makes it incomplete. We will only consider complete manifolds in this thesis.\n\n\\vspace{12pt}\n\\section{Lie algebra}\n\\vspace{2pt}\n\\defn{2.5.1} A {\\it left-invariant} vector field $X$ for a Lie group $G$ is defined as the vector field which is $l_g$-related to itself for all $g\\in G$ i.e. \n\\begin{equation*}\n    l_{g*}X \\= X \\quad \\mathrm{or}\\quad l_{g*}X_{g'} = X_{gg'}\n\\end{equation*}\nfor all $g' \\in G$.\n\n\\defn{2.5.2} The set of all left-invariant vector fields (a vector space) denoted as $L(G)$ together with the Lie bracket $[\\cdot,\\cdot]$ i.e. $\\left(L(G),[\\cdot,\\cdot]\\right)$ is known as the {\\it Lie algebra} of $G$\\footnote{Remark 2.4.4 is crucial here.}.\n\n\\thm{2.5.1} We have a Lie algebra isomorphism\\footnote{It is an invertible map that preserves the Lie algebraic structure.} $(T_eG,[\\cdot,\\cdot]) \\cong (L(G),[\\cdot,\\cdot])$ defined by\n\\begin{equation*}\n    j:\\ T_eG \\rightarrow L(G)\\ ,\\quad A\\mapsto j(A)\\equiv L^A\\ ,\n\\end{equation*}\nwhere the left-invariant vector field $L^A$ is defined by\n\\begin{equation*}\n    L^A_g\\ :=\\ l_{g*}A \\in T_gG\\ .\n\\end{equation*}\n\n\\rem{2.5.1} Note that the Lie bracket on $T_eG$ is induced from the one on $L(G)$ via the map $j$ i.e.\n\\begin{equation*}\n    [A,B]\\ :=\\ j^{-1}[L^A,L^B] \\= l_{g{-1}*}[L^A_g,L^B_g]\n\\end{equation*}\nfor any $g\\in G$. This takes the form of usual matrix commutator for matrix Lie groups that we are dealing with. We thus have the fact that $\\textrm{dim}L(G) = \\textrm{dim}T_eG = \\textrm{dim}G$.\n\n\\ex{2.5.1} We note down below in Table \\ref{LieTable} the Lie algebras $T_eG$ of some of the Lie groups $G$ that we encountered before.\n\\begin{table}[!htbp]\n    \\centering\n    \\begin{tabular}{|p{0.15\\textwidth}|p{0.6\\textwidth}|}\n    \\hline\n        $G$ & $T_eG$ \\\\\n    \\hline\\hline\n        $GL(n,\\mathds R\/\\mathds{C})$ & $M(n,\\mathds R\/\\mathds{C}) := $ The set of $n\\times n$ real\/complex matrices \\\\\n    \\hline\n        $SO(n)$ & $so(n) := \\{ A \\in M(n,\\mathds R)\\ |\\ A^T = -A \\}$  \\\\\n    \\hline\n        $SU(n)$ & $su(n) := \\{ A \\in M(n,\\mathds{C})\\ |\\ \\bar{A} = -A\\ \\&\\ \\mathrm{tr}{A} = 0 \\}$ \\\\\n    \\hline\n    \\end{tabular}\n    \\caption{A list of Lie groups and their Lie algebras.}\n    \\label{LieTable}\n\\end{table}\n\n\\thm{2.5.2} A Lie group homomorphism $\\varphi : G \\rightarrow H$ between Lie groups $G$ and $H$ induces a Lie algebra homomorphism\n\\begin{equation*}\n    \\mathrm{d}{\\varphi}\\equiv \\varphi_*:\\ T_eG \\rightarrow T_eH\\ , \\quad \\mathrm{i.e.}\\quad \\varphi_*[A,B] \\= [\\varphi_*A,\\varphi_*B]\\ .\n\\end{equation*}\n\n\\rem{2.5.2} Given a basis $\\{ E_1,\\ldots, E_n \\}$ of $L(G)\\cong T_eG$ for an $n$-dimensional Lie group $G$, its structure constants $f_{ij}^{\\ \\ k}$ are defined by\n\\begin{equation*}\n    [E_i,E_j] \\= \\sum\\limits_{k=1}^n f_{ij}^{\\ \\ k}\\,E_k\\ .\n\\end{equation*}\n\n\\ex{2.5.2} The structure constants for the Lie algebra $su(2)$ are given by $f_{ij}^{\\ \\ k} = 2\\,\\varepsilon_{ij}^{\\ \\ k}$, where $\\varepsilon_{ij}^{\\ \\ k}$ is the $3$-dimensional Levi-Civita symbol.\n\n\\thm{2.5.3} A left-invariant vector field $X$ on a Lie group $G$ is complete.\n\n\\rem{2.5.3} A consequence of the above theorem is that there exist a unique integral curve\n\\begin{equation*}\n    t \\mapsto \\sigma_{L^A}(t)\n\\end{equation*}\nfor all $t\\in \\mathds R$ and left-invariant vector field $L^A$ constructed from a given $A\\in T_eG$ of the Lie group $G$.\n\n\\defn{2.5.3} The {\\it exponential map} for a Lie group $G$ is defined by\n\\begin{equation*}\n    \\mathrm{exp}:\\ T_eG \\rightarrow G\\ , \\quad A \\mapsto \\mathrm{exp}(A)\\ :=\\ \\sigma_{L^A}(t{=}1)\\ .\n\\end{equation*}\n\n\\rem{2.5.4} The exponential map is locally diffeomorphic. Furthermore, it lifts $T_eG$ to a connected component of $G$ (connected to $e$) and is a surjection when $G$ is compact.\n\n\\defn{2.5.4} A {\\it one-parameter subgroup} of a Lie group $G$ is a smooth homomorphism $\\mu :\\ \\mathds R \\rightarrow G$ from the additive group $\\mathds R$ into $G$ i.e.\n\\begin{equation*}\n    \\mu(t_1+t_2) \\= \\mu(t_1)\\mu(t_2).\n\\end{equation*}\n\n\\thm{2.5.4} If $\\mu: \\mathds R \\rightarrow G$ is a one-parameter subgroup of $G$ then, for all $t\\in \\mathds R$,\n\\begin{equation*}\n    \\mu(t) \\= \\mathrm{exp}(tA) \\quad\\textrm{with}\\quad  A\\ :=\\ \\mu_*\\sfrac{\\mathrm{d}}{\\mathrm{d}{t}}\\big|_0\\ .\n\\end{equation*}\n\n\\rem{2.5.5} The above theorem shows that there exist a one-to-one correspondence between one-parameter subgroups of a Lie group $G$ and its Lie algebra $T_eG$. \n\n\\defn{2.5.5} Given a vector field $X\\in \\vecF{M}$ one defines the {\\it flow generated by X} as the one-parameter group $\\{ \\varphi_t : t \\in \\mathds R \\}$ where the set of maps $\\varphi_t : M \\rightarrow M$ are nothing but the integral curves\n\\begin{equation*}\n    \\varphi_t(m) \\= \\sigma(t)\\ .\n\\end{equation*}\nOne can define the {\\it Lie derivative of $Y$ along $X$}, for $Y\\in \\vecF{F}$, by\n\\begin{equation}\n    {\\cal L}_X Y \\= \\frac{\\mathrm{d}}{\\mathrm{d} t}(\\varphi_t^{-1})_*(Y)\\Big|_{t=0}\\ .\n\\end{equation}\n\n\\rem{2.5.6} It can be shown that ${\\cal L}_XY = [X,Y]$.\n\n\\vspace{12pt}\n\\section{Tensor fields}\n\\vspace{2pt}\n\\label{covectorFields}\n\\defn{2.6.1} A {\\it (p,q)-tensor} $T^{p,q}(V)$ for a vector space $V$ and its dual $V^*$ is the space of multilinear functions\n\\begin{equation*}\n    f:\\ V^*\\times\\overset{p}{\\cdots}\\times V^*\\times V\\times\\overset{q}{\\cdots}\\times V \\xrightarrow{\\sim} \\mathds R\n\\end{equation*}\nand is, alternatively, denoted using tensor products as $V\\otimes\\overset{q}{\\cdots}\\otimes V\\otimes V^*\\otimes\\overset{p}{\\cdots}\\otimes V^*$.\n\n\\rem{2.6.1} Note that $T^{0,0}(V):=\\mathds R$. Moreover, we have the following results\n\\begin{equation*}\n    T^{0,1}(V)\\ \\cong\\ V^* \\quad\\textrm{and}\\quad T^{1,0}(V) \\= (V^*)^*\\ \\cong\\ V\n\\end{equation*}\nfor any finite-dimensional vector space $V$. Elements of $T^{p,q}(V)$ can be easily expanded in terms of a given basis of $V$ and the corresponding dual basis of $V^*$.\n\n\\defn{2.6.2} A {\\it (p,q)-tensor bundle} $(T^{p,q}M,M,\\pi)$ over a manifold $M$ is the following disjoint union of tensors:\n\\begin{equation*}\n    T^{p,q}M \\= \\bigcup\\limits_{m\\in M}T^{p,q}(T_mM)\n\\end{equation*}\nwhere the projection $\\pi$ associates the base point $m\\in M$ of a given vector in $T^{p,q}M$.\n\n\\rem{2.6.2} The tangent bundle $TM$ is isomorphic to $T^{1,0}M$, while the bundle $T^{0,1}M$ is isomorphic to the so called {\\it cotangent bundle} $T^*M$ defined analogously to Definition 2.4.1 before.\n\n\\defn{2.6.3} The {\\it exterior algebra} over a vector space $V$ denoted as $\\bigwedge V$ is the algebra\\footnote{Plainly speaking, it is a linear combination of all possible (finitely many) anti--symmetrized tensor products of vectors in $V$.} generated by the so called {\\it wedge product} $\\wedge$ satisfying\n\\begin{equation*}\n    v_1\\wedge v_2 \\= - v_2\\wedge v_1\n\\end{equation*}\nfor all $v_1,v_2 \\in V$. A subspace of $\\bigwedge V$ consisting of linear combinations of $k$--fold wedge products of vectors in $V$ is denoted $\\bigwedge^k (V)$.\n\n\\subsection{Differential forms}\n\n\\defn{2.6.4} A {\\it $k$-form} $\\omega$ is a $(0,k)$-tensor field $\\omega \\in \\Gamma^\\infty(M,T^{0,k}M)$ such that $\\omega_m \\in \\bigwedge^k(T_mM)$ for all $m\\in M$.\n\n\\rem{2.6.3} The space of $k$-forms is denoted $\\Omega^k(M)$. Note that $\\Omega^0(M) := C^\\infty(M)$ and the only member of $\\Omega^n(M)$ (upto to a scalar multiple), known as the {\\it volume form}, is denoted $\\mu$ or $\\mathrm{d}{V}$. The space $\\Omega^n(M)$ splits into two equivalence classes $\\{ [\\mu_+], [\\mu_-] \\}$ with the relation $\\mu \\sim \\mu'$ defined by $\\mu = \\lambda \\mu'$ such that $\\mu \\in [\\mu_+]$ if $\\lambda > 0$ and $\\mu \\in [\\mu_-]$ if $\\lambda < 0$.\n\n\\rem{2.6.4} On a chart $(U,\\varphi)$ of $M$ with coordinates $\\varphi(m)=(x^1,\\ldots,x^n)$ one can expand a $k$--form $\\omega$ as \n\\begin{equation*}\n    \\omega \\= \\sfrac{1}{k!}\\sum\\limits_{i_1,\\ldots,i_k = 1}^n\\omega_{i_1,\\ldots,i_k}\\,\\mathrm{d}{x}^{i_1}\\wedge\\ldots\\wedge\\mathrm{d}{x}^{i_k}\n\\end{equation*}\nwhere the coefficients $\\omega_{i_1,\\ldots,i_k} \\in C^\\infty(M)$ are totally anti--symmetric in its indices.\n\n\\defn{2.6.5} The {\\it exterior derivative} $\\mathrm{d}$ is the linear map $\\mathrm{d} : \\Omega^k(M) \\xrightarrow{\\sim} \\Omega^{k+1}(M)$ defined, for all $\\omega \\in \\Omega^k(M)$, by\n\\begin{equation*}\n\\mathrm{d}{\\omega} \\= \\sfrac{1}{k!}\\sum\\limits_{i_1,\\ldots,i_k = 1}^n\\mbox{$\\partial$}_i(\\omega_{i_1,\\ldots,i_k})\\,\\mathrm{d}{x}^i\\wedge\\mathrm{d}{x}^{i_1}\\wedge\\ldots\\wedge\\mathrm{d}{x}^{i_k}\\ ,\n\\end{equation*}\nwhere the partial derivative $\\mbox{$\\partial$}_i$ is taken with respect to coordinate $x^i$. It can also be defined in a coordinate independent way, for all $X_1,\\ldots,X_{k+1} \\in \\vecF{M}$, as\n\\begin{equation*}\n \\begin{aligned}\n    \\mathrm{d}{\\omega}(X_1,\\ldots,X_{k+1}) &\\= \\sum\\limits_{i=1}^{k+1} (-1)^{i+1}\\, X_i(\\omega(X_i,\\ldots,\\hat{X}_i,\\ldots,X_{k+1})) \\\\\n    &\\qquad + \\sum\\limits_{1\\leq i < j \\leq n} (-1)^{i+j}\\, \\omega([X_i,X_j],X_1,\\ldots,\\hat{X}_i,\\ldots,\\hat{X}_j,\\ldots,X_{k+1})\\ ,\n \\end{aligned}\n\\end{equation*}\nwhere the circumflex means that the symbol beneath it is omitted.\n\n\\rem{2.6.5} The action of exterior derivative follow {\\it graded Leibniz rule}:\n\\begin{equation*}\n    \\mathrm{d}{(\\alpha\\wedge\\beta)} \\= \\mathrm{d}{\\alpha}\\wedge\\beta + (-1)^p\\,\\alpha\\wedge\\mathrm{d}{\\beta}\n\\end{equation*}\nfor all $\\alpha\\in \\Omega^p(M)$ and $\\beta\\in\\Omega^q(M)$. Moreover, it can easily be shown (using the first definition above) that\n\\begin{equation*}\n    \\mathrm{d}^2\\ \\equiv\\ \\mathrm{d}\\circ\\mathrm{d} \\= 0\\ .\n\\end{equation*}\n\n\\defn{2.6.6} For a map $\\varphi : M \\rightarrow N$ and a $k$-form $\\omega \\in \\Omega^k(N)$ its {\\it pull-back} $\\varphi^*\\omega \\in \\Omega^k(M)$ is defined by\n\\begin{equation*}\n    (\\varphi^*\\omega)_m(X_1,\\ldots,X_k)\\ :=\\ \\omega_{\\varphi(m)}(\\varphi_{*}X_1,\\ldots,\\varphi_{*}X_k)\n\\end{equation*}\nfor all $X_1,\\ldots,X_k \\in \\vecF{M}$. \n\n\\rem{2.6.6} It can be shown that the exterior derivative is {\\it natural} i.e. it is compatible with the pull-back:\n\\begin{equation*}\n    \\varphi^*(\\mathrm{d}{\\omega}) \\= \\mathrm{d}{(\\varphi^*\\omega)}\\ ,\n\\end{equation*}\nfor any $\\omega\\in \\Omega^k(M)$. Furthermore, one can prove that\n\\begin{equation*}\n    \\varphi^*(\\alpha\\wedge\\beta) = \\varphi^*\\alpha\\wedge\\varphi^*\\beta \\quad\\textrm{and}\\quad (\\varphi\\circ\\tilde{\\varphi})^*\\omega = \\tilde{\\varphi}^*(\\varphi^*\\omega)\\ .\n\\end{equation*}\n\n\\defn{2.6.7} The {\\it Lie derivative of $\\omega$ along $X$}, for $\\omega\\in\\Omega^k(M)$ and $X\\in \\vecF{M}$, can be defined analogous to Definition 2.5.5:\n\\begin{equation*}\n    {\\cal L}_X\\omega \\= \\frac{\\mathrm{d}}{\\mathrm{d} t}(\\varphi_t^{-1})^*(\\omega)\\Big|_{t=0}\\ .\n\\end{equation*}\n\n\\rem{2.6.7} There exist a nice formula by \\'{E}lie Cartan for the Lie derivative of differential forms given, for any $X\\in \\vecF{M}$, by\n\\begin{equation*}\n    {\\cal L}_X \\= \\mathrm{d}\\circ\\iota_X + \\iota_X\\circ\\mathrm{d}\\ ,\n\\end{equation*}\nwhere the linear map $\\iota_X: \\Omega^p(M) \\xrightarrow{\\sim} \\Omega^{p-1}(M)$ is the {\\it interior product} defined by  \n\\begin{equation*}\n    (\\iota_X\\omega)(X_1,\\ldots,X_{p-1}) \\= \\omega(X,X_1,\\ldots,X_{p-1})\n\\end{equation*}\nfor $\\omega \\in \\Omega^p(M)$ and $X_1,\\ldots,X_{p-1} \\in \\vecF{M}$.\n\n\\subsection{Metric}\n\n\\defn{2.6.8} A {\\it metric} $g$ is a $(0,2)$-tensor field $g \\in \\Gamma^\\infty(T^{0,2}M)$ where $g_m$ for every $m\\in M$ defines an inner product on the vector space $T_mM$. If this inner product is positive definite then the manifold is called {\\it Riemannian}. If the metric $g$ has an underlying inner product that is indefinite then the manifold is called {\\it pseudo-Riemannian}.\n\n\\rem{2.6.8} A metric $g$ on an $n$-dimensional manifold $M$ in local coordinates can be written\n\\begin{equation*}\n    g \\= \\sum\\limits_{i=1}^n\\, g_{ij}\\,\\mathrm{d}{x}^i\\otimes\\mathrm{d}{x}^j \\quad\\textrm{with}\\quad g_{ij} := g(\\mbox{$\\partial$}_i,\\mbox{$\\partial$}_j)\n\\end{equation*}\nbeing the {\\it metric components} for the basis vector fields $\\mbox{$\\partial$}_i,\\mbox{$\\partial$}_j \\in \\vecF{M}$. Often times in physics literature the tensor product sign is ignored. The fact that $g$ is nondegenerate means that the matrix $g_{ij}$ can be inverted to yield another matrix $g^{-1}$ with components $g^{-1}_{ij} =: g^{ij}$ that in turn defines a $(2,0)$-tensor field called the {\\it induced metric}. These can be used to raise or lower indices of any tensor component e.g.\n\\begin{equation*}\n    T_{i'}^{\\ j'k'} := g_{ii'}g^{jj'}g^{kk'}T^i_{\\ jk}\\quad \\forall \\quad T \\in \\Gamma^\\infty(M,T^{1,2}M)\\ .\n\\end{equation*}\n\n\\ex{2.6.1} A standard example of a Reimannian manifold is $\\mathds R^n$ with metric\n\\begin{equation*}\n    g \\= (\\mathrm{d}{x}^1)^2 + \\ldots + (\\mathrm{d}{x}^n)^2\\ .\n\\end{equation*}\nA prominent example of a psuedo-Riemannian manifold is {\\it Minkowski space} $\\mathds R^{1,n}$ with metric\n\\begin{equation*}\n    g \\= -(\\mathrm{d}{x}^1)^2 + (\\mathrm{d}{x}^2)^2 + \\ldots + (\\mathrm{d}{x}^n)^2\\ .\n\\end{equation*}\n\n\\defn{2.6.9} The {\\it signature} of a pseudo-Riemannian manifold $(M,g)$ of dimension $n$ is a count of the number of positive and negative eigenvalues of the matrix $g_{ij}$ and is denoted as an $n$-tuple of $+$ and $-$. A $4$-dimensional Lorentzian manifold has one of the signature different from the rest three.\n\n\\ex{2.6.2} The signature of the Minkowski space $\\mathds R^{1,3}$ as presented above is $(-,+,+,+)$, and is thus a Lorentzian manifold.\n\n\\defn{2.6.10} Two (pseudo-)Riemannian manifolds $(M,g_M)$ and $(N,g_N)$ are called {\\it conformal} if their metrices are related by some smooth function $\\Omega^2 \\in C^\\infty(N)$: \n\\begin{equation*}\n    g_M \\= \\Omega^2\\,g_N\\ .\n\\end{equation*}\n\n\\defn{2.6.11} A locally {\\it orthonormal basis} of $1$-forms $\\{ e^i \\}$ (aka coframe) with $i=1,\\ldots,n$ for an $n$-dimensional Riemannian manifold $M$ satisfy, for all $e^i,e^j \\in \\Omega^1(M)$,\n\\begin{equation*}\n    g(e^i,e^j) \\= g^{ij} = \\delta^{ij}\\ .\n\\end{equation*}\nSimilarly, one defines locally orthonormal basis of vector fields (aka frame) $\\{ X_1,\\ldots,X_n \\}$ by demanding that they satisfy\n\\begin{equation*}\n    g_{ij} \\= g(X_i,X_j) \\= \\delta_{ij}\n\\end{equation*}\nfor all $X_i,X_j \\in \\vecF{M}$.\n\n\\subsection{Maurer--Cartan form}\n\\defn{2.6.12} A $k$-form $\\omega \\in \\Omega^k(G)$ on a Lie group $G$ is said to {\\it left-invariant} if, for all $g\\in G$,\n\\begin{equation*}\n    l_g^*\\omega \\= \\omega\\ ,\\quad \\mathrm{i.e.}\\ ,\\quad l_g^*(\\omega_{g'}) \\= \\omega_{g^{-1}g'}\\ \\forall\\ g' \\in G\\ .\n\\end{equation*}\nThe set of all left-invariant one-forms on $G$ is denoted $L^*(G)$. \n\n\\rem{2.6.9} From Remark 2.6.6 we see that if $\\omega$ is left-invariant then so is its exterior derivative:\n\\begin{equation*}\n    l_g^*(\\mathrm{d}{\\omega}) \\= \\mathrm{d}{l_g^*\\omega}\\ .\n\\end{equation*}\n\n\\rem{2.6.10} Similar to $j$ of Theorem 2.5.1, there exist an isomorphism between $T^*_eG$ and $L^*(G)$:\n\\begin{equation*}\n    \\tilde{j}:\\ T^*_eG \\rightarrow L^*(G)\\ ,\\quad \\omega \\mapsto \\tilde{j}(\\omega) \\equiv \\lambda^\\omega\\ ,\n\\end{equation*}\nwhere the left-invariant one-forms $\\lambda^\\omega$ is defined as\n\\begin{equation*}\n    \\lambda^\\omega_g\\ :=\\ l_{g-1}^*(\\omega) \\in T^*_gG\\ .\n\\end{equation*} \nNotice that these $1$-forms are dual to the left-invariant vector fields $L^A$ for $A \\in T_eG$ i.e.\n\\begin{equation*}\n    \\< \\lambda^\\omega, L^A \\>_g \\= \\< \\omega, A\\>\n\\end{equation*}\nfor all $g \\in G$.\n\n\\rem{2.6.11} For a basis of $\\{ E_1,\\ldots,E_n \\}$ of $L(G)$ with dim$G=n$ (see Remark 2.5.2) we can define the dual basis $\\{ \\omega^1,\\ldots,\\omega^n \\}$ of $L^*(G)$ by\n\\begin{equation*}\n    \\< \\omega^i, E_j \\> \\= \\delta^i_j\\ ,\n\\end{equation*}\nwhich satisfy the Maurer--Cartan equation\n\\begin{equation*}\n    \\mathrm{d}{\\omega}^i + \\frac{1}{2} f_{jk}^{\\ \\ i}\\,\\omega^j\\wedge\\omega^k\\ .\n\\end{equation*}\n\n\\defn{2.6.13} The Maurer--Cartan $1$-form $\\Omega_l$ is the $L(G)$-valued $1$-form on $G$ that for any $A \\in T_gG$ gives a left-invariant vector field as follows\n\\begin{equation*}\n    \\< \\Omega_l, A \\>_{g'}\\ :=\\ l_{g'*}(l_{g^{-1}*}A) \n\\end{equation*}\nfor any $g' \\in G$.\n\n\\rem{2.6.12} It is more useful to consider the Maurer--Cartan $1$-form to be valued in $T_eG \\cong L(G)$, which yields a nice result:\n\\begin{equation*}\n    \\< \\Omega_l, L^A \\> \\= A\\ .\n\\end{equation*}\nIt can be shown that the Maurer-Cartan $1$-form takes the following explicit form for the matrix Lie groups that we consider here: \n\\begin{equation*}\n    \\Omega_l^{ij} \\= \\sum\\limits_{k=1}^n (g^{-1})^{ik}\\, \\mathrm{d}{g}^{kj}\\ ,\n\\end{equation*}\nwhere $g^{ij}$ are the coordinates on the matrix Lie group $G$ in a given chart. \n\n\\vspace{12pt}\n\\section{Integration} \n\\vspace{2pt}\n\\defn{2.7.1} A manifold $M$ is called {\\it orientable} if it is equipped with a nowhere vanishing volume form $\\mu$. An {\\it orientation} of $M$ refers to a choice of one of the the two equivalence classes of Remark 2.6.3; this is usually chosen to be the positive one.\n\n\\ex{2.7.1} The standard volume form on $\\mathds R^n$, which is an oriented manifold, is given by \n\\begin{equation*}\n    \\mu \\= \\mathrm{d}{x}^1\\wedge\\ldots\\wedge\\mathrm{d}{x}^n\\ . \n\\end{equation*}\nM\\\"{o}bius strip provides a classic example of a nonorientable manifold.\n\n\\defn{2.7.2} {\\it Integration} of a compactly supported volume form $\\mu$ (i.e. it vanishes outside of a compact subset of $M$) of an oriented manifold $M$ that is covered with charts $(U_1,\\phi_1),\\ldots,(U_N,\\phi_N)$ can be defined, using a partition of unity\\footnote{It is a set of smooth functions $f_i \\in C^\\infty(M)$ that vanish outside of $U_i$, lies within $[0,1]$, and satisfies the condition: $\\sum_{i=1}^Nf_i(m) = 1$.} $\\{ f_i \\}$, as \n\\begin{equation*}\n    \\int_M \\mu \\= \\sum\\limits_{i=1}^N \\int_{\\phi(U_i)} (\\phi_i^{-1})^*(f_i\\mu)\\ .\n\\end{equation*}\n\n\\rem{2.7.1} It can be shown that this definition of integration is independent of the choice of a chart. \n\n\\thm{2.7.1} For an oriented manifold $M$ with boundary $\\mbox{$\\partial$} M$ ( that inherits an induced orientation from $M$) the integral of a compactly supported $(n{-}1)$-form $\\omega \\in \\Omega^{n{-}1}(M)$ is related to an integral of its exterior derivative $\\mathrm{d}{\\omega}$:\n\\begin{equation*}\n    \\int_M \\mathrm{d}{\\omega} \\= \\int_{\\mbox{$\\partial$} M} \\omega\\ .\n\\end{equation*}\n\n\\rem{2.7.2} This is the famous {\\it Stockes' theorem}. Notice that if the manifold $M$ has no boundary then the above integral vanishes.\n\n\\defn{2.7.3} For an $n$-dimensional, oriented, Riemannian manifold $(M,g)$ the {\\it Riemannian volume form} is given by\n\\begin{equation*}\n    \\mu_g \\= \\sqrt{|g|}\\,\\mathrm{d}{x}^1\\wedge\\cdots\\wedge\\mathrm{d}{x}^n\\ ,\n\\end{equation*}\nwhere $|g| := |\\mathrm{det}(g_{ij})|$ for metric compoents $g_{ij}$.\n\n\\rem{2.7.3} Integration on Riemannian manifolds are performed using volume form $\\mu_g$. \n  \n\\vspace{12pt}  \n\\section{Hodge duality}\n\\vspace{2pt}\n\\rem{2.8.1} On an $n$-dimensional pseudo-Riemannian manifold $(M,g)$ one can induce an inner product on $\\Omega^k(M)$, generated by the $k$-forms $\\{ \\mathrm{d}{x}^{i_1}\\wedge\\cdots\\wedge\\mathrm{d}{x}^{i_k} \\}$ with $i_1,\\ldots,i_k \\in \\{ 1,\\ldots,n \\}$, that is given by\n\\begin{equation}\n    \\Big\\< \\mathrm{d}{x}^{i_1}\\wedge\\cdots\\wedge\\mathrm{d}{x}^{i_k}, \\mathrm{d}{x}^{j_1}\\wedge\\cdots\\wedge\\mathrm{d}{x}^{j_k} \\Big\\> \\= g^{i_1j_1}\\cdots g^{i_kj_k}\\ ,\n\\end{equation}\nwhere $g^{ij} := g(\\mathrm{d}{x}^i,\\mathrm{d}{x}^j)$.\n\n\\defn{2.8.1} The {\\it Hodge star operator} $\\ast : \\Omega^k(M) \\xrightarrow{\\sim} \\Omega^{n-k}(M)$ is a linear map that is defined on an oriented $n$-dimensional pseudo-Riemannian manifold $(M,g)$ with volume form $\\mu$ given by\n\\begin{equation}\n    \\alpha\\wedge\\ast\\beta \\= \\< \\alpha,\\beta \\>\\, \\mu\n\\end{equation}\nfor all $\\alpha,\\beta \\in \\Omega^k(M)$. Here $\\ast\\beta$ is called the {\\it Hodge dual} of $\\beta$.\n\n\\rem{2.8.2} For an $n$-dimensional oriented pseudo-Riemannian manifold $(M,g)$ with signature $(-,\\overset{s-2}{\\ldots},-,+,\\overset{n-s-2}{\\ldots},+)$ the following condition holds true on $\\Omega^p(M)$:\n\\begin{equation*}\n    \\ast^2 \\= (-1)^{p(n-p)+s}\\ .\n\\end{equation*}\n\n\\ex{2.8.1} For the Minkowski space $\\mathds R^{1,3}$ with signature $(-,+,+,+)$, coordinates $ ~ \\\\(x^0, x^1, x^2, x^3) =: (t,x,y,z)$ and volume form $\\mathrm{d}{V} = \\mathrm{d}{t}\\wedge\\mathrm{d}{x}\\wedge\\mathrm{d}{y}\\wedge\\mathrm{d}{z}$ we have the following results:\n \\begin{align*}\n    &\\ast\\mathrm{d}{z} \\= -\\mathrm{d}{t}\\wedge\\mathrm{d}{x}\\wedge\\mathrm{d}{y}\\ , &\n    &\\ast\\,\\mathrm{d}{t}\\wedge\\mathrm{d}{x} \\= -\\mathrm{d}{y}\\wedge\\mathrm{d}{z}\\ , & &\\ast\\,\\mathrm{d}{t}\\wedge\\mathrm{d}{y} \\= \\mathrm{d}{x}\\wedge\\mathrm{d}{z}\\ ,\\\\\n    &\\ast\\mathrm{d}{t}\\wedge\\mathrm{d}{z} \\= -\\mathrm{d}{x}\\wedge\\mathrm{d}{y}\\ , & &\\ast\\,\\mathrm{d}{x}\\wedge\\mathrm{d}{y} \\= -\\mathrm{d}{t}\\wedge\\mathrm{d}{z}\\ , & &\\ast\\,\\mathrm{d}{x}\\wedge\\mathrm{d}{z} \\= -\\mathrm{d}{t}\\wedge\\mathrm{d}{y}\\ ,\\\\\n    &\\ast\\mathrm{d}{y}\\wedge\\mathrm{d}{z} \\= \\mathrm{d}{t}\\wedge\\mathrm{d}{x}\\ , & &\\ast\\,\\mathrm{d}{t}\\wedge\\mathrm{d}{x}\\wedge\\mathrm{d}{y} \\= -\\mathrm{d}{z}\\ , & &\\ast\\,\\mathrm{d}{t}\\wedge\\mathrm{d}{x}\\wedge\\mathrm{d}{z} \\= \\mathrm{d}{y}\\ ,\\\\\n    &\\ast\\mathrm{d}{t}\\wedge\\mathrm{d}{y}\\wedge\\mathrm{d}{z} \\= -\\mathrm{d}{x}\\ , & &\\ast\\,\\mathrm{d}{x}\\wedge\\mathrm{d}{y}\\wedge\\mathrm{d}{z} \\= -\\mathrm{d}{t}\\ , & &\\ast\\,\\mathrm{d}{t}\\wedge\\mathrm{d}{x}\\wedge\\mathrm{d}{y}\\wedge\\mathrm{d}{z} \\= -1\\ .\n \\end{align*}\n\n\\vspace{12pt}\n\\section{Representations}\n\\vspace{2pt}\n\\defn{2.9.1} A {\\it representation of a Lie group} $G$ is a Lie group homomorphism\n\\begin{equation*}\n    \\Pi :\\ G \\rightarrow \\mathrm{GL}(V)\n\\end{equation*}\nfor a finite dimensional vector space $V$. \n\n\\defn{2.9.2} A {\\it representation of a Lie algebra} $\\mathfrak{g}$, for a finite dimensional vector space $V$, is a Lie algebra homomorphism\n\\begin{equation*}\n    \\pi :\\ \\mathfrak{g} \\rightarrow \\mathrm{gl}(V)\\ ,\n\\end{equation*}\nwhere $\\mathrm{gl}(V) := \\mathrm{End}(V)$\\footnote{The endomorphism of $V$ denoted $\\mathrm{End}(V)$ is the space of linear maps $V\\xrightarrow{\\sim} V$.}.\n\n\\ex{2.9.1} The {\\it standard representation} of a Lie group $G \\ni g$ is $\\Pi(g) = g$ and of a Lie algbra $\\mathfrak{g} \\ni X$ is $\\pi(X) = X$. \n\n\\ex{2.9.2} For matrix Lie groups $G$ with Lie algebra $\\mathfrak{g}$, its {\\it adjoint representation} is the following homomorphism\n\\begin{equation*}\n    \\mathrm{Ad}:\\ G \\rightarrow \\mathrm{GL}(\\mathfrak{g})\\ ,\\quad g \\mapsto \\mathrm{Ad}_g\\ ,\n\\end{equation*}\nwhere the {\\it adjoint map} $\\mathrm{Ad}_A$ is defined by\n\\begin{equation*}\n    \\mathrm{Ad}_g:\\ \\mathfrak{g} \\xrightarrow{\\sim} \\mathfrak{g}\\ ,\\quad X \\mapsto \\mathrm{Ad}_g(X)\\ :=\\ gXg^{-1}\\ .\n\\end{equation*}\n\n\\rem{2.9.1} It can be shown that the map $\\mathrm{Ad}$ induces (see Theorem 2.5.2) the following lie algebra homomorphism\n\\begin{equation*}\n    \\mathrm{Ad}_* \\equiv \\mathrm{ad}:\\ \\mathfrak{g} \\rightarrow \\mathrm{gl}(\\mathfrak{g})\\ ,\\quad X \\mapsto \\mathrm{ad}_X\\ ,\n\\end{equation*}\nwhere the action of the map $\\mathrm{ad}_X$ can be shown to be the following \n\\begin{equation*}\n    \\mathrm{ad}_X:\\ \\mathfrak{g}\\rightarrow \\mathfrak{g}\\ , \\quad Y\\mapsto \\mathrm{ad}_X(Y)\\ :=\\ [X,Y]\\ .\n\\end{equation*}\nThis is known as the {\\it adjoint representation} of a finite-dimensional Lie algebra $\\mathfrak{g}$. \n\n\\vspace{12pt}\n\\section{Maxwell equations}\n\\vspace{2pt}\n\\rem{2.10.1} For the nice\\footnote{Here it means a connected and simply connected manifold.} manifolds that we are interested in this thesis the Maxwell theory boils down to a choice of the gauge potential $A$, which is a $1$-form on the manifold.\n\n\\defn{2.10.1} For Minkowski space $\\mathds R^{1,3}$ we define the gauge potential $A$ by\n\\begin{equation*}\n    A \\= A_0\\,\\mathrm{d}{t} + A_i\\,\\mathrm{d}{x}^i\\ ,\n\\end{equation*}\nand the field strength $F$ by\n\\begin{equation*}\n    F\\ :=\\ \\mathrm{d}{A} \\= E_i\\,\\mathrm{d}{x}^i\\wedge\\mathrm{d}{t} + \\frac{1}{2} B_i\\, \\varepsilon^{i}_{\\ jk}\\,\\mathrm{d}{x}^j\\wedge\\mathrm{d}{x}^k\n\\end{equation*}\nwith electric field ${\\bf E}$ and magnetic field ${\\bf B}$.\n\n\n\\rem{2.10.2} The source free Maxwell equations viz.\n\\begin{equation*}\n    \\nabla\\cdot {\\bf B} \\= 0 \\quad\\textrm{and}\\quad \\nabla\\times{\\bf E} + \\frac{\\mbox{$\\partial$} {\\bf B}}{\\mbox{$\\partial$} t} \\= 0\n\\end{equation*}\nare given by \n\\begin{equation*}\n    \\mathrm{d}{F} \\= 0\\ .\n\\end{equation*}\nNotice that, due to Remark 2.6.5, the above condition is trivially satisfied here.\n\n\\rem{2.10.3} The other two Maxwell equations with source viz.\n\\begin{equation*}\n    \\nabla\\cdot{\\bf E} \\= \\rho \\quad\\textrm{and}\\quad \\nabla\\times{\\bf B} - \\frac{\\mbox{$\\partial$} {\\bf E}}{\\mbox{$\\partial$} t} \\= {\\bf J}\n\\end{equation*}\nwith {\\it charge density} $\\rho$ and {\\it current density} ${\\bf J}$ is given by \n\\begin{equation*}\n    \\delta F \\= j \\quad\\textrm{with}\\quad j \\= -\\rho\\,\\mathrm{d}{t} + J_i\\,\\mathrm{d}{x}^i \\in \\Omega^1(\\mathds R^{1,3})\\ , \n\\end{equation*}\nwhere the {\\it codifferential} $\\delta := *\\mathrm{d}*$.\n\n\\rem{2.10.4} An important feature of the Maxwell theory is that it possesses gauge symmetry i.e. the transformation \n\\begin{equation*}\n    A \\rightarrow A + \\mathrm{d}{f}\n\\end{equation*}\nfor all $f\\in C^\\infty(\\mathds R^{1,3})$ leaves the field strength $F$ invariant. There are many ways to fix this redundancy by the so called gauge-fixing. On Minkowski space we can always work in the so called ``temporal gauge\" where $A_0 = 0$ (see Chapter 6 of \\cite{Baez&Muniain}).\n\n\\rem{2.10.5} Noticing that $\\delta^2 = \\pm *\\mathrm{d}^2* = 0$ and applying $\\delta$ on the Maxwell equations with source we arrive at the following {\\it continuity equation}:\n\\begin{equation*}\n    0 \\= \\delta^2 F \\= \\delta j \\quad \\implies \\quad \\frac{\\mbox{$\\partial$} \\rho}{\\mbox{$\\partial$} t} + \\nabla\\cdot{\\bf J} \\= 0\\ .