data_all_eng_slimpj/shuffled/split2/finalzfmp

{"text":"\\section{Introduction: The Bloch sphere and its extension}\nIn the study of the dynamics of a spin-1\/2 particle, a visual metaphor that has played a powerful role is that of the ``Bloch sphere\" \\cite{Schleich}. Pure states of the system are represented by the tip of a vector from the origin to the surface of such a unit sphere S$^2$. In the field of nuclear magnetic resonance (nmr) \\cite{nmr} and elsewhere, transformations between states are then viewed as rotations of that vector, described by the Bloch equation of motion, $\\dot {\\vec m} = -2 {\\vec B} \\times {\\vec m}$, for a magnetic moment in a magnetic field ${\\vec B}$. Thus, various sequences of nmr manipulations can be pictured in a nice geometrical way as successive rotations, and this has now become central to our intuition of spin dynamics. The relevant group of unitary transformations is SU(2), a rank-one, three-parameter group that is the double covering group of the three-dimensional rotation group SO(3) \\cite{groups}. The three operators of angular momentum, $(J_x, J_y, J_z)$, are the generators of these groups. A canonical set of parameters of SO(3) are the Euler angles. Integer values $j=0, 1, \\ldots $ provide various $(2j+1)$-dimensional  representations, while for SU(2), the half-odd integers occur as well. The two coordinates on S$^2$, together with a phase, provide the three parameters describing the full state. \n\nThis latter phase is often not accessible as, for instance, when dealing with the density matrix $\\rho$. Mixed states also are naturally accommodated in this picture. They are represented by points inside the sphere so that the vector is of length less than unity. Correspondingly, Tr $\\rho ^2 < {\\rm Tr}\\, \\rho$, which constitutes a definition of a mixed state \\cite{mixedstate}. States of light polarization, also a two-valued object, map onto the same mathematics and geometry through the ``Poincare\" sphere \\cite{Poincare}.\n\nIt would be of interest to have analogous geometrical pictures for multiple spins, especially in today's fields of quantum computation, cryptography, and teleportation, because the fundamental elements of these subjects are built up of a few qubits \\cite{Nielsen}. Thus, all logic gates for quantum computation can be built up from qubit pairs, while teleporting one qubit state requires an entangled pair held by the sender and receiver, for a total of three qubits. With SU(2$^p$) being the relevant group for $p$ qubits, this calls for a similar geometrical description of higher SU($N$). In this paper, we develop such a picture, through an easily accessible procedure which iteratively descends from $N$ to $N-n$, with $n < N$, in a manner that closely follows the description of SU(2). \n\nOur procedure also applies when $N$ is odd, a situation that does not arise with qubits but elsewhere widely in physics (for example, qutrits \\cite{qutrit}, neutrino oscillations \\cite{neutrino}, the quark model and quantum chromodynamics (QCD), etc.) The $N \\times N$ matrices of Hamiltonians and evolution operators are viewed as built up of $2 \\times 2$ block matrices through this $N=(N-n)+n$ decomposition, the block matrices then described in terms of the Pauli spinors of SU(2). Each step of this iterative reduction introduces an analog of the Bloch sphere, albeit of higher dimension and more complex structure, and constructs the effective Hamiltonians of dimension $(N-n)$ and $n$ for the next step. In this manner, using no more than the operations familiar from the SU(2) case, the full construction for SU($N$) is achieved. \n\nThe philosophy behind such a construction may be seen as generalizing Schwinger's philosophy for representations of SU(2) or SO(3), where higher $j$-representations are constructed from those of the fundamental, $j=1\/2$ \\cite{Schwinger}. We now do the analogous step of using SU(2) as the template for solving larger SU($N$). In particular, for the important case of SU(4) for two qubits, we give a complete description of the manifolds and phases involved and analytical expressions for them. Note again, as with light polarization and spin-1\/2, that the mathematics of $N$-level systems in quantum optics, atomic and molecular physics, and elsewhere, is the same as that we describe in the language of multiple qubits. This provides an even wider context for our results.\n\nThe arrangement of this paper is as follows. Section II describes the basic iterative decomposition of the evolution operator for SU($N$), mimicking the familiar procedure for spin-1\/2.  With $N=4$, and $n=2$, Section III specializes the results to SU(4), the case of two qubits, when all the manipulations involved are in terms of Pauli spinors. It also applies these results to Hamiltonians involving a restricted set of operators of the full group. An interesting one is SO(5), which can be described by a $5 \\times 5$ antisymmetric matrix that is the analog of the $3 \\times 3$ antisymmetric one for the magnetic field ${\\vec B}$ in the Bloch equation. Section IV then considers Hamiltonians requiring the full SU(4) group for their description. Linear equations, analogous to the Bloch equation, are derived in terms of vectors $\\vec {m}$, five- and six-dimensional vectors, respectively, for the SO(5) and full SU(4) cases. The latter also correspond to so-called ``Pl\\\"{u}cker coordinates\" \\cite{Plucker} which are also presented. Appendix A deals with the generalization to non-Hermitian Hamiltonians, and Appendix B presents the isomorphism between SU(4) and the groups Spin(6) and SO(6) which we exploit.\n\n\\section{Iterative construction of evolution operator in $N$ dimensions}\n\nWe wish to obtain the evolution operator ${\\bf U}^{(N)}(t)$ for the $N$-dimensional time-dependent Hamiltonian ${\\bf H}^{(N)}$:\n\n\\begin{equation}\n{\\bf H}^{(N)}(t) =\\left(\n\\begin{array}{cc}\n{\\bf H}^{(N-n)}(t) & {\\bf V}(t) \\\\\n{\\bf V}^{\\dagger}(t) & {\\bf H}^{(n)}(t)\n\\end{array}\n\\right).\n\\label{eqn1}\n\\end{equation}\nWe have blocked the Hamiltonian into $(N-n)$- and $n$-dimensional blocks, the diagonal blocks being square matrices while the off-diagonal ${\\bf V}$ is $(N-n) \\times n$ and ${\\bf V}^{\\dagger}$ is $n \\times (N-n)$. Although our discussion is for Hermitian ${\\bf H}^{(N)}$, the procedure can also apply more generally, in which case the off-diagonal blocks will not be simply related as adjoints (see Appendix A). We will also assume ${\\bf H}^{(N)}$ to be traceless, again a restriction that can be easily relaxed, the time integral of the trace becoming an overall phase of ${\\bf U}^{(N)}$.\n\nTo solve the evolution equation, with an over-dot denoting derivative with respect to time,\n\n\\begin{equation}\ni\\dot {\\bf U}^{(N)}(t) =   {\\bf H}^{(N)}(t){\\bf U}^{(N)}(t), \\,\\, {\\bf U}^{(N)}(0) ={\\bf I},\n\\label{eqn2}\n\\end{equation}\nwe similarly block the unitary matrix, writing it also as a product of three factors, the first two further grouped as $\\tilde{\\bf U}_1$ and the second, $\\tilde{\\bf U}_2$, block-diagonal in form:\n\n\\begin{eqnarray}\n{\\bf U}^{(N)}(t) & = & \\tilde{\\bf U}_1 \\tilde{\\bf U}_2, \\,\\,\\,\\, \\tilde{\\bf U}_1= e^{{\\bf z}(t) A_{+}} e^{{\\bf w}^{\\dagger} (t)A_{-}}, \\nonumber \\\\\n\\tilde{\\bf U}_1 & = & \\left(\n\\begin{array}{cc}\n{\\bf I}^{(N-n)} & {\\bf z}(t) \\\\\n{\\bf 0}^{\\dagger} & {\\bf I}^{(n)}\n\\end{array} \\right) \\left(\n\\begin{array}{cc}\n{\\bf I}^{(N-n)} & {\\bf 0} \\\\\n{\\bf w}^{\\dagger}(t) & {\\bf I}^{(n)}\n\\end{array} \\right), \\nonumber \\\\\n\\tilde{\\bf U}_2 & = &  \\left(\n\\begin{array}{cc}\n\\tilde{\\bf U}^{(N-n)} (t) & {\\bf 0} \\\\\n{\\bf 0}^{\\dagger} & \\tilde{\\bf U}^{(n)} (t)\n\\end{array} \\right),\n\\label{eqn3}\n\\end{eqnarray}\nwhere $A_{\\pm}$ are matrix generalizations of the Pauli spin step-up\/down $\\sigma _{\\pm}$, and ${\\bf z}$ and ${\\bf w}^{\\dagger}$ are rectangular matrices of complex parameters.\n \n\nThe above structure, with $\\tilde{\\bf U}_1$ having blocks of zero in the lower and upper off-diagonal blocks of its matrix factors, is crucial in our method. For the case of spin-1\/2 and SU(2), the form of a product of three factors, each an exponentiation of one of the Pauli spinors, is well known \\cite{groups}. Their Cartesian form, with Euler angles in the exponents, is the familiar choice but we choose instead the triplet, $(\\sigma _{\\pm}, \\sigma_z)$, when the first two factors have zero off-diagonal entries. This introduces complex ${\\bf z}$ and ${\\bf w}^{\\dagger}$ in place of the Euler angles, and makes the individual factors in Eq.~(\\ref{eqn3}) not separately unitary although our construction ensures unitarity of the full ${\\bf U}^{(N)}(t)$. Further, for non-Hermitian $H$ when $U$ is non-unitary, our construction still applies. The specific structure of an upper and lower triangular matrix and a diagonal one proves fruitful, giving simpler equations for ${\\bf z}$ and ${\\bf w}^{\\dagger}$, which will have at most quadratic nonlinearity in these parameters and not more complicated  trigonometric dependences as with the Euler angle decomposition \\cite{Rau98, Uskov}. They also yield more naturally to a geometrical picture of the manifolds they describe.\n\nA remark about notation. We will use the symbol tilde when the corresponding Hamiltonians or evolution operators may not be Hermitian or unitary, respectively. Unitarity of the full ${\\bf U}^{(N)}(t)$ leads to relations between ${\\bf z}$ and ${\\bf w}^{\\dagger}$ which would otherwise be independent for evolution under a non-Hermitian Hamiltonian (see Appendix A),\n\n\\begin{eqnarray}\n{\\bf z} = - {\\bf w}\\, {\\mbox {\\boldmath $\\gamma$}_2} & = & {-\\mbox {\\boldmath $\\gamma$}_1} \\,{\\bf w}, \\nonumber \\\\\n{\\mbox {\\boldmath $\\gamma$}_1} \\equiv  \\tilde{\\bf U}^{(N-n)} \\tilde{\\bf U}^{(N-n)\\dagger} & = & {\\bf I}^{(N-n)} +{\\bf z}{\\bf z}^{\\dagger}, \\nonumber \\\\\n{\\mbox {\\boldmath $ \\gamma$}_2}^{-1} \\equiv \\tilde{\\bf U}^{(n)} \\tilde{\\bf U}^{(n)\\dagger} & = & ({\\bf I}^{(n)} +{\\bf z}^{\\dagger} {\\bf z})^{-1}.\n\\label{eqn4}\n\\end{eqnarray}\n\nWith ${\\bf U}=\\tilde{\\bf U}_1 \\tilde{\\bf U}_2$, Eq.~(\\ref{eqn2}) formally reduces to the evolution of $\\tilde{\\bf U}_2$ alone with an effective Hamiltonian \\cite{Uskov, Heff},\n\n\\begin{equation}\ni\\dot{\\tilde{\\bf U}}_2 = \\tilde {\\bf H}_{\\rm eff} \\tilde{\\bf U}_2, \\,\\,\\tilde{\\bf H}_{\\rm eff}=\\tilde{\\bf U}_1^{-1}{\\bf H}\\tilde{\\bf U}_1 -i\\tilde{\\bf U}_1^{-1}\\dot{\\tilde{\\bf U}}_1.\n\\label{eqn5}\n\\end{equation}\nA key element of our construction lies in this effective Hamiltonian and corresponding evolution for the reduced problem. Since $\\tilde{\\bf U}_2$ and this equation are block diagonal, the off-diagonal blocks in ${\\bf H}_{\\rm eff}$ on the right-hand side must vanish. This condition leads to the defining equation for ${\\bf z}$,\n\n\\begin{equation}\ni\\dot{\\bf z}={\\bf H}^{(N-n)} {\\bf z} + {\\bf V}-{\\bf z}({\\bf V}^{\\dagger}{\\bf z}+{\\bf H}^{(n)}).\n\\label{eqn6}\n\\end{equation}\n\n\nFor SU(2), when $N=2, n=1$, all the matrices above reduce to single numbers and Eq.~(\\ref{eqn6}) is a Riccati equation for the complex $z$. More generally, it is a matrix Riccati equation \\cite{Reid}, and its solutions are involved in the subsequent construction. With the off-diagonal blocks of Eq.~(\\ref{eqn5}) accounted for, the diagonal ones defining the Hamiltonians for the $(N-n)$ and $n$ problems remain, and are given by $({\\bf H}^{(N-n)} -{\\bf z}{\\bf V}^{\\dagger})$ and $({\\bf H}^{(n)}+{\\bf V}^{\\dagger}{\\bf z})$, respectively. Although the overall trace is preserved in our construction and remains zero, these individual Hamiltonians are neither traceless nor Hermitian. The equations for ${\\bf z}$ need to be solved numerically in general but form a smaller set than the $N^2$ elements in the original Eq.~(\\ref{eqn2}).\n\nTo set up the process for iteration, the above individual Hamiltonians in $(N-n)$- and $n$-dimensional subspaces must be rendered Hermitian and traceless. The latter is easily achieved, by subtracting Tr $({\\bf H}^{(N-n)} -{\\bf z}{\\bf V}^{\\dagger})$ and Tr $({\\bf H}^{(n)}+{\\bf V}^{\\dagger}{\\bf z})$ from them. These traces being equal and opposite, this translates into the introduction of a phase, the integral of the trace, in ${\\bf U}^{(N)}$, representing a relative phase between the two subspaces. \n\nThere are alternative methods for rendering the Hamiltonians Hermitian, the most accessible one being through \n\n\\begin{equation}\n\\tilde{\\bf U}_1^{\\dagger} \\tilde{\\bf U}_1 =  \\left(\n\\begin{array}{cc}\n{\\mbox {\\boldmath $ \\gamma$}_1}^{-1} & {\\bf 0} \\\\\n{\\bf 0}^{\\dagger} & {\\mbox {\\boldmath $ \\gamma$}_2}\n\\end{array} \\right)\n\\equiv \\left(\n\\begin{array}{cc}\n{\\bf g}_1 {\\bf g}_1^{\\dagger} & {\\bf 0} \\\\\n{\\bf 0}^{\\dagger} & {\\bf g}_2 {\\bf g}_2^{\\dagger}\n\\end{array} \\right)^{-1}.\n\\label{eqn7}\n\\end{equation}\nThe first part of this equation is the observation that $\\tilde{\\bf U}_1^{\\dagger} \\tilde{\\bf U}_1$ is block diagonal. This suggests the second part of the equation, namely, the definition of an inverse through two ``Hermitian square-root\" matrices ${\\bf g}_i$. Together, they serve as a gauge factor to unitarize according to\n\n\\begin{equation}\n{\\bf U}_1 =\\tilde{\\bf U}_1 \\left(\n\\begin{array}{cc}\n{\\bf g}_1 & {\\bf 0} \\\\\n{\\bf 0}^{\\dagger} & {\\bf g}_2\n\\end{array} \\right).\n\\label{eqn8}\n\\end{equation} \nWith that, the second factor, $\\tilde{\\bf U}_2$, in Eq.~(\\ref{eqn3}) is also unitarized,\n\n\\begin{equation}\n{\\bf U}_2= \\left(\n\\begin{array}{cc}\n{\\bf g}_1^{-1} & {\\bf 0} \\\\\n{\\bf 0}^{\\dagger} & {\\bf g}_2^{-1}\n\\end{array} \\right)\\tilde{\\bf U}_2.\n\\label{eqn9}\n\\end{equation}\n\nAfter some algebra, the explicitly Hermitian forms of the two diagonal block Hamiltonians of dimension $(N-n)$ and $n$ are \n\n\\begin{eqnarray}\n{\\bf H}^{(N-n)}\\!\\!\\! & =\\! \\!\\!& \\frac{i}{2} [\\frac{d}{dt} {\\bf g}_1 ^{-1}, {\\bf g}_1]\\! +\\!\\! \\frac{1}{2}\\! \\left( {\\bf g}_1^{-1} ({\\bf H}^{(N-n)}\\!\\!-{\\bf z}{\\bf V}^{ \\dagger})\\, {\\bf g}_1 \\!\\! +\\!{\\rm hc} \\right)\\!, \\nonumber \\\\\n\\!\\!{\\bf H}^{(n)}\\!\\!\\! & =\\!\\!\\! & \\frac{i}{2} [\\frac{d}{dt} {\\bf g}_2 ^{-1}, {\\bf g}_2 ]  \\!\\!+\\!\\! \\frac{1}{2}\\! \\left( {\\bf g}_2 ^{-1} ({\\bf H}^{(n)}\\!\\!+{\\bf z}^{\\dagger}{\\bf V}) \\, {\\bf g}_2 \\!\\! + \\! {\\rm hc} \\right)\\!,\n\\label{eqn10}\n\\end{eqnarray}\nwith commutator brackets in the first term, and hc in the second term denoting the Hermitian conjugate of the preceding expression. Again, the trace of each Hamiltonian in Eq.~(\\ref{eqn10}) can be subtracted to render them traceless; as clear by inspection, this is the same trace discussed just above. These Hamiltonians in Eq.~(\\ref{eqn10}) can now be treated further as SU($N-n$) and SU($n$) problems.\n\nThe ${\\mbox {\\boldmath $\\gamma$}}$ matrices in Eq.~(\\ref{eqn4}) are Hermitian with non-negative eigenvalues because of their origin from $\\tilde{\\bf U}_1^{\\dagger} \\tilde{\\bf U}_1$. This permits their decomposition into ${\\bf g}$ as shown in Eq.~(\\ref{eqn7}). The ${\\bf g}$ matrices and their inverses in Eq.~(\\ref{eqn7})-Eq.~(\\ref{eqn10}), are square roots of them, and because any power, including fractional ones, are Hermitian term by term in a formal power-series expansion, we can choose ${\\bf g}$ also as Hermitian. The use of identities such as\n\n\\begin{equation}\n{\\bf z}^{\\dagger} {\\mbox {\\boldmath $\\gamma$}_1}^p={\\mbox {\\boldmath $\\gamma$}_2}^p {\\bf z}^{\\dagger}, \\,\\,\\, {\\mbox {\\boldmath $\\gamma$}_1}^p {\\bf z}={\\bf z} {\\mbox {\\boldmath $\\gamma$}_2}^p, \\,\\,\n\\label{eqn11}\n\\end{equation}\nserves to express all ${\\bf g}$ in terms of the linearly independent set of matrices of dimension $(N-n)$ or $n$, whichever is smaller. With $n=2$, this means that all the algebra of calculating such square-root matrices and the subsequent evaluation of the effective Hamiltonian in Eq.~(\\ref{eqn10}) reduces to manipulation of Pauli matrices.       \n\nA count of the parameters is instructive. The original SU($N$) evolution involves $(N^2-1)$ elements and, therefore, grows quadratically with $N$. These are divided in the above construction into the $2n(N-n)$ elements in ${\\bf z}$, which for small $n$ grows only linearly with $N$. The rest are contained in the elements of the SU($N-n$) and SU($n$) and the single phase between those two subspaces. Our construction of higher SU($N$) evolution in terms of smaller ones, with the template in Eq.~(\\ref{eqn3}) of three factors as in SU(2), resembles the Schwinger scheme of generating higher $j$ representations of SU(2) or SO(3) from the fundamental one of $j=1\/2$ \\cite{Schwinger}. Whereas that scheme was for higher representations but of the same group, SU(2), our procedure extends in the direction of larger groups SU($N$).\n\nIn mathematical language of base manifolds and fiber bundles \\cite{fiber}, the SU(2) and its Bloch sphere are seen as the bundle [SU(2)\/U(1)] $\\times $ U(1), the former the two-sphere S$^2$ base and the latter U(1) phase the fiber. Likewise, our construction is in terms of the base manifold [SU($N$)\/(SU($N-n)$ $\\times $ SU($n$) $\\times $ U(1))] and the fiber (SU($N-n)$ $\\times $ SU($n$) $\\times $ U(1)). For SU(2), there is a single complex $z$ that defines the base manifold. The Bloch sphere of a unit three-dimensional vector $\\vec{m}$ corresponding to $z$ is then constructed by inverse stereographic projection from R$^2$ to S$^2$. Similar structures of a $\\vec{m}$ associated with the larger ${\\bf z}$ will be considered in the next sections. \n\n\\section{The case of SU(4), with application to its sub-groups}\nAn important case is of $N=4$. Four-level systems are commonly considered in quantum optics and molecular systems and, of course, in today's quantum computation where they describe two qubits \\cite{Nielsen}. Since all logic gates can be built up from such qubit pairs, the study of the evolution operator for such $N=4$ problems is of current interest. As a combined description of spin and isospin, SU(4) also has central importance in the study of nuclei and particles \\cite{Close}. The group also occurs in the description of unusual magnetic phases of $f$ electron states in CeB$_6$ \\cite{Ohkawa}. Both choices $n=1,2$ in the general procedure of Section II lead to interesting decompositions, with the latter the more natural for qubit applications. We now turn to this case. \n\nIn physics terms, a 4-level Hamiltonian has three real parameters along the diagonal to fix the energy positions of the levels. (One overall element, represented by the trace, can be subsumed as an uninteresting definition of the zero energy reference level, leading also to an irrelevant overall phase in the evolution operator.) In addition, six off-diagonal couplings, which are complex, make for a total of 15 parameters to describe the full Hamiltonian. Symmetries often reduce this number so that the Hamiltonian involves only a smaller number as a closed sub-algebra. For two identical qubits, there are indeed such symmetries which reduce the number of independent energies and couplings of a 4-level system.