\n\\end{equation*}\n\n\n\n\n\n\n\\chapter{Gauge theory}\n\\label{Chapter3}\n\n\\justifying\n\nAll the four known forces of nature viz. gravity, electromagnetism, weak and strong nuclear forces can be described (at least classically) in terms of a gauge theory. The study of modern gauge theory requires the notion of principal bundles and subsequent structures on it. While it is possible to study a lot of physics, including Yang--Mills theory, with just vector bundles alone, e.g. as in \\cite{Baez&Muniain}, a lot of deep physics arising from the underlying topology of the base manifold can not be fully appreciated without making reference to the principle bundle. We review in this chapter the construction of gauge theory from principal bundles without bothering about proofs of the statements, which can be found in \\cite{Isham}. For a quick review of Yang--Mills theory one may refer to \\cite{Bleecker} or a nice review article by Daniel and Viallet \\cite{DV80}.\n\n\\vspace{12pt}\n\\section{Principal $G$-bundles}\n\\vspace{2pt}\n\\defn{3.1.1} A bundle $(P,M,\\pi)$ is called a {\\it principle $G$-bundle} and denoted diagrammatically as \n\\begin{tikzcd}\n   G \\arrow[r, \"r_g\"] &P \\arrow[d, \"\\pi\"] \\\\\n                               &M\n\\end{tikzcd}\nif there exist a free right action of the Lie group $G$ on $P$ and if there exist a bundle isomorphism between $(P,M,\\pi)$ and $(P,P\/G,\\pi_0)$ with the canonical projection $\\pi_0$ on the coset space $P\/G$, meaning that the diagram\n\\begin{tikzcd}\n   P \\arrow[d, \"\\pi\"] \\arrow[r, \"\\varphi\"] &P \\arrow[d, \"\\pi_0\"] \\\\\n   M \\arrow[r, \"\\psi\"]  &P\/G\n\\end{tikzcd}\ncommutes.\n\n\\rem{3.1.1} Notice that the structure group in this case is $G$. Furthermore, the fibre $\\pi^{-1}(\\{m\\})$ for any $m \\in M$ is diffeomorphic to $G$, but it does not have a canonical group structure. Another way to put this would be to say that the Manifold $M$ has a $G$ fibre attached to all its points except that the identity element is forgotten. Also, notice here that the projection map is insensitive to the $G$-action.\n\n\\ex{3.1.1} The simplest example of a principle bundle is $(G\\times M, M,\\pi)$ with the right action given by $(x,g_0)g := (x,gg_0)$.\n\n\\ex{3.1.2} An important example of a principle $G$-bundle is the so called {\\it frame bundle} $LM$ over an $n$-dimensional manifold $M$, which is a collection of basis-frames:\n\\begin{equation*}\n    L_mM\\ :=\\ \\{ (b_1,\\ldots,b_n) ~|~ \\<b_1,\\ldots,b_n\\> = T_mM \\} \\cong GL(n,\\mathds R)\n\\end{equation*}\ncorresponding to the tangent bundle $TM$ attached at each point $m$ of the manifold i.e.\n\\begin{equation*}\n    LM\\ :=\\ \\bigcup\\limits_{m\\in M} L_mM\\ .\n\\end{equation*}\nThe free right action of any $g \\in GL(n,\\mathds R)$ on a given $(b_1,\\ldots,b_n) \\in L_mM$ is defined as\n\\begin{equation*}\n    (b_1,\\ldots,b_n)g\\ :=\\ (b_i\\,g^i_{\\ 1},\\ldots,b_i\\,g^i_{\\ n})\\ .\n\\end{equation*}\n\n\\ex{3.1.3} Another important example of a principle bundle is $(G,G\/H,\\pi)$ where $H$ acts freely from right on $G$ via group multiplication resulting into orbits of cosets $G\/H$. A famous example of this kind is the {Hopf bundle} represented diagrammatically as follows: \n\\begin{tikzcd}\n   U(1) \\arrow[r] &SU(2) \\arrow[d] \\\\ \n                  &SU(2)\/U(1)\n\\end{tikzcd} \n$\\equiv$\n\\begin{tikzcd}\n   S^1 \\arrow[r] &S^3 \\arrow[d] \\\\\n                 &S^2\n\\end{tikzcd}\n\n\\defn{3.1.2} A {\\it principle morphism} between a pair of bundles $(P,M,\\pi)$ and $(P',M',\\pi')$ is a bundle morphism $(\\varphi,\\psi)$ that satisfies \n\\begin{equation*}\n    \\varphi(pg) \\= \\varphi(p)g \\quad\\forall\\quad p \\in P \\quad\\textrm{and}\\quad g\\in G\\ .\n\\end{equation*}\n\n\\rem{3.1.2} The notion of trivialization, both local $(U,\\varphi)$ as well as global $(G\\times M,M,\\pi_1)$, follows similar to the general theory of fibre bundles that we saw before, albeit with this extra condition of principle morphism. In a similar way, the idea of transition functions $g_{ij}$ between any two such overlapping local trivializations $(U_i,\\varphi_i)$ and $(U_j,\\varphi_j)$ and the corresponding structure group carries over. One can also define smooth sections on a principle bundle $P$ just like before.\n\n\\rem{3.1.3} The set of all principal morphisms between a bundle $P$ to itself forms a group $Aut(P)$ called the {\\it automorphism group} of the principle bundle $P$. In the case of a trivial bundle $P=G\\times M$ we have that $Aut(P)\\cong C^\\infty (M,G)$, where the latter is the group of {\\it gauge transformations}. \n\n\\thm{3.1.1} A principle $G$-bundle $(P,M,\\pi)$ is trivial if and only if it possesses a smooth section $\\sigma: M \\rightarrow P$.\n\n\\rem{3.1.4} An illustrative counter-example of the above theorem is the fact that the frame bundle $LS^2$ over the $2$-sphere is not trivial because there does not exist nowhere vanishing smooth vector fields, and hence a basis, on $TS^2$ (look at the north or south poles). This result is famously summarized as ``sphere can not be combed\".\n\n\\rem{3.1.5} The topological properties, such as twisting, of the base manifold $M$ is intrinsically linked with that of the principal bundle $P$ and also carries over to the below defined associated bundles.\n\n\\vspace{12pt}\n\\section{Associated bundles}\n\\vspace{2pt}\n\n\\defn{3.2.1} The $G$-product of two spaces $X$ and $Y$ admitting right action of $G$, denoted $X\\times_G Y$, is the space of orbits under this action on the Cartesian product $X\\times Y$. In other words it is defined via the following equivalence relation\n\\begin{equation*}\n    (x,y) \\sim (x',y')\\quad \\mathrm{iff}\\quad \\exists\\ g \\in G :\\ x' \\= xg \\quad\\textrm{and}\\quad y' \\= yg\\ ,\n\\end{equation*}\nwhere the equivalence class is denoted as $[x,y]$.\n\n\\defn{3.2.2} For a given principle $G$-bundle $(P,M,\\pi)$ and a manifold $F$ one defines the {\\it associated bundle} $P_F$ by\\footnote{Notice how we have employed left $G$-action to define its right action (see Remark 2.3.7).}\n\\begin{equation*}\n    P_F\\ :=\\ P\\times_G F \\quad \\mathrm{where} \\quad (p,v)g\\ :=\\ (pg,g^{-1}v)\n\\end{equation*}\nand the projection\n\\begin{equation*}\n    \\pi_F :\\ P_F \\to M\\ , \\quad \\pi_F([p,v])\\ :=\\ \\pi(p)\\ .\n\\end{equation*}\n\n\\rem{3.2.1} It can be shown that the associated bundle $(P_F,M,\\pi_F)$ has the structure of a fibre bundle with typical fibre $F$. \n\n\\ex{3.2.1} An important example of a fibre associated with the frame bundle $LM$ with Lie group $GL(n,\\mathds R)$ is the tensor bundle $T^{p,q}M$ with fibres $F = (\\mathds R^n)^{\\times p}\\times (\\mathds R^{n*})^{\\times q}$ where the left action of a given $g \\in GL(n,R)$ on some $v \\in F$ is given by the following representation $\\rho$\n\\begin{equation*}\n    (\\rho(g)v)^{i_1,\\ldots,i_p}_{\\qquad j_1,\\ldots,j_q}\\ :=\\ v^{i_1',\\ldots,i_p'}_{\\qquad j_1',\\ldots,j_q'}\\,(g)^{i_1}_{\\ i_1'}\\ldots(g)^{i_p}_{\\ i_p'}\\,(g^{-1})^{j_1'}_{\\ j_1}\\ldots(g^{-1})^{j_q'}_{\\ j_q}\\ .\n\\end{equation*}\nAnother very useful generalization of this is the so called {\\it tensor of density $\\omega$} that admits a left-action via the following representation\n\\begin{equation*}\n    (\\rho(g)v)^{i_1,\\ldots,i_p}_{\\qquad j_1,\\ldots,j_q}\\ :=\\ (\\mathrm{det} g)^\\omega\\,v^{i_1',\\ldots,i_p'}_{\\qquad j_1',\\ldots,j_q'}\\,(g)^{i_1}_{\\ i_1'}\\ldots(g)^{i_p}_{\\ i_p'}\\,(g^{-1})^{j_1'}_{\\ j_1}\\ldots(g^{-1})^{j_q'}_{\\ j_q}\\ .\n\\end{equation*}\n\n\\defn{3.2.3} Given a principle morphism $(\\varphi,\\psi)$ between principal $G$-bundles $(P,M,\\pi)$ and $(P',M',\\pi')$ one defines an {\\it associated bundle morphism} between the associated bundles $P_F\\times_G F$ and $P'\\times_G F$ by\n\\begin{equation*}\n    \\varphi_F([p,v])\\ :=\\ [\\varphi(p),v]\\ ,\n\\end{equation*}\nwhich is well-defined since\n\\begin{equation*}\n    \\varphi_F([pg,g^{-1}v]) \\= [\\varphi(pg),g^{-1}v] \\= [\\varphi(p)g,g^{-1}v] \\= [p,v]\\ .\n\\end{equation*}\n\n\\rem{3.2.2} An associated bundle $P_F$ is called {\\it trivial} if the underlying principal bundle $P$ is trivial. An important point to note here is that a trivial associated bundle is a trivial fibre bundle, but the converse is not true.\n\n\\defn{3.2.4} Let $H \\subset G$ be a closed subgroup of $G$ while $P$, resp., $P'$ are principal $G$-bundle, resp., principal $H$-bundle defined over the same base space. If there exist a principal morphism $\\varphi$ with respect to $H$ i.e. \n\\begin{equation*}\n    \\varphi(ph) \\= \\varphi(p)h\n\\end{equation*}\nfor all $h\\in H$ then $P$ is called a $G$-extension of $H$ while $P'$ is called a $H$-restriction of $P$.\n\n\\rem{3.2.3} While there always exists an extension $P$ of a given $P'$ as defined above, the converse is not always true. This has important ramifications both in Riemannian geometry as well as in Yang--Mills theory; for the latter this is related to the important question of the spontaneous breakdown of the internal symmetry group from $G$ down to $H$.\n\n\\thm{3.2.1} A principle $G$-bundle $(P,M,\\pi)$ can be restricted to a closed subgroup $H\\subset G$ iff the bundle $(P\/H,M,\\pi_0)$ admits a smooth section.\n\n\\rem{3.2.4} An important application of the above theorem is that in (pseudo-)Riemannian geometry one can always have a Riemannian metric defined on any $n$-diemensional manifold $M$ as $LM\/SO(n)$ for the closed subgroup $SO(n)\\subset GL(n,\\mathds R)$ always admits a smooth section; the same is not always true for a pseudo-Riemannian metric as, e.g., $LM\/SO(1,n{-}1)$ for the closed subgroup $SO(1,n)\\subset GL(n,\\mathds R)$ does not always admits a smooth section because of some topological restrictions.\n\n\\thm{3.2.2} There exists a one-to-one $G$-equivariant correspondence between sections of an associated bundle $(P_F,M,\\pi_F)$ and map $\\phi: P \\to F$ satisfying\n\\begin{equation*}\n    \\phi(pg) \\= g^{-1}\\phi(p) \\quad\\forall\\quad p \\in P \\quad\\textrm{and}\\quad g \\in G\n\\end{equation*}\nwhere the section $\\sigma_\\phi$ corresponding to $\\phi$ arises as\n\\begin{equation*}\n    \\sigma_\\phi(x)\\ :=\\ [p,\\phi(p)] \\quad\\textrm{for}\\quad p\\in \\pi^{-1}(x)\\ .\n\\end{equation*}\n\n\\defn{3.2.5} Given two local trivializing sections\\footnote{This refers to the existence of a local trivialization $(U_i,\\varphi_i)$ such that $\\varphi_i^{-1}(x,g) := \\sigma_i(x)g$.} $\\sigma_i: U_i\\subset M \\to P$ and $\\sigma_j: U_j\\subset M \\to P$ on a given principal $G$-bundle $P$ with $U_i\\cap U_j \\neq 0$ there exists some local {\\it gauge functions} $\\Lambda_{ij}: U_i\\cap U_j \\to G$ such that \n\\begin{equation*}\n    \\sigma_j(x) \\= \\sigma_i(x)\\Lambda_{ij}(x) \\quad\\forall\\quad x\\in U_i\\cap U_j\\ .\n\\end{equation*}\nOne defines {\\it local representatives} $s_i: U_i \\to F$ for a section $s$ of $P_F$ corresponding to these local sections $\\sigma_i$ by \n\\begin{equation*}\n    s_i(x)\\ :=\\ \\phi_{s_i}(\\sigma(x))\\ .\n\\end{equation*}\n\n\\rem{3.2.5} The local representatives also satisfy the same gauge transformation rule as above i.e.\n\\begin{equation*}\n    s_j(x) \\= s_i(x)\\Lambda_{ij}(x)\\ .\n\\end{equation*}\nIt turns out that these gauge functions $\\Lambda_{ij}$ are nothing but the transitions functions $g_{ij}$ for the local trivializations arising from $\\sigma_i$ and $\\sigma_j$. For this thesis, we will identify the gauge group with the structure group.\n\n\\vspace{12pt}\n\\section{Connections}\n\\vspace{2pt}\n\n\\rem{3.3.1} Let $(P,M,\\pi)$ be a principal $G$-bundle. Then there exists a Lie algebra homomorphism between $L(G)\\cong T_eG$ and $\\Gamma^\\infty(TP)$ given by\n\\begin{equation*}\n    i :\\ T_eG \\to \\Gamma^\\infty(TP)\\ ,\\quad A \\to X^A\\ ,\n\\end{equation*}\nwhere the induced vector-field $X^A$ arises from the right action of $G$ on $P$ as follows\n\\begin{equation*}\n    X^A[f]\\ :=\\ f'(p\\,\\mathrm{exp}(tA))(t{=}0)\\ .\n\\end{equation*}\n\n\n\\defn{3.3.1} For a given principal $G$-bundle $(P,M,\\pi)$ one defines the {\\it vertical subspace} at $p\\in P$, denoted by $V_pP$, as follows\n\\begin{equation*}\n    V_pP\\ :=\\ \\{X \\in T_pP ~|~ \\pi_*(X) \\= 0  \\}\n\\end{equation*}\nThe {\\it horizontal subspace} $H_pP$ arises as the orthogonal complement of $V_pP$ in the tangent space $T_pP$:\n\\begin{equation*}\n    T_pP \\= V_pP \\oplus H_pP\\ .\n\\end{equation*}\n\n\\rem{3.3.2} It can be shown that\n\\begin{equation*}\n    X^A_p \\in V_pP \\quad\\forall\\quad p\\in P\\ .\n\\end{equation*}\nThis means that the above map $i$ induces an isomorphism $i_p$ between $T_eG$ and $V_pP$.\n\n\n\\defn{3.3.2} A {\\it connection} on a principle $G$-bundle $(P,M,\\pi)$ is a smooth assignment of $H_pP$ to each point $p\\in P$ such that\n\\begin{enumerate}[(a)]\n    \\item $T_pP \\= V_pP \\oplus H_pP$\n    \\item $(r_g)_*(H_pP) \\= H_{pg}P$\n    \\item every $X_p \\in T_pP$ has a unique decomposition according to (a) as follows\n    \\begin{equation*}\n        X_p \\= ver(X_p) + hor(X_p)\\ ,\n    \\end{equation*}\n    where $ver(X_p) \\in V_pP$ and $hor(X_p) \\in H_pP$.\n\\end{enumerate}\n\n\\rem{3.3.3} A more technically convenient way to deal with connections is by associating them with a Lie-algebra valued one-form $\\omega$ in the following way\n\\begin{equation*}\n    \\omega_p(X) \\= i_p^{-1}(ver(X))\\ ,\n\\end{equation*}\nwhich, in turn, imposes following conditions on $\\omega$:\n\\begin{enumerate}[(i)]\n    \\item $\\omega_p(X^A) = A$ for all $p\\in P$ and $A\\in L(G)$\n    \\item $(r_g)^*\\omega = \\mathrm{Ad}_{g^{-1}}\\omega$, i.e., $(r_g^*\\omega)_p(X) = \\mathrm{Ad}_{g{-1}}(\\omega_p(X))$ for all $X\\in T_pP$\n    \\item $X\\in H_pP$ iff $\\omega_p(X) = 0$.\n\\end{enumerate}\n\n\\defn{3.3.3} For a local trivializing section $\\sigma: U\\subset M \\to P$ of a principal $G$-bundle $P$ its local {\\it gauge} fields arises from the following local representative of a Lie-algebra valued one-form $\\omega$:\n\\begin{equation*}\n    \\omega^U\\ :=\\ \\sigma^*\\omega\\ .\n\\end{equation*}\nWe label the $1$-form components of such local  gauge fields in Yang--Mills theory as\n\\begin{equation*}\n    A_\\mu \\equiv (\\omega^U)_\\mu\\ ,\n\\end{equation*}\nand in general relativity as \n\\begin{equation*}\n    \\Gamma_\\mu \\equiv (\\omega^U)_\\mu\\ .\n\\end{equation*}\n\n\\thm{3.3.1} An explicit form of the local Yang--Mills field $\\omega^U$ for the local trivialization $\\varphi: \\pi^{-1}(U) \\to U\\times G$ arising from $\\sigma$ with $(\\alpha,\\beta) \\in T_{(x,g)}(U\\times G) \\cong T_xU \\oplus T_gG$ is given by\n\\begin{equation*}\n    (\\varphi^*\\omega)_{(x,g)} \\= \\mathrm{Ad}_{g^{-1}} (\\omega^U_x(\\alpha)) + \\< \\Omega_l, \\beta \\>_g\\ ,\n\\end{equation*}\nwhere $\\Omega_l$ is the Maurer--Cartan form.\n\n\\ex{3.3.1} An important example of such a local representative $\\omega^U$ is in the case of the frame bundle $LM$ for an $n$-dimensional manifold $M$ with a given chart $(U,\\phi)$. For a given local section $\\sigma: U \\subset M \\to LM$ defined by\n\\begin{equation*}\n    \\sigma(m)\\ :=\\ ((\\mbox{$\\partial$}_1)_m,\\ldots,(\\mbox{$\\partial$}_n)_m,) \n\\end{equation*}\nthe corresponding $\\omega^U$ has components (the three indices below are divided into two indices $i,j$ for the Lie-algebra $L(GL(n,\\mathds R))$ and one $1$-form index $\\mu$)\n\\begin{equation*}\n    (\\omega^U)^i_{\\ j\\mu}\\ =:\\ \\Gamma^i_{\\ jk} \\quad\\textrm{with}\\quad i,j,\\mu = 1,\\dots,n\n\\end{equation*}\nwhere $\\Gamma^i_{\\ jk}$ is the famous {\\it Christoffel symbol} of this {\\it Levi-Civita or affine connection}\\footnote{This relies on a choice of the natural metric on $M$, which is akin to choosing a basis $G^{\\ a}_b$ of the Lie-algebra $L(GL(n,\\mathds R))$ with components $G^{\\ a}_b)^c_{\\ d} := \\delta^c_b\\delta^a_d$.} that is widely used in Riemannian geometry and general relativity.\n\n\\thm{3.3.2} Let $(P,M,\\pi)$ be a principal $G$-bundle with $U_i,U_j \\subset M$ such that $U_i\\cap U_j \\neq 0$. Further, let $A_\\mu^{(i)}$ and $A_\\mu^{(j)}$ be local gauge functions arising from given local trivializing sections $\\sigma_i: U_i \\to P$ and $\\sigma_j: U_j \\to G$ respectively. The transformation of these fields under the action of gauge functions $\\Lambda_{ij}: U_i\\cap U_j \\to G$ is, for any $m\\in M$, given by \n\\begin{equation*}\n    A_\\mu^{(j)}(m) \\= \\mathrm{Ad}_{\\Lambda_{ij}(m)^{-1}}\\left(A_\\mu^{(i)}(m)\\right) + (\\Lambda_{ij}^*\\Omega_l)_\\mu(m)\\ .\n\\end{equation*}\n\n\\rem{3.3.4} We notice that for matrix Lie groups the above transformation rule takes the following simple form:\n\\begin{equation*}\n    A_\\mu^{(j)}(m) \\= \\Lambda_{ij}(m)^{-1}\\left(A_\\mu^{(i)}(m)\\right)\\Lambda_{ij}(m) + \\Lambda_{ij}(m)^{-1}\\mbox{$\\partial$}_\\mu \\Lambda_{ij}(m)\\ .\n\\end{equation*}\n\n\\rem{3.3.5} This gauge transformation behaviour is what prevents a gauge field $\\omega^U$ from being global i.e. $A ~\\mathrm{or}~ \\Gamma \\not\\in \\Omega^1(M)$. In particular, this explains why the Christoffel symbol is not a tensor! This is because of the following transformation rule between any two given charts $(U,\\phi)$ and $(U',\\phi')$ on a $4$-dimensional manifold $M\\ni m$ with $\\phi(m) = \\{ x^0,\\ldots,x^4 \\}$ and $\\phi'(m) = \\{ x'^{0},\\ldots,x'^{4} \\}$ where the indices like $\\alpha$ below takes temporal $\\alpha {=} 0$ and spatial $\\alpha {=} 1,2,3$ values:\n\\begin{equation*}\n    \\Gamma^{'\\alpha}_{\\ \\beta\\mu} \\= \\frac{\\mbox{$\\partial$} x'^{\\alpha}}{\\mbox{$\\partial$} x^{\\tilde{\\alpha}}}\\,\\frac{\\mbox{$\\partial$} x^{\\tilde{\\beta}}}{\\mbox{$\\partial$} x'^{\\beta}}\\,\\frac{\\mbox{$\\partial$} x^{\\tilde{\\mu}}}{\\mbox{$\\partial$} x'^{\\mu}}\\,\\Gamma^{\\tilde{\\alpha}}_{\\ \\tilde{\\beta}\\tilde{\\mu}} + \\frac{\\mbox{$\\partial$} x'^{\\alpha}}{\\mbox{$\\partial$} x^{\\lambda}}\\,\\frac{\\mbox{$\\partial$} x^\\lambda}{\\mbox{$\\partial$} x'^{\\beta}\\mbox{$\\partial$} x'^{\\mu}}\\ .\n\\end{equation*}\nAn explicit expression for $\\Gamma^{\\alpha}_{\\ \\beta\\mu}$ for a given orthonormal basis of vector fields $\\{ \\mbox{$\\partial$}_0,\\ldots,\\mbox{$\\partial$}_4 \\}$ on $M$ equipped with a pseudo-Riemannian metric $g$ with components $g(\\mbox{$\\partial$}_\\mu,\\mbox{$\\partial$}_\\nu) = g_{\\mu\\nu}$ is given by\n\\begin{equation*}\n    \\Gamma^{\\alpha}_{\\ \\beta\\mu} \\= \\frac{1}{2}\\,g^{\\alpha\\lambda} \\left( g_{\\beta\\lambda,\\mu} + g_{\\mu\\lambda,\\beta} - g_{\\beta\\mu,\\lambda} \\right)\\ ,\n\\end{equation*}\nwhere $g_{{\\beta\\mu,\\lambda}} := \\mbox{$\\partial$}_\\lambda[g_{\\beta\\mu}]$.\n\n\\vspace{12pt}\n\\section{Parallel transport}\n\\vspace{2pt}\n\n\\defn{3.4.1} Owing to the fact that $\\pi_*: H_pP \\to T_{\\pi(p)}M$ is an isomorphism there exists the notion of a unique vector field for a given $X\\in \\vecF{M}$ known as the {\\it horizontal lift} of $X$ and is denoted as $X^\\uparrow$. This satisfies, for all $p\\in P$, the following conditions\n\\begin{enumerate}[(i)]\n    \\item $\\pi_*(X_p^\\uparrow) = X_{\\pi(p)}$\n    \\item $ver(X_p^\\uparrow) = 0$.\n\\end{enumerate}\n\n\\rem{3.4.1} The act of horizontal lifting is $G$-equivariant i.e. $(r_g)_*(X_p^\\uparrow) = X_{pg}^\\uparrow$.\n\n\\defn{3.4.2} A {\\it horizontal lift} of a smooth path $\\gamma: [a,b]\\subset \\mathds R \\to M$ is another path $\\gamma^\\uparrow: [a,b] \\to P$ which is horizontal i.e. $hor(\\gamma^\\uparrow) = 0$ such that $\\pi(\\gamma^\\uparrow(t)) = \\gamma(t)$ for all $t\\in [a,b]$.\n\n\\thm{3.4.1} For each point $p\\in \\pi^{-1}(\\{\\gamma(a)\\})$ there exist a unique horizontal lift $\\gamma^\\uparrow: [a,b] \\to P$ of $\\gamma$ such that $\\gamma^\\uparrow(a)=p$.\n\n\\rem{3.4.2} Given a path $\\gamma: [a,b] \\to M$ and another path $\\beta: [a,b] \\to P$ which projects down to $\\gamma$ i.e. $\\pi(\\beta(t)) = \\gamma(t)$ for all $t\\in [a,b]$, there exits some unique function $g: [a,b]\\to G$ such that \n\\begin{equation*}\n    \\gamma^\\uparrow(t) \\= \\beta(t)\\,g(t) \\quad\\forall\\quad t\\in [a,b]\\ .\n\\end{equation*}\n\n\\thm{3.4.2} The unique path $g: [a,b] \\to G$ defined above satisfies the following first order ODE in terms of a Lie-algebra valued $1$-form $\\omega$:\n\\begin{equation*}\n    \\mathrm{Ad}_{g(t)^{-1}*}\\left(\\omega_\\beta(t)(X_{\\beta,\\beta(t)}) \\right) + \\< \\Omega_l,X_g\\>_{g(t)}\\ ,\n\\end{equation*}\nwhere $X_{\\beta,\\beta(t)}$ is the tangent vector to the curve $\\beta$ at point $\\beta(t)$.\n\n\\rem{3.4.3} For a matrix Lie group $G$ and a local gauge field $\\omega^U = \\sigma^*\\omega$ arising from a local trivializing section $\\sigma: U\\subset M \\to P$ the above ODE takes the following simple form\n\\begin{equation*}\n    \\dot{g}(t) \\= - A_\\mu(\\gamma(t))\\,\\dot{\\gamma}^\\mu(t)\\,g(t)\\ ,\n\\end{equation*}\nwhere the components of the curve $\\gamma$ in a local chart has been denoted as $\\gamma^\\mu$. The solution to this ODE, for an initial condition $g(0)=g_0$, is obtained as a path-ordered exponential in the following way\n\\begin{equation*}\n \\begin{aligned}\n    g(t) &\\= \\left({\\bf P}\\,\\mathrm{exp}\\left( -\\int\\limits_a^t A_\\mu(\\gamma(s))\\,\\dot{\\gamma}^\\mu(s)\\mathrm{d}{s} \\right) \\right)g_0 \\\\\n    &\\= g_0 - \\left(\\int\\limits_a^t A_\\mu(\\gamma(s))\\,\\dot{\\gamma}^\\mu(s)\\mathrm{d}{s}\\right)g_0 \\\\\n    &\\qquad\\qquad + \\left(\\int\\limits_a^t\\mathrm{d}{s_1}\\int_a^{s_1}\\mathrm{d}{s_2}\\, A_{\\mu_1}(\\gamma(s_1))\\,A_{\\mu_2}(\\gamma(s_2))\\,\\dot{\\gamma}^{\\mu_1}(s_1)\\,\\dot{\\gamma}^{\\mu_2}(s_2)\\right)g_0 - \\ldots\\ .\n \\end{aligned}\n\\end{equation*}\n\n\\rem{3.4.4} Form the above result we observe that the local expression for the horizontal lift $\\gamma^\\uparrow$ of the path $\\gamma: [a,b] \\to U \\subset M$ is given by\n\\begin{equation*}\n    \\gamma^\\uparrow(t) \\= \\sigma(\\gamma(t))\\left({\\bf P}\\,\\mathrm{exp}\\left( -\\int\\limits_a^t A_\\mu(\\gamma(s))\\,\\dot{\\gamma}^\\mu(s)\\mathrm{d}{s} \\right) \\right)g_0\\ .\n\\end{equation*}\n\n\\defn{3.4.3} The {\\it parallel transport} along a path $\\gamma: [a,b] \\to M$ is defined by the following map\n\\begin{equation*}\n    T_\\gamma :\\ \\pi^{-1}(\\gamma(a)) \\to \\pi^{-1}(\\gamma(b))\\ ,\\quad p \\mapsto \\gamma^\\uparrow_p(b)\\ ,\n\\end{equation*}\nwhere $\\gamma^\\uparrow$ is the unique horizontal lift of $\\gamma$ passing through $p\\in \\pi^{-1}(\\gamma(a))$.\n\n\\rem{3.4.5} We observe that $T_\\gamma$ is a bijection on fibres and thus on $G$. An interesting thing happens when $\\gamma: [a,b] \\to M$ is a loop i.e. $\\gamma(a) = \\gamma(b)$; one obtains a natural map from loops based at $\\gamma(a) \\in M$ to elements of $G$. The subgroup of all elements of $G$ that can be obtained in this way is called the {\\it holonomy} group of the principle bundle $P$. This plays an important role in understanding the relation between certain topological properties of $M$ with the connection.\n\n\\defn{3.4.4} Let $(P,M,\\pi)$ be a principal $G$-bundle equipped with a connection $1$-form $\\omega$. Furthermore, let $(P_F,M,\\pi_F)$ be associated with $P$ via the left action of $G$ on $F$. A {\\it vertical subspace} of $T_{[p,v]}P_F$ is defined analogous to that of principal bundle i.e.\n\\begin{equation*}\n    V_{[p,v]}P_F\\ :=\\ \\{ X \\in T_{[p,v]}P_F ~|~ \\pi_{F*}X = 0 \\}\\ .\n\\end{equation*}\nSimilarly, the {\\it horizontal subspace} $H_{[p,v]}P_F$ can be defined as the orthogonal complement of $V_{[p,v]}P_F$.\n\n\\defn{3.4.5} The {\\it horizontal lift} of a path $\\gamma: [a,b] \\to M$ to the associated bundle $P_F$ and passing through $[p,v] \\in \\pi_F^{-1}(\\gamma(a))$ is defined as\n\\begin{equation*}\n    \\gamma_F^\\uparrow(t) \\ :=\\ [\\gamma^\\uparrow(t),v]\n\\end{equation*}\nwhere $\\gamma^\\uparrow(a) = p$.\n\n\\rem{3.4.6} One can define {\\it parallel transport} $T_\\gamma$ along a path $\\gamma: [a,b] \\to M$ on the associated bundle $P_F$ analogous to the principal bundle case using the above definition of the horizontal lifting. For associated vector bundles $P_V$ with the vector space $V$ admitting a linear representation of $G$, this notion then facilitates the following definition of the covariant derivative. \n\n\\subsection{Covariant derivative}\n\n\\defn{3.4.6} The covariant derivative of a section $\\psi: M \\to P_V$ of an associated vector bundle $P_V$ along a path $\\gamma: [0,\\varepsilon] \\to M$ with $\\varepsilon>0$ at $m_0=:\\gamma(0)$ is defined by\n\\begin{equation*}\n    D_{X_{\\gamma,\\gamma(0)}}\\psi\\ :=\\ \\lim_{t\\to 0}\\left( \\frac{T_\\gamma(\\psi(\\gamma(t))) - \\psi(m_0)}{t} \\right)\\in \\pi_V^{-1}(m_0)\\ .\n\\end{equation*}\n\n\\rem{3.4.7} The covariant derivative $D_X$, for $X\\in T_mM$, has following algebraic properties for any section $\\psi,\\widetilde{\\psi}\\in \\Gamma^\\infty(P_V)$:\n\\begin{enumerate}[(i)]\n    \\item $D_{fX+Y}\\psi \\= f\\,D_X\\psi + D_Y\\psi$ for all $f\\in C^\\infty(M)$ and $X,Y\\in T_mM$\n    \\item $D_X(\\psi + \\widetilde{\\psi}) \\= D_X\\psi + D_X\\widetilde{\\psi}$ for all $X\\in T_mM$\n    \\item $D_X(f\\psi) \\= X[f]\\,\\psi + f\\,D_X\\psi$ for all $f\\in C\\infty(M)$ and $X\\in T_mM$.\n\\end{enumerate}\nThis following more generic notion of a covariant derivative is related to this one, as we will see below.\n\n\\defn{3.4.7} The {\\it exterior covariant derivative} of a $k$-form $\\omega \\in \\Omega^k(P)$ on a principal bundle $P$ is a horizontal $(k{+}1)$-form defined by\n\\begin{equation*}\n    D\\omega(X_1,\\ldots,X_{k+1})\\ :=\\ \\mathrm{d}{\\omega}(hor(X_1),\\ldots,hor(X_{k+1}))\n\\end{equation*}\nfor a given set of vector fields $X_1,\\ldots,X_{k+1} \\in \\vecF{P}$.\n\n\\rem{3.4.8} This notion of covariant derivative can be extended to an associated vector bundle $P_V$ discussed before with the aid of a given $G$-equivariant map $\\phi: P \\to F$ that has an associated section $\\sigma_\\phi \\in \\Gamma^\\infty(P_V)$ (see Theorem 3.2.2) as follows\n\\begin{equation*}\n    D\\phi\\ :=\\ \\mathrm{d}{\\phi}\\circ hor\\ .\n\\end{equation*}\nOne can show, for any $X\\in \\vecF{P}$ and a given connection $\\omega$ on $P$, that\n\\begin{equation*}\n    F\\ni D\\phi(X) \\equiv D_X\\phi\\ :=\\ \\mathrm{d}{\\phi}(X) + \\omega(X)\\phi\\ ,\n\\end{equation*}\nwhere we notice that $\\phi$ are $F$-valued functions on $P$. Furthermore, one can pull this definition back to $M$ using any local trivializing map $\\sigma: U\\subset M\\to P$ and noticing the fact the pull-back operation is natural:\n\\begin{equation}\n    \\sigma^*(D\\phi)(X)\\ :=\\ \\mathrm{d}{\\sigma^*\\phi}(X) + \\sigma^*(\\omega)(X)(\\sigma^*\\phi) \\quad\\forall\\quad X \\in \\vecF{P}\\ .\n\\end{equation}\n\n\\ex{3.4.1} For a given local orthonormal coframe of vector fields $\\{\\mbox{$\\partial$}_\\mu \\}$ with $\\mu=0,1,2,3$ on a $4$-dimensional Lorentzian manifold $M$, e.g. the Minkowski space $\\mathds R^{1,3}$, the covariant derivative $D_{\\mbox{$\\partial$}_\\mu}=:D_\\mu$ of the above-mentioned section $\\phi$ (or $\\sigma_\\phi$ to be precise) is given in terms of local gauge fields $A_\\mu$ as\n\\begin{equation*}\n    D_\\mu\\phi\\ :=\\ \\mbox{$\\partial$}_\\mu\\phi + A_\\mu\\phi\\ .\n\\end{equation*}\n\n\\ex{3.4.2} The covariant derivative $D_\\mu =: \\nabla_\\mu$ of the tensor bundle $T^{p,q}(M)$ associated with the frame bundle $LM$ on $M$ is given in terms of the Levi--Civita connection and can be expressed in terms of the Chritoffel symbols. For example, covariant derivative of $T \\in T^{1,2}(M)$ with components $T^\\mu_{\\alpha\\beta} := T(\\mathrm{d}{x}^\\mu,\\mbox{$\\partial$}_\\alpha,\\mbox{$\\partial$}_\\beta)$ is given by\n\\begin{equation*}\n    D_\\rho\\,T^\\mu_{\\alpha\\beta} \\= \\mbox{$\\partial$}_\\rho T^\\mu_{\\alpha\\beta} + \\Gamma^\\mu_{\\lambda\\rho}T^\\lambda_{\\alpha\\beta} - \\Gamma^\\lambda_{\\rho\\alpha}T^\\mu_{\\lambda\\beta} - \\Gamma^\\lambda_{\\rho\\beta}T^\\mu_{\\lambda\\alpha}\\ .