\n\nWith $N=4, n=2$, all the matrices involved in the previous section can be rendered in terms of Pauli spinors and the unit $2 \\times 2$ matrix. A general Hamiltonian of SU(4) has 15 independent operators and time-dependent parameters multiplying them. A standard, explicit rendering of the 15 $4 \\times 4$ matrices is given in \\cite{Rau2, Hilbers}. ${\\bf z}$ comprises four complex quantities, $(z_4, z_i)$, and the matrix Riccati equation reduces to coupled first-order equations in them with quadratic nonlinearity. Deferring this general case to the next section, we consider first the smaller sets of operators of various sub-groups of SU(4). \n\n{\\it su(2) $\\times$ su(2) sub-algebra:} Consider first a Hamiltonian consisting of only six of the 15 operators. Since our construction is representation independent, in a suitable representation, the six may be viewed as two independent, mutually commuting, triplets that obey su(2) algebra. Clearly, each then may be expected to have its own geometrical description in terms of a Bloch sphere and phase. In our above, general formulation, this result is realized as follows. Thus, consider two independent magnetic moments, characterized by the standard Pauli matrices $\\sigma$, in time-varying magnetic fields $A(t)$ and $B(t)$ which may also be independent, with Hamiltonian $H = \\vec{\\sigma}^{(1)} \\cdot {\\vec A} + \\vec{\\sigma}^{(2)} \\cdot {\\vec B}$. Using a standard set of $ 4 \\times 4$ matrices \\cite{Rau2} to cast this Hamiltonian in the form of Eq.~(\\ref{eqn1}), we have ${\\bf V}=(A_x-iA_y){\\bf I}$ and ${\\bf H}^{(1,2)}= \\vec{\\sigma} \\cdot {\\vec B} \\pm A_z {\\bf I}$. The ${\\bf z}$ in Eq.~(\\ref{eqn3}) also reduces, as with ${\\bf V}$, to a unit operator with a single complex coefficient $z_4$ obeying a Riccati equation in Eq.~(\\ref{eqn6}). The gamma matrices in Eq.~(\\ref{eqn4}) are also proportional to the unit operator, thus simplifying Eq.~(\\ref{eqn10}), the ${\\bf g}$ dropping out. As a result, the Hermitian matrices in the block-diagonal effective Hamiltonian take the form of the same $\\vec{\\sigma} \\cdot {\\vec B}$ plus\/minus a term proportional to a unit matrix. The first term is viewed as for a single spin with a Bloch sphere and a phase, the second represents a phase between the two $2 \\times 2$ spaces. The complex $z_4$ can again be inverse stereographically projected into another two-sphere as in the Bloch construction. We arrive, therefore, at the same initial expectation, that a simultaneous viewing in terms of two Bloch vectors in individual two-spheres, along with their fibers, provides the geometrical picture for all such qubit-pair systems. A specific physical example occurs in the construction of optimal quantum NOT operations \\cite{Novotny}.\n\n{\\it su(2) $\\times$ su(2) $\\times$ u(1) sub-algebra:} Another sub-algebra, involving seven of the 15 operators, has been considered before \\cite{Rau2, Ganesh}. It has the symmetry of SU(2) $\\times $ SU(2) $\\times $ U(1). In a suitable representation, such a Hamiltonian can be cast as a diagonal form in Eq.~(\\ref{eqn1}) plus a term which is proportional to the unit operator in both diagonal blocks but with equal and opposite sign. Such an operator commutes with all the other six, themselves comprised of two mutually commuting triplets of $4 \\times 4$ matrices \\cite{Rau2}. With ${\\bf V}=0$, ${\\bf z}$ also vanishes and we reduce trivially to the two independent SU(2) and a phase between the two spaces, together accounting for the 7 parameters of this problem. An example is provided by the CNOT gate constructed with two Josephson junctions \\cite{Nakamura}. Many such sets of seven operators, one of which commutes with all the remaining six, have been identified through a general procedure in footnote 11 of \\cite{Ganesh}.\n\n{\\it so(5) sub-algebra:} Proceeding further to other sub-groups, a non-trivial example is provided by a $H$ that involves ten operators satisfying an so(5) sub-algebra of su(4). Again, there are many such sets of ten operators\/matrices which close under commutation within the full set of 15 as noted in footnote 11 of \\cite{Ganesh}. As a physical example, a four-level system of two symmetric pairs, as naturally so with two identical qubits, has only two real parameters along the diagonal in its $H$. Selection rules often restrict the off-diagonal coupling between the levels from six to four, thus introducing four complex, or eight real, parameters. The net result of such symmetric four-level systems is a ten-parameter problem \\cite{Legare}. Such $H$ fall into this so(5) sub-algebra. The corresponding group is the so-called spin group Spin(5) which is the double-covering group of SO(5), the group of five-dimensional rotations, much as Spin(3), isomorphic to SU(2), is the covering group of SO(3) \\cite{spin}. All such Spin(5) or SO(5) will themselves have a Spin(4) or SO(4) sub-group, which in turn has the two mutually commuting SU(2) or SO(3) discussed above so that the ten matrices can be conveniently viewed as two sets of commuting triplets plus four more which transform like a four-dimensional vector under SO(4). For completeness here in this paper, we briefly summarize results on this so(5) sub-algebra that were published elsewhere \\cite{Uskov}; see also \\cite{Rangan}. \n\nIn a convenient representation that uses Pauli matrices for two spins \\cite{Rau2}, we have $H(t)=F_{21}\\sigma^{(2)}_z-F_{31}\\sigma^{(2)}_y+F_{32}\\sigma^{(2)}_x-F_{4i}\\sigma^{(1)}_z\\sigma^{(2)}_i+F_{5i}\\sigma^{(1)}_x\\sigma^{(2)}_i-F_{54}\\sigma^{(1)}_y$, where the ten arbitrarily time-dependent coefficients $F_{\\mu \\nu}(t)$ form a $5 \\times 5$ antisymmetric real matrix. (We will use $\\mu,\\nu=1-5$ and $i,j,k=1-3$ and summation over repeated indices.) Several quantum optics and multiphoton problems of four levels driven by time-dependent electric fields have such a Hamiltonian. It has also been considered extensively in coherent population transfer in many molecular and solid state systems \\cite{Legare}.  Casting this Hamiltonian in the form of Eq.~(\\ref{eqn1}), we have\n\n\\begin{equation}\n{\\bf H}^{(1,2)}=(\\mp F_{4k} -\\frac{1}{2} \\epsilon_{ijk} F_{ij}) \\sigma_k, {\\bf V}=iF_{54} {\\bf I}^{(2)}+F_{5i} \\sigma_i.\n\\label{eqn12}\n\\end{equation}\n\nWith the matrix Riccati equation in Eq.~(\\ref{eqn6}) cast in terms of Pauli spinors together with coefficients $ z_{\\mu}=z_4,z_i$: ${\\bf z}=z_4 {\\bf I}^{(2)} -iz_i \\sigma_i$, it takes the form \n\n\\begin{equation}\n\\dot{z}_{\\mu}=F_{5\\mu}(1-z_{\\nu}^2)+2F_{\\mu \\nu}z_{\\nu}+2F_{5\\nu}z_{\\nu}z_{\\mu}.\n\\label{eqn13}\n\\end{equation}\n(As an alternative, ${\\bf V}$ and $ {\\bf z}$ can also be rendered in terms of quaternions $(1,-i\\sigma_i)$.) ${\\mbox {\\boldmath $ \\gamma $}_1}$ and ${\\mbox {\\boldmath $ \\gamma $}_2}$ in Eq.~(\\ref{eqn4}) become equal and proportional to a unit matrix, $(1+z_{\\mu}z_{\\mu}){\\bf I}^{(2)}$. The structure of Eq.~(\\ref{eqn13}) admits to the four quantities $z$ being real. The effective Hamiltonian in Eq.~(\\ref{eqn10}) in terms of these $z$ becomes\n\n\\begin{equation}\n{\\bf H}_{\\rm eff}^{(1,2)}={\\bf H}^{(1,2)}-\\epsilon_{ijk} z_i F_{5j} \\sigma_k \\mp F_{5j} z_4\\sigma_j \\pm F_{54 }z_i \\sigma_i.\n\\label{eqn14}\n\\end{equation}\n\n\nWe can now construct a five-dimensional unit vector $\\vec{m}$ out of the four $z$,\n\n\\begin{equation}\nm_{\\mu}=\\frac{-2z_{\\mu}}{(1+z_{\\nu}^2)}, \\,m_5=\\frac{(1-z_{\\nu}^2)}{(1+z_{\\nu}^2)},\\,\\,\\,\\, \\mu,\\nu=1-4. \n\\label{eqn15}\n\\end{equation}\nThe nonlinear Eq.~(\\ref{eqn6}), or Eq.~(\\ref{eqn13}) in $z$, becomes of simple, linear Bloch-like form, \n\n\\begin{equation}\n\\dot{m}_{\\mu}=2F_{\\mu\\nu}m_{\\nu},\\,\\,\\,\\,\\, \\mu,\\nu=1-5.\n\\label{eqn16}\n\\end{equation}\n\nAs in the single spin case, this represents an inverse stereographic projection, now from the four-dimensional plane $z \\in R^4$ to the four-sphere $S^4$. It provides a higher-dimensional polarization vector for describing such two spin problems. With ${\\bf z}$ so described, the two effective SU(2) Hamiltonians in Eq.~(\\ref{eqn14}), when solved in turn, give the complete solution. In all, such Hamiltonians possessing Spin(5) symmetry are, therefore, described by the geometrical picture of one S$^4$ and two S$^2$ spheres along with two phases.\n\n{\\it su(3) sub-algebra:} Four-level systems with only two independent energy parameters along the Hamiltonian's diagonal and three complex off-diagonal couplings constitute a su(3) sub-algebra with 8 parameters. A general three-level system, embedded into four with the fourth level completely uncoupled, constitutes a trivial example of such an su(3) sub-algebra but less trivial examples can also occur. The ${\\bf z}$ now has two non-zero complex $z$ for a total of four parameters. The description of this four-dimensional manifold, as well as the remaining SU(2) and a U(1) phase, parallel the discussion of the general SU(4) in the next section, and will be presented elsewhere \\cite{Sai}. Therefore, we omit details except to note that setting $z_4=-iz_1$ and $z_3=-iz_2$ in Section IV reduces to such a SU(3) symmetry. \n\n\\section{The general SU(4) Hamiltonian involving all fifteen operators}\n\nInstead of the Hamiltonians considered in Section III which involve sub-algebras of the full two-qubit system, consider an arbitrary $4 \\times 4$ Hamiltonian with its entire complement of 15 operators\/matrices. Such a $H$ is obtained by adding to the previous Spin(5) Hamiltonian considered above the five additional terms, $F_{65} \\sigma ^{(1)}_z +F_{64} \\sigma ^{(1)}_x +F_{6i} \\sigma ^{(1)}_y \\sigma ^{(2)}_i$. Correspondingly, Eq.~(\\ref{eqn11}) gets an additional term $\\pm F_{65} {\\bf I}^{(2)}$ in the diagonal ${\\bf H}^{(1,2)}$ while in ${\\bf V}$, the $F_{5 \\mu}$ are replaced by $F_{5 \\mu} -iF_{6 \\mu}$. Thus, the full SU(4) amounts to a simple modification of the previously considered Spin(5) by adding a term proportional to the unit operator to the diagonal blocks and making the four $F_{5 \\mu}$ complex, with $F_{6 \\mu}$ absorbed as their imaginary parts.             \n\nThe Riccati Eq.~(\\ref{eqn13}), now for complex $z$, becomes\n\n\\begin{eqnarray}\n\\dot{z}_{\\mu} & = & F_{5\\mu}(1-z_{\\nu}^2) -iF_{6 \\mu}(1+z_{\\nu}^2)+2F_{\\mu \\nu}z_{\\nu} \\nonumber \\\\\n \\!\\!& + &\\!\\! 2(F_{5\\nu}+iF_{6\\nu})z_{\\nu}z_{\\mu}-2iF_{65}z_{\\mu},  \\mu,\\nu=1-4.\n\\label{eqn17}\n\\end{eqnarray}\nThe two gammas in Eq.~(\\ref{eqn4}) are given by\n\n\\begin{eqnarray}\n{\\mbox {\\boldmath $\\gamma$}_{1,2}} & = &  (1+z_{\\mu}^2){\\bf I}^{(2)} +i(z_i^{*}z_4-z_4^{*}z_i)\\sigma_i \\nonumber \\\\\n & \\pm & \\frac{1}{2} i \\epsilon_{ijk} (z_i z_j^{*}-z_j z_i^{*}) \\sigma_k.\n\\label{eqn18}\n\\end{eqnarray}\nTheir square-root matrices $g_{1,2}$ can also be evaluated in terms of the Pauli matrices and the two SU(2) effective Hamiltonians then constructed in explicitly traceless and Hermitian form.\n\nJust as the very structure of Eq.~(\\ref{eqn13}) suggests that $z_{\\mu}$ and $(1-z_{\\nu}^2)$ with suitable normalization define a five-dimensional unit vector $\\vec{m}$ in Eq.~(\\ref{eqn15}), the occurrence of $z_{\\mu}, (1-z_{\\nu}^2), (1+z_{\\nu}^2)$ in Eq.~(\\ref{eqn17}) suggests now the introduction of six quantities according to\n\n\\begin{equation}\nm_{\\mu}=\\frac{-2z_{\\mu}}{De^{i\\phi}}, \\,m_5=\\frac{(1-z_{\\nu}^2)}{De^{i\\phi}},\\, m_6=-i\\frac{(1+z_{\\nu}^2)}{De^{i\\phi}},\n\\label{eqn19}\n\\end{equation}\nwith \n\n\\begin{eqnarray}\nD & \\equiv & (1+2|z_{\\nu}|^2+z_{\\mu}^2 {z^{*}_{\\nu}}^2)^{1\/2}, \\nonumber \\\\\n\\dot{\\phi} & = & \\!\\!-2F_{65}+iF_{5\\mu}(z^{*}_{\\mu}\\!-\\!z_{\\mu})+F_{6\\mu}(z^{*}_{\\mu}\\!+\\!z_{\\mu}).\n\\label{eqn20}\n\\end{eqnarray}\n\n\nAs with the so(5) case in Section III, with such a set of six complex quantities $\\vec{m}$, the nonlinear Riccati equation for the four complex $z_{\\mu}$ in Eq.~(\\ref{eqn17}) becomes a linear Bloch-like equation as before,\n\n\\begin{equation}\n\\dot{m}_{\\mu}=2F_{\\mu\\nu}m_{\\nu}, \\,\\,\\,\\mu,\\nu=1-6.\n\\label{eqn21}\n\\end{equation}\nOnce again, the $m_{\\mu}$ obey a first-order equation with an antisymmetric matrix which describes rotations. Since the 15 $F_{\\mu\\nu}$ are real, the real and imaginary parts of the six $m_{\\mu}$ each obey such a rotational transformation. These six-dimensional rotations reflect the isomorphism between the groups SU(4) and SO(6) (more accurately, its covering group  Spin(6)) and suggest a mapping between their generators (see Appendix B). \n\nTo get a geometrical picture of the manifold $m$, we note first the relations, \n\n\\begin{equation}\nm_{\\mu}^2=0, \\,\\, |m_{\\mu}|^2=2,\n\\label{eqn22}\n\\end{equation}\nwhich amount to three constraints. In addition, only the derivative, not the value, of $\\phi$ is determined in Eq.~(\\ref{eqn20}). Thereby, the number of independent parameters in $m_{\\mu}$ is eight just as in the complex $z_{\\mu}$, themselves built from ${\\bf z}$. The description of such an eight-dimensional manifold will be taken up in the next sub-section but we note here the reduction to the previous so(5) example. This follows upon setting $F_{65}=0, F_{6\\mu}=0$ which makes $\\phi =0$ and $D=(1+z_{\\nu}^2)$ in Eq.~(\\ref{eqn20}), and reduces $m_{\\mu}$ and $m_5$ to the values in Eq.~(\\ref{eqn15}) whereas $m_6=-i$. This, of course, makes $\\vec{m}$ a five-dimensional unit vector and its manifold the four-sphere S$^4$. The first relation in Eq.~(\\ref{eqn22}), of the vanishing of a square, hints at Grassmannian elements, to be discussed further below. \n\n\\subsection{Nature of the manifold describing $(z,m)$ for general SU(4)}\n\nOur construction of the evolution operator for $(N=4, n=2)$ in Eq.~(\\ref{eqn3}) is in terms of the eight-dimensional base manifold ${\\bf z}$ and a fiber consisting of two residual SU(2) along its diagonal blocks and a U(1) phase between them: SU(4) $\\rightarrow$ [SU(4)\/SU(2) $\\times$ SU(2) $\\times$ U(1)] $\\times$ [SU(2) $\\times$ SU(2) $\\times$ U(1)]. To describe the former base manifold, consider first [SU(4)\/SU(2) $\\times$ SU(2)], which is a nine-dimensional manifold. It can also be described in terms of spin-groups as Spin(6)\/Spin(4). The six complex $m_{\\mu}$ in Eq.~(\\ref{eqn18}) with the three constraints in Eq.~(\\ref{eqn22}) constitute such a manifold called a Stiefel manifold {\\bf St}(6, 2, {\\sf R}) $\\cong \\Re ^9$, this name being given to manifolds consisting of $n$ orthogonal vectors from an $N$-dimensional space $\\Re^N$ \\cite{Stiefel}. Geometrically, the second relation in Eq.~(\\ref{eqn22}) states that the real and imaginary parts of $m$ are six-dimensional unit vectors while the first relation expresses their mutual orthogonality. Therefore, one can view the manifold as a five-sphere S$^5$ with another four-sphere S$^4$ attached at each point on it. The absolute value of the phase parameter $\\phi$ in Eq.~(\\ref{eqn19}) and Eq.~(\\ref{eqn20}) being undefined, reduces such a manifold by one dimension to [SU(4)\/SU(2) $\\times$ SU(2) $\\times$ U(1)], which is equivalent to the reduction from the Stiefel to a Grassmannian manifold {\\bf G}(4, 2, {\\sf C}) according to {\\bf St}(6, 2, {\\sf R}) $\\cong {\\bf G}(4, 2, {\\sf C}) \\times $ U(1). Such a Grassmannian manifold, which has eight dimensions, thereby describes the ${\\bf z}$ in Eq.~(\\ref{eqn3}) or its equivalent $z_{\\mu}$ in Eq.~(\\ref{eqn17}) or $m_{\\mu}$ in Eq.~(\\ref{eqn19}).       \n\nA more accessible geometrical picture is to consider a single five-sphere S$^5$ embedded in six-dimensional space and two six-dimensional unit vectors from the origin to the surface to represent the real and imaginary parts of $m_{\\mu}$. The two vectors are always taken as orthogonal, so that one views such an orthogonally-coupled pair rotating within the sphere \\cite{circle}. This nine-dimensional object, combined with the zero reference of $\\phi$ being undefined, is our eight-dimensional manifold of interest.\n\n\\subsection{Description in Pl\\\"{u}cker coordinates}\n\nAn alternative view of these manifolds is provided in terms of what are termed Pl\\\"{u}cker coordinates, defined as a set of six complex parameters $(P_{12}, P_{13}, P_{14}, P_{23}, P_{24}, P_{34})$ formed as minors of the $2 \\times 4$ sub-matrix of the last two columns of an arbitrary, unitary SU(4) matrix \\cite{Plucker},\n\n\\begin{equation}\n{\\bf U}=\\left( \n\\begin{array}{cccc}\nu_{11} & u_{12} & u_{13} & u_{14} \\\\ \nu_{21} & u_{22} & u_{23} & u_{24} \\\\ \nu_{31} & u_{32} & u_{33} & u_{34} \\\\\nu_{41} & u_{42} & u_{43} & u_{44}\n\\end{array}\n\\right).\n\\label{eqn23}\n\\end{equation}\nThey obey the relations\n\n\\begin{equation}\nP_{12}P_{34}-P_{13}P_{24}+P_{14}P_{23} =0, \\,\\, \\sum |P_{ij}|^2=1.\n\\label{eqn24}\n\\end{equation}\nThey are combinations of the $m_{\\mu}$ according to \n\n\\begin{equation}\n\\left( \n\\begin{array}{c}\nP_{12} \\\\ \nP_{13} \\\\ \nP_{14} \\\\\nP_{23} \\\\\nP_{24} \\\\\nP_{34}\n\\end{array}\n\\right) = \\frac{1}{2} \\left(\n\\begin{array}{c}\nim_6 -m_5 \\\\\nim_1 +m_2 \\\\\n-im_3 +m_4 \\\\\n-im_3 -m_4 \\\\\n-im_1 +m_2 \\\\\nim_6 +m_5\n\\end{array} \\right).\n\\label{eqn25}\n\\end{equation}\n\nThe linear equations for $m_{\\mu}$ in Eq.~(\\ref{eqn20}) translate into a similar linear equation \n\n\\begin{equation}\ni\\dot{\\bf P}={\\bf H}{\\bf P}, \\,\\, {\\bf P} \\equiv (P_{12},-P_{13}, P_{14}, P_{23}, P_{24}, P_{34}),\n\\label{eqn26}\n\\end{equation}\nwith\n\n\\begin{equation}\n{\\bf H}_P \\! =\\!\\! \\left( \n\\begin{array}{cccccc}\nH_{11,22} & H_{41} & H_{31} & -H_{42} & H_{32} & 0 \\\\ \nH_{14} & H_{11,33} & -H_{34} & -H_{12} & 0 & -H_{32} \\\\ \nH_{13} & -H_{43} & H_{11,44} & 0 & H_{12} & H_{42} \\\\\n-H_{24} & -H_{21} & 0 & H_{22,33} & H_{34} & -H_{31} \\\\\nH_{23} & 0 & H_{21} & H_{43} & H_{22,44} & -H_{41} \\\\\n0 & -H_{23} & H_{24} & -H_{13} & -H_{14} & H_{33,44}\n\\end{array}\n\\right),\n\\label{eqn27}\n\\end{equation}\nwhere we have adopted the notation for the diagonal entries: $H_{ii,jj}=H_{ii}+H_{jj}$.\n\n\nActually, the above equations for ${\\bf P}$ can be arrived at directly from the evolution equation $i\\dot{\\bf U}={\\bf H}{\\bf U}$ because the elements of ${\\bf P}$ are quadratic in the elements of ${\\bf U}$ in Eq.~(\\ref{eqn23}): $P_{ij}=i \\varepsilon_{ijkl}u^{(3)}_k u^{(4)}_l$, and $i\\dot{u}^{(3)}_k=H_{kj}u^{(3)}_j$. Also, ${\\bf z}$ can be defined in terms of the two minors on the right in Eq.~(\\ref{eqn23}):\n\n\\begin{equation}\n{\\bf z} = \\left(\n\\begin{array}{cc}\nu_{13} & u_{14} \\\\\nu_{23} & u_{24}\n\\end{array} \\right) \/\n\\left( \n\\begin{array}{cc}\nu_{33} & u_{34} \\\\\nu_{43} & u_{44}\n\\end{array} \\right),\n\\label{eqn28}\n\\end{equation} \nthe matrix in the denominator assumed to be non-singular. Writing $U$ in Eq.~(\\ref{eqn23}) in the form in Eq.~(\\ref{eqn3}), the first factor $\\tilde{U}_1$ involving ${\\bf z}$ is a map of the Grassmannian manifold {\\bf G}(4, 2, {\\sf C}) onto ${\\sf C}^4$, and provides a partial coordinization of that manifold. Elements of {\\bf G}(4, 2, {\\sf C}) are two-dimensional complex hyperplanes spanned by vectors ${\\bf u}_3 =(u_{13}, u_{23}, u_{33}, u_{43})^T$ and ${\\bf u}_4 =(u_{14}, u_{24}, u_{34}, u_{44})^T$. The Pl\\\"{u}cker coordinates provide a unique identification of such planes. They are an analog of the coordinization of the $n$-dimensional sphere S$^n$ by an $(n+1)$-dimensional unit vector $\\vec{m}$ as in Section III.\n\nThe matrix ${\\bf H}_P$ in Eq.~(\\ref{eqn27}) being Hermitian, ${\\bf P}^{\\dagger}{\\bf P}$ = constant = 1. This can be verified by the relation between $P$'s and $m$'s in Eq.~(\\ref{eqn25}) which involves a unitary matrix so that ${\\bf P}^{\\dagger}{\\bf P}=\\frac{1}{2} m^{\\dagger}m$, and combining with Eq.~(\\ref{eqn22}). Further, a symplectic structure can be introduced. \nDefining a $6 \\times 6$ matrix ${\\bf \\Omega} \\equiv {\\bf \\delta}_{i,7-j}$ with non-zero entries of 1 only along the anti-diagonal, the first relation in Eq.~(\\ref{eqn24}) can be rendered as ${\\bf P}^T {\\bf \\Omega} {\\bf P}=0$, and the matrix ${\\bf H}_P$, a generator of the symplectic group {\\bf Sp}(6, \\sf C),\n\n\\begin{equation}\n{\\bf H}_P {\\bf \\Omega} + {\\bf \\Omega}{\\bf H}_P^T = {\\rm Tr} ({\\bf H}_P){\\bf \\Omega}=0. \n\\label{eqn29}\n\\end{equation}\nAny two vectors ${\\bf P}_i$, evolving according to Eq.~(\\ref{eqn26}), satisfy ${\\bf P}_1^T (t) {\\bf \\Omega} {\\bf P}_2 (t)$ =constant. If ${\\bf P}_1 {\\bf \\Omega} {\\bf P}_2 =0$, then the two hyperplanes defined by ${\\bf P}_i$ intersect, and if $|{\\bf P}_1 {\\bf \\Omega} {\\bf P}_2| =1$, they do not.\n\nGeometrically, the set ${\\bf P}^{\\dagger}{\\bf P}=1$ is a sphere S$^{11}$, the algebraic relation ${\\bf P}{\\bf \\Omega}{\\bf P}=0$ determining a 9-dimensional sub-manifold, an intersection between S$^{11}$ and the affine variety of roots of the polynomial equation ${\\bf P}{\\bf \\Omega}{\\bf P}=0$. This manifold may be denoted $\\Re ^9$. Multiplication by a phase acts as a transformation group on this manifold, that is, if ${\\bf P} \\in \\Re ^9$, then ${\\bf P}e^{i\\phi} \\in \\Re ^9$. Therefore, {\\bf G}(4, 2, {\\sf C}) is a quotient space $\\Re ^9\/$U(1) and has eight dimensions. The connection to SU(4) is, as noted before, $\\Re ^9 \\cong $SU(4)\/(SU(2) $\\times$ SU(2)) $\\cong$ Spin(6)\/ Spin(4). The stability sub-group of a vector ${\\bf P} \\in \\Re ^9$ is SU(2) $\\times$ SU(2) while the stability sub-group of $\\Re ^9$\/U(1) is SU(2) $\\times$ SU(2) $\\times$ U(1). Since Spin(6)\/Spin(5) $\\cong$ S$^5$ and Spin(5)\/Spin(4) $\\cong$ S$^4$, we can identify the fibration of $\\Re ^9$ with S$^5 \\times$ S$^4$.\n\n\\section{Summary}\n\nWe have presented a complete analysis of the evolution operator for SU($N$), setting up its construction in a hierarchical way in terms of those for smaller SU($N-n$) and SU($n$), with $n<N$ and arbitrary. The evolution operator is written as a product of two $N \\times N$ matrices, the second of which is block diagonal in $(N-n) \\times (N-n)$ and $n \\times n$ of the smaller groups. The first factor is obtained through a ${\\bf z}$, which is an $(N-n) \\times n$ complex matrix obeying a matrix Riccati equation. Its solutions determine both the first factor as well as the Hermitian matrices for the subsequent $N-n$ and $n$ evolution problems. \n\nThis general constructive method is applied especially to a four-level system with special emphasis on two qubits. The general symmetry is of SU(4), a 15-parameter group. Our procedure expresses the evolution operator as a product of two 4 $\\times$ 4 matrices, the second of which is block diagonal, each block an SU(2) problem. The ${\\bf z}$ is also a $2 \\times 2$ matrix with complex entries in general and obeys a matrix Riccati equation. Alternatively, we transform ${\\bf z}$ into a six-dimensional complex vector $\\vec{m}$, whose real and imaginary parts both separately undergo linear, six-dimensional rotational transformations. This is exactly analogous to the linear Bloch equation for real three-dimensional rotations of a vector to represent the evolution operator for a single spin in a magnetic field.  \n\nJust as a Bloch sphere describes the three-dimensional vector $\\vec{m}$ for a single spin (and, together with a phase, the complete SU(2)), we also present the geometrical manifold describing ${\\bf z}$ or its equivalent six-dimensional complex vector $\\vec{m}$. Together with two residual SU(2) problems and a phase, this provides a complete description of the quantum evolution operator for SU(4). For certain sub-algebras of SU(4), the manifold is an analogous higher-dimensional sphere; a four-sphere, for example, for an so(5) sub-algebra. For the most general SU(4), we have an eight-dimensional Grassmannian manifold. We provide a picture of it as two five-spheres with an orthogonality and phase constraint. These geometrical objects may serve for all possible four-level and two qubit systems the useful purpose that the Bloch sphere has for two-level and single qubit problems in physics. \n\n\\section*{APPENDIX A: EXTENSION TO NON-UNITARY EVOLUTION FOR A NON-HERMITIAN HAMILTONIAN}\n\nThe iterative method of Section II for the evolution operator in Eq.~(\\ref{eqn2}) through writing it as in Eq.~(\\ref{eqn3}) applies also when $H$ in Eq.~(\\ref{eqn1}) is not Hermitian and, therefore, the evolution not unitary. However, ${\\bf z}$ and ${\\bf w}$ in Eq.~(\\ref{eqn3}) are no longer simply related as in Eq.~(\\ref{eqn4}) but obey independent equations, the former still in Riccati form but the latter given in terms of ${\\bf z}$. Thus, instead of Eq.~(\\ref{eqn1}), consider\n\n\\begin{equation}\n\\tilde{\\bf H}^{(N)}(t) =\\left(\n\\begin{array}{cc}\n\\tilde{\\bf H}^{(N-n)}(t) & {\\bf V}(t) \\\\\n{\\bf Y}^{\\dagger}(t) & \\tilde{\\bf H}^{(n)}(t)\n\\end{array}\n\\right),\n\\label{eqnA1}\n\\end{equation}\nwhere we have again indicated by tildes non-Hermiticity, and ${\\bf V}$ and ${\\bf Y}$ are not equal but independent.\n\nWriting $\\tilde{\\bf U}^{(N)}(t)$ again as in Eq.~(\\ref{eqn3}), Eq.~(\\ref{eqn6}) now becomes\n\n\\begin{eqnarray}\ni\\dot{{\\bf z}} & = & (\\tilde{\\bf H}^{(N-1)}{\\bf z} -{\\bf z}\\tilde{\\bf H}^{(n)} -{\\bf z}{\\bf Y}^{\\dagger}{\\bf z} +{\\bf V}, \\nonumber \\\\\ni\\dot{{\\bf w}}^{\\dagger}\\!\\! & = & \\!\\! {\\bf w}^{\\dagger}({\\bf z}{\\bf Y}^{\\dagger}\\!\\!-\\tilde{\\bf H}^{(N-1)} \\!) \\!+(\\!\\tilde{\\bf H}^{(n)}\\!-{\\bf Y}^{\\dagger}{\\bf z}){\\bf w}^{\\dagger}\\!\\!+\\! {\\bf Y}^{\\dagger}.\n\\label{eqnA2}\n\\end{eqnarray}\nThe residual problems of $(N-n)$ and $n$ dimension then become\n\n\\begin{equation}\n\\left( \\begin{array}{cc}\ni\\dot{\\tilde{\\bf U}}^{(N-n)} \\! & \\!\\! {\\bf 0} \\\\\n{\\bf 0}^{\\dagger} \\! & \\!\\! i\\dot{\\tilde{\\bf U}}^{(n)}\n\\end{array} \\right) \\! = \\! \\left(\n\\begin{array}{cc}\n\\tilde{\\bf H}^{(N-1)} -{\\bf z} {\\bf Y}^{\\dagger} \\! & \\!\\! {\\bf 0} \\\\\n{\\bf 0}^{\\dagger} \\! & \\!\\! \\tilde{\\bf H}^{(n)} + {\\bf Y}^{\\dagger}{\\bf z}\n\\end{array} \\right) \\tilde{\\bf U}_2.\n\\label{eqnA3}\n\\end{equation}\n\n\\section*{APPENDIX B: DESCRIPTION OF EVOLUTION AS SIX-DIMENSIONAL ROTATIONS}\n\nFor a single spin or qubit, the rewriting of the quantum evolution operator, which is complex, as rotational transformations of a real, unit vector in three dimensions given by the Bloch equation, rests on the isomorphism of the group SU(2) to SO(3) (or its double covering Spin(3)). A similar isomorphism between the groups SU(4) and SO(6) (or its extension Spin(6)) underlies the construction in Sections III and IV of the complex evolution operator for two qubits in terms of rotations of a vector in six dimensions. Both groups are described by 15 real parameters through an antisymmetric $F_{\\mu\\nu}, \\mu, \\nu=1, 2, \\ldots 6$. In Sections III and IV, explicit expressions are given for the Hamiltonian with each of these parameters multiplying one of the 15 complex generators of SU(4) in a standard representation of Pauli matrices, $\\vec{\\sigma}^{(1)} \\otimes \\mathcal{I}^{(2)}, \\mathcal{I}^{(1)} \\otimes \\vec{\\sigma}^{(2)}, \\vec{\\sigma}^{(1)} \\otimes \\vec{\\sigma}^{(2)}$. An alternative rendering in terms of the 15 generators of SO(6) is useful and recorded here.\n\nThe Hamiltonian in Section IV, apart from a factor of $\\frac{1}{2}$, can be cast in terms of a matrix array\n\n\\begin{equation}\n\\!\\!\\left(\\!\\! \n\\begin{array}{cccccc}\n\\!\\!0 & \\!\\!\\sigma^{(2)}_z & \\!\\!-\\sigma^{(2)}_y & \\!\\!-\\sigma^{(1)}_z \\sigma^{(2)}_x & \\!\\!\\sigma^{(1)}_x \\sigma^{(2)}_x & \\!\\!\\sigma^{(1)}_y\\sigma^{(2)}_x \\\\ \n\\!\\!-\\sigma^{(2)}_z & \\!\\!0 & \\!\\!\\sigma^{(2)}_x & \\!\\!-\\sigma^{(1)}_z \\sigma^{(2)}_y & \\!\\!\\sigma^{(1)}_x \\sigma^{(2)}_y & \\!\\!\\sigma^{(1)}_y \\sigma^{(2)}_y \\\\ \n\\!\\!\\sigma^{(2)}_y & \\!\\!-\\sigma^{(2)}_x & \\!\\!0 & \\!\\!-\\sigma^{(1)}_z \\sigma^{(2)}_z & \\!\\!\\sigma^{(1)}_x \\sigma^{(2)}_z & \\!\\!\\sigma^{(1)}_y \\sigma^{(2)}_z \\\\\n\\!\\!\\sigma^{(1)}_z \\sigma^{(2)}_x & \\!\\!\\sigma^{(1)}_z \\sigma^{(2)}_y & \\!\\!\\sigma^{(1)}_z \\sigma^{(2)}_z & \\!\\!0 & \\!\\!-\\sigma^{(1)}_y & \\sigma^{(1)}_x \\\\\n\\!\\!-\\sigma^{(1)}_x \\sigma^{(2)}_x & \\!\\!-\\sigma^{(1)}_x \\sigma^{(2)}_y & \\!\\!-\\sigma^{(1)}_x \\sigma^{(2)}_z & \\!\\!\\sigma^{(1)}_y & \\!\\!0 & \\!\\!\\sigma^{(1)}_z\\\\\n\\!\\!-\\sigma^{(1)}_y \\sigma^{(2)}_x & \\!\\!-\\sigma^{(1)}_y \\sigma^{(2)}_y & \\!\\!-\\sigma^{(1)}_y \\sigma^{(2)}_z & \\!\\!-\\sigma^{(1)}_x & \\!\\!-\\sigma^{(1)}_z & \\!\\!0\n\\end{array}\\!\\!\n\\right),\n\\label{eqnB1}\n\\end{equation}\nwhich is explicitly anti-symmetric. We thus have $H=2F_{\\mu\\nu}L_{\\nu\\mu}$. Analogous to the familiar triplet of angular momentum generators, six-dimensional generators of SO(6) are given by $L_{\\mu\\nu} =-il_{\\mu\\nu}$, where the $l$ are 15 real antisymmetric $6 \\times 6$ matrices with only two non-zero entries, $+1$ in the $(\\mu\\nu)$ and $-1$ in the $(\\nu\\mu)$ position:\n\n\\begin{equation}\n(l_{\\mu\\nu})_{\\rho\\sigma} = \\delta_{\\mu\\rho} \\delta_{\\nu\\sigma} - \\delta_{\\mu\\sigma} \\delta_{\\nu\\rho}.\n\\label{eqnB2}\n\\end{equation}\nTheir commutators close:\n\n\\begin{equation}\n[l_{\\mu\\nu}, l_{\\rho\\sigma}]=\\delta_{\\nu\\rho}l_{\\mu\\sigma} +\\delta_{\\mu\\sigma} l_{\\nu\\rho} -\\delta_{\\nu\\sigma} l_{\\mu\\rho} -\\delta_{\\mu\\rho} l_{\\nu\\sigma},\n\\label{eqnB3}\n\\end{equation}\nso that $L_{\\mu\\nu}$ form an so(6) algebra.\n\n\nThe array in Eq.~(\\ref{eqnB1}) is also a convenient display of the generators of the various sub-groups of SO(6) of lower-dimensional rotations. Either upper left or lower right corner $2 \\times 2$ blocks describe the SO(2) generator of one of the qubits. Adding a third row and column gives the full triplet of SO(3) generators. To this can be added a next row and column of three non-zero entries to give the six generators of SO(4). For this purpose, any of the three remaining row\/column can be employed, each giving an SO(4), the three added entries transforming as a vector under SO(3). This continues. Adding another row and column's four new entries, which transform as a vector under SO(4) (further subdividing into three components that transform as a vector and one as a scalar under the previous SO(3)), gives the ten SO(5) generators. The final sixth row\/column adds five entries, an SO(5) vector, to give the full 15 generators of SO(6). This hierarchical nesting of SO sub-groups, together with the corresponding Clifford structure with Pauli matrices in Eq.~(\\ref{eqnB1}), accounts for the richness of the structures in the isomorphic groups SU(4) and SO(6), one we have exploited in Sections III and IV. Note that the linear Bloch-like equation for $\\vec{m}$ in Eq.~(\\ref{eqn21}) for a general SU(4) Hamiltonian reduces to the same antisymmetric form for its sub-groups such as in Eq.~(\\ref{eqn16}), all the way down to the standard Bloch equation for a single qubit, whose SO(3) antisymmetric $F_{ij}$ is usually written as a vector product with a magnetic field.       \n\n\n  \n           \n    \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\\label{sec:Introduction}\n\nFrom atoms and molecules under visible-to-midinfrared laser fields of an intensity $\\gtrsim 10^{14}\\,{\\rm W\/cm}^2$ emerge highly nonlinear strong-field phenomena, e.g., above-threshold ionization, tunneling ionization, high-harmonic generation (HHG), and nonsequential double ionization (NSDI) \\cite{Protopapas1997RPP,Brabec2000RMP}. \nIn particular, HHG is more and more widely used as an ultrashort (down to attoseconds) coherent light source in the extreme-ultraviolet (XUV) and soft x-ray spectral ranges \\cite{Popmintchev2012Nature,Chang2011,AttosecondPhysics}. \nIn addition, free-electron lasers are now in operation as another type of ultrashort, intense, coherent XUV and x-ray sources.\nSuch a rapid progress in experimental techniques for ultrafast intense laser science has opened new research areas including ultrafast molecular probing\n\\cite{Itatani2004Nature,Haessler2010NPhys,Salieres2012RPP}, attosecond science \\cite{Agostini2004RPP,Krausz2009RMP,Gallmann2013ARPC}, and XUV nonlinear optics \\cite{Sekikawa2004Nature,Nabekawa2005PRL}, with the ultimate goal to directly observe, and even manipulate ultrafast electronic motion in atoms, molecules, and solids.\n\nFurther advances in these areas require first-principles methods to numerically simulate the real-time dynamics of multielectron atoms and molecules in ultrafast intense laser pulses, or {\\it ab initio strong-field physics}. \nAlthough the time-dependent Schr\\\"odinger equation (TDSE) [see Eq.~(\\ref{eq:TDSE}) below] rigorously describes these phenomena in principle,\nits numerical integration in the real space for systems with more than two electrons \\cite{Pindzola1998PRA,Pindzola1998JPB,Colgan2001JPB,\nParker2001,Laulan2003PRA,Piraux2003EPJD,Laulan2004PRA,ATDI2005,\nFeist2009PRL,Pazourek2011PRA,He_TPI2012PRL,Suren2012PRA,He_TPI2013AS,\nVanroose2006PRA,Horner2008PRL,Lee2010JPB} poses a major challenge.\n\nA promising class of approaches is time-dependent multiconfiguration self-consistent field (TD-MCSCF) methods\\index{Time-dependent multiconfiguration self-consistent field (TD-MCSCF) method}\\cite{Ishikawa2015JSTQE,Loestedt2017PUILS}, where the total electronic wave function is expressed as a superposition of different electronic configurations or Slater determinants built from a given number of single-electron spin orbitals [see Eq.~(\\ref{eq:general-mcwf}) and Fig.~\\ref{fig:MC} below].\nIn the multiconfiguration time-dependent Hartree-Fock (MCTDHF) method\\index{Multiconfiguration time-dependent Hartree-Fock (MCTDHF) method}\\cite{Zanghellini2003LP,Kato2001CPL,Caillat2005PRA}, both the expansion coefficients [configuration-interaction (CI) coefficients] and orbital functions are varied in time, and all the possible realizations to distribute the electrons among the spin orbitals (full CI expansion) are included.\nThough pioneering and powerful, the computational cost of MCTDHF increases factorially with the number of electrons.\n\nTo overcome this difficulty, we have recently developed and successfully implemented a TD-MCSCF method called the time-dependent complete-active-space self-consistent-field (TD-CASSCF) method \\cite{Sato2013PRA,Sato2016PRA,Orimo2018PRA}, which is the topic of the present Chapter. \nTD-CASSCF classifies the spatial orbitals into doubly occupied and time-independent frozen core (FC), doubly occupied and time-dependent dynamical core (DC), and fully correlated active orbitals.\nThanks to this classification, the number of configurations used in simulations and the computational cost are significantly reduced without sacrificing accuracy.\nThe classification can be done flexibly, based on simulated physical situations and desired accuracy. \nThrough comparison of the results from various subspace decompositions, one can analyze the contribution from different shells, the effect of electron correlation, and the mechanism underlying the simulated phenomena.\nIn this sense, TD-CASSCF is even more useful than merely numerically exact black-box simulations.\n\nThis Chapter proceeds as follows. In Sec.~\\ref{sec:Problem Statement} we describe the statement of the problem that we are going to treat, i.e., the time-dependent Schr\\\"odinger equation for many electron systems in a driving laser field within the dipole and fixed-nuclei approximations.\nWe also briefly mention an important concept of gauge transformation.\nSection \\ref{sec:TD-CASSCF} explains the formulation of the TD-CASSCF method, the equations of motion for CI coefficients and orbital functions, and its important features of gauge invariance and size extensivity.\nIn Sec.~\\ref{sec:Initial-State Preparation and Simulation Boundary} we describe how to prepare the initial wave function and absorb the electron wave packet that reaches the simulation box boundary without unphysical reflection.\nSection \\ref{sec:Numerical examples} presents how to extract relevant physical quantities from the wave function obtained by TD-CASSCF simulations, along with representative numerical examples.\nSummary is given in Sec.~\\ref{sec:Summary}.\nHartree atomic units are used throughout unless otherwise stated.\n\n\n\\section{Problem Statement}\n\\label{sec:Problem Statement}\n\n\\subsection{Time-Dependent Schr\\\"odinger Equation}\n\nWe consider an atom or molecular system consisting of $N$ electrons subject to an external laser field. Within the electric dipole approximation of laser-electron interaction and the fixed-nuclei or clamped-nuclei approximation that treats nuclei as classical point charges fixed in space, the dynamics of the laser-driven multielectron system is described by the time-dependent Schr\\\"odinger equation (TDSE)\\index{Time-dependent Schr\\\"odinger equation (TDSE)},\n\\begin{equation}\n\\label{eq:TDSE}\ni\\frac{\\partial\\Psi (t)}{\\partial t} = \\hat{H}(t)\\Psi (t),\n\\end{equation}\nwhere the time-dependent Hamiltonian,\n\\begin{equation}\n\\hat{H}(t)=\\hat{H}_1(t)+\\hat{H}_2,\n\\end{equation}\nis decomposed into the one-electron part (kinetic energy, nuclear Coulomb energy, and laser-electron interaction),\n\\begin{equation}\n\\label{eq:H1}\n\\hat{H}_1(t) = \\sum_i \\hat{h}({\\bf r}_i,t) \n\\end{equation}\nand the two-electron part,\n\\begin{equation}\n\\hat{H}_2 = \\sum_{i=1}^N \\sum_{j < i} \\frac{1}{|{\\bf r}_i - {\\bf r}_j|},\n\\end{equation}\nfor the interelectronic Coulomb interaction. \nThe laser-electron interaction can be expressed either in the length gauge (LG)\\index{Length gauge} or velocity gauge (VG)\\index{Velocity gauge}: $\\hat{h}({\\bf r},t)$ in Eq.~(\\ref{eq:H1}) is given by,\n\\begin{equation}\n\\label{eq:length-gauge}\n\\hat{h}({\\bf r},t) = \\frac{\\hat{{\\bf p}}^2}{2}+{\\bf r}\\cdot {\\bf E}(t)-\\sum_\\alpha \\frac{Z_\\alpha}{|{\\bf r} - {\\bf R}_\\alpha|},\n\\end{equation}\nin the length gauge, with $\\hat{{\\bf p}}=-i\\nabla$, and, \n\\begin{equation}\n\\label{eq:velocity-gauge}\n\\hat{h}({\\bf r},t) = \\frac{\\left[\\hat{{\\bf p}}+{\\bf A}(t)\\right]^2}{2}-\\sum_\\alpha \\frac{Z_\\alpha}{|{\\bf r} - {\\bf R}_\\alpha|},\n\\end{equation}\nin the velocity gauge, with ${\\bf A}(t) = -\\int {\\bf E}(t)dt$ being the vector potential.\n\n\\subsection{Gauge Transformation}\n\\label{subsec:Gauge Transformation}\n\nThe wave functions $\\Psi_{\\rm L} (t)$ and $\\Psi_{\\rm V} (t)$ expressed in the length and velocity gauges, respectively, are transformed into each other through the gauge transformation\\index{Gauge transformation},\n\\begin{equation}\n\\label{eq:gauge-transformation}\n\\Psi_{\\rm V}(t) = \\hat{\\mathcal{U}}(t) \\Psi_{\\rm L}(t),\n\\end{equation}\nwith the unitary operator,\n\\begin{equation}\n\\label{eq:gauge-transformation-operator}\n\\hat{\\mathcal{U}}(t) = \\exp \\left[-i{\\bf A}(t)\\cdot \\sum_{i=1}^{N}{\\bf r}_i\\right].