\n\\end{equation*}\n\n\\rem{3.4.9} The Levi--Civita connection $\\Gamma$ is metric compatible $\\nabla g = 0$ i.e.\n\\begin{equation*}\n    \\nabla_\\mu\\, g_{\\alpha\\beta} \\= 0\\ .\n\\end{equation*}\nMoreover this connection is torsion-free i.e.\n\\begin{equation*}\n    0 \\= T(X,Y)\\ :=\\ \\nabla_XY - \\nabla_Y X - [X,Y]\n\\end{equation*}\nfor all $X,Y \\in \\vecF{M}$. Choosing vector fields $X=\\mbox{$\\partial$}_\\alpha$ and $Y=\\mbox{$\\partial$}_\\beta$ the torsion-free condition ensures that the  Christoffel symbol is symmetric in the subscript indices:\n\\begin{equation*}\n    \\Gamma^\\mu_{\\alpha\\beta} \\= \\Gamma^\\mu_{\\beta\\alpha}\\ .\n\\end{equation*}\n\n\n\\subsection{Curvature}\n\n\\defn{3.4.8} If $\\omega$ is a connection $1$-form on a principal $G$-bundle $P$ then $D\\omega=:\\Omega$ is the Lie-algebra valued {\\it curvature} $2$-form of $\\omega$.\n\n\\rem{3.4.10} It can be shown that the curvature $2$-form $\\Omega$ satisfies the Bianchi identity:\n\\begin{equation*}\n    D\\Omega \\= 0\\ .\n\\end{equation*}\n\n\\thm{3.4.3} For arbitrary pair of vector fields $X,Y \\in \\vecF{P}$ the curvature $2$-form $D\\omega$ satisfies the following Cartan structure equation\n\\begin{equation*}\n    D\\omega(X,Y)\\ :=\\ \\mathrm{d}{\\omega}(X,Y) + [\\omega(X),\\omega(Y)]\\ ,\n\\end{equation*}\nwhere we have employed the Lie bracket on $L(G)$.\n\n\\ex{3.4.3} For the Levi--Civita connection $\\Gamma$ the curvature $2$-form $D\\Gamma=:R$, valued in $L(GL(n,\\mathds R))$ for an $n$-dimensional pseudo-Riemannian manifold $M$, is known as the Riemann curvature and is given by\n\\begin{equation*}\n    R \\= \\mathrm{d}{\\Gamma} + \\Gamma\\wedge \\Gamma\\ .\n\\end{equation*}\nThis turns out to be a tensor $R \\in T^{1,3}(M)$ and in local coframe $\\{ \\mbox{$\\partial$}_\\mu \\}$ and frame $\\{ \\mathrm{d}{x}^\\mu \\}$ with $\\mu =0,1,2,3,$ of $M$ admits the following expression in terms of Christoffel symbol\n\\begin{equation*}\n    R^\\rho_{\\ \\sigma\\mu\\nu} \\= \\mbox{$\\partial$}_\\mu \\Gamma^\\rho_{\\sigma\\nu} - \\mbox{$\\partial$}_\\nu \\Gamma^\\rho_{\\sigma\\mu} + \\Gamma^\\rho_{\\mu\\lambda}\\,\\Gamma^\\lambda_{\\sigma\\nu} - \\Gamma^\\rho_{\\nu\\lambda}\\,\\Gamma^\\lambda_{\\sigma\\mu}\\ .\n\\end{equation*}\n\n\\rem{4.3.11} Riemann curvature tensor gives rise to the so called Ricci tensor $R_{\\mu\\nu} := R^\\rho_{\\ \\mu\\rho\\nu} \\in T^{0,2}(M)$ through index contraction, which in turn gives rise to the scalar curvature $R:= g^{\\mu\\nu}R_{\\mu\\nu}$ via the metric. With these tools and the insight of {\\it equivalence principle} that states \\\\ ``For any point $m\\in M$ there always exists a local chart $(U,\\phi)$ with $m\\in U$ admitting a smooth local section of orthonormal frame $\\sigma \\in \\Gamma^\\infty(LU)$ on the frame bundle $LU$; in other words every (spacetime) manifold is locally flat in a Minkowski sense\" \\\\ Albert Einstein was able to construct the following field equation relating the curvature of spacetime with its matter content\n\\begin{equation*}\n    G_{\\mu\\nu}\\ :=\\ R_{\\mu\\nu} - \\ \\frac{1}{2}g_{\\mu\\nu}R \\= \\frac{8\\pi G}{c^4} T_{\\mu\\nu}\\ ,\n\\end{equation*}\nwhere $G$ is the Newton constant, $c$ is the speed of light and $T_{\\mu\\nu}$ is the stress-energy tensor arising from the following variation of the matter action $S_m$ with respect to the matric:\n\\begin{equation*}\n    T_{\\mu\\nu} \\= -\\frac{2}{\\sqrt{-\\mathrm{det}\\,g}} \\frac{\\delta S_m}{\\delta g^{\\mu\\nu}}\\ .\n\\end{equation*}\n\n\\ex{3.4.4} A trivial solution of the Einstien equation is the Minkowski metric $\\eta_{\\mu\\nu}$ which is globally flat. Another very important solution is the FLRW metric for homogeneous and isotropic cosmology:\n\\begin{equation*}\n    g \\= -c^2\\mathrm{d}{t}^2 + a(t)^2\\left(\\frac{\\mathrm{d}{r}^2}{1-\\kappa r^2} + r^2\\mathrm{d}{\\theta}^2 + r^2\\sin^2\\theta\\mathrm{d}{\\phi}^2 \\right)\\ ,\n\\end{equation*}\nwhere $\\{r,\\theta,\\phi \\}$ are coordinates on celestial spheres with respect to an observer (like us), $a(t)$ is a scalar function called {\\it scale factor} and the parameter $\\kappa = -1$, $0$ or $1$ denotes the topology of the $3$-dimensional Euclidean space as being open, flat or closed respectively. \n\n\\ex{3.4.5} For local Yang--Mills field $A:=\\sigma^*\\omega$ the curvature $2$-form $D\\omega=:F$ is given by\n\\begin{equation*}\n    F \\= \\mathrm{d}{A} + A\\wedge A\\ .\n\\end{equation*}\n\n\\vspace{12pt}\n\\section{Yang-Mills equation}\n\\vspace{2pt}\n\n\\rem{3.5.1} A local expression for the Yang--Mills curvature $F$ on a Lorentzian manifold $M$ with orthonormal coframe $\\{\\mbox{$\\partial$}_\\mu \\}$ is given by\n\\begin{equation*}\n    F_{\\mu\\nu} \\= \\mbox{$\\partial$}_\\mu A_\\nu - \\mbox{$\\partial$}_\\nu A_\\mu - [A_\\mu,A_\\nu]\\ .\n\\end{equation*}\n\n\\rem{3.5.2} The Yang--Mills curvature transforms under a local gauge transformation $\\Lambda_{ij}(m)$ with $m \\in U_i\\cap U_j \\neq 0$ arising from two local sections $\\sigma_i: U_i \\to P$ and $\\sigma_j: U_j \\to P$ related by $\\sigma_j(m) = \\sigma_i(m)\\Lambda_{ij}(m)$ in the following way\n\\begin{equation*}\n    F^{(j)}_{\\mu\\nu}(m) \\= \\Lambda_{ij}(m)\\,F^{(i)}_{\\mu\\nu}(m)\\,\\Lambda_{ij}(m)\\ ,\n\\end{equation*}\nand is thus global i.e. $F\\in \\Omega^2(M)$.\n\n\\rem{3.5.3} The Yang--Mills equation on a Lorentzian manifold $M$ is given by\n\\begin{equation*}\n    *D(*F) \\= J\\ ,\n\\end{equation*}\nwhere the $1$-form $J \\in \\Omega^1(M)$ is the {\\it current} that arises from the presence of any source ``charge\" on the manifold.\n\n\\rem{3.5.4} The Yang--Mills action (coupling constant $g$)\n\\begin{equation*}\n    S_{YM} \\= \\frac{1}{2g^2}\\int_M \\mathrm{Tr}(F\\wedge *F)\n\\end{equation*}\ntogether with the action for the source \n\\begin{equation*}\n    S_J \\= \\frac{1}{g^2} \\mathrm{Tr}(A \\wedge *J)\n\\end{equation*}\ngives rise to the above Yang--Mills equation by variational principle.\n\n\\rem{3.5.5} Maxwell's theory of electromagnetism that we saw in the last chapter is a special case of Yang--Mills theory where the gauge group is $U(1)$.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n     \\label{introduction}\n     Helioseismology uses the observed solar surface acoustic wavefield to construct \nimages of the subsurface structure of the Sun. Of particular interest\nhas been the three-dimensional (3D) modeling of time-distance \n\\cite{DJHP93} observations\nof travel-time shifts to deduce the subsurface structure of active regions \n\\cite{KDS00,ZK03}.  Assuming the travel-time shifts are due to perturbations in\nthe sound speed below the spot, a general consensus in the models has\nemerged consistent with sound-speed reductions (relative to surrounding quiet\nSun) near the surface ($\\lesssim 4$ Mm) and enhancements up to 15 Mm below \nsunspots \\cite{KDS00,CBK06}.  Using ring diagram analysis \n\\inlinecite{BAB04}  also find a lower sound speed immediately below the \nsurface and an increase in the sound speed below 7 Mm.\n\nA comparison between Fourier-Hankel analysis and time-distance results \nby \\inlinecite{B97} first prompted caution in the interpretation of acoustic \noscillation signals within sunspots. \nThe influences of strong surface magnetic fields have not been explicitly\nincluded in most helioseismic models of active regions. \n \\citeauthor{LB05ii} (2005a) and \\citeauthor{LB05i} (2005b) have shown\nthat helioseismic phase shifts observed with helioseismic\nholography vanish below a depth of about 5 Mm, \nwhen a surface (``showerglass'') phase shift\nbased on photospheric magnetic flux density, is removed from  \nthe data.\n\nOther evidence supports the possibility of strong near-surface\ncontributions to the helioseismic phase (or travel-time) shifts.\nThese include the possible contamination of surface perturbations\ninto the 3D inversions (e.g. \\opencite{K06}; \n\\opencite{CR07}). It has also been shown that\nthe reduction of $p$-mode amplitudes in magnetic regions\ncan cause travel-time shifts \\cite{RBDTZ06}. The\nsuppression of sources of wave excitation within sunspots\ncan also produce measurable shifts \\cite{HCRB07}.\n\n\\inlinecite{SBCL05} and \\inlinecite{SBC07} have found that\nphase shifts obtained from seismic holography in sunspot\npenumbrae vary with the line-of-sight angle (from vertical) \nas projected into the plane containing the magnetic field and the \nvertical direction.  A similar effect has also been noted by \n\\inlinecite{ZK06} with time distance measurements.\n\\inlinecite{SBC07} find that the  effect is dependent upon the strength and\/or \ninclination of the magnetic field. \nIn the penumbra the magnetic field strength \ndecreases as the magnetic field angle from vertical increases, hence the two \nproperties of the magnetic field cannot be extricated. \nThe phase variation with line-of-sight \nviewing angle is demonstrated most substantially at frequencies \naround 5 mHz with a \nstrong, almost vertical magnetic field close to the umbra.  \n\\inlinecite{SBC07} also find that the total variation across\nall lines-of-sight increases with temporal frequency,\nparticularly in the stronger fields in the penumbrae.\n\nMode conversion of the acoustic waves in the near surface has been explored as \nthe physical cause of the observed absorption of acoustic waves by sunspots.  A \nfast acoustic wave, propagating towards the surface from the interior, \nencounters the depth at which the Alfv\\'en speed is equal to the sound speed \n($a\\approx c$) which is typically close to the surface in a  sunspot. Under \nthese conditions it is able to transmit to a slow acoustic mode and convert to \na fast magnetic mode \\cite{C05}. \\inlinecite{CC03} explore the mode \nconversion in two-dimensions with a uniform \\emph{inclined} magnetic field \nrelevant to sunspot penumbrae. The inclination of the magnetic field is found \nto have a significant dependence on the likelihood of conversion  and fits \nextremely well with the analysis of \\inlinecite{B95} \\cite{CCB03}. Further \nwork \\cite{C05,SC06} using ray theory has since established that it is the \nangle between the acoustic wave path and the magnetic field (the `attack \nangle') which is the crucial factor inducing conversion. With a wide attack \nangle at the $a\\approx c$ level there is maximum conversion from a fast \nacoustic to a fast magnetic mode. A fine attack angle encourages  transmission  to a slow acoustic mode, which is guided `up' the magnetic field lines to \nobservational heights in the atmosphere. A consequence of mode conversion\nmay be the observational signature of elliptical motion in regions\nof inclined magnetic field. \n\nThe aim of this paper is to model the observations of two\nsunspots previously analyzed by \\inlinecite{SBC07} in order to determine the properties of velocity ellipses consistent with the data.\nThese models are based on a least-squares-fit of the phase and \nmodulus information of the local ingression control correlation. \nWe explore the properties of these ellipses, observed\nwith waves at different temporal frequencies, as functions of\nthe magnetic field inclination angle. \nWe also examine the variation with line-of-sight angle\nof the phase shifts in the outgoing waves, using the local egression \ncontrol correlation, to assess the relation of the phase shift\nvariations between incoming and outgoing waves.\n\nIn the following sections we describe the data (Section \\ref{obs}), give an \noutline of the helioseismic holography technique used (Section 3), describe the \nresults (Section 4) and discuss our results in the\ncontext of mode conversion  (Section 5).\n\n\n     \n     \n\n \\section{Observations}\n      \\label{obs}      \n\nAs this is a continuation of studies by \\inlinecite{SBCL05} and \n\\inlinecite{SBC07} we use the same data, however, we provide a short \ndescription here to maintain coherence. The \\emph{Michelson Doppler Imager} \n(MDI) aboard the \\emph{Solar and Heliospheric Observatory} (SOHO) \\cite{MDI95} \nprovides the solar surface Doppler velocity information. The Dopplergrams are \nfull disk, have a 60 second cadence and  a resolution of  $\\approx 1.4$ Mm per \npixel. These full disk Dopplergrams are Postel projected and a $512\\times 512$ \npixel extract is taken centered on the active region. We analyze two sunspots: \nthe first in AR9026  observed over 10 days from 3rd - 12th June 2000 with a \nCarrington longitude (L0) of $75^\\circ$ and latitude (B0) of $20^\\circ$ and \npenumbral boundaries defined by inner and outer radii of 7 Mm and 16 Mm \nrespectively. The second is a sunspot in AR9057 observed over 9 days from 24th \nJune - 2nd July 2000 with $L0=158^\\circ$ and $B0=13^\\circ$, inner and outer\npenumbral boundaries given by 6 Mm and 13 Mm.  \nThe penumbral boundaries are \ndefined to be between 50\\% and 85\\% of the nearby quiet-Sun \ncontinuum intensity.\nThe sunspots in these active regions were chosen \nbased on the existence of continual MDI observation as they traversed the solar \ndisk, and for their relatively simple magnetic structure and evolution. \n\nMagnetograms from the \\emph{Imaging Vector Magnetograph} (IVM) at the \nUniversity of Hawaii Mees Solar Observatory \\cite{IVM96}  provide the \norientation and strength of the surface magnetic field in the sunspots chosen \nin AR9026 and AR9057. The IVM observations are made over a 28 minute interval: \nfor AR9026  starting at 18:29 Universal Time (UT) on 5th June 2000 and  for \nAR9057 starting at 16:19 UT  on 28th June 2000. \nThe IVM data reveals an azimuthally spreading magnetic field configuration \nfor both sunspots although they are not entirely symmetric.  It is \nassumed, supported largely by available line-of-sight magnetograms, that there \nis no significant evolution of the magnetic field in the sunspots during the \ntime of observation. Therefore, using only one vector magnetogram for the \nduration of the observation is reasonable. Rotation and scaling are applied to \nalign the IVM data to the line-of-sight MDI magnetograms.\n\nFigure~\\ref{g_vs_b} shows a strong correlation between the \nmagnetic field inclination (from vertical, $\\gamma$) and \nfield strength (the strong field is almost vertical whereas highly \ninclined field is relatively weak). In \nthis paper we use the inclination $\\gamma$ as the primary variable,\ndividing the penumbrae into three regions defined by the values\nof $\\gamma$ (Fig~~\\ref{g_vs_b}).\nIt is understood in this analysis that the magnetic field strength is \nimplicitly correlated with the inclination through Figure~\\ref{g_vs_b}, \nand that independent dependencies of observables with field strength \nand inclination are not extracted. The \ndependence of the phase shifts on the line-of-sight viewing angle of the \nmagnetic field is facilitated by knowing the full vector magnetic field.\n \n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{G_vs_B_9026_9057.eps}\n\\vspace{0cm}\n\\caption{Magnetic field strength, $| B |$, plotted against inclination from \nvertical, $\\gamma$, determined from IVM vector magnetograms. AR9026 (5th June \n2000) (bottom) and AR9057 (28th June 2000) (top). The different symbols divide \nthe penumbra into roughly equal regions of inclination: $\\gamma < 42^\\circ$ \n(asterisk); $42^\\circ < \\gamma < 66^\\circ$ (diamonds); $\\gamma > 66^\\circ$ \n(triangles). This corresponds to average magnetic field strengths of 1700 G, \n1000 G and 600 G for AR9057 and for AR9026 1900 G, 1400 G  and 600 G. In \ngeneral, the progression from the upper left portion of the distribution to the \nlower right portion represents increasing distance from the centre of the spot.}\n\\label{g_vs_b}\n\\end{center}\n\\end{figure}\n\n\\section{Helioseismic Holography}\n\n\nHelioseismic holography \\cite{LB00,BL00} is the phase coherent imaging of the \nsolar subsurface based on photospheric acoustic oscillations. The ingression is \nan assessment of the observed wavefield, $\\psi(\\mathbf{r}',t)$, converging to a \nselected focal point, $(\\mathbf{r},z,t)$, and the egression is the time reverse \n- an assessment of waves diverging from that point. In this case we calculate \nthe quantities at the surface, $z=0$. In practice, the observed wavefield used \nfor the calculation is usually an annulus surrounding the chosen focal point. \nThe pupil used here is identical to that described by \\inlinecite{SBCL05} and is \nconstructed for the  calculations with inner radius $a=20.7$ Mm  and outer \nradius $b= 43.5$ Mm, designed to be large enough that when the focal point is \nwithin the penumbra the area covered by the annulus does not include large \nareas of strong magnetic field.  At a frequency of 5 mHz this selects $p$-modes \nwith spherical harmonic degree and radial degree between $\\ell \\approx 450$ and \n$\\ell \\approx 700$.  The ingression at the surface is given by,\n\\begin{equation}\nH_{-}(\\mathbf{r},0,t)=\\int_{a < |\\mathbf{r}-\\mathbf{r'}|<b} d^2 \\mathbf{r}' \nG_{-}( |\\mathbf{r} - \\mathbf{r}' |, 0, t - t') \\psi(\\mathbf{r}',t),\n\\label{ingress}\n\\end{equation}\n$G_{-}$ is the ingression Green's function \\cite{LB00}. The egression, $H_{+}$ \nis simply the time reverse of  \\ref{ingress}, i.e. $t' - t$. \n\nThe ingression (\\ref{ingress}) is correlated with the observed wavefield in the \nspace-frequency domain,\n\\begin{equation}\nC_{-}(\\mathbf{r},\\nu) = \\langle \\hat{H}_{-} \n(\\mathbf{r},\\nu)\\hat{\\psi}^{*}(\\mathbf{r},\\nu ) \\rangle_{\\Delta \\nu} = \n|C_{-}|  \\rm{e}^{-i \\ \\delta \\phi_{-}},\n\\label{correl}\n\\end{equation}\nwhere $\\hat{H}_{-}(\\mathbf{r},\\nu)$ and $\\hat{\\psi}(\\mathbf{r},\\nu )$ are\nthe temporal Fourier transforms of $H_{-}$ and $\\psi$ respectively,\nand we have dropped the dependence on depth $z$.\nThe correlation has a modulus $|C_{-}|$ and \na phase, $\\delta \\phi_{-}$.   \nIn the frequency spectrum, with the pupil covering mostly quiet Sun, the local \ningression control correlation simply characterizes how the local magnetic \nphotosphere responds acoustically to upcoming waves, prescribed by $H_{-}$, \noriginating within the pupil. \n\n\nIn \\inlinecite{SBCL05} and \\inlinecite{SBC07} the phase shift caused by the \nsurface perturbations to the incident wave is calculated at various \nfrequencies. Here we extend that research and monitor the variation of the \ncorrelation amplitude within the sunspot penumbral region, which is then \ncombined with the phase information to form estimates of the surface velocity \nellipse. We then go on to examine the phase of the local egression control \ncorrelation, $\\delta \\phi_{+} = \\rm{Arg} [ C_{+}(\\mathbf{r},\\nu)]$, with the \nline-of-sight angle in the penumbra of sunspots in AR9057 and AR9026.\n\n\n\\section{Elliptical Representation of Surface Velocities}\n\n\nFor better statistics multiple days of observation of each sunspot are \ncombined and a least-squares-fit of the observations allows an estimation of \nthe velocity ellipse. To calculate the velocity ellipse  \nthe ingression correlation (eqn. \\ref{correl}) \nis used as a proxy for the local velocity. The modulus and phase\nof the correlation vary with respect \nto the line-of-sight angle and these variations are modeled to construct \nan ellipse representing the \nsurface velocity associated with a particular magnetic field element. \n\nA smeared ingression `flat field' takes out undesired contributions to the \ncorrelation modulus due to the temporal or spatial variations of the\ningression, due to the presence of magnetic\nregions in the pupil. This is achieved by dividing $|C_{-}|$ by the \nGaussian smear of the root-mean-square of the ingression to get \n$|C_{-}|_{flat}$. The resulting correlations are normalized so that \nthe ingression correlation modulus in the nearby quiet Sun \nhas a value of $\\cos(\\zeta)$, where $\\zeta$ is the heliocentric\nangle of the quiet region from disk center. A dependence of the\nquiet Sun (root-mean-squared) amplitude with  $\\cos(\\zeta)$ is\nexpected for predominantly vertically oscillating $p$-modes. \nThe normalization is achieved by dividing $|C_{-}|_{flat}$ \nby the quiet Sun average divided by the cosine of the heliocentric angle, i.e. \n\\begin{equation}\n|C_{-}|_{norm}=|C_{-}|_{flat} \\frac{\\cos(\\zeta)}{\\langle |C_{-}| \n\\rangle_{QS}}. \n\\end{equation}\nThe normalization corrects for variations in the modulus due to \nduty cycle variations, foreshortening, and other effects which \ncause undesired variations in the modulus from day to day. \nThese procedures assume that these \nfactors have the same relative effect on the (desired) correlations in the \npenumbra as they do to the quiet Sun. It is difficult to assess the validity \nof this, and so it is used as a reasonable working assumption subject to some \ncaution.\n\nWe use the same angle $\\theta_p$ as defined in \\inlinecite{SBCL05}, which is \nthe angle between the projection of the line-of-sight vector onto the plane \ncontaining the magnetic field vector and the radial vector. \nIt is a measure of the angle that the magnetic field is being viewed from. We \nnow  simply refer to $|C_{-}|_{norm}$ as $|C_{-}|$ and see how it varies with \n$\\theta_p$.\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth]{phase_mod_thetap_3mHz.ps} \\hspace{1.5cm}\n\\caption{The modulus of the correlation, $|C_{-}|$  (left column), and the \nphase, $\\delta \\phi_{-}$, in the penumbra of AR9026 at 3 mHz for all days of \nobservation are plotted against projected angle $\\theta_p$ for different values \nof magnetic field inclinations as indicated. The top panel (a) shows $\\gamma < \n42^\\circ$, where the mean field strength is $\\langle \\mathbf{B} \\rangle = 1900$ \nG, the middle panel (b) shows $42^\\circ < \\gamma < 66^\\circ$, where $\\langle \n\\mathbf{B} \\rangle = 1400$ G, and the bottom panel (c) shows  $\\gamma > \n66^\\circ$, where $\\langle \\mathbf{B} \\rangle = 600$ G. The horizontal dashed \nlines indicate the mean value of $|C_{-}|$ for each panel. The error bars \nindicate the standard deviation of the mean over bins of 20 measurements in \n$\\theta_p$. The solid line of fit is a fit for all the displayed data; the \ndotted line is a fit for the data from 3rd - 7th June 2000; the dashed line is \na fit for data from 8th - 12th June 2000.  }\n\\label{3mhz}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth]{phase_mod_thetap_4mHz.ps}\n\\caption{Same as Figure~\\ref{3mhz} except at 4 mHz.}\n\\label{4mhz}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth]{phase_mod_thetap_5mHz.ps}\n\\caption{Same as Figure~\\ref{3mhz} except at 5 mHz.}\n\\label{5mhz}\n\\end{center}\n\\end{figure}\n\n\n\n\n\\begin{figure}[ht]\n\\begin{center}\n\\vspace{1cm}\n\\includegraphics[width=0.7\\textwidth]{fig_ellipse.eps}\n\\caption{Least-squares-fit surface velocity ellipses as given by the phase and \namplitude of the local ingression control correlation in the penumbra of \nAR9026. The top row is when $\\gamma > 66^\\circ$, the middle row when $42^\\circ \n< \\gamma < 66^\\circ$ and the bottom row is when $\\gamma < 42^\\circ$. The \n$\\gamma$ listed in the plot is the angle that the magnetic field vector is \ndrawn at. $\\beta$ is the inclination angle of the semi-major axis of the \nvelocity ellipse. The left column is at frequencies of 3 mHz, the middle column \nat 4 mHz and the right column at 5 mHz.}\n\\label{ellipse9026}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{phase_mod_thetap_3mHz_AR9057.ps}\n\\caption{The modulus of the correlation, $|C_{-}|$ (left column) and phase, \n$\\delta \\phi_{-}$ (right column), in the penumbra of AR9057 at  3 mHz for all \ndays of observation are plotted against projected angle $\\theta_p$ for \ndifferent values of magnetic field inclinations as indicated.  The top panel \n(a) shows $\\gamma < 42^\\circ$, where the mean field strength is $\\langle \n\\mathbf{B} \\rangle = 1700$ G, the middle panel (b) shows $42^\\circ < \\gamma < \n66^\\circ$, where $\\langle \\mathbf{B} \\rangle = 1000$ G, and the bottom panel \n(c) shows  $\\gamma > 66^\\circ$, where $\\langle \\mathbf{B} \\rangle = 600$ G. The \nhorizontal dashed lines indicate the mean value of $|C_{-}|$ for each panel. \nThe error bars indicate the standard deviation of the mean over bins of 20 \nmeasurements in $\\theta_p$. The solid line of fit is a fit for all the \ndisplayed data; the dotted line is a fit for the data from 24th - 28th June \n2000; the dashed line is a fit for data from 29th June - 2nd July 2000.  }\n\\label{90573mhz}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{phase_mod_thetap_3mHz_AR9057.ps}\n\\caption{Same as for  Figure~\\ref{90573mhz} except at 4 mHz. }\n\\label{90574mhz}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{phase_mod_thetap_5mHz_AR9057.ps}\n\\caption{Same as for Figure~\\ref{90573mhz} except at 5 mHz.}\n\\label{90575mhz}\n\\end{center}\n\\end{figure}\n\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\vspace{1cm}\n\\includegraphics[width=0.7\\textwidth]{fig_ellipse_AR9057.eps}\n\\caption{Least-squares-fit surface velocity ellipses as given by the phase and \namplitude of the local ingression control correlation in the penumbra of \nAR9057. The top row is when $\\gamma > 66^\\circ$, the middle row when $42^\\circ \n< \\gamma < 66^\\circ$ and the bottom row is when $\\gamma < 42^\\circ$. The \n$\\gamma$ listed in the plot is the angle that the magnetic field vector is \ndrawn at. $\\beta$ is the inclination angle of the semi-major axis of the \nvelocity ellipse. The left column is at frequencies of 3 mHz, the middle column \nat 4 mHz and the right column at 5 mHz.}\n\\label{ellipse9057}\n\\end{center}\n\\end{figure}\n\n\nFigures~\\ref{3mhz} to \\ref{5mhz} and Figures~\\ref{90573mhz} to \\ref{90575mhz} \nshow the variation of $|C_{-}|$  with $\\theta_p$ for AR9026 and AR9057 in the \nleft columns at 3, 4 and 5mHz in the same three bins of inclination shown in \nFigure~\\ref{g_vs_b}. The right column shows $\\delta \\phi_{-}$. \nThe variations of $\\delta \\phi_{-}$ with $\\theta_p$ have been the subject\nof our previous analyses (\\opencite{SBCL05}; \\opencite{SBC07}). \nIn the absence of magnetic effects, we expect\nthe local oscillatory wave field, as assessed by the ingression correlation,\nto be consistent with purely vertical motion. This would predict a\n dependence of $|C_{-}|$ on $\\cos(\\theta_p)$ and no dependence  of  $\\delta \\phi_{-}$ on\n$\\cos(\\theta_p)$. \nDepartures from these expectation\nare clearly visible in Figs~\\ref{3mhz} - \\ref{5mhz} and \\ref{90573mhz} - \n\\ref{90575mhz}. A rough understanding of these results can be\ngained by noting that the net variation in the phase shift with\n$\\theta_p$ is related to the eccentricity of the ellipse, while\nthe value of $\\theta_p$ for maximum $|C_{-}|$ determines the \norientation of the semi-major axis.