\n\\end{equation}\nIf we substitute Eq.~(\\ref{eq:gauge-transformation}) into the TDSE with Eq.~(\\ref{eq:velocity-gauge}), we can easily show that $\\Psi_{\\rm L}$ indeed satisfies the TDSE with Eq.~(\\ref{eq:length-gauge}).\n\nWhile the operator $\\hat{{\\bf p}}$ corresponds to the kinetic momentum in the length gauge, it corresponds to the canonical momentum in the velocity gauge, and the kinetic momentum is given by $\\hat{{\\bf p}}+{\\bf A}(t)$. Then, a plane wave state with a kinetic momentum ${\\bf p}_{kin}$ is $e^{i {\\bf p}_{kin} \\cdot {\\bf r}}$ in the length gauge and $e^{i \\left[{\\bf p}_{kin}-{\\bf A}(t)\\right] \\cdot {\\bf r}}$ in the velocity gauge, which fulfills Eq.~(\\ref{eq:gauge-transformation}).\n\nThe gauge principle\\index{Gauge principle} states that all physical observables are gauge invariant\\index{Gauge invariance}, i.e., take the same values whether the laser-electron interaction may be represented in the length or velocity gauge \\cite{Bandrauk2013JPB}.\nFor example, the probability density is gauge invariant, $|\\Psi_{\\rm V}(t)|^2 = |\\Psi_{\\rm L}(t)|^2$.\n\nOne may be surprised to realize that the projection $\\langle \\Xi | \\Psi (t) \\rangle$ of the wave function $\\Psi (t)$ onto a field-free stationary state $\\Xi$ and the population $|\\langle \\Xi | \\Psi (t) \\rangle|^2$ are {\\it not} gauge invariant and, thus, {\\it not} a physical observable when ${\\bf A}(t) \\ne 0$, i.e., during the pulse. \nAs a consequence, the degree of ionization is {\\it not} gauge invariant during the pulse, either.\nLet us assume that a hydrogen atom under a laser field linearly polarized in the $z$ direction is in the ground state in the length gauge,\n\\begin{equation}\n\t\\psi_{\\rm L}({\\bf r},t) = \\frac{e^{-r}}{\\sqrt{\\pi}},\n\\end{equation}\nat some moment, e.g., after a complete Rabi oscillation cycle.\nThen, its velocity gauge wave function is,\n\\begin{equation}\n\\label{eq:HgsVG}\n\t\\psi_{\\rm V}({\\bf r},t) = e^{-iA(t)z} \\psi_{\\rm L}({\\bf r},t)\n\t= 2 e^{-r}\\sum_{l=0}^\\infty \\sqrt{2l+1}(-i)^lj_l(A(t)r)Y_{l0}(\\theta,\\phi),\n\\end{equation}\nwhich contains not only the $1s$ state but all the angular momenta $l$ including continuum levels unless $A(t)=0$.\n\n\\section{TD-CASSCF method}\n\\label{sec:TD-CASSCF}\n\n\\index{Time-dependent complete-active-space self-consistent-field (TD-CASSCF) method}\n\n\\subsection{Multicongifuration Expansion}\n\\label{subsec:Multicongifuration Expansion}\n\nIn order to simulate multielectron dynamics, as illustrated in Fig.~\\ref{fig:MC}, we expand the total wave function $\\Psi (t)$ as a superposition of different Slater determinants or configuration state functions\\index{Multiconfiguration expansion},\n\\begin{equation}\n\\label{eq:general-mcwf}\n\\Psi (t) = \\sum_{I}^{\\sf P} \\Phi_{I}(t)C_{I}(t),\n\\end{equation}\nwhere expansion coefficients $\\{C_{I}\\}$ are called configuration interaction (CI) coefficients\\index{Configuration interaction coefficient}\\index{CI coefficient} and bases $\\{\\Phi_{I}\\}$ are the Slater determinants  built from $N$ spin orbitals out of $2n$ spin orbitals $\\{\\psi_p (t);p=1,2,\\cdots,n\\}\\otimes\\{\\alpha,\\beta\\}$  (in the spin-restricted treatment) with $\\{\\psi_p\\}$ being spatial orbital functions and $\\alpha (\\beta)$ the up- (down-) spin eigenfunction. The summation in Eq.~(\\ref{eq:general-mcwf}) with respect to configurations $I$ runs through the\nelement of a CI space ${\\sf P}$, consisting of\na given set of determinants. \n\nMuticonfiguration expansion Eq.~(\\ref{eq:general-mcwf}) can represent a wide variety of different methods; whereas $\\{C_I\\}$ are usually taken as time-dependent, they can also be fixed \\cite{Miranda2011JCP}. $\\{\\psi_p\\}$, and thus $\\{\\Phi_I\\}$, can be considered either time-independent, as in the time-dependent configuration interaction singles (TDCIS) method \\cite{Greenman_2010}, or time-dependent, as in the TD-CASSCF, MCTDHF, and time-dependent Hartree-Fock (TDHF) \\cite{Pindzola1991PRL} methods described below. \nWhile orbital functions are usually assumed to fulfill orthonormality, it is not a necessary condition.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=0.75\\textwidth]{figs\/MC.pdf}\t\n\\end{center}\n\\caption{Schematic representation of the multiconfiguration expansion Eq.~(\\ref{eq:general-mcwf}). Each term on the right-hand side corresponds to a configuration $\\Phi_1, \\Phi_2, \\cdots$ with CI coefficients $C_1, C_2, \\cdots$. The first term corresponds to the Hartree-Fock configuration.}\n\\label{fig:MC}      \n\\end{figure}\n\n\\subsection{TD-CASSCF ansatz}\n\nIn the TD-CASSCF method, we use orthonormal time-dependent orbital functions. The $n$ occupied orbitals are classified into $n_c$ core orbitals\\index{Core orbitals} $\\{\\psi_i: i=1,2,\\cdot\\cdot\\cdot,n_c\\}$ that are doubly occupied all the time and $n_a(=n-n_c)$ active orbitals\\index{Active orbitals} $\\{\\psi_t: t=n_c+1,n_c+2,\\cdot\\cdot\\cdot,n\\}$.\nThis idea is based on a reasonable expectation that only high-lying electrons are strongly driven, while deeply bound core electrons remain nonionized. \nOn the other hand, we consider all the possible distributions of $N_a(=N-2n_c)$ electrons among $n_a$ active orbitals. It should be noticed that not only the active orbitals but also the core orbitals, though constrained to the closed-shell structure, vary in time, in general, responding to the field formed by the laser and the other electrons.\nThe use of time-dependent (especially active) orbitals that are initially localized near the nuclei but spatially expand in the course of time allows us to efficiently describe excitation and ionization.\n\nIt is also possible to further decompose core orbitals into $n_{fc}$ frozen-core (FC) orbitals\\index{Frozen-core (FC) orbitals} that do not vary in time and $n_{dc}$ time-dependent dynamical core (DC) orbitals ($n_c = n_{fc} + n_{dc}$)\\index{Dynamical-core (DC) orbitals}. The $N$-electron CASSCF wave function can be symbolically expressed as,\n\\begin{align}\n\\Psi_{\\rm CAS} : \\psi_1^2\\cdots\\psi_{n_{fc}}^2\\psi_{n_{fc}+1}^2(t)\\cdots\\psi_{n_{c}}^2(t) \\{\\phi_{n_c+1}(t)\\cdots\\phi_{n}(t)\\}^{N_A},\n\\end{align}\nand given by,\n\\begin{eqnarray}\\label{eq:casscf}\n\\Psi_\\textrm{CAS} = \\hat{A}\\left[\\Phi_\\textrm{fc}\\Phi_\\textrm{dc}(t)\\sum_I \\Phi_I(t) C_I(t)\\right],\n\\end{eqnarray}\nwhere $\\hat{A}$ is the antisymmetrization operator, $\\Phi_\\textrm{fc}$ and $\\Phi_\\textrm{dc}$ are the closed-shell\ndeterminants constructed with FC and DC orbitals, respectively, and\n$\\{\\Phi_I\\}$ are the determinants formed {\\color{black}by} active orbitals.\nIn the following, we will denote the level of the CAS approximation\nemployed in $\\Psi_{\\rm CAS}$\nby the integer triple  $(n_{fc}, n_{dc}, n_{a})$.\nHereafter, we use orbital indices $\\{i,j,k\\}$ for core ($\\mathcal{C}$),\n$\\{t,u,v,w,x,y\\}$ for active ($\\mathcal{A}$), and $\\{o,p,q,r,s\\}$ for arbitrary occupied (core and active)\n($\\mathcal{P}=\\mathcal{C}+\\mathcal{A}$)\norbitals (Fig.\\ \\ref{fig:CASSCF-concept}).\nThe FC and DC orbitals are distinguished explicitly only when necessary.\n\nThere are two limiting cases. On one hand, if we use a single configuration made up of only DC orbitals, i.e., $(0,N\/2,0)$, or equivalently $(0,0,N\/2)$, it corresponds to TDHF\\index{Time-dependent Hartree-Fock (TDHF) method}\\cite{Pindzola1991PRL}, where some orbitals can also be frozen in a broader sense.\nOn the other hand, the special case $(0,0,n)$ ($n>N\/2$), where all the orbitals are fully correlated or treated as active, corresponds to MCTDHF.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=\\textwidth]{figs\/CASSCF-concept.pdf}\t\n\\end{center}\n\\caption{Schematic illustration of the TD-CASSCF concept for a twelve-electron system with two frozen-core, two dynamical-core, and eight active orbitals. The classification of orbitals and the indices we use are also shown.}\n\\label{fig:CASSCF-concept}      \n\\end{figure}\n\n\\subsection{TD-CASSCF equations of motion}\n\nThe equations of motion (EOMs) that govern the temporal evolutions of the CI coefficients $\\{C_I(t)\\}$ and orbital functions $\\{\\psi_p(t)\\}$ have been derived on the basis of the time-dependent variational principle (TDVP)\\index{Time-dependent variational principle (TDVP)}\\cite{Frenkel,Loewdin1972CPL,Moccia1973IJQC}, which requires the action integral,\n\\begin{equation}\n\\label{eq:action-integral}\nS[\\Psi] = \\int_{t_0}^{t_1}\\langle\\Psi | \\left(\\hat{H}-i\\frac{\\partial}{\\partial t}\\right)|\\Psi\\rangle,\n\\end{equation}\nto be stationary, i.e.,\n\\begin{equation}\n\\label{eq:TDVP}\n\\delta S = \\delta\\langle\\Psi|\\hat{H}|\\Psi\\rangle-i\\left(\\langle\\delta\\Psi | \\frac{\\partial\\Psi}{\\partial t}\\rangle\n-\\langle\\frac{\\partial\\Psi}{\\partial t}|\\delta\\Psi \\rangle\\right) = 0,\n\\end{equation}\nwith respect to arbitrary variation of CI coefficients and orbitals. By substituting Eq.~(\\ref{eq:casscf}) into Eq.~(\\ref{eq:TDVP}) and after laborious algebra, one can derive the equations of motion for the CI coefficients and orbital functions.\n\nThe form of the resulting EOMs {\\color{black}is} not unique\nbut can be written in various equivalent ways \\cite{Sato2016PRA}. Here we present \nthe EOMs in the form convenient for numerical implementation \\cite{Sato2016PRA}. \nThe EOMs for the CI coefficients read,\n\\begin{eqnarray}\n\\label{eq:eom_split_cic}\ni \\frac{d}{dt}C_I (t) &=& \\sum_J \\langle \\Phi_I|\\hat{H}_2|\\Phi_J\\rangle C_J (t),\n\\end{eqnarray}\nwhich describes transitions among different configurations solely mediated by the interelectronic Coulomb interaction.\nThe EOMs of the orbitals are given by\n\\begin{eqnarray}\n\\label{eq:eom_split_orb}\ni\\frac{d}{dt}|{\\psi}_p\\rangle &=& \\hat{h}|\\psi_p\\rangle +\\hat{Q} \n\\hat{F} |\\psi_p\\rangle + \\sum_q |\\psi_q\\rangle R^q_p,\n\\end{eqnarray}\nwhere $\\hat{Q}=1-\\sum_p |\\psi_p\\rangle\\langle\\psi_p|$ is the projector\nonto the orthogonal complement of the occupied orbital space. $\\hat{F}$ is a mean-field operator that describes the contribution from the interelectronic Coulomb interaction, defined by\n\\begin{align}\\label{eq:fock2}\n\\hat{F} |\\psi_p\\rangle = \\sum_{oqsr} (D^{-1})_p^o P^{qs}_{or} \\hat{W}^r_s |\\psi_q\\rangle,\n\\end{align}\nwhere {\\color{black} $D$} and {\\color{black} $P$} are the one- and two-electron reduced\ndensity matrix (RDM) in the orbital representation, respectively\n(see Ref.~\\cite{Sato2013PRA} for their definition and the simplification\ndue to the core-active separation), \nand $\\hat{W}^r_s$ is the electrostatic potential of an orbital product (pair potential),\n\\begin{eqnarray}\\label{eq:meanfield}\n\\hat{W}^r_s({\\bf r}) &=&\n\\int d{{\\bf r}^\\prime}\n\\frac{\\psi^*_r({{\\bf r}^\\prime}) \\psi_s({{\\bf r}^\\prime})}\n{|{\\bf r} - {\\bf r}^\\prime|}.\n\\end{eqnarray}\nThe matrix element $R^q_p$,\n\\begin{eqnarray}\\label{eq:nonredundent}\nR^q_p \\equiv i \\langle\\psi_q|\\dot{\\psi}_p\\rangle-h^q_p,\n\\end{eqnarray}\nwith $h^q_p = \\braket{\\psi_q|\\hat{h}|\\psi_p}$, determines the components of the time derivative of orbitals\n{\\color{black}in the subspace spanned by the occupied orbitals}.\nThe elements within one subspace, i.e.{\\color{black},} $R^i_j$ and $R^u_t$, can be\narbitrary Hermitian matrix elements and are set to zero $R^i_j=R^u_t=0$\nin our implementation \\cite{Sato2016PRA}. The elements between the\ncore and active subspaces are given by,\n\\begin{align}\n\t\\label{eq:R_ti}\n\t\t&R^{t}_i = \\left(R^i_t\\right)^* =  \\left\\{\n\\begin{array}{cc}\n{-h^t_i} & {\\rm (LG)}\\\\\n{-h^t_i} - \\vec{E}(t) \\cdot \\vec{r}^{\\,t}_{\\,i} &  {\\rm (VG)}\\\\\n\\end{array}\n\\right.& (\\text{for } i \\in \\text{frozen core}), \\\\\n\t\t\\label{eq:R1}\n\t\t&R^{t}_i = \\left(R^i_t\\right)^*  =  \\sum_u [(2-D)^{-1}]^{t}_{u} (2F^u_i - \\sum_v D^u_v F^{i*}_v) \n\t\t& (\\text{for } i \\in \\text{dynamical core}), \n\\end{align}\nwhere $F^u_i = \\langle\\psi_u | \\hat{F} | \\psi_i \\rangle$, and $\\vec{r}^{\\,t}_{\\,i}$ denotes a matrix element of the position vector $\\vec{r}$.\nFor the sake of gauge invariance (see Sebsec.~\\ref{subsec:Gauge Invariance}), frozen core orbitals, which are time-independent in the length gauge, are to be varied in time in the velocity gauge as \\cite{Sato2016PRA},\n\\begin{equation}\n\\psi_i( \\vec{r}, t)  =  \n\t {\\rm e}^{- i \\vec{A}(t) \\cdot \\vec{r}} \\psi_i( \\vec{r}, 0) \\qquad (\\text{for } i \\in \\text{frozen core}),\n\\end{equation}\nin spite of their name. Nevertheless, the FC orbital electron density distribution $|\\psi_i( \\vec{r}, t)|^2=|\\psi_i( \\vec{r}, 0)|^2$ is still time-independent.\n\nIt is noteworthy that the laser-electron interaction is explicitly contained only in the first term of the orbital EOM Eq.~(\\ref{eq:eom_split_orb}) and does not directly drive temporal change of the CI coefficients in Eq.~(\\ref{eq:eom_split_cic}).\nThus, in the form presented here, we can say that dynamical correlation induced by the laser field manifests itself first in the orbital EOMs and then spreads to the CI coefficients via the temporal change of orbitals (and, thus, of Slater determinants) in Eq.~(\\ref{eq:eom_split_cic}).\n\n\\subsection{Numerical Implementation for Atoms}\n\\label{subsec:Numerical Implementation for Atoms}\n\nWe have recently numerically implemented the TD-CASSCF method for atoms irradiated by a linearly polarized laser pulse, as detailed in Ref.~\\cite{Sato2016PRA}.\nOur implementation employs a spherical harmonics expansion of orbitals with the radial coordinate discretized by a finite-element discrete variable representation \\cite{Rescigno:2000,McCurdy:2004,Schneider:2006,Schneider:2011}.\nThe computationally most costly operation is to evaluate the pair potentials [Eq.~(\\ref{eq:meanfield})] contributing to the mean-field [Eq.~(\\ref{eq:fock2})], for which we use a Poisson solver thereby achieving linear scaling with the number of basis functions (or equivalently, grid points) \\cite{McCurdy:2004, Hochstuhl:2011, Sato2013PRA, Omiste:2017, Erik:2018}. \nA split-operator propagator is developed with an efficient implicit method for stiff derivative operators which drastically stabilizes the temporal propagation of orbitals. \nThanks to the combination of these techniques, we can take full advantage of the TD-CASSCF method.\n\n\\subsection{Gauge Invariance}\n\\label{subsec:Gauge Invariance}\n\n\\index{Gauge invariance}\nThe TD-CASSCF method is gauge invariant. For a TD-MCSCF method to be gauge invariant, it must meet the following two requirements:\n\\begin{enumerate}\n\t\\item Any LG wave function $\\Psi_{\\rm L}(t)$ that satisfies a given multiconfiguration ansatz Eq.~(\\ref{eq:general-mcwf}) can be transformed to a VG wave function $\\Psi_{\\rm V}(t)$ that satisfies another multiconfiguration ansatz of the same form, and vice versa.\n\t\\item If a LG wave function $\\Psi_{\\rm L}(t)$ is optimized on the basis of the TDVP expressed in the length gauge, its VG counterpart $\\Psi_{\\rm V}(t)$ satisfies the TDVP in the velocity gauge, and vice versa.\n\\end{enumerate}\n\nTo discuss the first requirement, let us denote the orbital functions calculated with a given multiconfiguration ansatz Eq.~(\\ref{eq:general-mcwf}) within the length gauge by $\\{\\psi_p^{\\rm L}({\\bf r})\\}$. Equation (\\ref{eq:gauge-transformation}) is fulfilled if one constructs the wave function $\\Psi_{\\rm V}(t)$ of the same ansatz with the CI coefficients unchanged using the orbital functions $\\{\\psi_p^{\\rm V}({\\bf r})\\}$ defined by,\n\\begin{equation}\n\\label{eq:orbital-gauge-transformation}\n\\psi_p^{\\rm V}({\\bf r}) = \\exp \\left[-i{\\bf A}(t)\\cdot {\\bf r}\\right] \\psi_p^{\\rm L}({\\bf r}).\n\\end{equation}\nSince this tells us that at least one of $\\{\\psi_p^{\\rm L}({\\bf r})\\}$ and $\\{\\psi_p^{\\rm V}({\\bf r})\\}$ is necessarily time-dependent, TD-MCSCF methods that use time-independent orbital functions such as TDCIS are, in general, not gauge invariant, i.e., the values of the observables obtained within the length gauge are not equal to those within the velocity gauge. \nThis is because $\\Psi_{\\rm V}(t)$ does not necessarily belong to the subspace of the Hilbert space spanned by $\\{\\Phi_{\\bf I}\\}$, in which $\\Psi_{\\rm L}(t)$ is optimized.\nIt should be noticed that even if we could use an infinite number of orbitals, TDCIS would not be gauge-invariat;\nit follows from Eq.~(\\ref{eq:HgsVG}) that if we use time-independent orbitals and $\\Psi_{\\rm L}(t)$ is expressed as a single (Hartree-Fock) determinant, $\\Psi_{\\rm V}(t)$ involves up to $N$-tuple excitations.\n(see Ref.~\\cite{Sato2018AS} for a recently reported gauge-invariant formulation of TDCIS with time-dependent orbitals.)\n\nFor the second requirement, it should be noticed that the length- and velocity-gauge Hamiltonians $\\hat{H}_{\\rm L}(t)$ with Eq.~(\\ref{eq:length-gauge}) and $\\hat{H}_{\\rm V}(t)$ with Eq.~(\\ref{eq:velocity-gauge}), respectively, are related by \\cite{Bandrauk2013JPB},\n\\begin{equation}\n\\hat{H}_{\\rm V} = \\hat{\\mathcal{U}}\\hat{H}_{\\rm L}\\hat{\\mathcal{U}}^\\dag+i\\frac{d\\hat{\\mathcal{U}}}{dt}\\hat{\\mathcal{U}}^\\dag.\n\\end{equation}\nThen, using the unitarity of the gauge-transformation operator $\\hat{\\mathcal{U}}(t)$[Eq.~(\\ref{eq:gauge-transformation-operator})], we can show that the TDVP expressions Eq.~(\\ref{eq:TDVP}) in the two representations are equivalent. This guarantees that the wave function transformed via Eq.~(\\ref{eq:orbital-gauge-transformation}) from the wave function satisfying the length-gauge TDVP fulfills the velocity-gauge TDVP. \nTherefore, satisfying both of the above-mentioned conditions, TD-MCSCF methods with time-varying orbital functions, including TDHF, MCTDHF, TD-CASSCF, and the time-dependent occupation-restricted multiple active-space (TD-ORMAS) \\cite{Sato2015PRA} methods, are gauge invariant in general \\cite{Sato2013PRA,Miyagi2014PRA,Sato2015PRA,Ishikawa2015JSTQE}\n\n\\subsection{Size Extensivity}\n\\label{subsec:Size Extensivity}\n\\index{Size Extensivity}\n\nThe TD-CASSCF method is size extensive. Size extensivity\\footnote{It is not to be confused with a similar but different concept of size consistency, which, for the case of the ground-state energy, states ``if molecule AB dissociates to molecules A and B, the asymptote of molecule AB at infinite internuclear separation should be the sum of the energies of molecules A and B\" \\cite{Veszpremi} and ``is only defined if the two fragments are non-interacting\" \\cite{Jensen}.} states ``the method scales properly with the number of particles\" \\cite{Veszpremi} or, for the case of the ground-state energy, ``if we have $k$ number of noninteracting identical molecules, their total energy must be $k$ times the energy of one molecule\" \\cite{Jensen}.\nRoughly speaking, it can be understood as follows.\n\nLet us consider that we simulate photoionization of a He atom for such a laser parameter that He is substantially singly ionized but that double ionization is negligible. \nThen, what will happen if we simulate photoionization of a He dimer by the identical laser pulse, in which the two He atoms are sufficiently far apart from each other but the dipole approximation is still valid? \nPhysically, we would expect substantial single ionization of each atom, resulting in double ionization in total (Fig.~\\ref{fig:size-extensivity}).\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=\\textwidth]{figs\/size-extensivity.pdf} \t\n\\end{center} \n\\caption{Schematic illustration of size extensivity explained with a He atom and dimer (see text).}\n\\label{fig:size-extensivity}\n\\end{figure}\n\nThis seemingly obvious requirement is, in general, {\\it not} met by TD-MCSCF methods with truncated expansion such as TDCIS and TD-ORMAS.\nOn the other hand, TD-CASSCF as well as MCTDHF and TDHF fulfills size extensivity.\n\n\\section{Initial-State Preparation and Simulation Boundary}\n\\label{sec:Initial-State Preparation and Simulation Boundary}\n\nIn {\\it ab initio} simulation study of multielectron dynamics, we usually need to (i) prepare the initial state, (ii) propagate the wave function in time (Sec.~\\ref{sec:TD-CASSCF}), (iii) absorb electrons that leave the calculation region, and (iv) read out physically relevant information from the wave function (Sec~\\ref{sec:Numerical examples}). \nLet us discuss (i) and (iii) in this Section.\n\n\\subsection{Imaginary-Time Propagation}\n\\index{Imaginary-time propagation}\n\nWhile the initial state can also be obtained by a separate time-independent calculation of the ground state, a convenient alternative is imaginary-time propagation (or relaxation) \\cite{Flocard1978PRC}.\nThe solution of the field-free TDSE can be expressed as,\n\\begin{equation}\n\t\\Psi (t) = \\sum_{\\alpha=0}^\\infty c_\\alpha \\Xi_\\alpha e^{-iE_\\alpha t}= e^{-iE_0 t} \\left(c_0 \\Xi_0  + \\sum_{\\alpha=1}^\\infty c_\\alpha \\Xi_\\alpha e^{-i(E_\\alpha-E_0) t}\\right),\n\\end{equation}\nwith eigenstates $\\Xi_\\alpha$, of which $\\Xi_0$ is the ground state, and energy eigenvalues $E_\\alpha$.\nBy substituting {\\it imaginary} time $t = -is$ with $s$ being a real number, we obtain,\n\\begin{equation}\n\t\\Psi (-is) e^{E_0 s} =  c_0 \\Xi_0  + \\sum_{\\alpha=1}^\\infty c_\\alpha \\Xi_\\alpha e^{-(E_\\alpha-E_0) s} \\xrightarrow[s\\to \\infty]{} c_0 \\Xi_0,\n\\end{equation}\nsince $E_\\alpha-E_0>0$ ($\\alpha\\ge 1$). Thus, we can obtain the ground state by integrating the field-free EOMs in imaginary time and renormalizing the wave function after every several time steps.\nThe imaginary-time propagation is used for the results presented in this Chapter.\n\n\\subsection{Absorption Boundary}\n\nSince ionization is essential in the ultrafast intense laser science, it is one of the major issues how to treat electrons that leave the calculation region and suppress unphysical reflections.\nWe use either mask function or infinite-range exterior complex scaling in our numerical implementations.\n\n\\subsubsection{Mask Function and Complex Absorbing Potential (CAP)}\n\nOne common method is to multiply orbital functions outside a given radius (mask radius) $R_0$ by a function that decreases from unity and vanishes at the simulation box boundary after each time step\\index{Mask function} \\cite{Krause_1992}. \nTypical forms of the mask function include $\\cos^{1\/4}$ and $\\cos^{1\/8}$.\n\nAnother method is to add a complex absorbing potential (CAP), e.g., of the form\n\\begin{equation}\n\t-i\\eta W(r) = -i \\eta (r-R_0)^2,\n\\end{equation}\nwhere $\\eta$ denotes a CAP strength, to the Hamiltonian outside a given radius $R_0$. In the context of {\\it ab initio} simulations of strong-field processes, CAP has been used in combination with TDCIS \\cite{Greenman_2010}.\n\n\\subsubsection{Exterior Complex Scaling (ECS)}\n\nExterior complex scaling\\index{Exterior complex scaling (ECS)}\\cite{McCurdy_1991} analytically continues the wave function outside a given scaling radius $R_0$ into the complex plane as, for the case of the polar coordinate (Fig.~\\ref{fig:ECScontour}),\n\\begin{equation}\n\t\\label{eq:rchange}\n\tr \\rightarrow R(r) = \n\t\\begin{cases}\n\t\tr & (r < R_0) \\\\\n\t\tR_0 + (r-R_0) e^{\\lambda + i \\eta} & (r > R_0),\n\t\\end{cases}\n\\end{equation}\nwhere $\\lambda$ and scaling angle ${\\eta}$ are real numbers. Then, the orbital function is transformed via ECS operator $U_{\\eta R_0}$ as,\n\\begin{equation}\\label{eq:Ueta}\n\t(U_{\\eta R_0} \\psi) (\\vec{r}): = \n\t\\begin{cases}\n\t\t\\psi( \\vec{r} ) & (r < R_0) \\\\\n\t\te^{\\frac{\\lambda + i \\eta}{2}} \\dfrac{R(r)}{r} \\psi( \\vec{R}(r)) & (r > R_0),\n\t\\end{cases}\n\\end{equation}\nwhere $\\vec{R}(r) = \\frac{R(r)}{r}\\vec{r}$.\nIn actual simulations, we numerically store $(U_{\\eta R_0} \\psi) (\\vec{r})$ instead of $\\psi( \\vec{r} )$ in the scaled region $r > R_0$.\nWe can understand why this works as an absorbing boundary by considering a spherical wave $e^{ikr}\/r$. At $r>R_0$ it becomes $e^{\\frac{\\lambda + i \\eta}{2}}e^{ik[R_0 + (r-R_0) e^{\\lambda} \\cos\\eta]-k(r-R_0)\\sin\\eta}\/r$, which exponentially diminishes as $\\sim e^{-kr\\sin\\eta}\/r$ at large distance.\nIt should be noticed that ECS modifies neither the wave function nor the system Hamiltonian.\n\nWhile ECS is usually applied on a finite discretization range, one can infinitely extend the scaled region, thus moving the simulation box boundary to infinity, while using a finite number of exponentially damped basis functions \\cite{Scrinzi_2010}. This method, called infinite-range ECS (irECS)\\index{Infinite-range exterior complex scaling (irECS)}, significantly improves the accuracy and efficiency over standard ECS with a considerably smaller number of basis functions. It also has a conceptual advantage of being able to simulate the entire space.\n\nWhile irECS has originally been formulated for a single-electron system and found only limited use for strongly-driven multielectron systems, we have applied it to our TD-CASSCF numerical implementation, as detailed in Ref.~\\cite{Orimo2018PRA}.\nWe set $\\lambda=0$ and introduce Gauss-Laguerre-Radau quadrature points \\cite{Gautschi_2000,Weinmueller_2017} to construct discrete-variable-representation basis functions in the last finite element extending to infinity.\nAn essential difference from a single-electron case is the presence of interelectronic Coulomb interaction via mean-field operator Eq.~(\\ref{eq:fock2}). Its evaluation as well as that of $\\hat{Q}$ requires $\\hat{U}_{(-\\eta) R_0} \\Ket{\\psi_p}$, which is not available in the scaled region.\nSince the scaled region is far from the origin, it is reasonable to assume that the scaled part of the orbital functions hardly affects the electron dynamics close to the nucleus and that the interaction between electrons residing in the scaled region is negligible.\nThus, we neglect $\\hat{U}_{(-\\eta) R_0} \\Ket{\\psi_p}$ in the scaled region wherever their information is necessary. This treatment roughly corresponds to the neglect of the Coulomb force acting on electrons from scaled-region electrons ($r > R_0$). On the other hand, the Coulomb force acting on scaled-region electrons from unscaled-region electrons ($r < R_0$) is not neglected. Hence, the effect of the ionic Coulomb potential is properly taken into account in the dynamics of departing electrons.\n\nFigure \\ref{fig:radBe} compares the electron radial distribution functions after the pulse for the case of a Be atom exposed to a laser pulse with 800 nm wavelength and $3.0 \\times 10^{14} \\text{ W\/m$^2$}$ peak intensity, calculated with different absorbing boundaries listed in Table \\ref{tab:detailBe}.\nThe pulse has a $\\sin^2$ envelope with a foot-to-foot pulse width of five cycles. We use $(n_{fc},n_{dc},n_{a}) = (1,0,4)$.\nThe result of condition A, with $R_0=320$ a.u. much larger than the quiver radius 28.5 a.u., is converged and can be considered to be numerically exact. We can see that the irECS delivers much better results (C and E) inside $R_0$ than the mask function (F).\nIt is remarkable that the irECS works well even with the scaling radius ($R_0=28$ a.u.) comparable with the quiver radius.\nThe result of the simulation (condition D) similar to C but neglecting also the interelectronic Coulomb force from the unscaled (inner) to the scaled (outer) region is plotted with a blue dashed curve in Fig.~\\ref{fig:radBe}.\nWe find a large discrepancy from the exact result (A).\nThis indicates that proper account of the Coulomb force acting on scaled-region electrons from unscaled-region ones is crucial for accurate simulations, even though the total momentum of the system is not conserved due to imbalance in counting the interelectronic Coulomb interactions.\n\n\\begin{figure}[tb]\n\\sidecaption\n\\includegraphics[width=0.5\\textwidth]{figs\/ECS-contour.pdf}\n\\caption{Schematic illustration of radial exterior complex scaling contour $R(r)$ with scaling radius $R_0$ and scaling angle $\\eta$.}\n\\label{fig:ECScontour}\n\\end{figure}\n\n\\begin{figure}[tb]\n\\includegraphics[width=\\textwidth]{figs\/radial-distribution.pdf}\n\\caption{Electron radial distribution function $\\rho (r)$ after the laser pulse for the case of Be exposed to a laser pulse with 800 nm wavelength and $3.0 \\times 10^{14} \\text{ W\/m$^2$}$ peak intensity, calculated with different absorbing boundaries listed in Table \\ref{tab:detailBe}.}\n\\label{fig:radBe}\n\\end{figure}\n\n\\begin{table}[tb]\n\t\\caption{Absorbing boundaries tested for Be.}\n\t\t\\begin{tabular}{p{1cm}p{2cm}p{1cm}p{1cm}p{1cm}p{1cm}p{4cm}}\n\t\t\t\\hline\n\t\t\t& Absorber & $R_0$ & $n_\\text{ua}$ & $L_{\\text{a}}$ & $n_\\text{a}$ & Remark\\\\\n\t\t\t\\svhline\n\t\t\tA & mask & 320 & 1600 & 80 & 400 & nominally exact\\\\\n\t\t\tB & mask & 320 & 1600 & 80 & 400 & truncated at $28$ a.u. (see text in Sebsec.~\\ref{subsubsec:after-absorption})\\\\\n\t\t\tC & irECS & 28 & 140 & $ \\infty $ & 40 \\\\\n\t\t\tD & irECS & 28 & 140 & $ \\infty $ & 40 & unscaled-to-scaled Coulomb neglected \\\\\n\t\t\tE & irECS & 52 & 260 & $ \\infty $ & 40 \\\\\n\t\t\tF & mask & 52 & 260 & $ 8 $ & 40 \\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\\label{tab:detailBe\n\\end{table} \n\n\\subsubsection{Which part of the total wave function is propagated after one or more electrons are absorbed at the simulation boundary?}\n\\label{subsubsec:after-absorption}\n\nLet us specifically consider a He atom, which is a two-electron system. \nThe $(r_1,r_2)$ space can be divided into four regions, as shown in Fig.~\\ref{fig:He-regions}, A: $r_1<R_0, r_2<R_0$, B: $r_1 > R_0, r_2 < R_0$, C: $r_1 < R_0, r_2 > R_0$, D: $r_1 > R_0, r_2 > R_0$.\n\nFor the case of direct numerical simulation of the two-electron TDSE, e.g., by the time-dependent close-coupling method \\cite{Colgan2001JPB, Parker2001, ATDI2005, Feist2009PRL}, the wave function only in region A is stored and propagated. Hence, once one electron is absorbed, the dynamics of the other electron is no longer followed even if it is still inside the absorption radius $R_0$, and, as a consequence, the transition from B or C to D cannot be traced.\n\nIn great contrast, not only the two electrons in region A but also the inner electron in regions B and C is simulated in the TD-CASSCF, MCTDHF, and TDHF simulations.\nIn order to understand this prominent feature, we decompose the exact orbital $\\ket{\\psi_p}$, which would be obtained if we used an infinitely large simulation box, into the part numerically stored and propagated during actual simulation $\\ket{\\phi_p}$ and the remaining, i.e., absorbed and lost part $\\ket{\\chi_p}$:\n\\begin{equation}\n\t\\ket{\\psi_p} = \\ket{\\phi_p} + \\ket{\\chi_p}.\n\\end{equation}\nRoughly speaking, $\\ket{\\phi_p}$ is the part at $r<R_0$ and $\\ket{\\chi_p}$ at $r>R_0$. The TD-CASSCF equations of motion are derived on the assumption that $\\{\\ket{\\psi_p}\\}$ is orthonormal. On the other hand, $\\{\\ket{\\phi_p}\\}$ is {\\it not} orthonormal in general, and its norm decreases. By good use of absorption boundary, $\\ket{\\phi_p}$ expectedly reproduces $\\ket{\\psi_p}$ within $R_0$.\nIn region B, the two-electron wave function is generally expressed as,\n\\begin{equation}\n\t\\label{eq:He-expansion}\n\t\\Psi ({\\bf r}_1,{\\bf r}_2) = \\sum_{p,q} C_{pq}\\chi_p ({\\bf r}_1) \\phi_q ({\\bf r}_2),\n\\end{equation}\nwith an expansion coefficient $C_{pq}$. Here we neglect the spin part for simplicity. As Eq.~(\\ref{eq:He-expansion}) suggests, even after electron 1 is absorbed, the dynamics of electron 2, still entangled with electron 1, continues to be simulated, though approximately, as long as it stays inside the absorption radius.\n\nIn Fig. \\ref{fig:radBe} we have seen that the irECS works much better than the mask function. Nevertheless, the irECS results (conditions C and E in Table \\ref{tab:detailBe}) still deviate slightly from the nominally exact solution (condition A).\nIn the present case, Be is nearly totally ionized, and double ionization amounts to 50 \\%, as we will see below in Fig. \\ref{fig:ipxBe}. Hence, the deviation may be due to the neglect of the Coulomb interaction in and from the scaled region and\/or the loss of information on the wave function in the scaled region.\n\nIn order to reveal the effect of the latter, we have performed a simulation (condition B in Table \\ref{tab:detailBe}) with a sufficiently large domain as condition A but by truncating the interelectronic Coulomb interaction at $r=28\\,{\\rm a.u.}$ as in the irECS.\nThe result is plotted in Fig.~\\ref{fig:radBe}.\nThe ``truncated\" result (B) slightly deviates from the exact one (A) but overlaps with the irECS result (C) at $r<28$ a.u., which indicates that the difference between the exact and irECS results in Fig.~\\ref{fig:radBe} originates from the neglect of the Coulomb interaction in and from the scaled region, not from the loss of information.\n\nOne may be surprised that the loss of information on orbital functions at the absorption boundary hardly affects simulation results within the absorption radius.\nIt should be noticed that, even if the explicit form of $\\ket{\\chi_p}$ is unknown, some information on them is still available. At least, we can tell,\n\\begin{equation}\n\t\t\\langle \\phi_p | \\chi_q \\rangle = 0\t, \\qquad \\langle \\chi_p | \\chi_q \\rangle = \\delta_{pq} - \\langle \\phi_p | \\phi_q \\rangle,\n\\end{equation}\nfrom the orthonormality of $\\{\\ket{\\psi_p}\\}$. \nThis not only helps accurate simulations but also allows to extract useful information such as ionization yields and charge-state-resolved observables, as discussed in the next section. \n\n\\begin{figure}[tb]\n\\sidecaption\n\\includegraphics[width=0.5\\textwidth]{figs\/He-regions.pdf}\n\\caption{Four regions of the $(r_1,r_2)$ space of the two electrons in He.}\n\\label{fig:He-regions}      \n\\end{figure}\n\n\\section{Numerical examples}\n\\label{sec:Numerical examples}\n\nIn this Section, we present how to extract physical observables from the wave function and numerical results obtained with TD-CASSCF and TDHF simulations.\n\n\\subsection{Ionization Yield}\n\\label{subsec:Ionization Yield}\n\nOne might consider that the ionization yield for charge state $n$ could be obtained through the integration over the population of all possible $n$-electron continuum states (note that $n$ denotes the ionic charge state in this Section). \nUnfortunately, however, direct application of this naive idea would encounter difficulties. \nFirst, it is not trivial (even more difficult than TD-CASSCF itself) to prepare $n$-electron continuum wave functions. The ionic core is not necessarily in the ground state and may also be excited. \nSecond, we have to keep the entire wave function within the simulation box, without being absorbed. The computational cost would be prohibitive. \nThird, as discussed in Subsec.~\\ref{subsec:Gauge Transformation}, the population of each field-free stationary state is not gauge invariant during the pulse.\n\nInstead, we define ionization in terms of the spatial positions of electrons and introduce ionization probability $P_n$ as a probability\nto find $n$ electrons in the outer region $|{\\bf r}| >\nR_{\\rm ion}$ and the remaining $N - n$ \nelectrons in the inner region $|{\\bf r}| < R_{\\rm ion}$,\nwith a given distance $R_{\\rm ion}$ from the origin,\n\\begin{eqnarray}\n\\label{eq:ionp} \nP_n &\\equiv& \\binom{N}{n} \n\\int_> dx_1 \\cdot\\cdot\n\\int_> dx_n\n\\int_< dx_{n+1} \\cdot\\cdot\n\\int_< dx_N \\,\n\\left| \\Psi (x_1,\\cdot\\cdot,x_N) \\right|^2,\n\\end{eqnarray} \nwhere $\\int_<$ and $\\int_>$ denote integrations over a spatial-spin\nvariable $x = \\{{\\bf r}, \\sigma\\}$ with the spatial part restricted to the domains\n$|{\\bf r}| < R_{\\rm ion}$, and $|{\\bf r}| > R_{\\rm ion}$, respectively.\n${P_n}$ satisfies $\\sum_{n=0}^N P_n = 1$.\nThis spatial-domain-based ionization probability has an advantage of being gauge invariant. Moreover, it is consistent with our usual perception of ionization as a spatial separation of electron from the parent ion, such as ejection from the surface and arrival of electron at a detector.\n\nIf we introduce,\n\\begin{eqnarray}\n\\label{eq:ionp-aux} \nT_n &\\equiv& \\binom{N}{n} \n\\int dx_1 \\cdot\\cdot\n\\int dx_n\n\\int_< dx_{n+1} \\cdot\\cdot\n\\int_< dx_N \\,\n\\left| \\Psi (x_1,\\cdot\\cdot,x_N) \\right|^2,\n\\end{eqnarray} \nit is related to $P_n$ as,\n\\begin{eqnarray}\n\\label{eq:ionp-t}\nP_n = \\sum_{k=0}^n \\binom{N-n+k}{k}(-1)^k T_{n-k},\n\\end{eqnarray}\ndue to the orthonormality of orbitals with full-space integration \\cite{Sato2013PRA} (see also Subsec.~\\ref{subsubsec:after-absorption}), allowing to calculate the ionization probability only from the information of orbitals inside the radius $R_{\\rm ion}$ and CI coefficients.\nBy adopting the multiconfiguration expansion Eq.~(\\ref{eq:general-mcwf}), and making use of\nthe orthonormality of spin-orbitals in the full-space integration, we\nhave \n\\begin{eqnarray}\n\\label{eq:iont}\nT_n = \\sum_{IJ}^{\\sf P} C^*_I C_J D^{(n)}_{IJ},\n\\end{eqnarray}\nwhere,\n\\begin{eqnarray} \n\\label{eq:iond}\nD^{(0)}_{IJ} &=& \\sum_{ij}^N\n\\det\\left(S^<_{IJ}\\right), \\nonumber \\\\\nD^{(1)}_{IJ} &=& \\sum_{ij}^N \\epsilon^{IJ}_{ij} \n\\det\\left(S^<_{IJ}[i:j]\\right), \\nonumber \\\\\nD^{(2)}_{IJ} &=& \\sum_{i>j}^N \\sum_{k>l}^N \\epsilon^{IJ}_{ik}\n \\epsilon^{IJ}_{jl}\n\\det\\left(S^<_{IJ}[ij:kl]\\right),\n\t       \\end{eqnarray}\netc, and  $S^<_{IJ}$ is an $N\\times N$\nmatrix with its $\\{ij\\}$ element being the inner-region overlap integral,\n\\begin{eqnarray}\n\\label{eq:overlap}\n(S^<_{IJ})_{ij} = \\int_< dx \\phi^*_{p(i,I)}(x) \\phi_{q(j,J)}(x) \\equiv\n\\langle \\phi_p | \\phi_q \\rangle_<,\n\\end{eqnarray}\nwhere $\\phi_{p(i,I)}$ is the $i$-th (in a predefined order) spin orbital in\nthe determinant $I$. $S^<_{IJ}[ij\\cdot\\cdot:kl\\cdot\\cdot]$ is the\nsubmatrix of $S^<_{IJ}$ obtained after removing rows $i,j,\\cdot\\cdot$ and columns\n$k,l,\\cdot\\cdot$ from the latter, and,\n\\begin{eqnarray}\n\\label{eq:epsilon}\n\\epsilon^{IJ}_{ij} = \\delta^{p(i,I)}_{q(j,J)} (-1)^{i+j}.