\nWe perform a least-squares-fit to the observed $|C_{-}|$ \nand $\\delta \\phi_{-}$ to determine the elliptical motion \nconsistent with the data. The best-fit ellipses \nare shown in Figures~\\ref{ellipse9026} and \\ref{ellipse9057}. \nThe moduli and phase of $C_-$ as determined from the fits are plotted (as solid\nlines) with the data points in Figs 2-4 and 6-8. Separate fits were performed for\nindependent  5-day subsets of the data (also shown in the Figures by the dotted\nand dashed lines). There are no obvious systematic differences in the fits\nover time. The orientations and eccentricities are highly consistent across\nall frequency bands in both sunspots (Figure~\\ref{ellipse9026} \nand Figure~\\ref{ellipse9057}). Some systematic differences\nbetween the two spots are evident.\nFor AR9026, the ellipses are aligned slightly towards\nthe magnetic field direction at high field inclination \nbut `swing' over as the inclination becomes smaller (and the \nmagnetic field stronger). For AR9057, the motion is nearly vertical\nat low field inclination, but also tilts away from the field\nin the stronger, more vertical, fields. \nIn both spots, the eccentricity\nof the ellipses increases with decreasing field strength or increasing \ninclination.\n\n\n\\subsection{ELLIPSE PARAMETERS}\n\nWe define a deviation angle \nas the angle between the magnetic field vector and surface\nvelocity ellipse semi-major axis ($\\delta$ = $\\gamma - \\beta$, where $\\beta$ is the inclination of the semi-major axis from vertical).\nWe show the variation of\nthe deviation angle, length of the semi-major axis,\nand eccentricity with magnetic field strength and\/or inclination \nin Figure~\\ref{ellp}.\nAlso shown is the phase difference between the fits at \n$\\theta_p=-60^\\circ$ and $\\theta_p=+60^\\circ$. \nThis quantity is generally inversely correlated to the ellipse\neccentricity, but is a more \ndirect measure of the total variation of observed phase shift\nfor a specific penumbral region \\cite{SBCL05}.\nIn Figure~\\ref{ellp}, data from \nAR9026 is represented by an asterisk, AR9057 by a diamond and the frequencies \nare color-coded as following :- 5 mHz is black, 4 mHz is purple and 3 mHz is \nred.\n\\begin{figure}[ht]\n\\begin{center}\n\n\\includegraphics[width=0.4\\textwidth]{ellipse_gamma_gamma_minus_beta.eps}\n\\includegraphics[width=0.4\\textwidth]{ellipse_gamma_ecc.eps}\\\\\n\\includegraphics[width=0.4\\textwidth]{ellipse_gamma_diff.eps}\n\\includegraphics[width=0.4\\textwidth]{ellipse_amp.eps}\n\\caption{Properties of the ellipses; deviation angle, $\\delta$ (top left); the \nellipse eccentricity  for both sunspots at all frequencies against the three \naverage magnetic field inclinations (top right); the phase difference between \nthe least-squares-fit correlation phase at $\\theta_p=-60^\\circ$   and at  \n$\\theta_p=+60^\\circ$ ( $\\delta \\phi_{( \\theta_p=-60^\\circ) } -  \\delta \\phi_{( \n\\theta_p=+60^\\circ) }$ ) (bottom left);  the length of the semi-major axis of \neach ellipse in each region for both sunspots at all frequencies against the \nthree average magnetic field inclinations (bottom right). AR9026 is represented \nby the asterisk, and AR9057 by the diamond. 5 mHz is black, 4 mHz is purple and \n3 mHz is red.}\n\\label{ellp}\n\\end{center}\n\\end{figure}\n\nIn general, the trends shown in Figure~\\ref{ellp}, namely an increase\nin the deviation angle, eccentricity, and semi-major axis length,\nand a decrease in the phase variation, with increasing inclination\nare observed in both sunspots and at all frequencies. There are\nsome deviations from this. For example, at low inclinations, it is\nobserved that the eccenctricity (phase variation) decreases (increases)\nwith frequency. Note that the semi-major axis length is an indication\nof the total wave amplitude in the magnetic region. The trend observed\nin the lower-left panel of Figure~\\ref{ellp} is consistent with \na reduction in wave amplitude related to the field strength. \n\n\n\n\\section{Local Egression Control Correlation}\\label{eg}\nIn this section we \nexamine the phase of the local egression control correlation, \n\\begin{equation}\nC_{+}(\\mathbf{r},\\nu) = \\langle \\hat{H}_{+} \n(\\mathbf{r},\\nu)\\hat{\\psi}^{*}(\\mathbf{r},\\nu ) \\rangle_{\\Delta \\nu} = |C_+|  \n\\rm{e}^{-i \\ \\delta \\phi_{+}}.\n\\label{correl2}\n\\end{equation}\nSince the egression is simply the time reverse of the \ningression, we might expect \nto see a reversal of the phase change compared to the ingression phases.\n\nUsing all the days' of data, the egression correlation phase is plotted against \n$\\theta_p$ in Figures~\\ref{hephase1} and \\ref{hephase2}, along with\nthe fits for the phase variation due to elliptical motion.  We see similar  \ntrends in both sunspots, AR9026 and AR9787. \nFigure~\\ref{dphase} shows the phase \ndifference between the least-squares-fit correlation phase at \n$\\theta_p=-60^\\circ$   and at  $\\theta_p=+60^\\circ$ of the ingression plotted \nagainst that of the egression. The colors and symbols represent the same as \nbefore:  AR9026 is represented by an asterisk, AR9057 by a diamond and the \nfrequencies are 5 mHz (black), 4 mHz (purple) and 3 mHz (red) in addition the \nsize of the symbols represent the average inclination from vertical. The solid \nline has a slope of $-1$.  There is a reverse \nbehaviour of the ingression, compared to the previous egression\nresults, present for all frequencies at most magnetic field \ninclinations.  The exception is when the field is highly inclined, where \nwe observe a trend in the same sense as the ingression. \nThis is unexpected \nand warrants further study.\n\n\n\\begin{figure}[p]\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{fig_phi_vs_thetap_HE_9026.ps} \n\\vspace{1cm}\n\\caption{The 3 mHz (left column), 4 mHz (middle column) and 5 mHz (right \ncolumn) egression correlation phase ($\\delta \\phi_{+}$) vs. $\\theta_p$ within \nthe penumbra of sunspot AR9026 for different values of magnetic field \ninclination as indicated.  The three rows represent different portions of the \npenumbra as shown in Figure~\\ref{g_vs_b}(a). The top panel row (a) shows \n$\\gamma < 42^\\circ$, where the mean field strength is $\\langle \\mathbf{B} \n\\rangle = 1900$ G, the middle row (b) shows $42^\\circ < \\gamma < 66^\\circ$, \nwhere $\\langle \\mathbf{B} \\rangle = 1400$ G, and the bottom row (c) shows  \n$\\gamma > 66^\\circ$, where $\\langle \\mathbf{B} \\rangle = 600$ G. The horizontal \ndashed lines indicate the mean value of $\\delta \\phi_{+}$ for each panel. }\n\\label{hephase1}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[p]\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{fig_phi_vs_thetap_HEN_9057.ps}\n\\vspace{1cm}\n\\caption{The 3 mHz (left column), 4 mHz (middle column) and 5 mHz (left column) \negression correlation phase ($\\delta \\phi_{+}$) vs. $\\theta_p$ within the \npenumbra of sunspot AR9057 for different values of magnetic field strength as \nindicated.   The three different panels represent different portions of the \npenumbra, similar to Figure~\\ref{g_vs_b}(b). The top panel (a) shows $\\gamma < 42^\\circ$ where the mean field strength is $\\langle B \\rangle = 1700$ G the middle panel (b) \nshows $42^\\circ < \\gamma < 66^\\circ$, and $\\langle B \\rangle = 1000$ G and the \nbottom panel (c) shows $\\gamma > 66^\\circ$ and $\\langle B \\rangle = 600$ G. The horizontal dashed lines indicate the mean value of $\\delta \n\\phi_{+}$ for each panel. }\n\\label{hephase2}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[p]\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{ph_diff_eg_in.ps}\n\\vspace{1cm}\n\\caption{The phase difference between the least-squares-fit correlation phase \nat $\\theta_p=-60^\\circ$   and at  $\\theta_p=+60^\\circ$ of the ingression \nplotted against that of the egression. The solid line is a line of slope $-1$. \nAR9026 is represented by an asterisk, AR9057 by a diamond and the \ncolors indicate the following frequencies: 5 mHz (black), 4 mHz (purple) and \n3 mHz (red). The largest symbols represent $\\gamma > 66^\\circ$ and the \nsmallest symbols $\\gamma < 42^\\circ$. }\n\\label{dphase}\n\\end{center}\n\\end{figure}\n\n\n\n\\section{Discussion}\n\nMode Conversion predicts (among other things \\cite{C07}) that when the attack \nangle is small most of the observable energy will be in the slow acoustic mode, \nand when the attack angle is large most of the observable energy will be in the \nfast magnetic mode. In this case we will be seeing the line-of-sight effect of \na combination of waves coming from all directions impinging on magnetic field \nwith a particular orientation. The observations by MDI consist of line-of-sight Doppler signatures \nof the surface motion which are presumably caused mainly by pressure \nperturbations. This means that we would expect to be observing only the slow \nacoustic mode, however it is possible that we are observing a combination of \nthe acoustic and magnetic modes. \n\n\nThe dependence of the phase shift of the observed ingression correlations with azimuthal angle \naround sunspot penumbrae, as viewed from different observational vantages, shows that\nthe incoming phase shifts must be (at least partly) photospheric in origin and are\ninfluenced by the presence of inclined magnetic fields \\cite{SBCL05}. \nAnalysis of the variation of both the amplitude and phase of the surface velocities \nprovides an opportunity to characterize the magnetically influenced acoustic signature\nas an ellipse with properties determined by the magnetic fields.  \nWe found that the ellipses are either nearly vertical (for weaker, more inclined fields) or \ngenerally directed \\emph{away} from the magnetic field direction (for stronger,\nmore vertical fields).  Largely consistent results for two  active regions,  AR9026 and AR9057, are found.\nSome properties of the surface ellipses, e.g. their inclinations, are different for the two sunspots.\nSome of this variation may be due to  differences in the field properties. For\nexample, the field in AR9057 is on \naverage $\\sim 15\\%$ weaker than AR9026. \n\nFits of the elliptical motion in Figures~\\ref{ellipse9026} and \n\\ref{ellipse9057} depend critically on the correlation modulus which is prone \nto systematic uncertainties. But the trend is that a stronger, less \ninclined magnetic field produces elliptical motion with smaller amplitude,\neccentricity, and deviation angle, and a larger inclination from vertical.\nThese trends exist for both spots and, largely, at all frequencies.\nThe  shorter \nsemi-major axis at strong magnetic field strengths is consistent with previous \nknowledge of surface acoustic amplitude suppression in magnetic fields. It is \ncurious to note, however, that at 4 mHz the amplitude is consistently smaller \nthan even 3 mHz.\n\n\nThese are the first results to estimate the behaviour of the surface velocity \nellipse at the photosphere within sunspots.  \nThe results do not immediately suggest an observation of the slow wave as shown \nby \\inlinecite{C05} or \\inlinecite{SC06}, but are consistent with their \nexpectations of the behaviour of slow waves at this height in the atmosphere.  \nMode conversion theory states the alignment is dependent upon $a^2\/c^2$  which \nat the observational heights of $\\sim 200$km of the atmosphere may not be large \nenough to invoke alignment.  Since the ray analysis is somewhat unrealistic we \nwould expect to observe a combination of fast and slow waves at the surface, \nwhich will contribute to a clouded view of the surface velocities. \n\\inlinecite{RSWS07} are currently exploring the possibility that these apparent \nsurface effects are due to the changes in the radiative transfer within active \nregions and the formation height of the observational Ni 678 nm line. \nThis explanation requires an absorption mechanism, or else some other means\nof producing a difference between the amplitudes of upward and downward\npropagating waves. Thus mode conversion may still be important in this proposed\nmechanism. A test of the mechanism proposed by Rajaguru et al. (2007) would be to repeat\nthe observations performed here in a magnetically insensitive line, where the\nproposed radiative transfer effects would not be present. In terms of mode conversion, it is \nsuggested that the main effect occurs along the bright radial filaments of the \ninterlocking comb structure as presented in the penumbral models of \n\\inlinecite{WTBT04}. However,  observational helioseismic spatial resolution \ncannot currently resolve this.\n\n\nThis is also the first time that the variation of the phase of the local \negression correlation has been analysed in the penumbra. It is curious that the \negression correlation shows a reverse dependence when the magnetic field is \nweak and highly inclined.  This is evidence of a reverse ingression dependence \non the line-of-sight, but further investigation is required to understand the \nbehavior at high frequencies in the weaker, more inclined fields.\n\n\\vspace{2cm}\n\nThis work was supported in part by the \\textit{European Helio- and Asteroseismology Network} (HELAS).\n\n\n\n\n\n\n\n    \n\\bibliographystyle{spr-mp-sola}\n\n\n    \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\nThe Transformer~\\cite{attention-is-all-you-need} has become the dominant modeling paradigm in neural machine translation. \nIt follows the encoder-decoder framework using stacked multi-head self-attention and fully connected layers. Multi-head attention was shown to make more efficient use of the model's capacity: \nperformance of the model with 8 heads is almost 1 BLEU point higher than that of a model of the same size with single-head attention~\\cite{attention-is-all-you-need}.\nThe Transformer achieved state-of-the-art results in recent shared translation tasks \\cite{bojar-EtAl:2018:WMT1,iwslt18-overview}. Despite the model's widespread adoption and recent attempts to investigate the kinds of information learned by the model's encoder~\\cite{raganato-tiedemann:2018:BlackboxNLP}, \nthe analysis of multi-head attention and its importance for translation is challenging. \nPrevious analysis of multi-head attention considered the average of attention weights over all heads at a given position or focused only on the maximum attention weights~\\cite{voita18,tang-sennrich-nivre:2018:WMT}, but neither method explicitly takes into account the varying importance of different heads.\nAlso, this obscures the roles played by individual heads which, as we show, influence the generated translations to differing extents.\nWe attempt to answer the following questions:\n\\begin{itemize}\n    \\item To what extent does translation quality depend on individual encoder heads?\n    \\item Do individual encoder heads play consistent and interpretable roles? If so, which are the most important ones for translation quality?\n    \\item Which types of model attention (encoder self-attention, decoder self-attention or decoder-encoder attention) are most sensitive to the number of attention heads and on which layers?\n    \\item Can we significantly reduce the number of attention heads while preserving translation quality?\n\\end{itemize}\n\nWe start by identifying the most important heads in each encoder layer \nusing layer-wise relevance propagation~\\cite{lrp-ding-2017}. \nFor heads judged to be important, we then attempt to characterize the roles they perform. We observe the following types of role: positional (heads attending to an adjacent token), syntactic (heads attending to tokens in a specific syntactic dependency relation) and attention to rare words (heads pointing to the least frequent tokens in the sentence).\n\nTo understand whether the remaining heads perform vital but less easily defined roles, or are simply redundant to the performance of the model as measured by translation quality, we introduce a method for pruning heads based on~\\citet{louizos2018learning}. While we cannot easily incorporate the number of active heads as a penalty term in our learning objective (i.e.\\ the $L_0$ regularizer), we can use a differentiable relaxation. \nWe prune attention heads in a continuous learning scenario starting from the converged full model \nand identify the roles of those which remain in the model. \nThese experiments corroborate the findings of layer-wise relevance propagation;\nin particular, heads with clearly identifiable positional and syntactic functions are pruned last and hence shown to be most important for the translation task.\n\n\nOur key findings are as follows:\n\\begin{itemize}\n    \\item Only a small subset of heads are important for translation;\n    \\item Important heads have one or more specialized and interpretable functions in the model;\n    \\item The functions correspond to attention to neighbouring words and to tokens in specific syntactic dependency relations.\n\\end{itemize}\n\n\n\\section{Transformer Architecture}\nIn this section, we briefly describe the Transformer architecture~\\cite{attention-is-all-you-need} introducing the terminology used in the rest of the paper.\n\nThe Transformer is an encoder-decoder model that uses stacked self-attention and fully connected layers for both the encoder and decoder.\nThe encoder consists of $N$ layers, each containing two sub-layers: (a) a multi-head self-attention mechanism, and (b) a feed-forward network.\nThe multi-head attention mechanism relies on scaled dot-product attention, which operates on a query $Q$, a key $K$ and a value $V$:\n\\begin{equation}\n\\textnormal{Attention}(Q, K, V ) = \\textnormal{softmax}\\left(\\frac{QK^T}{\\sqrt{d_k}}\\right)V\n\\label{eq:mult_attention}\n\\end{equation}\nwhere $d_k$ is the key dimensionality. In self-attention, queries, keys and values come from the output of the previous layer.\n\nThe multi-head attention mechanism obtains $h$ (i.e.\\ one per head) different representations of ($Q$, $K$, $V$), computes scaled dot-product attention for each representation, concatenates the results, and projects the concatenation through a feed-forward layer. This can be expressed in the same notation as Equation~(\\ref{eq:mult_attention}):\n\\begin{equation}\n\\textnormal{head}_i = \\textnormal{Attention}(QW_i^Q , K W_i^K , V W_i^V )\n\\end{equation}\n\\vspace{-4ex}\n\\begin{equation}\n\\textnormal{MultiHead}(Q, K, V ) = \\textnormal{Concat}_i(\\textnormal{head}_i)W^O\n\\label{eq:concat_heads}\n\\end{equation}\nwhere the $W_i$ and $W^O$ are parameter matrices.\n\nThe second component of each layer of the Transformer network is a feed-forward network. The authors propose using a two-layer network with a ReLU activation. \n\nAnalogously, each layer of the decoder contains the two sub-layers mentioned above as well as an additional multi-head attention sub-layer. This additional sub-layer receives the output of the encoder as its keys and values.\n\nThe Transformer uses multi-head attention in three different ways: encoder self-attention, decoder self-attention and decoder-encoder attention. In this work, we concentrate primarily on encoder self-attention.\n\n\n\n\\section{Data and setting}\n\\label{sect:data_setting}\nWe focus on English as a source language and consider three target languages: Russian, German and French. For each language pair, we use the same number of sentence pairs from WMT data to control for the amount of training data and train Transformer models with the same numbers of parameters. We use 2{.}5m sentence pairs, corresponding to the amount of English--Russian parallel training data (excluding UN and Paracrawl). In Section~\\ref{sect:dependency_heads} we use the same held-out data for all language pairs; these are 50k English sentences taken from the WMT EN-FR data not used in training.\n\nFor English-Russian, we perform additional experiments using the publicly available OpenSubtitles2018 corpus~\\cite{LISON18.294} to evaluate the impact of domains on our results. \n\nIn Section~\\ref{sect:pruning_attention_heads} we concentrate on English-Russian and two domains: WMT and OpenSubtitles.\n\nModel hyperparameters, preprocessing and training details are provided in appendix~\\ref{app:exp}. \n\n\n\n\\begin{figure*}[t!h!]\n    \\centering\n    \\begin{subfigure}[b]{0.3\\textwidth}\n        \\includegraphics[width=\\textwidth]{.\/pict\/heads_lrp.png}\n        \\caption{LRP}\n        \\label{fig:heads_lrp}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.3\\textwidth}\n        \\includegraphics[width=\\textwidth]{.\/pict\/heads_mean_attn_max.png}\n        \\caption{confidence}\n        \\label{fig:heads_mean_attn_max}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.30\\textwidth}\n        \\includegraphics[width=\\textwidth]{.\/pict\/heads_functions_with_topic_ru_acc.png}\n        \\caption{head functions}\n        \\label{fig:heads_functions}\n    \\end{subfigure}\n    \\caption{Importance (according to LRP), confidence, and function of self-attention heads. In each layer, heads are sorted by their relevance according to LRP. Model trained on 6m OpenSubtitles EN-RU data.}\n   \n    \\label{fig:heads_all_info}\n\\end{figure*}\n\n\\begin{figure}\n    \\centering\n    \\begin{subfigure}[b]{0.22\\textwidth}\n       \n        \\includegraphics[width=\\textwidth]{.\/pict\/heads_lrp_de.png}\n        \\caption{LRP (EN-DE)}\n        \\label{fig:heads_lrp_de}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.22\\textwidth}     \\includegraphics[width=\\textwidth]{.\/pict\/heads_functions_with_topic_de_acc.png}\n        \\caption{head functions}\n        \\label{fig:heads_functions_de}\n    \\end{subfigure}\n    \\vskip\\baselineskip\n    \\begin{subfigure}[b]{0.22\\textwidth} \n        \\includegraphics[width=\\textwidth]{.\/pict\/heads_lrp_fr.png}\n        \\caption{LRP (EN-FR)}\n        \\label{fig:heads_lrp_fr}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.22\\textwidth}\n        \\includegraphics[width=\\textwidth]{.\/pict\/heads_functions_with_topic_fr_acc.png}\n        \\caption{head functions}\n        \\label{fig:heads_functions_fr}\n    \\end{subfigure}\n    \\caption{Importance (according to LRP) and function of self-attention heads. In each layer, heads are sorted by their relevance according to LRP. Models trained on 2{.}5m WMT EN-DE (a, b) and EN-FR (c, d).} \n   \n    \\label{fig:heads_all_info_de_fr}\n\\end{figure}\n\n\n\n\n\\section{Identifying Important Heads}\nPrevious work analyzing how representations are formed by the Transformer's multi-head attention mechanism focused on either the average or the maximum attention weights over all heads~\\cite{voita18,tang-sennrich-nivre:2018:WMT}, but neither method explicitly takes into account the varying importance of different heads.\nAlso, this obscures the roles played by individual heads which, as we will show, influence the generated translations to differing extents.\n\nWe define the ``confidence'' of a head as the average of its maximum attention weight excluding the end of sentence symbol,\\footnote{We exclude EOS on the grounds that it is not a real token.} where average is taken over tokens in a set of sentences used for evaluation (development set).\nA confident head is one that usually assigns a high proportion of its attention to a single token. Intuitively, we might expect confident heads to be important to the translation task.  \n\nLayer-wise relevance propagation (LRP)~\\cite{lrp-ding-2017} is a method for computing the relative contribution of neurons at one point in a network to neurons at another.\\footnote{A detailed description of LRP is provided in appendix \\ref{app:lrp}.} \nHere we propose to use LRP to evaluate the degree to which different heads at each layer contribute to the top-1 logit predicted by the model. Heads whose outputs have a higher relevance value may be judged to be more important to the model's predictions.\n\nThe results of LRP are shown in Figures~\\ref{fig:heads_lrp}, \\ref{fig:heads_lrp_de}, \\ref{fig:heads_lrp_fr}. \nIn each layer, LRP ranks a small number of heads as much more important than all others.\n\n\nThe confidence for each head is shown in Figure~\\ref{fig:heads_mean_attn_max}. \nWe can observe that the relevance of a head as computed by LRP agrees to a reasonable extent with its confidence. The only clear exception to this pattern is the head judged by LRP to be the most important in the first layer. It is the most relevant head in the first layer but its average maximum attention weight is low. We will discuss this head further in Section~\\ref{sec:topic_head}.\n\n\\section{Characterizing heads}\n\\label{sect:characterizing_heads}\n\nWe now turn to investigating whether heads play consistent and interpretable roles within the model.\n\n\nWe examined some attention matrices paying particular attention to heads ranked highly by LRP and identified three functions which heads might be playing:\n\\begin{enumerate}\n\\item positional: the head points to an adjacent token,\n\\item syntactic: the head points to tokens in a specific syntactic relation,\n\\item rare words: the head points to the least frequent tokens in a sentence. \n\\end{enumerate}\n\n\n\nNow we discuss the criteria used to determine if a head is performing one of these functions and examine properties of the corresponding heads. \n\n\n\n\n\\subsection{Positional heads}\\label{subsec:positional_heads}\nWe refer to a head as ``positional'' if at least 90\\% of the time its maximum attention weight is assigned to a specific relative position (in practice either -1 or +1, i.e.\\ attention to adjacent tokens). Such heads are shown in purple in Figures~\\ref{fig:heads_functions} for English-Russian, \\ref{fig:heads_functions_de} for English-German, \\ref{fig:heads_functions_fr} for English-French and marked with the relative position.\n\nAs can be seen, the positional heads correspond to a large extent to the most confident heads and the most important heads as ranked by LRP. In fact, the average maximum attention weight exceeds $0{.}8$ for every positional head for all language pairs considered here. \n\n\n\n\n\\subsection{Syntactic heads}\n\\label{sect:dependency_heads}\n\n\nWe hypothesize that, when used to perform translation, the Transformer's encoder may be responsible for disambiguating the syntactic structure of the source sentence. We therefore wish to know whether a head attends to tokens corresponding to any of the major syntactic relations in a sentence. \nIn our analysis, we looked at the following dependency relations: \nnominal subject (nsubj), direct object (dobj), adjectival modifier (amod) and adverbial modifier (advmod). These include the main verbal arguments of a sentence and some other common relations. They also include those relations which might inform morphological agreement or government in one or more of the target languages considered here. \n\n\n\n\\subsubsection{Methodology}\nWe evaluate to what extent each head in the Transformer's encoder accounts for a specific dependency relation by comparing its attention weights to a predicted dependency structure generated using CoreNLP~\\cite{manning-EtAl:2014:P14-5} \non a large number of held-out sentences. \nWe calculate for each head how often it assigns its maximum attention weight (excluding EOS) to a token with which it is in one of the aforementioned dependency relations. We count each relation separately and allow the relation to hold in either direction between the two tokens.\n\nWe refer to this relative frequency as the ``accuracy'' of head on a specific dependency relation in a specific direction.\nNote that under this definition, we may evaluate the accuracy of a head for multiple dependency relations.