\n\\end{eqnarray}\nThe matrix $S^<_{IJ}$ and its submatrices are block-diagonal due to\nthe spin-orthonormality, so that, e.g., $\\det\\left(S^<_{IJ}\\right) = \n\\det\\left(S^<_{I^\\alpha J^\\alpha}\\right) \n\\det\\left(S^<_{I^\\beta J^\\beta}\\right)$, where $I^\\sigma$ is the\n$\\sigma$-spin part of the determinant $I$.\n\nIn Fig.~\\ref{fig:ipxBe}, we show the temporal evolution of thus calculated single, double, and total ionization yields of Be for the same pulse and orbital subspace decomposition as for Fig.~\\ref{fig:radBe}.\nAs an absorption boundary, we have used irECS with $R_0 = 40\\,{\\rm a.u.}$. $R_{\\rm ion}$ is set to be $20\\,{\\rm a.u.}$.\nWe can see step-like evolution every half cycle typical of tunneling ionization. \nAfter the pulse, there is practically no neutral species left, and the double ionization yield is $\\sim 50\\%$.\nIt is remarkable that the neglect of the Coulomb interaction in and from the scaled region is a good approximation and that irECS works excellently even under such massive double ionization.\n\n\\begin{figure}[tb]\n\t\\begin{center}\n\t\t\\includegraphics[width=0.75\\textwidth]{figs\/ionization-evolution-Be.pdf}\n\t\\end{center}\n\t\\caption{Time evolution of spatial-domain-based single, double, and total ionization probabilities of Be exposed to a laser pulse with 800 nm wavelength and $3.0 \\times 10^{14} \\text{ W\/m$^2$}$ peak intensity (the same as for Fig.~\\ref{fig:radBe}). $R_{\\rm ion} = 20\\,{\\rm a.u.}$ is used.}\n\t\\label{fig:ipxBe}\n\\end{figure}\n\nFigure \\ref{fig:NSDI} presents the intensity dependence of the double ionization yields of He and Ne irradiated by a laser pulse whose wavelength is 800 nm. \nAlthough the results are not converged with respect to the number of orbitals yet, we can already clearly see knee structures in the TD-CASSCF results, but not in the TDHF ones.\nThus, the TD-CASSCF method can well reproduce non-sequential double ionization \\cite{Walker1994,Larochelle1998}, a representative strong-field phenomenon that witnesses electron correlation. \n\n\\begin{figure}[tb]\n\t\\begin{center}\n\t\t\\includegraphics[width=\\textwidth]{figs\/NSDI.pdf}\n\t\\end{center}\n\t\\caption{Double ionization yields of (a) He and (b) Ne as a function of intensity of a laser pulse with a wavelength of 800 nm, calculated by the TD-CASSCF and TDHF methods.}\n\t\\label{fig:NSDI}\n\\end{figure}\n\n\\subsection{Charge-State-Resolved Electron Density Distribution}\n\nThe usual electron density distribution,\n\\begin{equation}\n\\label{eq:EDD}\n\t\\rho ({\\bf r}) \\equiv N \\sum_{\\sigma}\n\\int dx_2 \\cdots\n\\int dx_N \\,\n\\left| \\Psi (x,x_2,\\cdots,x_N) \\right|^2,\n\\end{equation}\ncontains all the different charge states: neutral, singly ionized, doubly ionized, etc.\nTo discuss charge migration following attosecond photoionization, it will be useful to analyze, e.g., a hole distribution inside the cation.\nHence, we introduce a charge-state-resolved electron density distribution $\\rho^{(n)} ({\\bf r})$ as a probability to find an electron at ${\\bf r}$ on condition that $n$ out of the other $N-1$ electrons are at $|{\\bf r}| > R_{\\rm ion}$ and $N-1-n$ at $|{\\bf r}| < R_{\\rm ion}$,\n\\begin{equation}\n\\label{eq:EDD-chg}\n\t\\rho^{(n)} ({\\bf r}) \\equiv N\\binom{N-1}{n} \\sum_{\\sigma}\n\\int_> dx_2 \\cdots\n\\int_> dx_{n+1}\n\\int_< dx_{n+2} \\cdots\n\\int_< dx_N \\,\n\\left| \\Psi (x,x_2,\\cdots,x_N) \\right|^2.\n\\end{equation}\nNote that the electron density distribution in the neutral species is consistently given by,\n\\begin{equation}\n\\label{eq:EDD-neutral}\n\t\\rho^{(0)} ({\\bf r}) \\equiv N \\sum_{\\sigma}\n\\int_< dx_{2} \\cdots\n\\int_< dx_N \\,\n\\left| \\Psi (x,x_2,\\cdots,x_N) \\right|^2.\n\\end{equation}\nHere, again, we have used domain-based definition of ionization.\n\n$\\rho^{(n)} ({\\bf r})$ can be expressed in terms of orbitals and CI coefficients as well as $S_{IJ}^<$ introduced in the previous Subsection. For example, the electron density distribution of the cation can be calculated by,\n\\begin{equation}\n\t\\rho^{(1)}({\\bf r}) = \\sum_{IJ}^\\Pi C^*_I C_J \\sum_{i,j}^{N} \\phi^*_{p(i,I)}(x) \\phi_{q(j,J)}(x) (-1)^{i+j} \\left( \\sum_{k=1}^N \\epsilon_{ik}\\epsilon_{jk}S_{IJ}^<[ik;jk]-(N-1)S_{IJ}^<[i;j] \\right).\n\\end{equation}\n\nIn Fig.~\\ref{fig:Be-charge-oscillation}, we show snap shots of the electron density distribution in ${\\rm Be}^+$ produced by photoionization of Be by XUV pulses with a photon energy of 150 eV and a FWHM pulse width of 20 and 30 as.\nThe process is simulated with the TDHF method.\nAn isotropic charge density is formed by the superposition of $(1s)^{-1}$ and $(2s)^{-1}$ holes and oscillates with a period of ca.~35 as, consistent with the energy difference ($\\sim 120$ eV). We also see that its amplitude is larger for the 20 as pulse width than for 30 as, reflecting the wider spectrum of photon energy.\nThe present charge-state-resolved density can also be used to decompose physical observables to contributions from species of different ionic charges, e.g, charge-state-resolved HHG spectra \\cite{Tikhomirov2017PRL}. \n\n\\begin{figure}[tb]\n\\sidecaption\n\\includegraphics[width=\\textwidth]{figs\/Be-charge-oscillation}   \n\\caption{Snap shots of the electron density distribution in ${\\rm Be}^+$ produced by photoionization of Be by XUV pulses with a photon energy of 150 eV, a peak intensity of $10^{13}\\,{\\rm W\/cm}^2$, and a FWHM pulse width of (a) 20 as and (b) 30 as. The results of TDHF simulations.}\n\\label{fig:Be-charge-oscillation}      \n\\end{figure}\n\n\n\n\\subsection{Ehrenfest Expression for Dipole Acceleration and High-Harmonic Spectrum}\n\\label{subsec:Dipole Acceleration}\n\nHarmonic spectrum is usually extracted by Fourier transforming the dipole moment,\n\\begin{eqnarray}\\label{eq:ehrenfest_z}\n\\langle z \\rangle (t)  = \\langle\\Psi|z|\\Psi\\rangle,\n\\end{eqnarray}\nor the dipole acceleration\\index{Dipole acceleration} $\\langle a\\rangle(t)$,\n\\begin{eqnarray}\\label{eq:a_td}\n\\langle a\\rangle(t) = \\frac{d^2}{dt^2}\\langle z \\rangle (t).\n\\end{eqnarray}\nAs known as the Ehrenfest theorem, one can show, from the TDSE Eq.~(\\ref{eq:TDSE}), that,\n\\begin{eqnarray}\\label{eq:ehrenfest_acc}\n\\langle a\\rangle(t) = -\n\\langle\\Psi|\\left(\n\\frac{\\partial \\hat{V}_0}{\\partial z}+\n\\frac{\\partial \\hat{V}_{\\rm ext}}{\\partial z}\n\\right)|\\Psi\\rangle.\n\\end{eqnarray}\nThe right hand side of this equation is the expectation value of the force acting on the electrons from the nuclei and laser electric field.\nEquation~(\\ref{eq:ehrenfest_acc}), with smaller numerical noise than in Eq.~(\\ref{eq:ehrenfest_z}), is widely used in combination with TDSE simulations within the single-active-electron (SAE) approximation, with $\\hat{V}_0$ replaced by the effective potential.\n\nThe equivalence of Eqs.~(\\ref{eq:a_td}) and (\\ref{eq:ehrenfest_acc}) holds also for the TD-CASSCF methods with all the orbitals time-varying \\cite{Sato2016PRA}, and, hence, the Ehrenfest expression Eq.~(\\ref{eq:ehrenfest_acc}) can be safely used. \nHowever, the use of frozen-core orbitals requires a special care.\nWe have shown that, in the latter case, the following expression should be used instead of Eq.~(\\ref{eq:ehrenfest_acc}) \\cite{Sato2016PRA}:\n\\begin{eqnarray}\\label{eq:ehrenfest_fc_acc}\n\\langle a \\rangle_\\textrm{fc} (t) = -\n\\langle\\Psi|\\left(\n\\frac{\\partial \\hat{V}_0}{\\partial z}+\n\\frac{\\partial \\hat{V}_{\\rm ext}}{\\partial z}\n\\right)|\\Psi\\rangle +\n\\Delta(\\dot{p}_z).\n\\end{eqnarray}\nIn the length gauge and if we neglect the indistinguishability between core and active electrons, the additional term $\\Delta(\\dot{p}_z)$ can be approximated as \\cite{Sato2016PRA},\n\\begin{eqnarray}\\label{eq:gbf_fc_approx_p}\n\\Delta(\\dot{p}_z) &\\approx& \n \\langle\\Phi_\\textrm{fc}|\n \\frac{\\partial \\hat{V}_0}{\\partial z} +\n \\frac{\\partial \\hat{V}_{\\rm ext}}{\\partial z} +\n \\frac{\\partial \\hat{V}_a}{\\partial z}\n |\\Phi_\\textrm{fc}\\rangle,\n\\end{eqnarray}\nwhere,\n\\begin{eqnarray}\\label{eq:coulomb_act}\nV_a({\\bf r}) = \\int d{\\bf r}^\\prime \\frac{\\rho_a({\\bf r}^\\prime)}{|{\\bf r}-{\\bf r}^\\prime|},\n\\end{eqnarray}\nwith $\\rho_a$ being the density of active electrons.\n\nThe meaning of Eq.~(\\ref{eq:ehrenfest_fc_acc}) can be interpreted as follows: \nThe Ehrenfest theorem states that the dipole acceleration is given by the expectation value of the total force on the electronic system, made up of the laser electric force acting on the active, $f_{\\rm la}$, and core electrons, $f_{\\rm lc}$, the nuclear Coulomb force on the active, $f_{\\rm na}$, and core electrons, $f_{\\rm nc}$, and the interelectronic forces from the active electrons on the core, $f_{\\rm ac}$, and vice versa, $f_{\\rm ca}$.\nThen, we obtain the total force,\n\\begin{equation}\\label{eq:force_ehrenfest}\nf = (f_{\\rm na} + f_{\\rm nc}) + (f_{\\rm la} + f_{\\rm lc}) + (f_{\\rm ac} + f_{\\rm ca}) = (f_{\\rm na} + f_{\\rm nc}) + (f_{\\rm la} + f_{\\rm lc}),\t\n\\end{equation}\nwhere we have used the action-reaction law $f_{\\rm ac}=-f_{\\rm ca}$ in the second equality.\nWe can see correspondence of this expression to Eq.~(\\ref{eq:ehrenfest_acc}).\nHowever, if the core orbitals are frozen, we have to take account of an additional ``binding force'' $f_{\\rm b}$ to fix the frozen core, which is inherent in the variational procedure to derive the EOMs.\nSince the binding force $f_{\\rm b}$ cancels the forces acting on frozen-core electrons from the nuclei, laser field, and active electrons, it is given by,\n\\begin{eqnarray} \\label{eq:force_ehrenfest3}\nf_{\\rm b} = - f_{\\rm nc} - f_{\\rm lc} - f_{\\rm ac}.\n\\end{eqnarray}\nConsequently, the effective force in the presence of frozen core becomes, \n\\begin{eqnarray} \\label{eq:force_fc}\nf_{\\rm eff} = f + f_{\\rm b} = (f_{\\rm na} + f_{\\rm ca}) + f_{\\rm la}.\n\\end{eqnarray}\nThe comparison between Eqs.~(\\ref{eq:gbf_fc_approx_p}) and (\\ref{eq:force_ehrenfest3}) tells us that the additional term $\\Delta(\\dot{p}_z)$ in the former represents the binding force $f_{\\rm b}$, and Eq.~(\\ref{eq:ehrenfest_fc_acc}) is a quantum-mechanical expression of the effective force [Eq.~(\\ref{eq:force_fc})].\n\nFigure \\ref{fig:BeHHG} compares the HHG spectra\\index{High-harmonic generation (HHG)} from Be, calculated as the modulus squared of the Fourier transform of the dipole acceleration, extracted from the simulations with a dynamical and frozen core orbital.\nIf we calculate the frozen-core HHG spectrum using the modified formula Eq.~(\\ref{eq:ehrenfest_fc_acc}), it overlaps with the DC result almost perfectly.\nThis indicates that the use of FC is a good approximation for the circumstances considered here.\nHowever, the use of Eq.~(\\ref{eq:ehrenfest_acc}) with FC leads to an erroneous spectrum.\nThus, it is essential to use Eq.~(\\ref{eq:ehrenfest_fc_acc}) for the calculation of dipole acceleration and, then, HHG spectra from the simulation results with frozen core.\n\n\\begin{figure}[tb]\n\t\\begin{center}\n\t\t\\includegraphics[width=0.75\\textwidth]{figs\/BeHHG.pdf}\n\t\\end{center}\n\t\\caption{HHG spectra of Be exposed to a laser pulse with a wavelength of 800 nm, an intensity of 3$\\times$10$^{14}$ W\/cm$^2$, and a foot-to-foot pulse width of three cycles. Comparison between the simulations with DC $(n_{fc}, n_{dc},n_{a})=(0,1,5)$ and FC $(1,0,5)$. For the case of FC, we also compare the results extracted via Eqs.~(\\ref{eq:ehrenfest_acc}) and (\\ref{eq:ehrenfest_fc_acc}).}\n\t\\label{fig:BeHHG}\n\\end{figure}\n\n\\begin{figure}[tb]\n\t\\begin{center}\n\t\t\\includegraphics[width=0.75\\textwidth]{figs\/ArHHG.pdf}\n\t\\end{center}\n\t\\caption{HHG spectra of Ar exposed to a laser pulse with a wavelength of 800 nm, an intensity of 8$\\times$10$^{14}$ W\/cm$^2$, and a foot-to-foot pulse width of three cycles. Comparison between the TD-CASSCF $(n_{fc}, n_{dc},n_{a})=(5,0,13)$ and TDHF $(5,4,0)$.}\n\t\\label{fig:ArHHG}\n\\end{figure}\n\n\\begin{figure}[tb]\n\t\\begin{center}\n\t\t\\includegraphics[width=0.75\\textwidth]{figs\/KrHHG.pdf}\n\t\\end{center}\n\t\\caption{HHG spectra of Kr exposed to a laser pulse with a wavelength of 800 nm, an intensity of 8$\\times$10$^{14}$ W\/cm$^2$, and a foot-to-foot pulse width of three cycles. Comparison between the TD-CASSCF $(n_{fc}, n_{dc},n_{a})=(14,0,13)$ and TDHF $(14,4,0)$.}\n\t\\label{fig:KrHHG}\n\\end{figure}\n\nWe show in Fig.~\\ref{fig:ArHHG} the HHG spectra of Ar calculated with the TD-CASSCF and TDHF methods.\nThese results well reproduce a dip around 53 eV ($\\sim$34th order) that originates from the Cooper minimum and has been experimentally observed \\cite{Worner2009PRL}. \nWhereas the difference between the TD-CASSCF and TDHF is not large in this case, it is more prominent in the HHG spectrum of Kr shown in Fig.~\\ref{fig:KrHHG}; the TDHF overestimates the harmonic intensity near the cutoff more than one order of magnitude.\nSuch a quantitative difference is critical when we want to estimate the upper limit of the HHG pulse energy that can be generated with a given experimental setup.\nIt is wonderful that we can now achieve a converged simulation of high-harmonic generation from the thirty-six electron atom.\n\n\\subsection{Dipole Acceleration within the Single-Active-Electron Approximation}\n\\label{sec:DA within SAE}\n\nThe above discussion has important implications also for how to extract harmonic spectra from TDSE simulations of multielectron atoms and molecules within the single-active-electron approximation.\nAs stated above, Eq.~(\\ref{eq:ehrenfest_acc}) with $\\hat{V}_0$ replaced by the effective potential $V_{eff}$, corresponding to Eq.~(\\ref{eq:force_fc}), is usually used:\n\\begin{eqnarray}\\label{eq:ehrenfest_SAE_eff}\n\\langle a\\rangle(t) = -\n\\langle\\psi|\\left(\n\\frac{\\partial V_{eff}}{\\partial z}+\n\\frac{\\partial \\hat{V}_{\\rm ext}}{\\partial z}\n\\right)|\\psi\\rangle.\n\\end{eqnarray}\nOn the other hand, Gordon {\\it et al.} \\cite{Gordon2006PRL} have argued that one should rather use Eq.~(\\ref{eq:ehrenfest_acc}) as is, i.e., with the bare nuclear potential $\\hat{V}_0$ ($=-\\frac{Z}{r}$ for the atomic case):\n\\begin{eqnarray}\\label{eq:ehrenfest_SAE_bare}\n\\langle a\\rangle(t) = -\n\\langle\\psi|\\left(\n\\frac{\\partial \\hat{V}_0}{\\partial z}+\n\\frac{\\partial \\hat{V}_{\\rm ext}}{\\partial z}\n\\right)|\\psi\\rangle.\n\\end{eqnarray}\nThey have taken the action-reaction law into account but ignored the binding force. \nHowever, the observation that Eq.~(\\ref{eq:ehrenfest_fc_acc}) rather than Eq.~(\\ref{eq:ehrenfest_acc}) has to be used in the presence of frozen-core orbitals, also numerically confirmed in Fig.~\\ref{fig:BeHHG}, strongly suggests that, at the conceptual level, Eq.~(\\ref{eq:ehrenfest_SAE_eff}) is the correct choice.\n\n\\subsection{Photoelectron energy spectrum}\n\\label{subsec:PES}\n\nTime-resolved and angle-resolved photoelectron (photoemission) spectroscopy is becoming more and more important as a tool to probe ultrafast electron dynamics.\nIn principle, (angle-resolved) photoelectron energy spectrum can be calculated through projection of the departing wave packet onto plane waves or Coulomb waves (the difference in the results is usually negligibly small).\nTo apply this approach, however, we need to keep the wave function within the simulation box without being absorbed, which would lead to a huge computational cost.\nAs a new method that can be used with irECS, requiring a much smaller simulation box, the time-dependent surface flux (t-SURFF) method has recently been proposed \\cite{Tao2012NJP}.\nIn this method, spectra are computed from the electron flux through a surface, beyond which the outgoing electron wave packet is absorbed by irECS. Instead of analyzing spectra at the end of the simulation, one can record the surface flux in the course of time evolution. We have recently succeeded in applying t-SURFF, originally formulated for SAE-TDSE simulations, to the TD-CASSCF simulations, whose details will be presented in a separate publication.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=0.75\\textwidth]{figs\/photoionization-cross-section-Be.pdf} \t\n\\end{center} \n\\caption{Relative photoionization cross section of Be as a function of photon energy extracted 37, 75, and 112 fs after the pulse from TD-CASSCF simulations with $(n_{fc},n_{dc},n_{a}) = (1,0,4)$ for a ultrabroadband three-cycle (foot-to-foot) pulse with 22 eV central photon energy.}\n\\label{fig:photoionization-cross-section-Be}\n\\end{figure}\n\nFigure \\ref{fig:photoionization-cross-section-Be} presents the calculated photoionization cross section of Be.\nMaking use of a broadband nature of an ultrashort pulse, one can draw such a plot with a single run, by dividing the photoelectron spectrum by photon energy spectrum.\nThe results, in reasonable agreement with reported measurements \\cite{Wehlitz2003PRA}, well reproduce oscillating features due to the contribution from autoionizing states. \nWe plot three curves extracted at different delays (37, 75, 112 fs) after the pulse.\nWe see that peaks grow around 13 eV with increasing delay, reflecting the evolution of autoionization.\nThus, the TD-CASSCF method can properly describe the process induced by electron correlation.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=0.75\\textwidth]{figs\/bichromatic-Ne.pdf} \t\n\\end{center} \n\\caption{Angle-resolved photoelectron energy spectrum from Ne irradiated by a 10 fs bichromatic XUV pulse Eq.~(\\ref{eq:bichromatic}) with $\\omega=19.1\\,{\\rm eV}$ and (a) $\\delta=0$ and (b) $\\frac{3}{2}\\pi$, calculated with the TDHF simulation. The $\\omega$ and $2\\omega$ intensities are $10^{13}\\,{\\rm W\/cm}^2$ and $1.5\\times 10^{9}\\,{\\rm W\/cm}^2$, respectively.}\n\\label{fig:bichromatic-Ne}\n\\end{figure}\n\nAs a demonstration of capability to evaluate photoelectron angular distribution, let us consider a bichromatic XUV pulse whose electric field is of the form,\n\\begin{equation}\n\tE(t) = F_\\omega (t) \\cos\\omega t + F_{2\\omega} (t) \\cos (2\\omega t - \\delta),\n\t\\label{eq:bichromatic}\n\\end{equation}\nwhere $F_\\omega (t)$ and $F_{2\\omega} (t)$ denote the envelopes of the $\\omega$ and $2\\omega$ pulses, respectively, and $\\delta$ the relative phase.\nBecause of the interference between two-photon ionization by $\\omega$ and single-photon ionization by $2\\omega$, the photoelectron angular distribution is expected to vary with $\\delta$.\nThis is confirmed by TDHF simulations as shown in Fig.~\\ref{fig:bichromatic-Ne}.\nWhereas roughly the same number of photoelectrons are emitted to the upper ($\\sim 0^\\circ$) and lower ($\\sim 180^\\circ$) hemispheres at $\\delta=0$ (54\\% to the lower hemisphere), approximately two-thirds (63 \\%) of the electrons are emitted to the lower hemisphere at $\\delta = \\frac{3}{2}\\pi$.\nHence, such simulations will be useful to design and analyze, e.g., coherent control experiments that can be realized by use of high-harmonic and free-electron-laser XUV sources with temporal coherence \\cite{Prince2016NPhoton,Iablonskyi2017PRL}.\n\n\\section{Summary}\n\\label{sec:Summary}\n\nWe have compiled our recent development of the time-dependent complete-active-space self-consistent-field method to simulate multielectron dynamics in ultrafast intense laser fields along with numerical examples for atoms.