\n\nMany dependency relations are frequently observed in specific relative positions (for example, often they hold between adjacent tokens, see Figure~\\ref{fig:dep_posdiff_distribution}). We say that a head is ``syntactic'' if its accuracy is at least $10\\%$ higher than the baseline that looks at the most frequent relative position for this dependency relation. \n\n\n\n\\begin{figure}[t!]\n\\center{\\includegraphics[scale=0.18]{.\/pict\/dep_posdiff_distribution_wmt_no_compound.png}}\n\\caption{Distribution of the relative position of dependent for different dependency relations (WMT).}\n\\label{fig:dep_posdiff_distribution}\n\\end{figure}\n\n\n\n\n\\subsubsection{Results}\n\\label{sec:syntactic-heads:results}\n\n\n\\begin{table}[t!]\n\\centering\n\\begin{tabular}{lccc}\n\\toprule\n\\bf dep.\\!\\!&\\!\\!\\!\\! \\bf direction & \\multicolumn{2}{c}{\\bf best head \/ baseline}\\\\ \n &  & \\multicolumn{2}{c}{\\bf accuracy} \\\\ \n \\cmidrule{1-4}\n & & \\bf WMT  & \\bf OpenSubtitles\\!\\!\\!\\\\\n\\toprule\n\\multicolumn{3}{l}{nsubj}\\\\\n&\\!\\!\\!\\!v $\\rightarrow$ s & 45 \/ 35 & 77 \/ 45 \\\\\n&\\!\\!\\!\\!s $\\rightarrow$ v & 52 \/ 35 & 70 \/ 45\\\\\n\\cmidrule{1-4}\n\n\\multicolumn{3}{l}{dobj}\\\\\n&\\!\\!\\!\\!v $\\rightarrow$ o & 78 \/ 41 & 61 \/ 46\\\\\n&\\!\\!\\!\\!o $\\rightarrow$ v & 73 \/ 41 & 84 \/ 46\\\\\n\\cmidrule{1-4}\n\n\\multicolumn{3}{l}{amod}\\\\\n   &\\!\\!\\!\\!noun $\\rightarrow$ adj.m. & 74 \/ 72 & 81 \/ 80\\\\\n&\\!\\!\\!\\!adj.m. $\\rightarrow$ noun  & 82 \/ 72 & 81 \/ 80 \\\\\n\\cmidrule{1-4}\n\n\\multicolumn{3}{l}{advmod}\\\\\n   &\\!\\!\\!\\!v $\\rightarrow$ adv.m. & 48 \/ 46 & 38 \/ 33\\\\\n&\\!\\!\\!\\!adv.m. $\\rightarrow$ v   & 52 \/ 46 & 42 \/ 33\\\\\n\n\\bottomrule\n\\end{tabular}\\textbf{}\n\\caption{Dependency scores for EN-RU, comparing the best self-attention head to a positional baseline. Models trained on 2{.}5m WMT data and 6m OpenSubtitles data.} \\label{tab:dependency_head_scores}\n\\vspace{1ex}\n\\end{table}\n\n\n\\begin{figure}[t!]\n\\center{\\includegraphics[scale=0.18]{.\/pict\/dep_wmt_langs.png}}\n\\caption{Dependency scores for EN-RU, EN-DE, EN-FR each trained on 2{.}5m WMT data. \n}\n\\label{fig:wmt_deps_ru_de_fr}\n\\end{figure}\n\n\n\n\\begin{figure*}[t!]\n    \\centering\n    \\begin{subfigure}[b]{0.30\\textwidth}\n        \\includegraphics[width=\\textwidth]{.\/pict\/topic_wmt_tyumen.png}\n        \\caption{}\n        \\label{fig:topic_head_wmt_en_ru}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.30\\textwidth}\n        \\includegraphics[width=\\textwidth]{.\/pict\/topic_wmt_en_de_ruthless.png}\n        \\caption{}\n        \\label{fig:topic_head_wmt_en_de}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.30\\textwidth}\n        \\includegraphics[width=\\textwidth]{.\/pict\/topic_wmt_en_fr_collector.png}\n        \\caption{}\n        \\label{fig:topic_head_wmt_en_fr}\n    \\end{subfigure}\n    \\caption{Attention maps of the rare words head. Models trained on WMT: (a) EN-RU, (b) EN-DE, (c) EN-FR}\\label{fig:topic_head}\n\\end{figure*}\n\nTable~\\ref{tab:dependency_head_scores} shows the accuracy of the most accurate head for each of the considered dependency relations on the two domains for English-Russian. Figure~\\ref{fig:wmt_deps_ru_de_fr} compares the scores of the models trained on WMT with different target languages. \n\nClearly certain heads learn to detect syntactic relations with accuracies significantly higher than the positional baseline. This supports the hypothesis that the encoder does indeed perform some amount of syntactic disambiguation of the source sentence. \n\nSeveral heads appear to be responsible for the same dependency relation. These heads are shown in green in Figures~\\ref{fig:heads_functions}, \\ref{fig:heads_functions_de}, \\ref{fig:heads_functions_fr}. \n\n\n\nUnfortunately, it is not possible to draw any strong conclusions from these results regarding the impact of target language morphology on the accuracy of the syntactic attention heads although relations with strong target morphology are among those that are most accurately learned. \n\nNote the difference in accuracy of the verb-subject relation heads across the two domains for English-Russian. We hypothesize that this is due to the greater variety of grammatical person present\\footnote{First, second and third person subjects are encountered in approximately $6\\%$, $3\\%$ and $91\\%$ of cases in WMT data and in $32\\%$, $21\\%$ and $47\\%$ of cases in OpenSubtitles data.} in the Subtitles data which requires more attention to this relation. However, we leave proper analysis of this to future work. \n\n\n\n\\subsection{Rare words}\n\\label{sec:topic_head}\n\n\nIn all models (EN-RU, EN-DE, EN-FR on WMT and EN-RU on OpenSubtitles), we find that one head in the first layer is judged to be much more important to the model's predictions \nthan any other heads in this layer.  \n\nWe find that this head points to the least frequent tokens in a sentence. For models trained on OpenSubtitles, among sentences where the least frequent token in a sentence is not in the top-500 most frequent tokens, this head points to the rarest token in 66$\\%$ of cases, and to one of the two least frequent tokens in 83$\\%$ of cases. For models trained on WMT, this head points to one of the two least frequent tokens in more than 50$\\%$ of such cases. This head is shown in orange in Figures~\\ref{fig:heads_functions}, \\ref{fig:heads_functions_de}, \\ref{fig:heads_functions_fr}. Examples of attention maps for this head for models trained on WMT data with different target languages are shown in Figure~\\ref{fig:topic_head}.\n\n\n\n\n\n\\section{Pruning Attention Heads}\n\\label{sect:pruning_attention_heads}\n\nWe have identified certain functions of the most relevant heads at each layer and showed that to a large extent they are interpretable. What of the remaining heads? Are they redundant to translation quality or do they play equally vital but simply less easily defined roles? We introduce a method for pruning attention heads to try to answer these questions. Our method is based on \\citet{louizos2018learning}. Whereas they pruned individual neural network weights, we prune entire model components (i.e.\\ heads).  We start by describing our method and then examine how performance changes as we remove heads, identifying the functions of heads retained in the sparsified models. \n\n\n\n\n\n\\subsection{Method}\n\n\n\n\nWe modify the original Transformer architecture by multiplying the representation computed by each head$_i$ by a scalar gate $g_i$. Equation~(\\ref{eq:concat_heads}) turns into\\vspace{-2ex}\n\\begin{equation}\n\\nonumber\n\\textnormal{MultiHead}(Q,K,V )\\!=\\! \\textnormal{Concat}_i(g_i\\!\\cdot\\!\\textnormal{head}_i)W^O.\n\\end{equation}\nUnlike usual gates, $g_i$ are parameters specific to heads and are independent of the input (i.e.\\ the sentence).\nAs we would like to disable less important heads completely rather than simply downweighting them,  we would ideally apply $L_0$ regularization to the scalars $g_i$. The $L_0$ norm equals the number of non-zero components and would push the model to switch off less important heads:\n\\vspace{-1ex}\n$$\nL_0(g_1, \\ldots, g_h) = \\sum_{i=1}^{h} (1 - [[ g_i = 0 ]]),\n$$\nwhere $h$ is the number of heads, and $[[ \\quad ]]$ denotes the indicator function. \n\nUnfortunately, the $L_0$ norm is non-differentiable and so cannot be directly incorporated as a regularization term in the objective function. \nInstead, we use a stochastic relaxation: each gate $g_i$ is now a  random variable drawn independently from a head-specific distribution.\\footnote{In training, we resample gate values $g_i$ for each batch.} \nWe use the Hard Concrete distributions~\\cite{louizos2018learning},  a parameterized family of mixed discrete-continuous distributions over the closed interval $[0,1]$, see Figure~\\ref{fig:hard_concrete}. The distributions have non-zero probability mass at 0 and 1, $P(g_i = 0 | \\phi_i)$ and $P(g_i = 1 | \\phi_i)$, where $\\phi_i$ are the distribution parameters.  Intuitively, the Hard Concrete distribution is obtained by stretching the binary version of the Concrete (aka Gumbel softmax) distribution \\cite{maddison2017concrete,jang2017gumbel} from the original support of $(0, 1)$ to $(- \\epsilon, 1 + \\epsilon)$ and then collapsing the probability mass assigned to $(- \\epsilon, 1]$ and  $[1, 1 + \\epsilon)$ to single points, 0 and 1, respectively. These stretching and rectification operations yield a mixed discrete-continuous distribution over $[0, 1]$.\nNow the sum of the probabilities of heads being non-zero can be used as a relaxation of the $L_0$ norm:\n\\vspace{-1ex}\n$$\n\\vspace{-1ex}\nL_C(\\phi) = \\sum_{i=1}^{h} (1 - P(g_i = 0 | \\phi_i)).\n$$\nThe new training objective is\n$$\n\\vspace{-1ex}\nL(\\theta, \\phi) = L_{xent}(\\theta, \\phi) + \\lambda L_C(\\phi),\n$$\nwhere $\\theta$ are the parameters of the original Transformer, $L_{xent}(\\theta, \\phi)$ is cross-entropy loss for the translation model, and $L_C(\\phi)$ is the regularizer described above.  \nThe objective is easy to optimize: the reparameterization trick ~\\cite{kingma-welling-vae,pmlr-v32-rezende14} can be used to backpropagate through the sampling process for each $g_i$, whereas the regularizer and its gradients are available in the closed form.  \nInterestingly, we observe that the model converges to solutions where gates are either almost completely closed (i.e.\\ the head is pruned, $P(g_i = 0 | \\phi_i) \\approx 1$) or completely open ($P(g_i = 1 | \\phi_i) \\approx 1$), the latter not being explicitly encouraged.\\footnote{The `noise' pushes the network not to use middle values. The combination of noise and rectification has been previously used  to achieve discretization~(e.g., \\citet{kaiser2018discrete}).}\nThis means that at test time we can treat the model as a standard Transformer and use only a subset of heads.\\footnote{At test time, gate values are either 0 or 1 depending on which of the values $P(g_i = 0 | \\phi_i)$, $P(g_i = 1 | \\phi_i)$ is larger.}\n\n\n\\begin{figure}[t!]\n    \\centering\n    \\begin{subfigure}[b]{0.23\\textwidth}\n        \\includegraphics[width=\\textwidth]{.\/pict\/hard_concrete.png}\n        \\caption{}\n        \\label{fig:hard_concrete}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.23\\textwidth}\n       \\includegraphics[width=\\textwidth]{.\/pict\/hard_concrete_params.png}\n        \\caption{}\n        \\label{fig:hard_concrete_params}\n    \\end{subfigure}\n    \\caption{Concrete distribution: (a) Concrete and its stretched and rectified version (Hard Concrete); (b) Hard Concrete distributions with different parameters.}\n    \\label{fig:concrete}\n\\end{figure}\n\n\nWhen applying this regularizer, we start from the converged model trained without the $L_C$ penalty  (i.e.\\ parameters~$\\theta$ are initialized with the parameters of the converged model) and then add the gates and continue training the full objective.\nBy varying the coefficient $\\lambda$ in the optimized objective, we obtain models with different numbers of heads retained.\n\n\n\n\n\n\\subsection{Pruning encoder heads}\n\nTo determine which head functions are most important in the encoder \nand how many heads the model needs, we conduct a series of experiments with gates applied only to encoder self-attention. Here we prune a model by fine-tuning a trained model with the regularized objective.\\footnote{In preliminary experiments, we observed that fine-tuning a trained model gives slightly better results (0{.}2--0{.}6 BLEU) than applying the regularized objective, or training a model with the same number of self-attention heads, from scratch.} \nDuring pruning, the parameters of the decoder are fixed and only the encoder parameters and head gates are fine-tuned. By not fine-tuning the decoder, we ensure that the functions of the pruned encoder heads do not migrate to the decoder.\n\n\\subsubsection{Quantitative results: BLEU score}\n\n\n\\begin{figure}[t!]\n\\center{\\includegraphics[scale=0.28]{.\/pict\/heads_dying_enc_bleu_linear_x.png}}\n\\caption{BLEU score as a function of number of retained encoder heads (EN-RU). Regularization applied by fine-tuning trained model.}\n\\label{fig:enc_heads_dying_bleu}\n\\end{figure}\n\n\n\nBLEU scores are provided in  Figure~\\ref{fig:enc_heads_dying_bleu}. Surprisingly, for OpenSubtitles, we lose only $0{.}25$ BLEU when we prune all but 4 heads out of 48.\\footnote{If all heads in a layer are pruned, the only remaining connection to the previous layer is the residual connection.} For the more complex WMT task, 10 heads in the encoder are sufficient to stay within $0{.}15$ BLEU of the full model.\n\n\n\\subsubsection{Functions of retained heads}\n\nResults in Figure~\\ref{fig:enc_heads_dying_bleu} suggest that the encoder remains effective even with only \na few heads. In this section, we investigate the function of those heads that remain in the encoder during pruning. \nFigure~\\ref{fig:encoder_heads_dying} shows all heads color-coded for their function \nin a pruned model.\nEach column corresponds to a model \nwith a particular number of heads retained after pruning.\nHeads from all layers are ordered by their function. Some heads can perform several functions (e.g., $s\\rightarrow v$ and $v\\rightarrow o$); in this case the number of functions is shown.\n\n\\begin{figure}[t!]\n\\center{\\includegraphics[scale=0.17]{.\/pict\/encoder_heads_dying_syntactic.png}}\n\\caption{Functions of encoder heads retained after pruning. Each column represents all remaining heads after varying amount of pruning (EN-RU; Subtitles).}\n\\label{fig:encoder_heads_dying}\n\\end{figure}\n\nFirst, we note that the model with 17 heads retains heads with all the functions that we identified in Section~\\ref{sect:characterizing_heads}, even though \\sfrac{2}{3} of the heads have been pruned.\n\nThis indicates that these functions are indeed the most important. Furthermore, when we have fewer heads in the model, some functions ``drift'' to other heads: for example, we see positional heads starting to track syntactic dependencies; hence some heads are assigned more than one color at certain stages in Figure~\\ref{fig:encoder_heads_dying}.\n\n\n\n\n\n\n\\subsection{Pruning all types of attention heads}\nWe found our pruning technique to be efficient at reducing the number of heads in the encoder without a major drop in translation quality. Now we investigate the effect of pruning all types of attention heads in the model (not just in the encoder). This allows us to evaluate the importance of different types of attention in the model for the task of translation.\nIn these experiments, we add gates to all multi-head attention heads in the Transformer, i.e.\\ encoder and decoder self-attention and attention from the decoder to the encoder.\n\n\\subsubsection{Quantitative results: BLEU score}\n\n\\begin{table}[t!]\n\\centering\n\\begin{tabular}{lccc}\n\\toprule\n & \\bf attention & \\multicolumn{2}{c}{\\bf BLEU} \\\\ \n & \\bf heads & from & from \\\\ \n & (e\/d\/d-e) & trained & scratch \\\\ \n \\toprule\n\\multicolumn{3}{l}{\\bf WMT, 2{.}5m}\\\\\n\\cmidrule{1-4}\nbaseline   &  48\/48\/48 & \\multicolumn{2}{c}{\\bf 29{.}6} \\\\\n\\cmidrule{1-4}\nsparse heads   & 14\/31\/30 & 29{.}62 & 29{.}47 \\\\\n   & 12\/21\/25 & 29{.}36 & 28{.}95 \\\\\n   & 8\/13\/15 & 29{.}06 & 28{.}56 \\\\\n   & 5\/9\/12 & 28{.}90 & 28{.}41 \\\\\n\n\\toprule\n\\multicolumn{3}{l}{\\bf OpenSubtitles, 6m}\\\\\n \\cmidrule{1-4}\nbaseline   &  48\/48\/48 & \\multicolumn{2}{c}{\\bf 32{.}4} \\\\\n\\cmidrule{1-4}\nsparse heads   & 27\/31\/46 & 32{.}24 & 32{.}23 \\\\\n   & 13\/17\/31 & 32{.}23 & 31{.}98 \\\\\n   & 6\/9\/13 & 32{.}27 & 31{.}84 \\\\\n\\bottomrule\n\\end{tabular}\\textbf{}\n\\caption{BLEU scores for gates in all attentions, EN-RU. Number of attention heads is provided in the following order: encoder self-attention, decoder self-attention, decoder-encoder attention.}\n\\label{tab:bleu_all_enru}\n\\end{table}\n\nResults of experiments pruning heads in all attention layers are provided in Table~\\ref{tab:bleu_all_enru}. For models trained on WMT data, we are able to prune almost \\sfrac{3}{4} of encoder heads and more than \\sfrac{1}{3} of heads in decoder self-attention and decoder-encoder attention without any noticeable loss in translation quality (sparse heads, row 1). We can also prune more than half of all heads in the model and lose no more than 0{.}25 BLEU.\n\n\nWhile these results show clearly that the majority of attention heads can be removed from the fully trained model without significant loss in translation quality, it is not clear whether a model can be trained from scratch with such a small number of heads. In the rightmost column in Table~\\ref{tab:bleu_all_enru} we provide BLEU scores for models trained with exactly the same number and configuration of heads in each layer as the corresponding pruned models but starting from a random initialization of parameters. \nHere the degradation in translation quality is more significant than for pruned models with the same number of heads. This agrees with the observations made in works on model compression: sparse architectures learned through pruning cannot be trained from scratch to the same test set performance as a model trained with joint sparsification and optimization~\\cite{zhu2017prune,state-of-sparcity-2019}. In our case, attention heads are less likely to learn important roles when a model is retrained from scratch with a small number of heads.\n\n\n\n\n\n\n\\subsubsection{Heads importance}\nFigure~\\ref{fig:all_heads_dying_by_attn_type} shows  the number\nof retained heads for each attention type at different pruning rates.\nWe can see that the model prefers to prune encoder self-attention heads first, while decoder-encoder attention heads appear to be the most important for both datasets. Obviously, without decoder-encoder attention no translation can happen.\n\nThe importance of decoder self-attention heads, which function primarily as a target side language model, varies across domains. These heads appear to be almost as important as decoder-encoder attention heads for WMT data with its long sentences (24 tokens on average), and slightly more important than encoder self-attention heads for OpenSubtitles dataset where sentences are shorter (8 tokens on average).\n\n\\begin{figure}[t!]\n\\center{\\includegraphics[scale=0.28]{.\/pict\/heads_dying_by_attn_type_both.png}}\n\\caption{Number of active heads of different attention type for models with different sparsity rate}\n\\label{fig:all_heads_dying_by_attn_type}\n\\end{figure}\n\n\n\\begin{figure}[t!]\n\\center{\\includegraphics[scale=0.28]{.\/pict\/heads_dying_decoder.png}}\n\\caption{Number of active heads in different layers of the decoder for models with different sparsity rate (EN-RU, WMT)}\n\\label{fig:all_heads_dying_decoder}\n\\end{figure}\n\n\nFigure~\\ref{fig:all_heads_dying_decoder} shows the number of active self-attention and decoder-encoder attention heads at different layers in the decoder \nfor models with different sparsity rate\n(to reduce noise, we plot the sum of heads remaining in pairs of adjacent layers). It can be seen that self-attention heads are retained more readily in the lower layers, while decoder-encoder attention heads are retained in the higher layers. This suggests that lower layers of the Transformer's decoder are mostly responsible for language modeling, while higher layers are mostly responsible for conditioning on the source sentence. These observations are similar for both datasets we use.\n\n\n\n\n\n\\section{Related work}\n\n\nOne popular approach to\nthe analysis of NMT representations is to evaluate how informative they are for various linguistic tasks. Different levels of linguistic analysis have been considered including morphology~\\cite{P17-1080,dalvi-morph-decoder,bisazza-tump:2018:EMNLP}, syntax~\\cite{D16-1159} and semantics~\\cite{hill-et-al,belinkov-I17-1001,raganato-tiedemann:2018:BlackboxNLP}.\n\n\\citet{bisazza-tump:2018:EMNLP} showed that the target language determines which information gets encoded. This agrees with our results for different domains on the English-Russian translation task in Section~\\ref{sec:syntactic-heads:results}. There we observed that attention heads are more likely to track syntactic relations requiring more complex agreement in the target language (in this case the subject-verb relation).\n\nAn alternative method to study the ability of language models and machine translation models to capture hierarchical information is to test their sensitivity to specific grammatical errors  \\cite{linzen-Q16-1037,colorless-green,tran-importance-of-being-recurrent,E17-2060,tang18-self-attention}.\nWhile this line of work has shown that NMT models, including the Transformer, do learn some syntactic structures, our work provides further insight into the role of multi-head attention.\n\nThere are several works analyzing attention weights of different NMT models~\\cite{monz-attention,voita18,tang-sennrich-nivre:2018:WMT,raganato-tiedemann:2018:BlackboxNLP}. \\citet{raganato-tiedemann:2018:BlackboxNLP} use the self-attention weights of the Transformer's encoder to induce a tree structure for each sentence \nand compute the unlabeled attachment score of these trees. However they do not evaluate specific syntactic relations (i.e.\\ labeled attachment scores) or consider how different heads specialize to specific dependency relations. \n\nRecently \\citet{bau2019neurons-in-mt} proposed a method for identifying important individual neurons in NMT models. They show that similar important neurons emerge in different models. Rather than verifying the importance of individual neurons, we identify the importance of entire attention heads using layer-wise relevance propagation and verify our findings by observing which heads are retained when pruning the model.\n\n\n \n\\section{Conclusions}\n\n\nWe evaluate the contribution made by individual attention heads to Transformer model performance on translation. \nWe use layer-wise relevance propagation to show that the relative contribution of heads varies: only a small subset of heads appear to be important for the translation task.\nImportant heads have one or more interpretable functions in the model, including attending to adjacent words and tracking specific syntactic relations.\nTo determine if the remaining less-interpretable heads are crucial to the model's performance, we introduce a new approach to pruning attention heads. \n\n\n\nWe observe that specialized heads are the last to be pruned, confirming their importance directly.\nMoreover, the vast majority of heads, especially the encoder self-attention heads, can be removed without seriously affecting performance.\nIn future work, we would like to investigate how our pruning method compares to alternative methods of model compression in NMT. \n\n\n\\section*{Acknowledgments}\nWe would like to thank anonymous reviewers for their comments. We thank Wilker Aziz, Joost Bastings for their helpful suggestions. \nThe authors also thank Yandex Machine Translation team for helpful discussions and inspiration. Ivan Titov acknowledges support of the European Research Council (ERC StG BroadSem 678254) and the Dutch National Science Foundation (NWO VIDI 639.022.518). \n\n\n\\nocite{training-tips-transformer}\n\\nocite{adam-optimizer}\n\\nocite{sennrich-bpe}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nAt the fundamental level, quantum field theory is local. Still,  in many situations non-locality emerges as a derived phenomenon. This can happen already  at a purely classical level, when one\nintegrates out some fast degree of freedom to obtain an effective theory for the slow degrees of freedom, or at the quantum level, when one considers the effective action that takes into account loop corrections involving light or massless particles. In recent years there  has been a significant activity on effective non-local modifications of gravity, largely motivated by the aim of understanding the origin of dark energy.\nIn particular, in \\cite{Maggiore:2013mea} (elaborating on previous works related to\nthe degravitation idea~\\cite{ArkaniHamed:2002fu,Dvali:2006su,Dvali:2007kt}, as well as on attempts at writing massive gravity in non-local form~\\cite{Porrati:2002cp,Jaccard:2013gla})\nwe proposed a phenomenological  modification of gravity, based on\nthe non-local equation of motion\n\\be\\label{RT}\nG_{\\mu\\nu} -(1\/3)m^2\\(g_{\\mu\\nu}\\Box^{-1} R\\)^{\\rm T}=8\\pi G\\,T_{\\mu\\nu}\\, .\n\\ee\nThe superscript T denotes the operation of taking the transverse part of a tensor (which is itself a nonlocal operation), $\\Box$ is the covariant d'Alembertian computed with the curved-space metric $g_{\\mu\\nu}$, and its inverse\n$\\Box^{-1}$ is defined using the retarded Green's function, to ensure causality. The factor $1\/3$ is a convenient normalization of the parameter $m^2$ in $d=3$ spatial dimensions. The extraction of the transverse part ensures that energy-momentum conservation is  automatically satisfied. A closed form for the action corresponding to \\eq{RT} is not known. This model is however  closely related to another non-local model, proposed in \\cite{Maggiore:2014sia}, and defined by the action\n\\be\\label{RR}\nS_{\\rm NL}=\\frac{m_{\\rm Pl}^2}{2}\\int d^{4}x \\sqrt{-g}\\, \n\\[R-\\frac{1}{6} m^2R\\frac{1}{\\Box^2} R\\]\\, .\n\\ee\nThese two models are related by the fact that,  linearizing over flat space the equations of motion derived from the action (\\ref{RR}), one finds the same equations of motion as those obtained by linearizing \\eq{RT} over flat space. However, at the full non-linear level, or linearizing over a background different from Minkowski, the two theories are different. \n\nAn intriguing aspect of the  models (\\ref{RT}) and (\\ref{RR}) is that, in the comparison with cosmological observations,  they perform remarkably well. They both have the same number of parameters as $\\Lambda$CDM, with the mass $m$ replacing the cosmological constant. They dynamically generate  a dark energy and have a realistic background FRW evolution \\cite{Maggiore:2013mea,Maggiore:2014sia,Foffa:2013vma}. Their cosmological perturbations are well-behaved (a step that already ruled out several modified gravity models)\nand their quantitative effects are consistent with CMB, supernovae, BAO and structure formation data~\\cite{Nesseris:2014mea,Dirian:2014ara,Barreira:2014kra}.\\footnote{This should be contrasted with the non-local model proposed in \\cite{Deser:2007jk,Deser:2013uya,Woodard:2014iga}. This model does not involve a mass scale, and is rather \nconstructed adding to the Einstein-Hilbert action a term of the form $Rf(\\Box^{-1} R)$. The function $f(\\Box^{-1} R)$ is  tuned so that, at the level of background evolution, this model closely mimics $\\Lambda$CDM. One can then study its\ncosmological perturbations, and  it has been found in \n\\cite{Dodelson:2013sma} that the  model is ruled out  by the comparison with structure formation.\nNon-local long-distance modifications of GR have also been suggested in \\cite{Barvinsky:2003kg,Barvinsky:2011hd,Barvinsky:2011rk,Wetterich:1997bz}.}\nThis allowed  a more detailed comparison with $\\Lambda$CDM. Implementing the cosmological perturbations of the non-local models into a Boltzmann code and performing parameter estimation and a global fit to CMB, supernovae and BAO data, one finds that these models perform as well as $\\Lambda$CDM, with  comparable values of the $\\chi^2$   \\cite{Dirian:2014bma} (in fact for model (\\ref{RT}) the $\\chi^2$ is even better than for $\\Lambda$CDM, although not at a statistically significant level). Furthermore, parameter estimation provides a value of the Hubble constant $H_0$ slightly higher than in $\\Lambda$CDM, in better agreement with that obtained from the most recent local measurements \\cite{Rigault:2014kaa}. \nTo the best of our knowledge, among the large\nvariety of existing  alternatives to the standard $\\Lambda$CDM  paradigm,\nthe models defined by  eqs.~(\\ref{RT}) or (\\ref{RR}) are the only  ones\nthat are competitive with $\\Lambda$CDM from the point of view of fitting current observations (at a level of accuracy which tests not only the background evolution but also the cosmological perturbations of the model), without being just an extension of $\\Lambda$CDM with extra free parameters (see also \\cite{Ade:2015rim}). Further conceptual and phenomenological aspects of these models have been discussed in \\cite{Foffa:2013sma,Kehagias:2014sda,Cusin:2014zoa,Dirian:2014xoa,Modesto:2013jea,Conroy:2014eja,Barreira:2015fpa,Barreira:2015vra}.\n\nThese non-local models have been proposed on a purely phenomenological ground, and it is of course important to eventually understand how they could emerge from a fundamental theory. The purpose of this paper is to investigate the possibility that such terms emerge from   infrared (IR) effects in  gravity. We will first discuss  the idea of generating the dark-energy scale from the  running of asymptotically-free coupling constants in $R^2$ extensions of Einstein gravity. We will then explore the possibility of generating similar infrared effects in Einstein gravity, with no $R^2$ terms, just as a consequence of the higher-derivative terms generated by the conformal anomaly.\n\nThe plan of the paper is as follows. In Section~\\ref{sect:one} we recall known results on the running of the couplings associated to the ${\\cal O}(R^2)$ terms. In sect.~\\ref{sect:IR} we discuss the possible IR dynamics that could be generated by the running of these coupling constants, and the  connection with the non-local models (\\ref{RT}) and (\\ref{RR}). Just as in QCD, the running of a coupling that becomes strong in the IR generates,  by dimensional transmutation, a mass scale $\\Lambda_{\\rm RR}$, analogous to $\\Lambda_{\\rm QCD}$ for strong interaction. \nIn sect.~\\ref{sect:sol} we show that the dynamical emergence of  this mass scale can have remarkable consequences on several aspects of the ``cosmological constant\" problem. An important role in the picture that we will develop is played by the conformal mode of the metric. Then, \nin sect.~\\ref{sect:conf} we examine in more detail the dynamics of the conformal mode in $R^2$ extensions of GR. Finally,\nin sect.~\\ref{sect:anomaly} we will consider  Einstein gravity, without $R^2$ terms, and the possibility that similar effects arise in this case from the higher-derivative term provided by the anomaly-induced effective action. In sect.~\\ref{sect:concl} we summarize our results.\n\n\\section{Non-local loop corrections in $R^2$ gravity}\\label{sect:one}\n\n\\subsection{Notations and conventions}\n\nWe consider the theory with action \n\\bees\\label{SEH}\nS&=&S_{\\rm EH}+S_{\\rm HD}\\nonumber\\\\\n&=& \\int d^4x \\sqrt{-g}\\, \\[ \\frac{m_{\\rm Pl}^2}{2}(R-2\\Lambda)-  \\(a_1C^2+a_2R^2+a_3 E\\)\\]\\, ,\\label{SHD}\n\\ees\nwhere $m_{\\rm Pl}=1\/(8\\pi G)^{1\/2}$ is the (reduced) Planck mass,  $S_{\\rm EH}$ \nis the Einstein-Hilbert (EH) action with a cosmological constant, and $S_{\\rm HD}$ is the higher-derivative term.\nHere \n\\be\nC^2=R_{\\mu\\nu\\rho\\sigma}^2-2R_{\\mu\\nu}^2+(1\/3)R^2\\, \n\\ee \nis the square of the Weyl tensor, and\n\\be\\label{defE}\nE=R_{\\mu\\nu\\rho\\sigma}^2-4R_{\\mu\\nu}^2+R^2\\, \n\\ee \nis the  Gauss-Bonnet term. Observe that $a_1$, $a_2$ and $a_3$ are dimensionless. We use the MTW sign conventions \\cite{MTW}, so in particular $\\eta_{\\mu\\nu}=(-,+,+,+)$. The overall minus sign in front of the higher-derivative terms in \\eq{SHD} is part of the  definition of the $a_i$ coefficients, and is chosen so that these terms will appear with a plus sign in the euclidean action, see \\eq{Seucl} below. Loop computations are indeed  typically performed with euclidean signature, and it will be important for us to be careful about the signs in the passage from Minkowskian to euclidean signature. The rotation to euclidean signature $(+,+,+,+)$ is obtained performing the  Wick rotation, i.e. introducing euclidean time $t_{\\rm E}$ from  $t_{\\rm E}=it$. Then $d^4x\\rightarrow -i(d^4x)_{\\rm E}$, while $\\sqrt{-g}\\rightarrow g^{1\/2}$ and $R$ transforms into the Ricci curvature computed with the euclidean metric, $R_{\\rm E}$, without extra minus signs, and similarly for the Riemann and Ricci tensors.  \nThen, \n\\be\n(S_{\\rm EH}+S_{\\rm HD})_{\\rm Mink}=-i \\int (d^4x)_{\\rm E} \\,g^{1\/2}\\,\\[ \\frac{m_{\\rm Pl}^2}{2} (R_{\\rm E}-2\\Lambda)\n-\\(a_1C_{\\rm E}^2+a_2R_{\\rm E}^2+a_3 E_{\\rm E}\\)\\]\\, .\n\\ee\nDefining the euclidean action $S_{\\rm Eucl}$ from \n$e^{iS_{\\rm Mink}}=e^{-S_{\\rm Eucl}}$, i.e. $S_{\\rm Eucl}=-iS_{\\rm Mink}$, we therefore find\n\\be\\label{Seucl}\nS_{\\rm Eucl}=\\int (d^4x)_{\\rm E}\\, g^{1\/2}\\,\\[ -\\frac{m_{\\rm Pl}^2}{2} (R_{\\rm E}-2\\Lambda)+a_1C_{\\rm E}^2+a_2R_{\\rm E}^2+a_3 E_{\\rm E}\\]\\, .\n\\ee\nWe will henceforth drop the subscript $E$ from euclidean quantities. Whether an action is written in Minkowskian or euclidean signature can be understood from the presence of the factor $\\sqrt{-g}$ or $g^{1\/2}$, respectively.\nWe will generically denote by $R^2$ gravity  the theory containing the Einstein-Hilbert action plus all the  terms $R^2$, $C^2$ and $E$. \n\n\\subsection{$R^2$ gravity at low energies}\\label{4sect:npert}\n\nLet us first consider $R^2$ gravity in an effective field theory approach, valid in the limit $E\\ll M_{\\rm Pl}$. In this case the ${\\cal O}(R^2)$ terms are simply seen as the first corrections that we meet as we approach the Planck scale.  More precisely, if we linearize over flat space, $g_{\\mu\\nu}=\\eta_{\\mu\\nu}+h_{\\mu\\nu}$, the Lagrangian density reads,\nschematically (i.e. displaying only the powers of $h_{\\mu\\nu}$ and of derivatives, without writing explicitly any tensor structure nor numerical factors), \n\\be\\label{LEHLHD}\n{\\cal L}_{\\rm EH}+{\\cal L}_{\\rm HD}\\sim m_{\\rm Pl}^2 (h\\pa^2 h +h\\pa h\\pa h+\\ldots)\n-a \\[ (\\Box h)^2 +h  (\\Box h)^2 +\\ldots \\]\n\\ee\nwhere, for the purpose of this schematic counting,  $a$ denotes generically $a_1$ or $a_2$ (the Gauss-Bonnet term is a topological invariant and we will neglect it for most of the following discussion). In an effective field theory approach the degrees of freedom are read from the Einstein-Hilbert term, so in the spectrum we only have the massless graviton. Canonical normalization is also performed with respect to the Einstein-Hilbert term, i.e. rescaling $h_{\\mu\\nu} \\rightarrow h_{\\mu\\nu}\/m_{\\rm Pl}$, and therefore the Lagrangian density is rewritten as\n\\be\\label{schemL}\n{\\cal L}_{\\rm EH}+{\\cal L}_{\\rm HD}\\sim \\( h\\pa^2 h +\\frac{1}{m_{\\rm Pl}} h\\pa h\\pa h+\\ldots\\)\n-\\frac{a}{m_{\\rm Pl}^2} \\[ (\\Box h)^2 +\\frac{1}{m_{\\rm Pl}} h  (\\Box h)^2 +\\ldots \\]\\, .\n\\ee\nWe see that the contribution to any process from the higher-derivative terms is suppressed, compared to the contributions coming from to the EH action, by a factor of order $aE^2\/m_{\\rm Pl}^2$. The standard lore is therefore that these corrections \nare irrelevant in the infrared, where $E\\llm_{\\rm Pl}$.  \n\nThe idea that we want to explore in this paper is that, in an effective action that derives from a  full quantum theory of gravity,  $a_1$ and $a_2$ are dimensionless coupling constants that run under renormalization group, so  \n$a_i=a_i(E)$. Depending on the sign of their beta functions, their absolute values can either grow or decrease as we run toward the IR. If a coupling grows in the IR, i.e. if the sign of its beta function corresponds to asymptotic freedom in the UV, we  meet a situation conceptually similar to what happens in QCD. Running toward the IR this coupling will enter a strong coupling regime. The energy scale at which this happens defines a dynamically generated scale, analogous to $\\Lambda_{\\rm QCD}$. At energies below this new scale,  the functions $a_i(E)$ have a behavior which has nothing to do with the one computed in perturbation theory, and  can in principle even develop singularities as $E\\rightarrow 0$. In particular, if in this regime $a_2(E)\\propto E^{-4}$,  in coordinate space we would have in the action a term  $Ra_2(\\Box)R\\propto R\\Box^{-2}R$ (see sect.~\\ref{sect:NLoneloop} below), which would indeed reproduce the model (\\ref{RR}). \n\nIn this section we investigate this scenario. We will first discuss  the running of the couplings  associated to the ${\\cal O}(R^2)$ terms, and we will then give arguments, drawing  on the known behavior of singularities in the QCD Green's functions, as well as on the dynamics of the conformal mode in $R^2$ gravity, that suggest that a IR behavior $a_2(\\Box)\\propto \\Box^{-2}$ is indeed  plausible.  \nWe begin  by reviewing the known results  on perturbative loop corrections to the $a_i$ couplings. \n\n\\subsection{Non-local form factors at one loop}\\label{sect:NLoneloop}\n\n\nIn general in QFT the  running of coupling constants, which is more commonly expressed in momentum space, can also be expressed in terms of non-local form factors in the effective action. For instance, at the one-loop level the running of the electric charge in QED can be described in coordinate space by the effective action~\\cite{Barvinsky:1987uw} (see also \\cite{Dalvit:1994gf,Lombardo:1996gp})\n\\be\\label{qed}\nS_{\\rm eff}=-\\frac{1}{4}\\int d^4x\\, F_{\\mu\\nu}\\frac{1}{e^2(\\Box)}F^{\\mu\\nu}\\, ,\n\\ee\nwhere\n\\be\n\\frac{1}{e^2(\\Box)}=\\frac{1}{e^2(\\mu)}-\\beta_0\\log\\(\\frac{-\\Box}{\\mu^2}\\)\\, .\n\\ee\nHere $\\mu$ is the renormalization scale, $e(\\mu)$ is the renormalized charge at the scale $\\mu$  and, for a single massless fermion, $\\beta_0=1\/(12\\pi^2)$. The logarithm of the d'Alembertian can be defined for instance from \n\\be\n\\log\\(\\frac{-\\Box}{\\mu^2}\\)=\\int_0^{\\infty}dm^2\\, \\[\\frac{1}{m^2+\\mu^2}-\n\\frac{1}{m^2-\\Box}\\]\\, .\n\\ee\nSuch non-local actions generate non-local effective\nequations of motion  for the expectation values of the quantum fields. The corresponding equations of motion for the in-out matrix elements depend on the Feynman propagator and are acausal, but the effective classical equations of motion for the in-in expectation values, which can be computed using the Schwinger-Keldysh formalism, involve the retarded propagator and are therefore non-local but causal\\cite{Jordan:1986ug,Calzetta:1986ey}. In practice, using the in-in formalism turns out to be equivalent to performing the computation in euclidean space, and in the end rotating to Minkowskian signature while at the same time replacing the euclidean $\\Box^{-1}$ with the Minkowskian $\\Box^{-1}$ computed with the retarded Green's function, as discussed in \\cite{Barvinsky:1987uw} and recently verified explicitly, for the case of $R^2$ gravity, in \\cite{Donoghue:2014yha}.\n\nNon-local terms also appear in the renormalization of gravity with higher derivatives, as was first  revealed using the heat-kernel technique in \\cite{Barvinsky:1985an, Barvinsky:1987uw}. The result has also been  checked with standard Feynman diagram computations~\\cite{Gorbar:2002pw,Gorbar:2003yt}. These non-local terms appear  when we consider gravity semiclassically and quantize massless scalars, massless spinors or massless vector fields over a fixed curved background. Furthermore, they also appear when we quantize gravity itself, as a result of the contribution of gravitons to the quantum loops. The contribution from matter field is very well understood, and summarized in textbooks such as~\\cite{Birrell:1982ix,Buchbinder:1992rb}. In contrast, a consistent computation of the running due to graviton loops is a much more subtle issue. We examine these contributions separately.\n\n\\subsubsection{Loop corrections from matter fields}\n\nTo lowest perturbative order, the loop corrections induced by massless matter fields can be computed in semiclassical gravity, i.e. on a fixed curved background. \nUsing $R_{\\mu\\nu\\rho\\sigma}^2$, $R_{\\mu\\nu}^2$ and $R^2$ as  independent terms, the non-local correction to the euclidean action at one-loop takes the form\\footnote{We follow the notation in \\cite{Donoghue:2014yha}, except that we define the constants $\\alpha,\\beta,\\gamma$ extracting an overall factor $1\/[2(4\\pi)^2]$ in front of $S_{\\rm NL}^{\\rm 1-loop}$, which makes easier the comparison with most of the literature, e.g. with  refs.~\\cite{Gorbar:2002pw,Gorbar:2003yt}.}\n\\bees\nS_{\\rm NL}^{\\rm 1-loop}&=&\\frac{1}{2(4\\pi)^2}\\, \\int d^4x\\, g^{1\/2}\\, \\[\\alpha \\,R\\log\\(\\frac{\\Box}{\\mu^2}\\)R+\n\\beta R_{\\mu\\nu}\\log\\(\\frac{\\Box}{\\mu^2}\\)R^{\\mu\\nu}\\right. \\nonumber\\\\\n&&\\hspace{3.4cm}\\left.+\\gamma\\, R_{\\mu\\nu\\rho\\sigma}\\log\\(\\frac{\\Box}{\\mu^2}\\)R^{\\mu\\nu\\rho\\sigma}\\]\\, ,\n\\ees\nwhere $\\Box$ is the  generally covariant d'Alembertian in euclidean space. Switching to the basis of $R^2,C^2,E$,\nthis expression can be rewritten as\n\\bees\nS_{\\rm NL}^{\\rm 1-loop}&=&\\frac{1}{2(4\\pi)^2}\\, \\int d^4x\\, g^{1\/2}\\, \\[\\bar{\\alpha} \\,R\\log\\(\\frac{\\Box}{\\mu^2}\\)R+\n\\bar{\\beta}\\, C_{\\mu\\nu\\rho\\sigma}\\log\\(\\frac{\\Box}{\\mu^2}\\)C^{\\mu\\nu\\rho\\sigma}\\right. \\nonumber\\\\\n&&\\hspace*{-1.4cm}\\left.+\\bar{\\gamma}\n\\(  R_{\\mu\\nu\\rho\\sigma}\\log\\(\\frac{\\Box}{\\mu^2}\\)R^{\\mu\\nu\\rho\\sigma} -4  R_{\\mu\\nu}\\log\\(\\frac{\\Box}{\\mu^2}\\)R^{\\mu\\nu}\n+R\\log\\(\\frac{\\Box}{\\mu^2}\\)R \\)\n\\]\\, ,\\label{4SNL1loop}\n\\ees\nwhere $C_{\\mu\\nu\\rho\\sigma}$ is the Weyl tensor, while the last structure corresponds to the Gauss-Bonnet term (\\ref{defE}).\nThe relation between the coefficients is \\cite{Donoghue:2014yha}\n$\n\\alpha=\\bar{\\alpha}+\\frac{\\bar{\\beta}}{3}+\\bar{\\gamma}$, \n$\\beta=-2\\bar{\\beta}-4\\bar{\\gamma}$,\n$\\gamma=\\bar{\\beta}+\\bar{\\gamma}$.\nThe  value of the coefficients \n$\\{\\bar{\\alpha},\\bar{\\beta}, \\bar{\\gamma}\\}$  can be found e.g. in \\cite{Birrell:1982ix} (since the logarithmic terms are fixed by the $1\/\\epsilon$ divergences of the effective action). In Table~\\ref{tab1} we collect the values of $\\{\\bar{\\alpha},\\bar{\\beta}, \\bar{\\gamma}\\}$  due to a massless scalar with a generic non-minimal coupling $\\xi$ to the curvature, described by the euclidean action\n\\be\nS_s=\\frac{1}{2} \\int d^4x\\, g^{1\/2}\\, \\(g^{\\mu\\nu}\\pa_{\\mu}\\phi\\pa_{\\nu}\\phi+\\xi R\\phi^2\\)\\, ,\n\\ee\nas well as the contributions from massless spinors and  massless vector fields. The value of the scalar field is given in terms of the parameter $\\bar{\\xi}=\\xi-(1\/6)$, and $\\bar{\\xi}=0$ corresponds to a conformally coupled scalar field. \n\n\\begin{table}[t]\n\\begin{center}\n\\begin{tabular}{|l|c|c|c|} \n\\hline \n\\phantom{$\\[ \\frac{(A^2)}{B}\\]$}            &$\\bar{\\alpha}$         & $\\bar{\\beta}$   &$  \\bar{\\gamma}$\\\\\n             \\hline\nscalar\\phantom{$\\[ \\frac{(A^2)}{B}\\]$}       &$(1\/2) \\bar{\\xi}^2$ &   1\/120             & $-1\/360$\\\\ \\hline\nfermion                                                           & 0                              &  6\/120             & $-11\/360$\\\\ \\hline\nvector                                                              & 0                             &  12\/120           & $-62\/360$\\\\ \\hline\n\\end{tabular}  \n\\end{center}\n\\caption{The value of the parameters $\\{\\bar{\\alpha},\\bar{\\beta}, \\bar{\\gamma}\\}$ due to loops of   massless matter fields. \n\\label{tab1}}\n\\end{table}\n\nThe cosmological effect of these non-local terms has  been recently studied in \\cite{Donoghue:2014yha}. Even if enhanced by a logarithmic factor, a term $R\\log(-\\Box\/\\mu^2)R$ can compete with the Einstein-Hilbert term $m_{\\rm Pl}^2R$ only for nearly Planckian curvatures. Therefore, the inclusion of the UV form of the loop corrections can be relevant for addressing issues such as the resolution of cosmological singularities near the big-bang, or the emergence of the classical behavior from the full quantum theory, which are the issues addressed in  \\cite{Donoghue:2014yha}.\\footnote{See also \\cite{Rinaldi:2014gha}, where one-loop corrections to the $R^2$ theory computed expanding over de~Sitter space, and involving corrections of the form  $R^2\\log R\/\\mu^2$, are used to construct an early-universe  inflationary model. Non-local extension of the $R^2$ theory relevant in the UV have also  been studied, with different motivations, in several papers, see e.g. \\cite{Biswas:2011ar,Modesto:2011kw}.}\nHere we are rather interested in the opposite limit  of curvatures much smaller than $m_{\\rm Pl}$ and  very large distances, i.e. in the far IR, with the aim of understanding the origin of dark energy.\n\nIt is also  instructive to compute the one-loop renormalization of the $R^2$ and $C^2$ terms induced by massive (rather than massless) matter fields in a curved background, to understand the interplay between non-locality  and the  mass of the field. This computation has been \nperformed  in \\cite{Gorbar:2002pw,Gorbar:2003yt} for scalar, spinor and vector fields (neglecting the renormalization of the Gauss-Bonnet term), using a mass-dependent renormalization scheme that allows one to correctly recover the decoupling of massive fields in the limit of small momenta. \nConsider for instance a real scalar field  with mass $m_s$ and a generic non-minimal coupling $\\xi$, described  by the euclidean action\n\\be\nS_s=\\frac{1}{2} \\int d^4x\\, g^{1\/2}\\, \\(g^{\\mu\\nu}\\pa_{\\mu}\\phi\\pa_{\\nu}\\phi+m_s^2\\phi^2+\\xi R\\phi^2\\)\\, .\n\\ee\nThe corresponding one-loop contribution to the euclidean effective action \nis \\cite{Gorbar:2002pw,Gorbar:2003yt}\\footnote{This result was  first obtained in \\cite{Gorbar:2002pw}  using \na non-covariant computation based on an expansion of the action over flat space, and it was then checked in the same paper using the covariant heat-kernel technique.  In the covariant technique the quantity which is computed is $\\bar{\\Gamma}^{(1)}$, defined by \n$\\det^{-1\/2}( -\\Box+m^2+\\xi R) =\\exp\\{\\bar{\\Gamma}^{(1)}\\}$. Since the euclidean action enters as $e^{-S_{\\rm Eucl}}$, we have $S_{\\rm Eucl}^{\\rm 1-loop}=-  \\bar{\\Gamma}^{(1)}$. This is the origin of the overall sign difference between the result of the covariant computation (eqs.~(2.15) and (2.18) of  \\cite{Gorbar:2002pw}) and of the non-covariant computation, eq.~(3.9) of ref.~\\cite{Gorbar:2002pw}. I thank E.~Gorbar for clarifying me this point.\\label{foot:Gamma}}\n\\bees\nS^{\\rm 1-loop} &=& -\\frac{1}{2(4\\pi)^2}\\,\\int d^4x\n\\,g^{1\/2}\\, \\Big\\{\\,\\frac{m_s^4}{2}\\Big(\\frac{1}{\\epsilon}\n+\\frac{3}{2} \\Big) \\,+ \\,\\bar{\\xi}m_s^2R\\,\n\\Big(\\frac{1}{\\epsilon}+1\\Big) \\nonumber\\\\\n&+& \\frac{1}{2}\\,C_{\\mu\\nu\\rho\\sigma} \\,\\Big[\\frac{1}{60\\,\\epsilon}+k_W(\\Box)\\Big]\nC^{\\mu\\nu\\rho\\sigma} \\,+\n\\,R\\,\\Big[\\,\\frac{1}{2\\epsilon}\\,\\bar{\\xi}^2\\, +\nk_R(\\Box)\\,\\Big]\\,R\\,\\Big\\}\\, ,\\label{ShapSola29}\n\\ees\nwhere $1\/\\epsilon=2\/(4-D)+\\log (4\\pi\\mu^2\/m_s^2)-\\gamma_{\\rm E}$ comes from the dimensional regularization scheme in $D$ space-time dimensions,  and the form factors $k_W(\\Box)$ and $k_R(\\Box)$ are given by\n\\bees\nk_W(\\Box) &=& \\frac{8A}{15\\,a^4}\n\\,+\\,\\frac{2}{45\\,a^2}\\,+\\,\\frac{1}{150}\\,, \\label{kR}\\\\\nk_R(\\Box) &=&\n\\bar{\\xi}^2A\n+ \\(\\frac{2A}{3a^2}-\\frac{A}{6}+ \\frac{1}{18}\\)\\bar{\\xi}\n+ A\\( \\frac{1}{9a^4}- \\frac{1}{18a^2}\n+ \\frac{1}{144} \\)\\nonumber\\\\\n&&+ \\frac{1}{108\\,a^2} -\\frac{7}{2160}\n\\,, \\label{kW}\n\\ees\nwhere\n\\be\nA\\,=\\,1-\\frac{1}{a}\\log\\,\\Big(\\frac{2+a}{2-a}\\Big)\\,, \\qquad a^2 =\n\\frac{4\\Box}{\\Box - 4m_s^2}\\, .\n\\ee\nA few comments on this result are in order. First, observe that the first two terms in \\eq{ShapSola29} correspond to the renormalization of the cosmological constant and of Newton constant, respectively. Even if they depend on the renormalization scale $\\mu$ (through the parameter $\\epsilon$), they have no dependence on the $\\Box$ operator and, in this sense, no running in momentum space. This is a trivial consequence of the fact that $\\Box\\Lambda=0$, while $\\Box R$ is a total derivative, so we cannot  obtain a form factor by acting with the $\\Box$ operator on $\\Lambda$ or on $R$. In contrast, the \n$C_{\\mu\\nu\\rho\\sigma}^2$ and the $R^2$ terms acquire corrections of the form\n$C_{\\mu\\nu\\rho\\sigma} k_W(\\Box)C^{\\mu\\nu\\rho\\sigma} $ and\n$Rk_R(\\Box)R$, in full analogy with \\eq{qed}.\nIn the UV limit $\\Box\/m_s^2\\gg 1$ we have (see also \\cite{Avramidi:1989er,Dalvit:1994gf})\n\\bees\nk_W(\\Box)&\\simeq &-\\frac{1}{60}\\log\\frac{\\Box}{m_s^2}+ \\frac{23}{450} \n+{\\cal O}\\(\\frac{m_s^2}{\\Box}\\log\\frac{m_s^2}{\\Box} \\)\n\\, ,\\label{kWUV}\\\\\nk_R(\\Box)&\\simeq &-\\frac{1}{2}\\bar{\\xi}^2\\log\\frac{\\Box}{m_s^2}+\n\\(-\\frac{1}{1080}+\\frac{\\bar{\\xi}}{18}+\\bar{\\xi}^2\\)\n +{\\cal O}\\(\\frac{m_s^2}{\\Box}\\log\\frac{m_s^2}{\\Box} \\)\n\\, .\\label{kRUV}\n\\ees\nTherefore, after subtracting the divergent parts, in the limit $\\Box\/m_s^2\\gg 1$ we remain with\n\\be\nS^{\\rm 1-loop} = \\frac{1}{2(4\\pi)^2}\\,\\int d^4x\\,g^{1\/2}\\,\n\\[ \\frac{1}{2}\\bar{\\xi}^2 \\,R\\log\\(\\frac{\\Box}{m_s^2}\\)R+\n\\frac{1}{120}\\, C_{\\mu\\nu\\rho\\sigma}\\log\\(\\frac{\\Box}{m_s^2}\\)C^{\\mu\\nu\\rho\\sigma}\\]\\, ,\n\\ee\n(plus local terms independent of $\\Box$) in full agreement with \\eq{4SNL1loop} and Table~\\ref{tab1} (the renormalization of the Gauss-Bonnet term was not included in this computation).\nObserve also that, in the conformal case  $m_s=0, \\bar{\\xi}=0$, the form factor $k_R(\\Box)$ becomes local, $k_R(\\Box)=-1\/1080$, in agreement with the prediction that can be obtained directly from the conformal anomaly.\\footnote{Again, checking the sign is a bit tricky due to the many different conventions in the literature.  The standard anomaly result can be read from  eq.~6.124 and Table 1 on page  179 of the Birrell and Davies textbook ~\\cite{Birrell:1982ix}. One must however take care that Birrell and Davies use the signature $(+,-,-,-)$ and their definition of Riemann tensor is opposite to MTW. To go to MTW sign conventions we therefore must switch $g_{\\mu\\nu} \\rightarrow -g_{\\mu\\nu}$ (so $\\Box \\rightarrow -\\Box$) and $R \\rightarrow +R$.\nTaking into account that this flips also the sign of the trace $T^{\\mu}_{\\mu}$, we find that the correct result for the anomaly of a single massless conformally coupled scalar field, in MTW conventions, is\n$ T = + \\frac{1}{180 (4\\pi^2)} \\Box R$. From the signature $(-,+,+,+)$ we can then trivially rotate to euclidean space without getting extra minus signs. In  eq.~(3.12) of \\cite{Gorbar:2002pw} the sign of the anomaly in the euclidean space was incorrectly taken to be a minus, and the result was checked using $\\bar{\\Gamma}^{(1)}$ instead of $S_{\\rm Eucl}^{\\rm 1-loop}=-  \\bar{\\Gamma}^{(1)}$, see footnote~\\ref{foot:Gamma}, so these two sign errors compensated.}\n \nIn contrast,  in the opposite limit $\\Box\/m^2\\ll 1$, \\eqs{kR}{kW} give\n\\be\\label{dec}\nk_W(\\Box),k_R(\\Box)={\\cal O}(\\Box\/m^2)\\, , \n\\ee\ncorresponding to the decoupling of particles with mass large compared to the momentum scale, which is explicit in the mass-dependent subtraction scheme used in \\cite{Gorbar:2002pw,Gorbar:2003yt} (see also the discussion in \\cite{Manohar:1996cq}).\nFurthermore, beside being small, a term ${\\cal O}(\\Box\/m^2)$ is local. This shows explicitly that non-local terms appear in the effective action as a consequence of the running in the loops of particles that are either massless, or light with respect to the energy scale considered.\n\n\n\n\\subsubsection{Gravitational loop corrections in Einstein gravity}\n\nIn semiclassical gravity, $a_1,a_2,a_3$ are simply parameters of the effective action. \nWhen we go beyond the semiclassical approximation, and treat gravity itself at the quantum level, $a_1$ and $a_2$  become genuine coupling constants, and the perturbative expansion is organized in terms of the three coupling of the theory $G, a_1$ and $a_2$ (while  the Gauss-Bonnet term is a topological invariant, and will be neglected in the following). \nTheir renormalization group equations also receive contributions from the purely gravitational sector, i.e. from graviton loops. However, these contributions depends crucially on the UV completion of the theory, as we now review.  \n\nIf one starts from the Einstein-Hilbert action without higher-derivative terms and computes the one-loop corrections, one in principle finds $R^2$ and $C^2$ terms, as was first shown in the classic paper by 't~Hooft and Veltman \\cite{'tHooft:1974bx}. The original computation of \\cite{'tHooft:1974bx} gives, in our notation, $\\bar{\\alpha}=90\/360=1\/4$ and $\\bar{\\beta}=126\/360=7\/20$, which are the values quoted in~\\cite{Donoghue:2014yha}. However, these divergences have no physical meaning, since they  can be set to zero using the equations of motion~\\cite{'tHooft:1974bx}, so that pure gravity at one loop is finite on-shell. Equivalently, they can be set to zero with a field redefinition\n$g_{\\mu\\nu}\\rightarrow g_{\\mu\\nu}+\\epsilon (c_1R_{\\mu\\nu}+c_2 Rg_{\\mu\\nu})$. More generally, if one quantizes the pure Einstein-Hilbert action, the coefficients of the $R^2$ and $R_{\\mu\\nu}^2$ terms which are generated at one loop depend on the gauge used, and one can find gauges in which these divergences are absent, so that the theory is one-loop finite even off-shell~\\cite{Kallosh:1978wt}. Thus, in pure Einstein-Hilbert gravity there is no sense in which we can compute the gravitational contribution to the running of the coupling constants associated to the $R^2$ terms. The same is true if we  consider the  higher derivative theory (\\ref{SHD}) in an effective action approach, since the higher-derivative term add extra vertices, but does not change the structure of the propagator.\n\n\n\\subsubsection{Gravitational loop corrections in Stelle theory}\\label{sect:Stelleth}\n\nThe situation  changes in an interesting manner if, rather than using an effective action approach, in which the higher-derivative term is seen as a correction valid only in the low-energy limit, we try to take the action (\\ref{SHD})\nas a full UV complete theory, and quantize it. In this approach, the higher-derivative theory (\\ref{SHD}) is usually known as Stelle theory.  In this case the propagator is read from the full quadratic term in \\eq{LEHLHD} and, again schematically, is of the form \n\\be\\label{Gdik}\nG(k)\\sim \\frac{1}{m_{\\rm Pl}^2k^2+ak^4}\\, ,\n\\ee\n(before any canonical normalization) so that now it goes as $1\/k^4$ in the far UV. This improved behavior of the propagator allowed Stelle to prove that this theory is renormalizable~\\cite{Stelle:1976gc}.