\nIntroducing the concept of frozen core, dynamical core, and active orbital subspace decomposition, TD-CASSCF allows compact and, at the same time, accurate representation of correlated multielectron dynamics in strongly driven atoms and molecules.\nIt also has desirable features of gauge invariance and size extensivity.\nWe can now handle strong-field phenomena in systems containing tens of electrons from the first principles, which was merely a dream several years ago.\n\n\nWhile the present work has focused on the TD-CASSCF method, especially, for atoms, we have developed and been actively developing a variety of different {\\it ab initio} methods.\nWe have numerically implemented the MCTDHF method for molecules, based on a multiresolution Cartesian grid, without need to assume any symmetry of molecular structure \\cite{Sawada2016PRA}.\nWe have developed the TD-ORMAS method \\cite{Sato2015PRA}, which is more approximate and thus computationally even less demanding than TD-CASSCF, and allows one to handle general MCSCF wave functions with arbitrary CI spaces.\nWe have more recently formulated the time-dependent optimized coupled-cluster method \\cite{Sato2018JCP}, based not on multiconfiguration expansion but on coupled-cluster expansion.\nThis method is gauge invariant, size extensive, and polynomial cost-scaling.\nFurthermore, as an alternative that can in principle take account of correlation effects and extract any one- and two-particle observable while bypassing explicit use of the wave function, we have reported a numerical implementation of the time-dependent two-particle reduced density matrix method \\cite{Lackner2015PRA,Lackner2017PRA}.\nWhereas the above methods concentrate on the electron dynamics, we have also considered electron-nuclear dynamics and formulated a fully general TD-MCSCF method to describe the dynamics of a system consisting of arbitrary different kinds and numbers of interacting fermions and bosons \\cite{Anzaki2017PCCP}.\nAll these developments will open various, flexible new possibilities of highly accurate {\\it ab initio} investigations of correlated multielectron and multinucleus quantum dynamics in ever-unreachable large systems.\n\n\n\\begin{acknowledgement}\nThis research was supported in part by a Grant-in-Aid for Scientific Research (Grants No. 23750007, No. 23656043, No. 23104708, No. 25286064, No. 26390076, No. 26600111, No. 16H03881, and 17K05070) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan and also by the Photon Frontier Network Program of MEXT. \nThis research was also partially supported by the Center of Innovation Program from the Japan Science and Technology Agency, JST, and by CREST (Grant No. JPMJCR15N1), JST. \nY.~O.~gratefully acknowledges support from the Graduate School of Engineering, The University of Tokyo, Doctoral Student Special Incentives Program (SEUT Fellowship). \nO.~T.~gratefully acknowledges support from the Japanese Government (MEXT) Scholarship. \nWe thank I. B\\v{r}ezinov\\'a, F. Lackner, S. Nagele, J. Burgd\\\"orfer, and A. Scrinzi for fruitful collaborations that have greatly contributed to this work.\n\\end{acknowledgement}\n\n\\input{referenc}\n\n\\printindex\n\n\\end{document}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\nAd hoc wireless networks have found use in a plethora of applications ranging from environmental monitoring to vehicle-to-vehicle communications primarily due to their ability to increase coverage through multihop transmissions, and to autonomously organise and initiate communications in a decentralised manner.\nThese networks possess commonality insomuch as the number and spatial distribution of nodes in the network are often random which is also why they are usually modelled as random geometric graphs \\cite{penrose2003random}, granting access to theoretical analysis of network performance and ultimately engineering insight.\nIn practice, this understanding can lead to improved protocols and network deployment strategies \\cite{younis2008strategies}, for example, understanding how transmission range affects the underlying network topology can reduce the cost of distributed algorithms, save energy, and lower interference between nodes while maintaining high levels of network connectivity \\cite{santi2005topology} usually measured as the probability that an ad hoc network is \\emph{fully }connected, i.e. there exists at least one reliable multihop path connecting every two nodes in the network.\n\nFor dense networks, several analytical results on connectivity have been published particularly in the form of insightful scaling laws. For example, in \\cite{Gupta1998}, the authors derive a power scaling law that ensures full connectivity is achieved \\emph{almost surely} as the number of nodes in the network tends to infinity. \nIn \\cite{Xue2004}, the number of nearest neighbours required to achieve full connectivity asymptotically is studied. \n\n\nWhile early works considered unbounded (infinite) networks, more recent research has attempted to better quantify the role that boundaries play in finite networks. \nSimple confining geometries (e.g., cubes, spheres, rectangles) were studied in~\\cite{Jia2006,Mao2012,Khalid2014,estrada2015random}.  \nA more versatile framework based on a cluster expansion approach was recently detailed by the authors in~\\cite{Coon2012a} capable of treating more complicated geometrical network domains (convex polytopes) and encompasses several aspects of subsequently reported theories (cf.~\\cite{Khalid2014}). The framework has also been shown to yield more accurate results than conventionally accepted approximations in some cases~\\cite{Khalid2014} and may even accommodate directional antenna gains \\cite{georgiou2013connectivity}.\n\nIn this letter we build on the framework presented in \\cite{Coon2012a,georgiou2013connectivity} and derive for the first time connectivity scaling laws (see equations \\eqref{eq:MPT}, \\eqref{Mdc} and \\eqref{eq:Mb}) with regards to transmit power, and the number of antennas employed by transceiver pairs as well as the adopted transmission scheme: a) orthogonal space-time block coding (STBC) and b) beamforming (MIMO with maximum ratio combining - MIMO-MRC).\nWe show that the local network connectivity scales like $\\mathcal{O}(z^\\mathcal C)$ in terms of system parameters $z$, where $\\mathcal C$ is the ratio of the dimension of the network domain $d$ to the path loss exponent $\\eta$.\nWe conclude by discussing the merits of these transmission schemes with respect to antenna diversity gain, how the derived scaling laws can be used to mitigate boundary effects and how our work can be adapted to account for interference limited networks \\cite{location,Haenggi2009}.\n\n\n\n\\section{Network Definitions and System Model}\n\\label{sec:net_conn}\n\n\n\\textit{1) Node Deployment:}\nConsider a network of $N$ uniformly distributed nodes in a $d$-dimensional convex domain $\\mathcal{V}\\subseteq\\mathbb{R}^{d}$ with location coordinates\n$\\mathbf{r}_{i}\\in\\mathcal{V}$ for $i=1,2,\\ldots,N$. \nThis is equivalent to a Binomial Point Process (BPP) with node density $\\rho= N\/V$, where $V= |\\mathcal{V}|$, and is a common configuration for modelling ad hoc networks in confined geometries. \n\n\\textit{2) Path-loss and Fading:}\nThe signal-to-noise ratio (SNR) is a commonly used metric to quantify the quality and reliability of a communication link. \nDue to path loss the signal power received by a destination node is inversely proportional to the separation distance $r_{ij}=|\\mathbf{r}_i - \\mathbf{r}_j|$ between two nodes located at $\\mathbf{r}_i$ and $\\mathbf{r}_j$, and can be modelled by \n\\es{\ng(r_{ij})= (\\epsilon +r_{ij}^\\eta)^{-1}, \\qquad \\eta\\geq 2, \\quad \\epsilon > 0\n}\nwhere $\\eta$ is the path loss exponent and $\\epsilon \\ll 1$ ensures that the attenuation function $g(r_{ij})$ is non-singular at zero distance.\n\nDue to small-scale fading, it is a standard assumption that the coefficient $h_{kl}\\in \\mathbb{C}$ modelling the transfer characteristics of the MIMO channel between the $k$th transmit antenna $(1\\leq k \\leq m)$ and the $l$th receive antenna ($(1\\leq l \\leq n)$) of nodes $i$ and $j$ respectively is a circularly symmetric complex Gaussian random variable with zero mean and unit variance. \nConsequently, for $m=n=1$ the channel gain $X_{ij}$ between nodes $i$ and $j$ is an exponentially distributed random variable $X_{ij} =|h_{1,1}|^2\\sim \\exp(1)$ corresponding to the usual SISO Rayleigh fading model adopted in most wireless ad hoc network literature due to its mathematical tractability.\n\n\\textit{3) Pairwise Connectivity:}\nAssuming  negligible  inter-node  interference (e.g.  perfect CDMA\/TDMA) and lossless \nantennas, we define the pairwise connectivity through the relation \n\\es{\nH_{ij} = P(\\textrm{SNR}_{ij} \\geq \\wp)\n\\label{H1}}\ni.e. the complement of the outage probability, where  the average received signal-to-noise ratio is given by $\\textrm{SNR}_{ij}=  g(r_{ij})X_{ij} \/ \\beta$, and the parameter $\\beta \\propto P_T^{-1}$ depends on transmit power $P_T$, the center frequency of the transmission and the power of the long-time average background noise at the receiver ($\\beta$ defines the length scale).\nWe therefore have that $H_{ij}(r_{ij}) = 1- F_{X_{ij}}(\\wp \\beta (\\epsilon + r_{ij}^\\eta))$,\nwhere $F_{X_{ij}}$ is the CDF of the channel gain $X_{ij}$.\nConsequently, for $m=n=1$ we obtain\n$\nH_{ij}(r_{ij}) = \\exp(-\\wp \\beta (\\epsilon + r_{ij}^\\eta)) \n$.\n\n\n\\section{Connectivity Mass and Scaling Laws}\n\nScaling laws describe how network connectivity properties (both local and global) scale with various network parameters.\n\n\\textit{1) Local Connectivity:}\nA good measure of the local connectivity of a transmitter node located at $\\mathbf{r}_i$ is given by the spatial average of $H_{ij}$ over all possible receiver positions \n\\es{\nM_i(\\mathbf{r}_i) = \\int_\\mathcal{V} H_{ij}(r_{ij}) \\textrm{d} \\mathbf{r}_j \n\\label{MM}}\ndefining the likelihood that a node located at $\\mathbf{r}_i$ will connect (i.e. has SNR greater than $\\wp$) to some other arbitrary node in the network domain $\\mathcal{V}$.\nNote that $(N-1) M_i(\\mathbf{r}_i)\/V$ is the expected degree of node $i$.\nIn fact $M_i$ is related to a number of local and global network observables, and for that reason we call $M_i$ the \\textit{connectivity mass} \\cite{georgiou2013connectivity}.\n\n\\textit{2) Global Connectivity:}\nA global measure of network connectivity is the probability that every node can communicate with every other node in a multihop fashion. \nThis is captured by the notion of full connectivity $P_{fc}$ which at high node densities $\\rho$ can be expressed as (see \\cite{Coon2012a,georgiou2013connectivity} for more details)\n\\es{\nP_{fc} \\approx 1- \\rho \\int_\\mathcal{V} e^{- \\rho M_i(\\mathbf{r}_i) } \\textrm{d} \\mathbf{r}_i\n\\label{Pfc}}\nessentially saying that full connectivity is the complement of the probability of an isolated node.\nThe connection between $P_{fc}$ and $M_i$ is clear from \\eqref{Pfc}: as $M_i$ increases, the probability of an isolated node decreases and hence $P_{fc}$ monotonically increases to $1$.\nMoreover, in the dense regime of $\\rho\\to \\infty$ we have that the integral in \\eqref{Pfc} will be dominated by integration regions where $M_i$ is small, i.e. near the boundaries of $\\mathcal{V}$.\n\n\n\\textit{3) Boundary effects:}\nTaylor expanding $M_i$ at $\\mathbf r_i$ situated on the boundary of $\\mathcal{V}$ yields the leading-order expression\\footnote{Assuming that $\\mathbb E[X_{ij}^{\\mathcal C}] <\\infty$, with $\\mathbb E[\\cdot]$ denoting the expectation operator, the integral in~\\eqref{M} is bounded.\nMoreover, the upper limit of integration in \\eqref{M} is justified if $H_{ij}$ decays exponentially with increasing distance $x_{ij}$ and $\\mathcal{V}$ is larger than the typical transmission range. \n}\n\\es{\nM_i &\\approx \\omega \\int_0^\\infty \\!\\! r^{d-1} \\pr{1- F_{X_{ij}}(\\wp \\beta (\\epsilon + r^\\eta))} \\textrm{d} r \\\\\n&=   \\frac{\\omega}{\\eta (\\wp \\beta)^\\mathcal{C}} \\int_{\\wp \\beta \\epsilon}^\\infty \\!\\! (x-\\wp \\beta \\epsilon)^{\\mathcal{C}-1} \\pr{1- F_{X_{ij}}(x)} \\textrm{d} x\n\\label{M}\n}\nwhere we define $\\mathcal C = d\/\\eta$ to be the \\emph{connectivity exponent} and $\\omega = \\int \\mathrm d \\Omega$ is the solid angle as seen from $\\mathbf r_i$, with $\\Omega = 2\\pi^{d\/2}\/\\Gamma(d\/2)$ being the full solid angle in $d$ dimensions and $\\Gamma(\\cdot)$ is the standard gamma function. \nEquation \\eqref{M} implies that the connectivity mass $M_i$ of node $i$ is proportional to the available angular area $\\omega$ for other nodes to connect to.\nFor example if $\\mathbf{r}_i$ is located at the corner of a 2D regular $n$-gon, then $\\omega=\\pi(1-2\/n)$.\nHence, boundary effects have a direct (negative) impact on both local and global network properties through $\\omega$ motivating the study of how one can mitigate such effects by means of network design parameters.\n\n\n\\textit{4) Power law scaling:}\nTaking a leading order approximation of \\eqref{M} for small $\\epsilon$ we have that\n\\es{\nM_i &\\approx \\frac{\\omega}{ \\eta (\\wp \\beta)^\\mathcal{C}}  \\int_0^\\infty \\!\\! x^{\\mathcal{C}-1} \\pr{1- F_{X_{ij}}(x)} \\textrm{d} x  + \\ldots\n\\label{M1}\n}\nwhich suggests that we can write the following scaling law\n\\begin{equation}\\label{eq:MPT}\n  M_i \\approx K_1 P_T^{\\,\\mathcal C}\n\\end{equation}\nwith $K_1$ independent of the transmit power $P_T$.\nCorrection terms to \\eqref{M1} can be shown to be $\\mathcal{O}(\\epsilon^{\\min(\\mathcal{C},1)})$ but are omitted for the sake of brevity. \nNote that for $m=n=1$ it follows that $M_i \\approx \\frac{\\omega}{ \\eta (\\wp \\beta)^\\mathcal{C}}\\Gamma(\\mathcal{C})$.\nSignificantly, since $M_i$ is proportional to the mean degree of node $i$ it follows that doubling the transmit power will double the expected number of receivers whose SNR level is higher than $\\wp$, if $\\mathcal{C}=1$.\nIt is particularly interesting to note the conditions under which power scaling provides a progressive improvement to local connectivity, versus those conditions under which diminishing returns are experienced with an increase in $P_T$. \nFor example, a high-dimensional network ($d = 3$) operating in low path loss conditions $\\eta<3$ will benefit from the former behaviour as $P_T$ is increased; however, a low-dimensional network ($d=2$) located in a cluttered environment where high path loss conditions prevail will experience the latter. \n\nThe power law given by \\eqref{eq:MPT} provides useful insights into the behaviour of random geometric networks, which can be used to enhance network designs in practice. \nFor example, in the case of transmit power scaling \\eqref{eq:MPT}, we can use this analysis to deduce that, for some nominal transmit power $P_{T_{0}}$ that defines the target connectivity probability of a homogeneous network, we must choose $P_T$ for the bounded network to satisfy $P_T = (\\Omega\/\\omega)^{1\/\\mathcal C}P_{T_0}$.\nWe conclude this discussion by pointing out that the power law~\\eqref{eq:MPT} arises from the fact that $M_i \\propto \\beta^{-\\mathcal C}$.  Thus, one can infer that scaling laws in other key system parameters are affected by the connectivity exponent in the same way i.e. $\\mathcal{O}(z^\\mathcal{C})$ (e.g. frequency or antenna gain). \n\n\n\n\n\\section{\\clred{Multi-Antenna Systems}}\n\\label{sec:diversity}\n\nConsider the case where each node in the network possesses $m$ transmit antennas and $n$ receive antennas, and one of two signalling mechanisms is employed: diversity coding following the conventional STBC scheme derived from generalized complex orthogonal designs (GCODs)~\\cite{Tarokh1999}, and transmitter\/receiver beamforming, also known as MIMO-MRC~\\cite{Kang2003}. \n\n\\textit{1) Diversity Coding:}\nIt is well known that the performance of a point-to-point STBC system is governed by the Frobenius norm of the associated $n \\times m$ channel matrix $\\mathbf H$ with $h_{kl}$ as its entries, that is the channel gain $X_{ij} = |\\mathbf H\\|_F^2 = \\sum_{k,l}{|h_{kl}|^2}$. \nConsequently, $X_{ij}$ is chi-squared distributed with $2mn$ degrees of freedom, and its cumulative distribution function is given by $F_{X_{ij}}(x) = \\gamma(mn,x)\/\\Gamma(mn)$, where $\\gamma(\\cdot,\\cdot)$ is the lower incomplete gamma function.\nMoreover, for $m \\geq 1$, the post-processing received SNR can be expressed as $\\text{SNR}_{ij}= \\frac{\\zeta_m}{m} g(r_{ij}) X_{ij}\/\\beta$, where $\\zeta_m = 1$ if $m=1,2$ and $\\zeta_m = 2$ otherwise.  The factor of $\\zeta_m\/m$ arises from power normalization and the fact that the rate of a code derived from GCOD is $\\frac{1}{2}$ for $m > 2$~\\cite{Tarokh1999}. \nThe pairwise connectivity is thus\n\\es{\nH_{ij}= P\\pr{ X_{ij} \\geq \\frac{m\\wp\\beta}{\\zeta_m g(r_{ij})}}= 1-\\frac{\\gamma\\pr{m n , \\frac{m \\wp \\beta }{\\zeta_m g(r_{ij})} }}{\\Gamma(mn)}\n\\label{Hdc}}\nfrom which we evaluate the connectivity mass~\\eqref{M1} for $\\epsilon\\ll 1$\n\\es{\n  M_{\\text{dc}} \\approx \\frac \\omega d \\left(\\frac{\\zeta_m}{m\\wp\\beta}\\right)^{\\mathcal C}\\frac{\\Gamma(mn+\\mathcal C)}{\\Gamma(mn)}\n\\label{Mdc}}\nwhere we have used a ``dc'' subscript to indicate the relation to diversity coding.  For large $m$ and\/or $n$, we can use the Stirling formula $\\Gamma(x)\\sim \\sqrt{2\\pi}\\,x^{x+\\frac 1 2} e^{-x}$ to obtain the scaling~law\n\\begin{equation}\\label{eq:Mdc}\n  M_{\\text{dc}} \\approx \\frac{\\omega}{d}\\left(\\frac{\\zeta_m n}{\\wp\\beta}\\right)^{\\mathcal C}\\Big(1 + O\\Big(\\tfrac{1}{mn}\\Big)\\Big),\\quad m,n\\rightarrow\\infty.\n\\end{equation}\nNote that the connectivity exponent $\\mathcal C$ arises naturally in a manner similar to the power scaling law in \\eqref{eq:MPT}. \n\n\n\\textit{2) Beamforming:}\nFor MIMO-MRC transmissions, the received SNR (after MRC) is $\\text{SNR}_{ij}= \\lambda_{\\max}(\\mathbf H^\\dagger \\mathbf H) g(r_{ij})\/\\beta$, (i.e. $X_{ij}=\\lambda_{\\max}(\\mathbf H^\\dagger \\mathbf H)$) with $\\lambda_{\\max}(\\cdot)$ denoting the maximum eigenvalue of the argument~\\cite{Kang2003}. \nA closed form expression for $H_{ij}$ like in \\eqref{Hdc} is not possible here.\nTo make progress we look at the behaviour of $\\lambda_{\\max}$ in the limit of large $m,n$ and apply the following result due to Edelman~\\cite[Lemma 4.3]{Edelman1988}\\footnote{Edelman's result assumed the complex Gaussian entries of $\\mathbf H$ had unit variance \\emph{per dimension}, whereas the result we use assumes each entry has unit variance \\emph{in total}, and thus this lemma is slightly different from ~\\cite{Edelman1988}.}:\n\\begin{lemma}\\label{lem:edelman}\n  Let $x_n \\xrightarrow{p} x$ signify that for all $\\delta > 0$, $\\lim_{n\\rightarrow\\infty}{P(|x-x_n|> \\delta)}=0$.  Now, suppose the $n \\times m$ matrix $\\mathbf H$ has independent circularly symmetric complex Gaussian entries, each with zero mean and unit variance. Then $\\mathbf W = \\mathbf {H^\\dagger H}$ has a complex Wishart distribution and\n\\begin{equation}\n  (1\/n)\\lambda_{\\max}(\\mathbf W) \\xrightarrow{p} (1+\\sqrt y)^2,\\qquad \\frac m n \\rightarrow y,~0\\leq y < \\infty.