\\footnote{Of course, the full story is  more complex. The $R^2$ and $C^2$ terms also lead to vertices growing as $k^4$, rather than  $k^2$ as in Einstein-Hilbert gravity. In GR, since the propagator go as $1\/k^2$ and the vertices as $k^2$, the superficial degree of divergence is $D=4L-2I+2V$, where $L$ is the number of loops, $I$ of internal lines, and $V$ of vertices. Combined with the topological relation $L=1+I-V$ this gives $D=2L+2$, so the degree of divergence grows with the number of loops. In contrast, for ${\\cal L}_{\\rm HD}$, $D=4L-4I+4V$ which, together with \n$L=1+I-V$, gives $D=4$, independent of the number of loops~\\cite{Stelle:1976gc}. This improved behavior is at the basis of Stelle's proof of renormalizability. \nObserve also that, for  renormalizability, it is crucial to include both the $R^2$ and the $C^2$ terms in the action, since they contribute separately to the propagator of the spin-0 and massive spin-2 excitation of the theory. Otherwise,  the propagator of one of these excitations would still go as $1\/k^2$, while the vertices grow as $k^4$, leading to an even worse UV behavior than in Einstein-Hilbert gravity. \nFurthermore, even the Gauss-Bonnet term is required for multiplicative renormalizability (see the recent discussion in ~\\cite{Einhorn:2014gfa}).}\nHaving a renormalizable quantum theory of gravity is quite  remarkable. This however  comes at a very high price. Namely, the spectrum of $R^2$ gravity, which is now read from the full propagator, consists of the usual massless graviton, \na massive spin-2 particle and a massless spin-0 particle~\\cite{Stelle:1977ry,Fradkin:1981iu}.\nThe massive spin-2  state has negative kinetic energy, so is a ghost, and its squared mass is\n\\be\\label{m22}\nm_2^2=\\frac{m_{\\rm Pl}^2}{2 a_1}\\, .\n\\ee\nThe massive spin-0  state has a normal kinetic term, and squared mass\n\\be\\label{m02}\nm_0^2=-\\frac{m_{\\rm Pl}^2}{12 a_2}\\, ,\n\\ee\nand is a tachyon if $a_2>0$. Because of the ghost, and (in case) of the tachyon, $R^2$ gravity, treated non-perturbatively as a full UV complete theory, is not a consistent theory  (otherwise, the problem of finding a consistent theory of quantum gravity would have been solved in the late 1970s...). There have been attempts at exorcising the ghost based on the idea that radiative corrections shift its pole into the complex plane~\\cite{Tomboulis:1977jk,Antoniadis:1986tu}, so that the ghost becomes unstable and does not appear in the asymptotic states. To leading order in $1\/N$, where $N$ is the number of conformally coupled matter fields, the theory turns out indeed to be unitary. However, attempts at proving unitarity beyond leading order based on the gauge-dependence of the position of the ghost poles fail~\\cite{Johnston:1987ue}, so what happens  to all orders is basically unknown. In any case,\nthere is a long tradition of ignoring the ghost problem and see what the $R^2$ theory has to say at a full non-perturbative level, in the hope that this will at least help to unveil properties of a fully consistent theory of quantum gravity. In particular, one can compute the running of the coupling constants associated to the ${\\cal O}(R^2)$ terms. As we mentioned above, in Einstein-Hilbert gravity this computation leads to beta functions that are not even gauge-independent.\nIn contrast, in Stelle theory the computation is well-defined, and the  beta functions  associated to the $R^2$, $C^2$  and Gauss-Bonnet terms are physical and gauge-independent (while, for the Newton constant and the cosmological constant, only the dimensionless combination \n$G\\Lambda$  has a gauge-independent beta function)~\\cite{Julve:1978xn,Fradkin:1981iu,Avramidi:1985ki}. For Stelle theory,\nthe  computation of the gravitational contribution to the beta functions of the $a_i$ coupling was first performed correctly in \\cite{Avramidi:1985ki} (building on earlier work in  \\cite{Fradkin:1981iu,Julve:1978xn}), and the result was  confirmed in \\cite{deBerredoPeixoto:2004if}.  The result is conveniently expressed trading $a_1$ and $a_2$ for two couplings $\\lambda$ and $u$ defined by\n\\be\\label{a1a2lu}\na_1=\\frac{1}{\\lambda}\\, ,\\qquad a_2=-\\frac{u}{3\\lambda}\\, ,\n\\ee\nso that the Minkowskian action (\\ref{SEH}) reads (neglecting the Gauss-Bonnet term and the cosmological constant)\n\\be\nS=\\int d^4x \\sqrt{-g}\\, \\[ \\frac{m_{\\rm Pl}^2}{2}R-  \\frac{1}{\\lambda} C^2+ \\frac{u}{3\\lambda}R^2\\]\\, .\n\\ee\nThe corresponding  beta functions\nare given in eqs.~(19)-(24) of \\cite{Avramidi:1985ki}. In particular, introducing the logarithm of the energy-scale parameter $t=(4\\pi)^{-2}\\log E\/\\mu_0$, the purely gravitational contribution is\n\\bees\n\\frac{d\\lambda^{-1}}{dt}&=&\\bar{\\beta}_g\\, ,\\label{betal}\\\\\n\\frac{du}{dt}&=& -\\lambda \\( \\frac{10}{3}u^2 +\\frac{183}{10}u+\\frac{5}{12}\\)\\, ,\\label{betau}\n\\ees\nwhere $\\bar{\\beta}_g=133\/10$ is positive.  It is important to stress that  these beta functions have been computed in the UV limit $E\\ggm_{\\rm Pl}$. In the  regime $E\\llm_{\\rm Pl}$ which is interesting for us, these equations will receive corrections from the Einstein-Hilbert term, which have not been computed.\nThe solution of \\eq{betal} is\n$\\lambda^{-1}(t)=\\lambda^{-1}(t=0)+\\beta_g t$,\ni.e.\n\\be\\label{ldiE}\n\\lambda(E)=\\frac{\\lambda(\\mu)}{1+\\lambda(\\mu) \\beta_g\\log (E\/\\mu)  }\\, .\n\\ee\nwhere $\\beta_g=\\bar{\\beta}_g\/(4\\pi)^2$. Whether this corresponds to a function $\\lambda(E)$ that becomes large in the UV and small in the IR, or viceversa,  depends of course of the sign of $\\lambda(\\mu)$. Since $\\bar{\\beta}_g>0$, \nif $\\lambda(\\mu)>0$ we have  $\\lambda(E)\\rightarrow 0$ as $E\\rightarrow\\infty$, and conversely $\\lambda(E)$ gets large and formally diverges at a finite energy scale  in the IR.\nThe opposite situation takes place if $\\lambda(\\mu)<0$.  Observe from\nTable~\\ref{tab1} that the contribution to the beta function of $a_1=\\lambda^{-1}$ coming from each\ntypes of matter fields has the same sign as the gravitational contribution, so the same behavior takes place in the full theory with gravitational plus matter loops.\n\nThe evolution of $u$ is instead determined, at least in the super-Planckian regime in which \\eq{betau} holds, by the two roots of the \nequation $(10\/3)u^2 +(183\/10)u+(5\/12)=0$, which are given by $u_1\\simeq -5.46714$, $u_2\\simeq -0.02286$. Writing \n\\eq{betau} as \n\\be\n\\frac{du}{dt}=-\\frac{10}{3}\\lambda (u-u_1)(u-u_2)\\, ,\n\\ee\nwe can easily read the sign of $du\/dt$ and the direction of the RG flow, for different values of $u$, and a given sign of $\\lambda$. For $\\lambda >0$ the one-dimensional flow in the $u$ variable is depicted in the upper panel of Fig.~\\ref{fig:flow_u} and for $\\lambda <0$ is shown in\nthe lower panel. Observe in particular that, for $\\lambda>0$, $u=u_2$ is a UV fixed point, which attracts all initial values\n$u(t=0)>u_1$. For $\\lambda<0$ the value  $u=u_1$ is a UV fixed point at its attraction basin is given by $u(t=0)<u_2$.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.5\\columnwidth]{flow_u.pdf}\n\\caption{\\label{fig:flow_u} The RG flow of $u(t)$ for $\\lambda>0$ (upper panel) and  $\\lambda<0$ (lower panel). The direction of the arrows indicate the flow of $u$ as $t$ increases, i.e. as we move from the IR to the UV. For a given initial value in the UV, the flow as we run toward the IR regime is therefore obtained reversing the sign of the arrows.\n\\label{fig:flow_u}\n}\n\\end{figure}\n\n\n\\begin{figure}\n\\begin{center}\n\\begin{tikzpicture}[line width=1 pt, scale=2]\n\\draw[graviton] (1,0)--(2,0);\n\\node at (2.5,0) {+};\n\\draw[graviton] (3,0)--(4.02,0);\n\\draw[graviton] (5,0) arc (0:360:0.5);\n\\draw[graviton] (5,0)--(6,0);\n\\end{tikzpicture}\n\\caption{The graviton propagator and a one-loop correction. \\label{fig:loop}}\n\\end{center}\n\\end{figure}\n\n\nOne might wonder how is it possible that a computation which is well-defined in the full $R^2$  theory is not well-defined in Einstein-Hilbert gravity, since the latter is formally obtained taking the limit $a_1,a_2\\rightarrow 0$ in the action (\\ref{Seucl}).\nThe reason is that, when we take the $R^2$ theory as a full non-perturbative theory,  the structure of the divergences, which determine the beta functions, changes discontinuously if we set $a_1$ or $a_2$ to zero. Indeed, consider for instance a loop correction to the graviton propagator such as that shown in Fig.~\\ref{fig:loop}. Even when the momentum $p$ of the external line is sub-Planckian, $|p^2|\\llm_{\\rm Pl}^2$, in the loops we will have propagators such as $G(k)$, given in \\eq{Gdik}, and $G(k-p)$, where $k$ is the loop integration variable and $p$ the external momentum. The divergence is determined by the behavior of the propagator (and of the vertices) for $k\\rightarrow \\infty$. For any $a\\neq 0$ the propagator at $k\\rightarrow\\infty$ behaves as $1\/(ak^4)$, even when the external momenta are sub-Planckian, while for $a=0$ it behaves as $1\/k^2$ and the structure of the divergences is completely different. In other words, the limit  $a\\rightarrow 0$   does not commute with the integrations over $d^4k$ in the loop integrals. \nSo, despite the fact that $S_{\\rm HD}$ is a higher-derivative term, which naively should be irrelevant at low energies, the structure of the divergences of Stelle theory is completely different from that of GR, even when the external lines are sub-Planckian, and the limit $a\\rightarrow 0$ (or, more precisely $a_1\\rightarrow 0$ or $a_2\\rightarrow 0$) of the beta functions of $R^2$ gravity  is not smooth. This fact has already been stressed in sect.~3.1 of \\cite{Fradkin:1981iu}, where in particular it was observed that\nthe beta function of the pure Weyl theory cannot be obtained setting $a_2=0$ in the beta function of the theory with a higher-derivative term $a_1C^2+a_2R^2$.\n\n\n\\subsubsection{Gravitational running in a consistent theory of quantum gravity}\n\nThe result (\\ref{betal}, \\ref{betau}) cannot be applied directly to our problem, namely the running of $a_2$ at low energies in the actual universe, for two  reasons. First, the computation leading to \\eqs{betal}{betau} has been performed in the deep UV region, $E\\ggm_{\\rm Pl}$. In the  regime $E\\llm_{\\rm Pl}$, which is interesting for us, these equations will receive contributions from the Einstein-Hilbert term, which have not been computed. This is just a technical problem. However, a second and  more fundamental problem is that,\nbarring a resolution of the ghost problem, Stelle theory can anyhow only be taken as a toy model, rather than a theory describing  Nature.\n\nHowever what we learn from this toy model is that, in a  theory of gravity with a good UV behavior, the computation of the beta functions associated to the $a_i$ couplings is well defined, and these quantities run with energy. In the end we expect that, whatever will be the consistent fundamental quantum theory of gravity,  in that context it will be possible to compute consistently the running of the coupling constant associated to the $R^2$ and $C^2$ term in the corresponding low-energy effective action, including the gravitational contribution.\\footnote{For instance, in string theory one could in principle compute  the string-loop correction to gravitational scattering amplitudes directly at the world-sheet level, and reconstruct from it the corresponding loop corrections in the target-space effective action. Actually, in the effective action of type II string theory in 10 dimension there is no $R^2$ term because of supersymmetry. However, in general, the presence and the structure  of ${\\cal O}(R^2)$ corrections that one obtains in four dimensions will depend on the type of string theory considered, as well as on how supersymmetry is broken in the compactification to four dimensions. Observe that it is sometimes argued that, since the fundamental string theory is ghost-free, in the low-energy effective action can appear only the ghost-free combination given by the Gauss-Bonnet term. This argument is however wrong. Being ghost-free is a property of the full theory, and if we write the effective action derived\nfrom a ghost-free theory in an expansion in derivatives and truncate this  expansion to a finite order (e.g. in our case keeping only the terms up to ${\\cal O}(R^2)$) apparent ghost state will in general appear, with a mass of the order of the cutoff scale of the effective theory (in our case $m_{\\rm Pl}$ or the string mass scale), but they are just an artefact of the truncation. See e.g. \\cite{Burgess:2014lwa} for a nice explicit example.}\n\nFrom the above analysis we also learn that the running of the  couplings $a_i(E)$ depends on the precise form of the UV completion of the theory, even at $E\\llm_{\\rm Pl}$. Indeed, the running of $a_i(E)$\ncomputed in a consistent  UV completion of GR cannot reduce at low energies to that  computed in the effective low-energy theory (i.e. in the Einstein-Hilbert action supplemented by the higher-derivative terms, considered as corrections in the effective field theory sense), simply because there is no consistent result in the low-energy effective theory. The beta functions computed in the low-energy effective action based on the Einstein-Hilbert term, in which the $R^2$ terms are treated as corrections, are not even gauge-invariant. This of course does not happens e.g. in QCD, because QCD is renormalizable, which in the end means that, even if we eventually embed it in a larger theory, the details of the UV completion, much as the details of the regularization scheme, are irrelevant at low energies. This is not the case for Einstein-Hilbert gravity, which is not renormalizable.\nIn a sense, this phenomenon is an interesting example of UV\/IR mixing, since it implies that the IR physics of GR can sensitive\nto the UV completion.\n\n\n\nThus, the points that we will take home from the following analysis are the following. \n \n\n\\begin{enumerate}\n\n\\item In a fundamental and consistent quantum theory of gravity the couplings associated to the $R^2$ and $C^2$ term will in general  run with energy, and their beta functions will  well-defined and gauge-independent.\\footnote{Of course, in some special case, the beta functions can be zero, e.g. if they are protected by supersymmetry.} \n\n\\item  The precise form of the corresponding RG equation, even in the IR regime $E\\llm_{\\rm Pl}$, depends on the specific UV completion of the theory, and is therefore at present completely unknown. \nWe cannot use the results from GR plus higher-derivative terms, considered in an effective action approach, because the corresponding beta functions are non even gauge-invariant. We cannot use the results of Stelle theory (\\ref{betal}) and (\\ref{betau}), either, because, first of all, these results only hold in the regime $E\\ggm_{\\rm Pl}$ and, more importantly, Stelle theory  anyhow is not a consistent UV completion of gravity because of the ghost.\n\n\\end{enumerate}\n\n\nIt is also interesting to observe that, to obtain the model (\\ref{RR}) in the IR, it is not a priori necessary to match the sign of $a_2(\\mu)$ in the UV with its (positive) sign in the IR. Indeed, Fig.~\\ref{fig:flow_u} gives an explicit example of a RG flow that changes the sign of $u$, and therefore of $a_2$.\nA renormalization group flow that changes the sign of the coupling $a_2(\\Box)$, flowing from a value $a_2(\\mu)<0$ at some scale $\\mu$ corresponding to the energy scale of inflation,  to a value $a_2(\\Box)=+\\Lambda_{\\rm RR}^4\/\\Box^2$ in  the IR (where $\\Lambda_{\\rm RR}$ is a constant that we will discuss below), would be particularly intriguing since it would interpolate between the Starobinsky model~\\cite{Starobinsky:1980te} in the UV and the non-local model (\\ref{RR}) in the IR, thereby providing a unified description of inflation in the early universe and dark energy in the present epoch.\\footnote{I thank  Eugenio Bianchi for a discussion from which this idea emerged.} \n\n\n\n\\section{Infrared dynamics}\\label{sect:IR}\n\nEven if we do not know the form of the IR running of the couplings $a_i(E)$ in the fundamental theory of gravity, we can still analyze some general scenarios. In the end, there are basically two natural possibilities. Either $|a_2(E)|$ grows in the IR, or it decreases. The interesting situation, from our point of view, takes place when in the IR  $|a_2(E)|$ grows and, at least at sufficiently low energies,  $a_2(E)>0$   (since this sign allows us to eventually match $a_2(\\Box)$ with the sign of the model \\ref{RR}). In this case, the situation will be quite similar to that in QCD. To stress the similarity, let us use the notation $a_2(E)=g^2(E)$ and assume that $g^2(E)$ is asymptotically free in the UV so, as long as we are in the perturbative regime,\n\\be\\label{gdiE}\ng^2(E)=\\frac{g^2(\\mu)}{1+g^2(\\mu) \\alpha_0\\log (E\/\\mu)  }\\, ,\n\\ee\nwith $\\alpha_0>0$. This implies that $g^2(E)$ eventually becomes large in the IR. Similarly, we can write $a_1(E)=f^2(E)$ and study the scenario in which $f^2(E)$ is asymptotically free in the UV, and\n\\be\\label{gdiE}\nf^2(E)=\\frac{f^2(\\mu)}{1+f^2(\\mu) \\beta_0\\log (E\/\\mu)  }\\, ,\n\\ee\nwith $\\beta_0>0$. To understand what happens when  we enter more deeply in the IR regime is of course a  difficult problem.\nAs already observed recently \\cite{Einhorn:2014gfa,Alvarez-Gaume:2015rwa}\na pure $R^2$ theory, with no Einstein-Hilbert term, has a confining Newtonian potential $V(r)\\propto r$, because of the $1\/|{\\bf k}|^4$  behavior of the propagator in the static limit. Its infrared limit is therefore very different from Einstein gravity, and indeed one can even expect   that in this theory gravitons should be confined. When we  also add  the Einstein-Hilbert term, at energies $E\\llm_{\\rm Pl}$ the standard $m_{\\rm Pl}^2k^2$ term from Einstein-Hilbert action  dominates over the $k^4$ term and we recover the standard behavior of gravity. Still, in the infrared, from the point of view of the quantum theory of gravity, we are in a peculiar situation: loops associated to an expansion in  powers of  Newton's constant are suppressed by powers of $GE^2\\sim E^2\/m_{\\rm Pl}^2\\ll 1$ and are therefore irrelevant. However, if the the couplings $a_1(E)$ and\/or  $a_2(E)$  run toward a strong coupling regime and formally diverge at some energy in the IR, quantum loops associated to them can be important. As a result, the form factors associated to the $R^2$ and $C^2$ terms can become quite different in the IR. To determine the expression of these form factors  in the IR is of course a difficult non-perturbative problem. However,  experience with QCD allows us to understand some aspects of it. In particular, we know that the logarithmic running  of an asymptotically free coupling constant  generates  by dimensional transmutation  a renormalization group invariant mass scale, that we denote by $\\Lambda_{\\rm RR}$, analogous to $\\Lambda_{\\rm QCD}$.\\footnote{A similar scenario has been discussed  recently, in\n\\cite{Einhorn:2014gfa,Salvio:2014soa}, although in these cases dimensional transmutation is rather used to generate the Planck scale from a pure higher-derivative gravity. The idea has been around in different forms since a long time, see e.g.~\\cite{Zee:1983mj,Nepomechie:1983yq}.   In ref.~\\cite{Tong:2014era} similar ideas have been applied to the coefficient of the Gauss-Bonnet term, using arguments from supergravity.} More precisely, once we fix observationally the value of $f^2(\\mu)$  at some scale $\\mu$,  formally, as we run toward the IR, the one-loop expression for $f^2(E)$ will hit infinity at some scale\n$\\Lambda_f$ given by \n\\be\n\\Lambda_f=\\mu \\exp\\left\\{-\\frac{1}{\\beta_0 f^2(\\mu)}\\right\\}\\, .\n\\ee\nSimilarly, the one-loop expression for $g^2(E)$ will hit infinity at some scale\n\\be\n\\Lambda_g=\\mu \\exp\\left\\{-\\frac{1}{\\alpha_0 g^2(\\mu)}\\right\\}\\, .\n\\ee\nAs we run toward the IR, the first scale that we hit, where one of the two running couplings formally diverges, defines the strong-coupling scale of the theory, so $\\Lambda_{\\rm RR}={\\rm max}(\\Lambda_f,\\Lambda_g)$.\\footnote{One might wonder whether one of the couplings could still remain weakly coupled in the regime where the other is  strongly coupled. The answer is in general no. For instance, a pure Weyl-square theory, which only has the $f^2$ coupling, is not multiplicatively renormalizable, and generates a $R^2$ term already at the two-loop level~\\cite{Fradkin:1983tg}. Therefore, when $f^2$  enters the strong-coupling regime, it necessarily induces a strong coupling in  $g^2$. For this reason, even if for instance the coupling $a_2$ associated to the $R^2$ term should not have the appropriate sign for becoming strongly coupled according to its own one-loop RG equation, it could eventually be driven toward  strong coupling by the coupling $a_1$.}\nWe therefore have a mass scale that appears in the IR regime by dimensional transmutation.\nThen, in the strong coupling region,\n$f^2=f^2(\\Box\/\\Lambda^2_{\\rm RR})$ and $g^2=g^2(\\Box\/\\Lambda^2_{\\rm RR})$. In principle, it is possible to obtain a power-like dependence of \n$f^2$ and $g^2$ on $\\Box$, from a resummation of leading logs, of the form\\footnote{See \\cite{Hamber:2005dw} for similar considerations applied to the running of Newton's constant, and \\cite{Taylor:1989tm,Taylor:1989ua} for earlier related ideas on how  the large-distance behavior of  quantum gravity might affect the cosmological constant.}\n\\be\\label{resumnu}\n\\sum_{n=0}^{\\infty}\\, \\frac{(-\\nu)^n}{n!}\\, \\(\\log\\frac{-\\Box}{\\Lambda^2_{\\rm RR}}\\)^n=\n\\(\\frac{\\Lambda^2_{\\rm RR}}{-\\Box}\\)^{\\nu}\\, ,\n\\ee\nfor some value of $\\nu$. The model (\\ref{RR}) would then be obtained if $\\nu=2$. Again, to determine the precise form of $g^2(\\Box)$ in the IR is of course  a difficult perturbative problem. However,  it is interesting to observe that in QCD the introduction in the action of a non-local term\n\\be\\label{Fmn2}\n-\\frac{m^2}{2}\\int d^4x\\, F_{\\mu\\nu}^a \\[\\frac{1}{D^2}\\]^{ab}F_{\\mu\\nu}^b\\, ,\n\\ee\n(where  $D_{\\mu}^{ab}=\\delta^{ab}\\pa_{\\mu}-gf^{abc}A_{\\mu}^c$ is the covariant derivative and $m$ is a mass scale) correctly reproduces the results on the non-perturbative gluon propagator in the IR, obtained from operator product expansions and lattice QCD~\\cite{Boucaud:2001st,Dudal:2005na,Capri:2005dy,Dudal:2007cw,Dudal:2008sp} (see also the review~\\cite{Boucaud:2011ug}). In the gauge $\\pa_{\\mu} A_{\\mu}=0$ we have \n\\be\n-\\frac{m^2}{2}\\int d^4x\\, F_{\\mu\\nu}^a \\[\\frac{1}{D^2}\\]^{ab}F_{\\mu\\nu}^b=\nm^2\\int d^4x\\, A_{\\mu}^aA_{\\mu}^a +{\\cal O}(A^3)\\, ,\n\\ee\nand therefore (\\ref{Fmn2}) is a mass term for the gluon. Observe that  the expression (\\ref{Fmn2}) is  non-local but  gauge-invariant. \nPhysically, the introduction of this term in the effective action can  be seen as describing the emergence of a dimension-2 gluon condensate,\n$\\langle A_{\\mu}^aA_{\\mu}^a\\rangle\\neq 0$. The same  operator has been proposed in 3-dimensional YM theory \\cite{Jackiw:1995nf,Jackiw:1997jga}, to describe the generation of a mass gap due to IR fluctuations.\n\nThe situation is quite similar for the term $R\\Box^{-2}R$ in the gravitational case. Indeed, consider the conformal mode $\\sigma(x)$ of the metric, \ndefined choosing a fixed fiducial metric $\\bar{g}_{\\mu\\nu}$ and writing\n\\be\\label{gmnsigma}\ng_{\\mu\\nu}(x)=e^{2\\sigma(x)}\\bar{g}_{\\mu\\nu}(x)\\, .\n\\ee\nUsing for simplicity a flat fiducial metric $\\bar{g}_{\\mu\\nu}=\\eta_{\\mu\\nu}$, the Ricci scalar computed from the metric $g_{\\mu\\nu}=e^{2\\sigma(x)}\\eta_{\\mu\\nu}$ is\n\\be\nR=-6 e^{-2\\sigma}\\( \\Box\\sigma +\\pa_{\\mu}\\sigma\\pa^{\\mu}\\sigma\\)\\, .\n\\ee\nTherefore, to linear order in $\\sigma$,\n\\be\nR=-6\\Box\\sigma +{\\cal O}(\\sigma^2)\\, ,\n\\ee\nand (upon integration by parts)\n\\be\\label{m2s2}\n R\\frac{1}{\\Box^2} R=36 \\sigma^2 +{\\cal O}(\\sigma^3)\\, .\n\\ee\nThe  $\\sigma^2$ term is  just a mass term for the conformal mode, while the higher-order interaction terms on the right-hand side of \\eq{m2s2} (which are non-local even in $\\sigma$) are required to reconstruct a diff-invariant quantity.\\footnote{Actually, there is here a subtle  technical point, both in the gravitational and YM case. Indeed,  in \\eq{m2s2} we have implicitly using both the equality $\\Box\\Box^{-1}=1$, and\n$\\Box^{-1}\\Box=1$. While the former is correct by definition of $\\Box^{-1}$, the latter is not true since $\\Box$ has a non-trivial kernel, given by the functions $f$ such that $\\Box f=0$. On such functions the formal equality $\\Box^{-1}\\Box f=f$ is not true. Thus, to be accurate, we should rather write in \\eq{m2s2}\n$\\sigma\\Box^{-1}\\Box\\sigma$  instead of $\\sigma^2$.\nNote  that the expression  $\\sigma\\Box^{-1}\\Box\\sigma$ has an invariance under constant shifts of the conformal factor, $\\sigma\\rightarrow\\sigma+c$, which is lost if we write it as $\\sigma^2$. The equality $\\sigma\\Box^{-1}\\Box\\sigma=\\sigma^2$ becomes however correct for all functions $\\sigma$ such that $\\Box\\sigma\\neq 0$.}\nIn this sense the term\n$R\\Box^{-2}R$ is completely analogous to the term (\\ref{Fmn2}) in the Yang-Mills case, and its introduction describes a sort of ``condensation\" of the conformal mode. The fact that  the term $R\\Box^{-2}R$ has this special physical meaning, and the analogy with the YM case, \nmakes it more plausible its emergence in the IR limit of $R^2$ gravity.\\footnote{However, in $R^2$ gravity the interpretation in terms of the dynamics of the conformal mode is complicated by the fact that the conformal already gets a kinetic term, as well as a mass term,  at tree level from the $R^2$ term, as we discuss in sect.~\\ref{sect:conf}. This will eventually lead us to consider, in sect.~\\ref{sect:anomaly}, the dynamics of the conformal mode in Einstein gravity supplemented by the anomaly-induced effective action, where no such complication arises, and one can really have a dynamical mass generation for an otherwise massless mode.}\n\nObserve also that the requirement of general covariance does not fix uniquely the term that reduces to $\\sigma^2$ to quadratic order in $\\sigma$. Indeed, as we already mentioned,  the equations of motion of the \nthe models (\\ref{RT}) and (\\ref{RR}) have the same expansion\n to linear order over Minkowski space, so the corresponding actions are the same in an expansion to  quadratic order over Minkowski, and they both reproduce the term $\\propto\\sigma^2 $ in the action, although with different non-linear completions. Both models can therefore be in principle justified on the basis of the argument that a mass should be dynamically generated for the conformal mode. \n\nThese arguments therefore suggests that, in the IR\n\\be\na_2(\\Box)\\simeq \\frac{\\Lambda_{\\rm RR}^4}{\\Box^2}\\, ,\n\\ee\nwhere $\\Lambda_{\\rm RR}$ is a dynamically generated mass scale,  analogous of the QCD Lambda parameter $\\Lambda_{\\rm QCD}$.  The parameter $\\Lambda_{\\rm RR}$\nis related to the mass scale $m$ that appears in \\eq{RR} by \n\\be\\label{LRR}\n\\Lambda_{\\rm RR}^4=\\frac{1}{12}m^2m_{\\rm Pl}^2\\, .\n\\ee\nAs for the numerical value of $\\Lambda_{\\rm RR}$,\njust as in $\\Lambda$CDM the energy fraction $\\Omega_{\\Lambda}$ is a derived parameter, fixed by the flatness condition, in the non-local model the parameter $m$ is similarly fixed, again by the flatness condition. In $\\Lambda$CDM one finds in this way that, to reproduce the observations,  the cosmological constant  must be of order $H_0^2$; not surprisingly, in the non-local model one finds that  $m^2\\sim H^2_0$ \\cite{Maggiore:2013mea,Maggiore:2014sia,Foffa:2013vma}. Thus, from \\eq{LRR}, \n\\be\n\\Lambda^2_{\\rm RR}={\\cal O}(H_0m_{\\rm Pl})\\, . \n\\ee\nMore precisely, using the best-fit values $m\\simeq 0.67 H_0$ for model (\\ref{RT})~\\cite{Maggiore:2013mea}, or $m\\simeq 0.28 H_0$ for model (\\ref{RR})~\\cite{Maggiore:2014sia} we get\n$\\Lambda_{\\rm RR}\\simeq 0.4 (H_0m_{\\rm Pl})^{1\/2}$\nand \n$\\Lambda_{\\rm RR}\\simeq 0.3 (H_0m_{\\rm Pl})^{1\/2}$,\nrespectively.\nThus, numerically, \n\\be\n\\Lambda_{\\rm RR}\\simeq 0.8\\, {\\rm meV}\\hspace{15mm}  {\\rm [model\\,\\, (\\ref{RT})]}\\, ,\n\\ee\nand \n\\be\n\\Lambda_{\\rm RR}\\simeq 0.5\\, {\\rm meV}\\hspace{15mm}  {\\rm [model\\,\\, (\\ref{RR})]}\\, .\n\\ee\nFinally, we should also in general take into account the effect of the running of the Weyl-squared term in the IR. The argument on the dynamical mass generation for the conformal mode says nothing about the behavior of the form factor for the Weyl-squared term, since the Weyl tensor vanishes for a conformally flat metric. If the form factor $f^2(\\Box)$ associated to the \nWeyl-squared term   has no significant IR enhancement, and more generally if it grows slower than $1\/\\Box^2$ in the far IR,  its contribution to the cosmological dynamics will be negligible  compared to the $R\\Box^{-2} R$ term, and we end up exactly with the non-local models of the form (\\ref{RT}) or (\\ref{RR}) (depending on the precise form of the non-linear terms, since the two models are equivalent at linear order over flat space). In contrast, if also $f^2(\\Box)$ develops a singularity proportional to $\\Box^{-2}$, a corresponding term proportional to $C_{\\mu\\nu\\rho\\sigma}\\Box^{-2}C^{\\mu\\nu\\rho\\sigma}$ should be included in the cosmological model. This would not affect the background FRW evolution, since $C^{\\mu\\nu\\rho\\sigma}$ vanishes in FRW, but would  give a contribution at the level of cosmological perturbations.\n\n\\section{A solution to the naturalness problem}\\label{sect:sol}\n\nThe above mechanism for the emergence of a term $m^2 R\\Box^{-2}R$ in the long-distance effective action gives a new perspective on the naturalness problem of the cosmological constant. As is well known, there are two aspects in the cosmological constant problem: the ``old\" cosmological constant problem, namely why we do not observe a huge value for the vacuum energy; and the ``new\" cosmological constant problem; namely, assuming that some mechanism sets to zero the vacuum energy, what fixes in a technically natural way the observed energy scale associated to dark energy? Here we have nothing to add to the ``old\" cosmological constant problem, since we have simply fixed the renormalized value of the cosmological constant to zero.\\footnote{A mechanism that could naturally set to zero the value of the vacuum energy density is proposed in \\cite{Maggiore:2010wr}.} In contrast, concerning the naturalness problem of the scale associated to dark energy, the above scenario provides a new and interesting viewpoint. In the mechanism that we have proposed,  in $R^2$ gravity the scale $\\Lambda_{\\rm RR}$ emerges from dimensional transmutation, similarly to $\\Lambda_{\\rm QCD}$. There is no issue of naturalness for such a quantity, which is determined by the logarithmic running of a dimensionless coupling constant. Of course, the value of the scale generated dynamically in this way cannot be predicted, just as we cannot predict the value of $\\Lambda_{\\rm QCD}$. The fact that $\\Lambda_{\\rm RR}$ turns out to be of the order of a  milli-eV is simply a fact of nature, just as $\\Lambda_{\\rm QCD}$ turns out to be of order 200~MeV.\nHowever, quantities such as  $\\Lambda_{\\rm QCD}$ and $\\Lambda_{\\rm RR}$\ndo not have a power-like dependence on the masses of the particles in the theory, and are therefore technically natural. \n\nAnother attractive aspect of the above construction is that, to explain dark energy,  we do not need to introduce a fundamental particle with an astonishingly small mass $m\\sim H_0\\sim 10^{-33}$~eV, as is the case for instance in massive gravity. \nIn the above construction  the fundamental mass scale of the theory, emerging from dimensional transmutation, is $\\Lambda_{\\rm RR}$, and in terms of it the effective action  in the IR reads\n\\be\\label{Sg2RinIR}\nS_{\\rm Mink}\\simeq \\int d^4x \\sqrt{-g}\\, \\[ \\frac{m_{\\rm Pl}^2}{2}\\,R-\\Lambda_{\\rm RR}^4R\\frac{1}{\\Box^2}R\\]\\, .\n\\ee\nEven if, just as in QCD,  the  value of $\\Lambda_{\\rm RR}$ cannot be predicted by the model, still the  numerical value required by the comparison with the observation, which turns out to be of the order of the meV, does not look particularly surprising. An extremely small mass scale only emerges when, in \\eq{Sg2RinIR}, we factorize  $m_{\\rm Pl}^2\/2$, rewriting the action as\n\\be\\label{Sg2RinIRbis}\nS_{\\rm Mink}\\simeq  \\frac{m_{\\rm Pl}^2}{2} \\int d^4x \\sqrt{-g}\\, \\[\\,R-\n \\frac{2\\Lambda_{\\rm RR}^4}{m_{\\rm Pl}^2}R\\frac{1}{\\Box^2}R\\]\\, ,\n\\ee\nso the mass scale $m$ defined in \\eq{RR} is given by\n\\be\nm=\\sqrt{12}\\,\\, \\frac{\\Lambda_{\\rm RR}^2}{m_{\\rm Pl}}\\, ,\n\\ee\nwhich is suppressed with respect to $\\Lambda_{\\rm RR}$ by an extra factor of order $\\Lambda_{\\rm RR}\/m_{\\rm Pl}$, and is then\nof order $10^{-33}$~eV. However, the fundamental mass scale generated by the theory is $\\Lambda_{\\rm RR}$,  not $m$.\n\n\n\\section{Conformal mode dynamics}\\label{sect:conf}\n\nIt is interesting to understand in more detail the consequences on the conformal mode of a RG flow that runs from \n $a_1R^2$ at high energies, with $a_1=a_1(\\mu)$ a constant, to $\\Lambda_{\\rm RR}^4 R\\Box^{-2}R$ in the IR. \n We expands on the technical detail in app.~\\ref{app:Bard}, and here we summarize the main results. First of all, let us linearize for simplicity over flat space, and restrict to the scalar sector of perturbations. Then the metric can be written in terms of the gauge-invariant Bardeen potentials $\\Phi$ and $\\Psi$ as \n\\be\\label{PsiPhitext}\nds^2=-(1-2\\Psi) dt^2+(1+2\\Phi) d{\\bf x}^2\\, .\n\\ee\nEquivalently, we can introduce  the conformal factor $\\sigma$ and another independent perturbation $\\varphi$, defined writing the metric perturbation in the scalar sector in the form\n\\be\\label{sigmavarphi}\nds^2=e^{2\\sigma}\\[ -(1-2\\varphi) dt^2+(1+2\\varphi) d{\\bf x}^2\\]\\, .\n\\ee\nAt the level of linearized theory, $e^{2\\sigma}=1+2\\sigma +{\\cal O}(\\sigma^2)$, so $2\\sigma=\\Phi-\\Psi$ and\n$2\\varphi=\\Phi+\\Psi$. In GR, without $R^2$ terms, of course only the tensor modes are dynamical, and scalar perturbations are non-dynamical constrained fields that satisfy a Poisson equation, rather than a Klein-Gordon equation. Indeed, in the presence of matter with energy-momentum tensor $T_{\\mu\\nu}$, in GR the Bardeen potentials satisfy\n\\bees\n\\n^2\\Phi &=&-4\\pi G\\rho\\, ,\\label{n2Phirhotext}\\\\\n\\n^2\\Psi &=&-4\\pi G (\\rho-2\\n^2\\Sigma)\\, ,\\label{n2Psirhotext}\n\\ees\nwhere $T_{00}=\\rho$ is the energy density, and $\\Sigma$ the scalar part of the anisotropic stress tensor, see \\eqst{T00}{Tij}.\nTherefore the conformal mode satisfies\n\\be\\label{n2sigmaSigmatext}\n\\n^2(\\sigma+4\\pi G\\Sigma)=0\\,  .\n\\ee\nImposing the boundary condition that the $\\sigma$ and $\\Sigma$ vanish at infinity, the Laplacian is invertible, so we get\n$\\sigma=-4\\pi G\\Sigma$. The conformal mode is therefore a constrained field, determined by the matter content, and vanishes if $\\Sigma=0$.\n\nThe situation changes drastically in $R^2$ gravity. In this case, as we show in app.~\\ref{app:Bard}, linearizing over flat space the conformal mode satisfies the equation\n\\be\\label{Boxsigmatext}\n(\\Box-m_0^2)\\(\\sigma +4\\pi G\\Sigma\\)=\\frac{4\\pi G}{3}\\, (\\rho-3P)\\, .\n\\ee\nThus the conformal mode $\\sigma$, shifted by $4\\pi G\\Sigma$ as in \\eq{n2sigmaSigmatext}, now satisfies a massive KG equation, where  $m_0^2$ is just the same as \\eq{m02}. Thus, the conformal mode becomes dynamical, and describe the scalar particle that appears in Stelle theory.\nAs we see from \\eq{Boxsigmatext}, its  source is given by the trace of the energy-momentum tensor, $T=\\eta^{\\mu\\nu}T_{\\mu\\nu}=-(\\rho-3P)$, as we expect indeed for the conformal mode. \n\nConsider now the theory with action\n\\be\nS=\\int d^4x\\sqrt{-g}\\, \\[ \\frac{m_{\\rm Pl}^2}{2} R-Ra_1(\\Box)R\\]\\, , \n\\ee\nwhere, \nat some UV scale $\\mu$, $a_1(\\Box)= a_1$ is a constant, while in the IR \n$a_1(\\Box)= \\Lambda_{\\rm RR}^4 \\Box^{-2}$. In this case in the UV the conformal mode is described by \\eq{Boxsigmatext}. In the IR, in contrast, the model reduces to \\eq{RR}. The dynamical content of this theory, in the scalar sector, has been studied\nin Sect.~3 of ref.~\\cite{Maggiore:2014sia}, where is has been shown that $\\Phi$ and $\\Psi$ are non-radiative field that satisfy Poisson equations, as in GR. The same is therefore true for the conformal mode $\\sigma=(\\Phi-\\Psi)\/2$. We therefore conclude that a RG flows in which $a_1(\\Box)$ evolves from a constant in the UV toward $a_1(\\Box)= \\Lambda_{\\rm RR}^4 \\Box^{-2}$ in the IR corresponds to a flow in which the conformal mode, which in the UV is dynamical, and is just the spin-0 mode  of $R^2$ theory with mass given by  \\eq{m02}, becomes a constrained field as we flow toward the IR.\n\n\n\n\n\n\\section{Dynamical mass generation for the conformal mode in Einstein gravity}\\label{sect:anomaly}\n\nThe discussion   of the previous sections suggests that, because of IR effects, a mass term could be dynamically generated for the conformal mode.\nThe introduction of non-local operators of the type $R\\Box^{-2}R$ \nallows us just to describe such a  mass term, in a diffeomorphism-invariant manner.\n\nThis leads us to investigate more closely the dynamics of the conformal mode in Einstein gravity, without $R^2$ corrections, to see if a similar ``condensation\" of the conformal mode might take place. Indeed, there are several indications for the relevance of the conformal mode in GR in the \nIR~\\cite{Antoniadis:1991fa,Antoniadis:2006wq}. In classical general relativity the conformal mode is a non-propagating degree of freedom. At the quantum level, when gravity is coupled to massless conformally-invariant fields, it however acquires  a dynamics through the conformal anomaly. As stressed in \\cite{Antoniadis:2006wq}, an anomaly implies that the effect of quantum fluctuations can remain large at the longest length and time scales, so quantum fluctuations can affect the behavior of the classical theory even in the far IR.\nThis is most evident in $D=2$ space-time dimension. In this case, \nthe trace anomaly takes the form (see e.g. \\cite{Birrell:1982ix}) \n\\be\\label{traceT}\n\\langle T^{\\mu}_{\\mu}\\rangle=\\frac{N}{24\\pi} R\\,,\n\\ee\nwhere $N=N_S+N_F$ is the total number of massless scalar and Dirac fermion fields. The trace anomaly can be derived from the non-local Polyakov action\\footnote{The coefficient $N$ in front of  $S_{\\rm anom}$  becomes $N-25$ if one also takes into account the metric fluctuations themselves, beside the fluctuations due to matter fields.}\n\\be\nS_{\\rm anom}=-\\frac{N}{96\\pi}\\int d^2x\\sqrt{-g}\\, R\\frac{1}{\\Box}R\\, ,\n\\ee\nwhose variation gives the right-hand side of \\eq{traceT}, and which can therefore be added to the classical Einstein-Hilbert action, to provide an effective action which takes into account  the quantum effect of the trace anomaly. In $D=2$ one can always write the metric in the form (\\ref{gmnsigma}), without truncating the theory, and in terms of the conformal mode the Polyakov action becomes local, \n\\be\nS_{\\rm anom}=\\frac{N}{24\\pi}\\int d^2x\\sqrt{-\\bar{g}}\\, (-\\sigma\\overline{\\square}\\sigma+\\overline{R}\\sigma)\\, ,\n\\ee\nwhere the bars denote quantities computed with the reference metric $\\bar{g}_{\\mu\\nu}$. In $D=2$ the effect of the anomaly is especially important, since the Einstein-Hilbert term is a topological invariant and the dynamics is entirely contained in $S_{\\rm anom}$. Observe that, in $D=2$,  the propagator of the $\\Box^{-1}$ operator is logarithmic, $G(x,x')\\propto\\log (x-x')^2$, and therefore grows with distance.\n\nThe situation is conceptually similar in four dimensions.  In this case the trace anomaly is given by (see e.g. \\cite{Birrell:1982ix,Buchbinder:1992rb,Shapiro:2008sf} for pedagogical introductions)\n\\be\\label{traceT4d}\n\\langle T^{\\mu}_{\\mu}\\rangle=b C^2+ b'\\(E-\\frac{2}{3}\\Box R\\)+b''\\Box R\\, ,\n\\ee\nwhere, as in \\eq{SHD}, $C^2$ is the square of the Weyl tensor, $E$ the Gauss-Bonnet term, and the coefficients $b,b',b''$ depends on the number of massless conformally-coupled scalars, massless fermions and massless vector fields. The $\\Box R$ term  on the right-hand side of\n\\eq{traceT4d}\ncan be derived from the variation of a local action proportional to $\\int d^4x\\sqrt{-g}\\,R^2$. The remaining combination is obtained from the variation of a non-local action\n\\be\nS_{\\rm anom}=-\\frac{1}{8}\\int d^4x\\sqrt{-g}\\(E-\\frac{2}{3}\\Box R\\) \\Delta_4^{-1}\n\\[ b' \\(E-\\frac{2}{3}\\Box R\\)-2b C^2\\]\\, ,\n\\ee\nwhere\n\\be\n\\Delta_4=\\Box^2+2R^{\\mu\\nu}\\n_{\\mu}n_{\\nu}-\\frac{2}{3} R\\Box +\\frac{1}{3}(\\n^{\\mu}R)\\n_{\\mu}\n\\, .\n\\ee\nOnce again, this non-local action becomes local in terms of the conformal mode. Writing the metric as in \\eq{gmnsigma},  using as a fiducial metric $\\bar{g}_{\\mu\\nu}=\\eta_{\\mu\\nu}$ (and choosing appropriately the local $R^2$ counterterm) one finds \n(see e.g. \\cite{Antoniadis:1991fa,Antoniadis:2006wq})\n\\be\\label{Sanom4D}\nS_{\\rm anom}=-\\frac{Q^2}{16\\pi^2}\\int d^4x\\, (\\Box\\sigma)^2\\, ,\n\\ee\nwhere \n\\be\\label{Q2} \nQ^2=\\frac{1}{180}(N_S+\\frac{11}{2}N_F+62 N_V-28)+Q^2_{\\rm grav}\\, .\n\\ee\nHere $N_S, N_f$ and $N_V$ are the number of massless conformally coupled scalars, massless Weyl fermions and massless vectors, respectively, while $Q^2_{\\rm grav}$ is the contribution from the gravitons, which is not unambiguously known, and $-28$ is the contribution from the $\\sigma$ field itself~\\cite{Antoniadis:1991fa,Antoniadis:1996pb}.\n\n\n Of course, a difference from the two-dimensional case is that in $D=4$ writing the metric in the form (\\ref{gmnsigma}) is a truncation of the theory. However, we see from \\eq{Sanom4D} that the propagator of the $\\sigma$ field in momentum space is $\\propto 1\/k^4$, which in coordinate space, in $D=4$, gives  again a propagator growing logarithmically, as in $D=2$,\n\\be\\label{Gx2m0}\nG(x,x')=-\\frac{1}{2Q^2}\\log\\[\\mu^2 (x-x')^2\\]\\, ,\n\\ee\nwhere $\\mu$ is an IR regulator.\nThis means again that the fluctuations of the $\\sigma$ field are large at long distances, and justifies a posteriori the approximation of keeping only the conformal mode when studying the quantum effects of gravity in the IR.\nThe fact that the conformal mode dominates the quantum theory in the IR is also confirmed by the fact that it gives the most infrared divergent contribution to the propagator in de~Sitter space~\\cite{Antoniadis:1986sb}. \nObserve that the fourth-order operator $\\Delta_4$  in $D=4$ is the analogue of the second-order operator $\\Box$  in $D=2$, since it is conformally invariant, $\\sqrt{-g}\\,\\Delta_4=\\sqrt{-\\bar{g}}\\,\\bar{\\Delta}_4$.\n\nAs discussed in ref.~\\cite{Antoniadis:1996pb}, the large fluctuations in the two-point function of the $\\sigma$ field due to its logarithmic growth create a situation that looks very similar to what happens in the two-dimensional XY model, with the Berezinsky-Kosterlitz-Thouless (BKT) transition. Let us recall the argument of ref.~\\cite{Antoniadis:1996pb}. The four-dimensional theory  for the conformal factor has solutions of the form\n\\be\n\\sigma(x)=q\\ln |x-x_0|\\, ,\n\\ee\nwhich are analogous to the vortices of the 2-dimensional theory. The euclidean action for $\\sigma$, evaluated on this configuration, takes the value~\\cite{Antoniadis:1996pb}\n\\be \nS_E=\\frac{1}{2}Q^2q^2\\ln\\(\\frac{L}{a}\\)\\, ,\n\\ee\nwhere $L$ is the IR cutoff and the ``lattice spacing\" $a$ is the UV cutoff. The parameter $x_0$ is a collective coordinate corresponding to the center of the solution. The number of ways of placing this solution in a volume $V$ is proportional to $V$, so the  corresponding entropy is $4\\log L\/a$, and the free energy is \n\\be\nF=\\(\\frac{1}{2}Q^2q^2-4\\)\\ln\\(\\frac{L}{a}\\)\\, .\n\\ee\nIf $Q^2>8\/q$ the free energy diverges as we remove the IR cutoff ($L\\rightarrow\\infty$) or the UV cutoff ($a\\rightarrow 0$), so these configurations are irrelevant. In contrast, if $Q^2<8\/q$ they dominate the partition function, so at the critical value of $Q$ we expect a phase transition. For the solution with $q=1$,  $Q_c^2=8$. The situation is indeed completely analogous to the BKT phase transition in two dimensions, which is also triggered by the logarithmic growth of the vortex-vortex interaction. The role of the coupling constant to be tuned to get the phase transition is here played by $Q$, which depends on the number of massless conformally-coupled particles. If the unknown graviton contribution in \\eq{Q2} is not too large, the theory with just a single massless photon is in the region where $Q^2<Q_c^2$, and the non-trivial configurations dominate.\n\nIn ref.~\\cite{Antoniadis:1996pb} it has been suggested that  $Q^2<Q_c^2$ corresponds to a phase where the space-time geometry is no longer smooth. Actually, pushing further the analogy with the BKT transition can suggest a different scenario. Indeed, in the two-dimensional case, in the regime where the vortices dominate, the correlation length becomes finite and a mass gap is generated. \nIt is plausible to expect that the same happens in the four-dimensional case. \nWe therefore expect the generation of a term $\\propto \\sigma^2$ in the effective action for $\\sigma$, whose coefficient is given by a dynamically-generated mass scale. Of course, this mass term must be constructed in such a way that it does not spoil the underlying diffeomorphism invariance, which leads us again to a  term $\\Lambda_{\\rm RR}^4R\\Box^{-2}R$, see \\eq{m2s2}.\n\n\n\nAdding this term to the  anomaly-induced effective action  (\\ref{Sanom4D}), to quadratic order  we get an effective action for $\\sigma$ given by\n\\be\\label{Sanom4Dmass}\nS^{(2)}_{\\rm eff}=-\\int d^4x\\,\\[ \\frac{Q^2}{16\\pi^2} (\\Box\\sigma)^2+\\Lambda_{\\rm RR}^4\\sigma^2\\]\n\\, .\n\\ee\nThe corresponding euclidean propagator is \n\\be\\label{Gxmu4}\nG(x)=\\frac{8\\pi^2}{Q^2}\\int \\frac{d^4k}{(2\\pi)^4}\\, \\frac{1}{k^4+\\mu^4} e^{-ikx}\\, ,\n\\ee\nwhere $\\mu^4=16\\pi^2\\Lambda_{\\rm RR}^4\/Q^2$. Carrying out the angular integrals, we get\n\\be\nG(x)=\\frac{2}{Q^2}\\int_0^{\\infty} dy \\frac{y^2 J_1(y)}{y^4+ (\\mu^2 x^2)^2}\\, ,\n\\ee\nwhere $J_1(y)$ is a Bessel function.\nThe final integral can in principle be carried out in term of a (not very illuminating) Meijer's G-function. However we already see from this expression, together with the expansion $J_1(y)\\simeq y\/2$ for small $y$, that the presence of the term $\\mu^2 x^2$ regularizes  the integral near the integration limit $y=0$. In the limit $\\mu^2 x^2\\rightarrow 0$ we recover \\eq{Gx2m0}, except that the artificial IR regulator $\\mu$ of \\eq{Gx2m0} is  replaced by the quantity $\\mu$ that appears in\n\\eq{Gxmu4}, which now is\na physical quantity related to $\\Lambda_{\\rm RR}$. In the opposite limit \n$\\mu^2 x^2\\gg 1$, introducing the notation $u=(\\mu^2x^2)^{1\/2}$ and using as integration variable $z=y\/u$, we have\n\\bees\nG(x)&=&\\frac{2}{Q^2}\\, \\frac{1}{u}\\int_0^{\\infty}dz\\, \\frac{z^2}{1+z^4} J_1(uz)\\nonumber\\\\\n&\\simeq &\\frac{2}{Q^2}\\, \\sqrt{\\frac{2}{\\pi}} \\, \\frac{1}{u^{3\/2}}\n\\int_0^{\\infty}dz\\, \\frac{z^{3\/2}}{1+z^4}\\, \\cos\\(uz-\\frac{3\\pi}{4}\\)\\, ,\n\\ees\nwhere in the second line we have used the asymptotic expansion of the Bessel function.\\footnote{Observe that, as $u\\rightarrow\\infty$, the regime $uz$ fixed corresponds to $z\\rightarrow 0$, whose contribution to the integral is suppressed by the factor $z^2$ in the integrand. The dominant contribution therefore comes from the region $uz\\rightarrow\\infty$, where we can use the asymptotic expansion\n$J_1(x)\\simeq (2\/\\pi x)^{1\/2}\\cos (x-3\\pi\/4)$ valid for large $x$.} The integrand is an oscillating function, which oscillates faster and faster as $u$ increases, so the Green's function vanishes faster than $1\/u^{3\/2}$, i.e faster than $1\/(\\mu^2 x^2)^{3\/2}$.\n\n\nThe physical picture that emerges from these considerations is that, when the dynamics of the conformal mode is described by the action (\\ref{Sanom4D}),  fluctuations grow larger and larger at long distances, as shown by the logarithmic growth of the propagator in \\eq{Gx2m0}. This is a sign of an IR instability of the vacuum state. When a mass for the conformal mode is dynamically generated, the long-distance correlations  grow only until the separation $|x-x'|$ reaches a  length-scale  of order $1\/\\mu$, and after that they decay faster than $1\/|x-x'|^3$. Therefore, there is no IR instability. The dynamical generation of a mass for the conformal mode is the mechanism by which the vacuum restructures itself, as a response to the IR instability triggered by the anomaly-induced effective action. \n\n\\section{Conclusions}\\label{sect:concl}\n\nIn this paper we have explored the possibility of generating non-local terms, such as those given in \\eqs{RT}{RR}, through strong IR effects in gravity. We have first of all considered the running of the coupling associated to the $R^2$ term in higher-derivative gravity. We have stressed that a peculiar aspect of the running of this coupling is that, even in the deep IR, its calculation requires a knowledge of the UV completion of GR. Indeed, in GR the corresponding beta function is not even gauge-independent, and has no physical meaning. Stelle theory provides an example of  an extension of GR where, thanks to the improved UV behavior, the beta functions of the ${\\cal O}(R^2)$ couplings are physical and gauge-independent. However, the result from Stelle theory cannot be taken literally either, because the theory is plagued by a ghost (and, furthermore, existing computations of the RG flow have been performed only in the regime $E\\ggm_{\\rm Pl}$). In the absence of a concrete reliable result for the RG flow, we have simply explored the consequences of a scenario in which the coupling $a_2$ associated to the $R^2$ term grows in the IR. In this case, similarly to QCD, we expect the dynamical generation of a mass scale, $\\Lambda_{\\rm RR}$, and of a non-trivial form factor $a_2(\\Box)$. We have provided arguments, based on the analogy with the known non-perturbative behavior  of the gluon propagator in QCD, that suggest that in the strong coupling regime the emergence of a pole-like behavior\n$a_2(\\Box)\\propto\\Box^{-2}$ is quite possible. \n\nThe emergence of a dynamical scale $\\Lambda_{\\rm RR}$ would have far reaching consequences for solving long-standing puzzles concerning dark energy. In particular, this scale emerges from the logarithmic running of a dimensionless coupling constant, similarly to $\\Lambda_{\\rm QCD}$, and does not receive large loop corrections, contrary to what happens to the vacuum energy. It is therefore technically natural. We found that, to explain dark energy today, \nthe dynamically-generated scale $\\Lambda_{\\rm RR}$ should be of the order of a  milli-eV. Of course we cannot predict its numerical value, just as we \ncannot  predict the value of $\\Lambda_{\\rm QCD}$. However,  it is interesting that, to obtain a dynamical explanation of dark energy,  we do not need to introduce a fundamental particle with an astonishingly small mass $m\\sim H_0\\sim 10^{-33}$~eV, nor a cosmological constant $\\Lambda$ with the amazingly small value $\\Lambda\\sim H_0^2$. In our scenario, \nin the IR the action  is given by \\eq{Sg2RinIR}, with a fundamental mass scale $\\Lambda_{\\rm RR}\\sim {\\rm meV}$, which is by no means a particularly surprising value.\nThe connection with the present value of the Hubble parameters comes out because, in a cosmological context, derivatives of the metric are of order $H$, so $\\Box\\sim H^2$ and $\\Box^{-1}\\sim H^{-2}$, while\n$R\\sim H^2$. Thus \nthe term \n$\\Lambda^4_{\\rm RR}R\\Box^{-2}R$ becomes competitive with $m_{\\rm Pl}^2R$ when $H\\sim \\Lambda^2_{\\rm RR}\/m_{\\rm Pl}$ which, for $\\Lambda_{\\rm RR}\\sim\\, {\\rm meV}$, gives $H\\sim H_0\\sim 10^{-33}$~eV. However, in this scenario the fundamental scale emerging dynamically in the theory is $\\Lambda_{\\rm RR}$, and there is no fundamental particle with the astonishingly small mass $10^{-33}$~eV.\n\nAnother interesting aspect of the above analysis is that it draws attention to the dynamics of the conformal mode of the metric, since the running of the coupling $a_2$ associated to $R^2$, from a constant value in the UV toward   $a_2(\\Box)\\propto\\Box^{-2}$ in the IR, has an interesting interpretation in terms of the conformal mode, as we discussed in sect.~\\ref{sect:conf}. \nThis stimulates to look in more detail into the dynamics of the conformal mode in Einstein gravity, in which the higher-derivative term is provided by the anomaly-induced effective action, i.e. by the loops of the massless matter fields in the theory. Once again, the conformal mode seem to play an interesting role. In Einstein gravity supplemented by the anomaly-induced effective action the emergence of a term $R\\Box^{-2}R$ can be interpreted as  the dynamical generation a mass term for the conformal mode of the metric (an interpretation that becomes blurred in Stelle theory, where a mass term for the conformal mode is already present at tree level because of the $R^2$ term). We\nhave argued that such  a dynamical mass generation can take place as a consequence of the strong IR fluctuations generated by the two-point function of the conformal mode, which grows logarithmically at large distances and, as already suggested in \nin ref.~\\cite{Antoniadis:1996pb}, can trigger a phase transition similar to  the Berezinsky-Kosterlitz-Thouless transition in two dimensions. If this results in a dynamical mass generation for the conformal mode, one  again obtains in the IR a non-local model of the type (\\ref{RT}) or (\\ref{RR}).\n\n\\vspace{5mm}\n\\noindent\n{\\bf Acknowledgments.}\nI am grateful to  Eugenio Bianchi, Giulia Cusin,\nEduard Gorbar, Alex Kehagias,  Ilya Shapiro, Sergey Sibiryakov, Julian Sonner and Arkady Tseytlin  for very useful discussions.\nThis work is supported by the Fonds National Suisse and by the SwissMAP National Center for Competence in Research.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}