\n\\end{equation}\n\\end{lemma}\nWe therefore draw inspiration from this lemma and write \n\\begin{equation}\n  H_{ij}(r_{ij}) \\approx \\left\\{\n    \\begin{array}{ll}\n  \t1, & r_{ij} < \\Big(\\frac{(1+\\sqrt y)^2n}{\\wp\\beta}\\Big)^{\\frac{1}{\\eta}} \\\\ \n  \t0, & \\text{otherwise} \\\\\n    \\end{array}\n   \\right.\n\\end{equation}\nfor large $n$.  \nLetting $\\mu(n) = (1+\\sqrt y)^2n$, we write\n\\begin{equation}\n  \\frac{M_{\\text{b}}}{ \\omega} \\approx \\int_{0}^{ (  \\frac{\\mu}{\\wp\\beta} )  ^{\\frac{1}{\\eta}}}\\!\\!r^{d-1}   \\,\\mathrm{d}r + \\varepsilon(n) = \\frac 1 d \\left(\\frac{(1+\\sqrt y)^2n}{\\wp\\beta}\\right)^{\\mathcal C}+ \\varepsilon(n)\n\\end{equation}\nwhere a ``b'' subscript is used to indicate the relation to beamforming.  The error term is given~by\n\\est{\n\\varepsilon (  n )  = \\int_{ (\\frac{\\mu}{\\wp\\beta} )  ^{\\frac{1}{\\eta}}}^{\\infty}r^{d-1}\n  H (r )  \\,\\mathrm{d}r\n  - \\int_{0}^{ (  \\frac{\\mu}{\\wp\\beta} )  ^{\\frac{1}{\\eta}}}r^{d-1} \n  (1-  H (  r )) \\,\\mathrm{d}r\n}\nwhich grows like $\\mathcal{O}(n^{\\mathcal C - \\frac 2 3})$. We omit the proof of this due to space restrictions. Thus, we arrive at the following scaling law\n\\es{\\label{eq:Mb}\n  M_{\\text{b}} \\approx \\frac{\\omega}{d} \\left(\\frac{(1+\\sqrt{y})^2 n}{\\wp\\beta}\\right)^{\\mathcal C}\\left(1 + \\mathcal{O}\\!\\left(n^{-\\frac 2 3}\\right)\\right)\n}\nvalid for the limit of $m,n\\rightarrow\\infty$ where $m\/n \\rightarrow y$.\n\n\\section{Comparison of the Two Multi-Antenna Schemes}\n\n\\textit{1) The case of $m=2$:}\nSuppose the number of transmit antennas per node is fixed at $m=2$, in which case $\\zeta_m = 1$ and $y = 0$, and thus the leading order of $M_{\\text{dc}}$ is the same as that of $M_{\\text{b}}$ (c.f. \\eqref{eq:Mdc} and \\eqref{eq:Mb}). However, we see that the first-order corrections for the two observables differ. This is illustrated in the left panel of Fig.~\\ref{fig1}, where exact results for the connectivity masses of the two systems are presented along with leading-order terms as a function of $n$.  In the figure, the solid lines show the leading-order behaviour, while the marker data was obtained from the direct calculation of~\\eqref{Mdc} for diversity coding and numerically calculating~\\eqref{M1} for beamforming. \n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=8.85cm]{fig12.jpg}\n\\caption{Connectivity mass vs.~$n$ for $d = 3$ and $m = 2$ (left panel) and $m\\approx y_c n$ (right panel), and various values of $\\eta$, corresponding to connectivity exponents $\\mathcal C= \\frac{d}{\\eta} = \\frac 3 2, 1, \\frac 3 4, \\frac 3 5$. The solid lines correspond to the leading-order term given in~\\eqref{eq:Mb} (equivalently~\\eqref{eq:Mdc}), whereas the data represented by markers was obtained from~\\eqref{Mdc} for the diversity coding scenario and by numerically calculating~\\eqref{M1} for the beamforming case. \n}\n\\label{fig1}\n\\end{center}\n\\end{figure}\n\n\nThree observations can be made from this example.  The first is that the difference in first-order corrections is apparent, and beamforming actually provides a connectivity benefit over diversity coding for finite numbers of receive antennas i.e. $M_{\\text{dc}}  \\lesssim M_{\\text{b}}$. Yet, convergence to the leading order can be seen for both schemes as $n\\to\\infty$. \nThe second observation is that the leading-order expression well approximates the exact connectivity mass, even for small numbers of antennas, particularly for $M_{dc}$. \nThe third observation is that the derivative of the connectivity mass satisfies $M'(n) \\propto n^{\\mathcal C -1}$ implying that progressive improvements are obtained for $\\mathcal C > 1$ and diminishing returns for $\\mathcal C < 1$.\n\n\n\\textit{2) The case of $m>2$:}\nFor any other fixed $m$ greater than two, $M_{\\text{dc}} > M_{\\text{b}}$ to leading order by a factor of $2^{\\mathcal C}$.  However, STBC suffers from a lower rate than MIMO-MRC in this case.  Consequently, it is informative to consider a modified view of the connectivity mass based on pairwise mutual information outage.  This can be easily achieved by redefining $H_{ij}$ to be\n\\es{\n    H_{ij}(r_{ij}) = P\\left(\\log_2(1+ g(r_{ij}) X_{ij} \/ \\beta) \\geq \\zeta_m R \\right)\n\\label{H2}}\nwhere $R$ is a target rate threshold and $\\zeta_m = 1$ if STBC is employed and $m \\leq 2$ or if MIMO-MRC is considered, and $\\zeta_m =2$ otherwise.  \nRearranging this it is clear that all previous calculations and results follow accordingly, but with $\\wp$ replaced by $(2^{\\zeta_m R} - 1)$ to explicitly account for the difference in rate characterised by $\\zeta_m$.  Thus, we deduce from~\\eqref{eq:Mdc} and~\\eqref{eq:Mb} that for the rate based connectivity metric of \\eqref{H2} we have that $M_{\\text{dc}} < M_{\\text{b}}$ since $(2^{2R}-1) > 2(2^R-1)$.  \n\n\n\\textit{3) The case of $m\/n\\to y$:}\nNow, let $m$ and $n$ scale such that their ratio approaches $y > 0$, then the relation $M_{\\text{dc}} < M_{\\text{b}}$ is maintained when considering mutual information outage \\eqref{H2}. \nNeglecting the rate differences and focusing only on SNR outage (a proxy for reliability in delay-tolerant networks) as originally given in~\\eqref{H1}, we see that when $y = (\\sqrt 2 -1)^2 \\approx 0.172$ the leading orders of the two schemes are equal. Denoting this critical value as $y_c$ we arrive at the conclusion that for $y < y_c$, diversity coding is favoured over beamforming to leading order, with the opposite being true for $y > y_c$. \nAgain, we must be careful for finite $n$ since the correction terms differ. \nTo draw further conclusions, we have plotted the connectivity mass against $n$ with $y \\approx 0.172$ in the right panel of Fig.~\\ref{fig1}.  \nWe see from Fig.~\\ref{fig1} that diversity coding is preferred over beamforming for small~$n$ but not overly so. \n\n\nWe have thus shown that for the chosen metric of outage probability \\eqref{H1} (reliability being the main concern relevant to delay tolerant networks \u2013 a paradigm of the IoT) beamforming results in superior network connectivity to diversity coding for $m=2$, and asymptotically when $y>y_c$. \nFor the metric of pairwise mutual information \\eqref{H2}, beamforming results in superior network connectivity to diversity coding for $m>2$.\n\n\n\n\n\n\n\n\n\\section{Design Implications}\nThe scaling laws developed herein can be exploited to mitigate the deleterious effects that boundaries present in confined networks using multiple antennas. \nLet our reference point be given by the connectivity mass $M_i$ corresponding to a homogeneous network connected by single-input single-output pairwise links, which can be computed to be $\\Omega\\Gamma(1+\\mathcal C)\/((\\wp\\beta)^{\\mathcal C}d)$ using~\\eqref{Mdc}.  In a bounded network, we can focus on a particular feature of solid angle $\\omega$ and use, for example,~\\eqref{eq:Mb} to obtain the \\clred{antenna scaling} law that will ensure the effect that this feature has on local connectivity is mitigated.  Specifically, we see that\n$wn \\approx \\big(\\tfrac \\Omega \\omega \\Gamma(1+\\mathcal C)\\big)^{1\/\\mathcal C}$, where $w = \\zeta_m$ for diversity coding and $w = (1+\\sqrt y)^2$ for beamforming.  Noting that these calculations pertain to nodes situated near boundaries, we can infer the possibility of designing sophisticated optimisation methods in practice.\nMoreover, through the efficient utilisation of the derived scaling laws for the connectivity mass $M_i$, one can predict and indeed optimise both local (e.g. mean degree) and global (e.g. full connectivity) network performance.\n\n\n\n\\section{Conclusions}\n\nIn this letter we studied how scaling the per-node transmit power and the number of transmit\/receive antennas $m\/n$ affect both local and global network connectivity properties. \nBy defining the \\emph{connectivity exponent} $\\mathcal C$ to be the ratio of the dimension of the network domain $d$ to the path loss exponent $\\eta$, we showed that the connectivity mass $M_i$ controlling both local and global network properties scales like $\\mathcal{O}(z^\\mathcal C)$ in several parameters of interest $z$ (see for example equations \\eqref{eq:MPT}, \\eqref{Mdc}, \\eqref{eq:Mb}).\nSignificantly, we analysed two MIMO signalling mechanisms (STBC and MRC) and showed that while both schemes scale like $\\mathcal{O}( n^\\mathcal{C})$, optimality depends on the relevant pre-factor and second-order correction terms.\nWe have also shown how antenna diversity scaling laws can be used to mitigate boundary effects in finite networks.\n\n\n\n\nWe have assumed negligible inter-node interference which for large scale ad hoc networks in low path loss environments becomes increasingly difficult to achieve.\nHence, interference effects must be included in future scaling law models and analysis, something which can be facilitated through the use of stochastic geometry tools \\cite{Haenggi2009}.\nInterestingly, very little work on bounded interference limited ad hoc networks has been published to date \\cite{location}; a natural extension of the present work on MIMO transmission schemes and antenna diversity gain.\n\n\n\n\n\n\n\n\n\\section*{Acknowledgment}\nThe authors would like to thank the FP7 DIWINE project (Grant Agreement CNET-ICT-318177), and the directors of the Toshiba Telecommunications Research Laboratory.\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Intoduction}\n\nWith almost 30 plenary talks and 70 parallel session talks the task of\nsummarizing all of them all into a single talk is an impossible\nmission. I will try instead simply to highlight some aspects of the\ntalks that touched my own prejudices.\n\nThe progress in the physics of neutrino oscillations in the last few\nyears has been truly remarkable.\nOscillations are now established, implying that neutrinos have masses,\nas first suggested by theorists in the early eighties, both on general\ngrounds~\\cite{Weinberg:1980bf,schechter:1980gr} and on the basis of\nvarious versions of the seesaw mechanism~\\cite{Valle:2006vb}.\nThis is a profound discovery that marks the beginning of a new age in\nneutrino physics. \n\nA gold rush towards precision results has been initiated, whose aim is\nto probe $\\theta_{13}$, to study leptonic CP violation and determine\nthe nature of neutrinos. Hopefully this will shed light on the\nultimate origin the universe and certainly that of neutrino mass.\n\n\\section{Data}\n\n\nThanks to the accumulation of events over a wide range of energy, and\nto the measurement of the dip in the L\/E (neutrino flight length L\nover neutrino energy E) distribution of the muon neutrino\ndisappearance probability, the interpretation of the atmospheric data\nhas finally turned into an unambiguous signal of $\\nu_\\mu\n\\leftrightarrow \\nu_\\tau$ oscillations, marking the beginning of a new\nera.\n\nThe interpretation of solar data {\\sl per se} is still ambiguous, with\nviable alternative explanations involving spin flavour\nprecession~\\cite{barranco:2002te,miranda:2000bi} or non-standard\nneutrino interactions~\\cite{guzzo:2001mi}.\nResults on (or relevant to) solar neutrinos were presented here by\nBroggini, Maneira, Pulido, Raghavan, Ranucci, Serenelli and Smy.\nReactor neutrino data from KamLAND not only confirm the solar neutrino\ndeficit but also observe the spectrum distortion as expected for\noscillations.  Reactors have played an important role in establishing\nthe robustness of the neutrino flavor oscillation interpretation {\\sl\n  vis a vis} the existence of solar density\nfluctuations~\\cite{Burgess:2003su} in the solar radiative zone as\nproduced by random magnetic fields~\\cite{Burgess:2003fj}, and also\nwith respect to the effect of convective zone magnetic fields, should\nneutrinos posses nonzero neutrino transition magnetic\nmoments~\\cite{Miranda:2003yh}.  Within the oscillation picture KamLAND\nhas also identified large mixing angle oscillation as its ``unique''\nsolution, ``solving'', in a sense, the solar neutrino problem. Note\nhowever that the interpretation of solar data {\\sl vis a vis} neutrino\nnon-standard interactions~\\cite{Miranda:2004nb} is not yet so robust.\n\nA lot more is to come from the upcoming reactor experiments starting\nwith Double-Chooz, and a new series of proposed experiments such as\nDaya-bay, RENO, Kaska, Angra, nicely reviewed in the talk by Cabrera.\n\nAccelerators K2K \\& MINOS confirm the atmospheric neutrino deficit as\nwell as a distortion of the energy spectrum consistent with the\noscillation hypothesis. More is to come from MINOS and the upcoming\nexperiments CNGS\/OPERA, T2K, NOVA, as reported here by Gugliemi,\nKajita, Kato, Kopp, Rebel, Sioli, and others.\n\n\\section{Oscillation parameters}\n\nThe basic tool to interpret neutrino data is the lepton mixing matrix,\nwhose simplest unitary 3-dimensional form is given as a product of\neffectively complex $2\\times 2$ matrices~\\cite{schechter:1980gr}\n\\begin{equation}\n  \\label{eq:2227}\nK =  \\omega_{23} \\omega_{13} \\omega_{12}\n\\end{equation}\nwhere each factor is given as\n\\begin{equation}\n   \\label{eq:w13}\n\\omega_{13} = \\left(\\begin{array}{ccccc}\nc_{13} & 0 & e^{i \\Phi_{13}} s_{13} \\\\\n0 & 1 & 0 \\\\\n-e^{-i \\Phi_{13}} s_{13} & 0 & c_{13}\n\\end{array}\\right)\\,,\n \\end{equation}\n in the most convenient ordering chosen in the PDG~\\cite{Yao:2006px}\n (here $c_{ij} \\equiv \\cos\\theta_{ij}$ and $s_{ij} \\equiv\n \\sin\\theta_{ij}$).\n The two Majorana phases associated to 12 and 23 can be eliminated\n insofar as neutrino oscillations are concerned, since they only\n affect lepton-number violating processes, like $0\\nu\\beta\\beta$ . Thus one can\n take the 12 and 23 factors as real, $ \\omega_{23} \\to r_{23}$ and $\n \\omega_{12} \\to r_{12}$. There is then a unique CP phase $\\Phi_{13}$,\n analogous to the KM phase $\\delta$ of quarks, that will be studied in\n future oscillation experiments, such as T2K and NOVA.\n Currently, however, oscillations have no sensitivity to this phase\n and we will neglect it in the analysis of current data.\n\n In such approximation oscillations depend on the three mixing\n parameters $\\sin^2\\theta_{12}, \\sin^2\\theta_{23}, \\sin^2\\theta_{13}$\n and on the two mass-squared splittings $\\Dms \\equiv \\Delta m^2_{21}\n \\equiv m^2_2 - m^2_1$ and $\\Dma \\equiv \\Delta m^2_{31} \\equiv m^2_3 -\n m^2_1$ characterizing solar and atmospheric neutrinos.  The fact that\n $\\Dms \\ll \\Dma$ implies that one can set $\\Dms \\to 0$ in the analysis\n of atmospheric and accelerator data, and $\\Dma$ to infinity in the\n analysis of solar and reactor data.\n\n\\subsection{Present status}\n \\label{sec:present-status}\n\n The current three--neutrino oscillation parameters are summarized in\n Fig.~\\ref{fig:global}. Equivalent results by the Bari group are in\n excellent agreement with those reported here, both\n pre~\\cite{Fogli:2005cq} and post-MINOS~\\cite{Fogli:2006yq}.  The\n analysis employs the latest Standard Solar\n Model~\\cite{Bahcall:2005va} which we heard here in the talk by\n Serenelli, and includes all new neutrino oscillation data from SNO\n salt~\\cite{Aharmim:2005gt}, K2K~\\cite{Ahn:2006zz} and\n MINOS~\\cite{Michael:2006rx}, described in Appendix C of\n hep-ph\/0405172 (v5)~\\cite{Maltoni:2004ei}~\\footnote{ In addition to\n   good calculations of the neutrino\n   fluxes~\\cite{Bahcall:2004fg,Honda:2004yz}, cross sections and\n   response functions, we need an accurate description of neutrino\n   propagation in the Sun and the Earth, including matter\n   effects~\\cite{mikheev:1985gs,wolfenstein:1978ue}.  }.\n\n The upper panels of the figure show $\\Delta \\chi^2$ as a function of\n the three mixing parameters $\\sin^2\\theta_{12}, \\sin^2\\theta_{23}, \\,\n \\sin^2\\theta_{13}$ and two mass squared splittings $\\Delta m^2_{21},\n \\Delta m^2_{31}$, minimized with respect to the undisplayed\n parameters. The lower panels give two-dimensional projections of the\n allowed regions in five-dimensional parameter space.  In addition to\n a confirmation of oscillations with $\\Dma$, accelerator neutrinos\n provide a better determination of $\\Dma$ as one can see by comparing\n dashed and solid lines in Fig.~\\ref{fig:global}.  The recent MINOS\n data~\\cite{Michael:2006rx} lead to an improved determination and a\n slight increase in $\\Dma$.\nAs already mentioned, reactors~\\cite{araki:2004mb} have played a\ncrucial role in selecting large-mixing-angle (LMA)\noscillations~\\cite{pakvasa:2003zv} out of the previous ``zoo'' of\npossible solar neutrino oscillation\nsolutions~\\cite{Gonzalez-garcia:2000sq,fogli:2001vr}.\n\\begin{figure}[t] \\centering\n    \\includegraphics[width=\\linewidth,height=7cm]{F-fig.summary06-comp.eps}\n    \\caption{\\label{fig:global} %\n      Current 90\\%, 95\\%, 99\\%, and 3$\\sigma$ \\CL\\ neutrino\n      oscillation regions for 2 \\dof\\, as of summer 2006,\n      from~\\cite{Maltoni:2004ei}. In top panels $\\Delta \\chi^2$ is\n      minimized with respect to undisplayed parameters.}\n\\end{figure}\nTable~\\ref{tab:summary} gives the current best fit values and allowed\n3$\\sigma$ ranges of oscillation parameters.\n\\begin{table}[t] \\centering    \\catcode`?=\\active \\def?{\\hphantom{0}}\n      \\begin{tabular}{|l|c|c|}        \\hline        parameter & best\n      fit & 3$\\sigma$ range         \\\\  \\hline\\hline        $\\Delta\n      m^2_{21}\\: [10^{-5}~\\eVq]$        & 7.9?? & 7.1--8.9 \\\\\n      $\\Delta m^2_{31}\\: [10^{-3}~\\eVq]$ & 2.6?? &  2.0--3.2 \\\\\n      $\\sin^2\\theta_{12}$        & 0.30? & 0.24--0.40 \\\\\n      $\\sin^2\\theta_{23}$        & 0.50? & 0.34--0.68 \\\\\n      $\\sin^2\\theta_{13}$        & 0.00 & $\\leq$ 0.040 \\\\\n      \\hline\n\\end{tabular}    \\vspace{2mm}\n\\caption{\\label{tab:summary} Neutrino oscillation parameters as of Summer 2006, \nfrom Ref.~\\cite{Maltoni:2004ei}.}\n\\end{table}\n\nNote that CP violation disappears in a three--neutrino scheme when two\nneutrinos become degenerate or when one of the angles\nvanishes~\\cite{schechter:1980bn}.  As a result CP violation is doubly\nsuppressed, first by $\\alpha \\equiv \\Dms\/\\Dma$ and also by the small\nvalue of $\\theta_{13}$.\n\\begin{figure}[t] \\centering\n  \\includegraphics[height=4cm,width=.48\\linewidth]{F-fcn.alpha06.eps}\n\\includegraphics[height=4cm,width=.48\\linewidth]{th13-06.eps}\n\\caption{\\label{fig:alpha}%\n  $\\alpha \\equiv \\Dms \/ \\Dma$ and $\\sin^2\\theta_{13}$ bound from the\n  updated analysis given in Ref.~\\cite{Maltoni:2004ei}.}\n\\end{figure}\nThe left panel in Fig.~\\ref{fig:alpha} gives the parameter $\\alpha$,\nwhile the right panel shows the impact of different data samples on\nconstraining $\\theta_{13}$.  One sees that for larger $\\Dma$ values\nthe bound on $\\sin^2\\theta_{13}$ is dominated by CHOOZ, while for low\n$\\Dma$ the solar and KamLAND data become quite relevant.\n\n\n\\subsection{Robustness}\n\\label{sec:robustness}\n\n\nReactor neutrino data have played a crucial role in testing the\nrobustness of solar oscillations vis a vis astrophysical\nuncertainties, such as magnetic fields in the\nradiative~\\cite{Burgess:2003su,Burgess:2003fj} and convective\nzone~\\cite{miranda:2000bi,guzzo:2001mi,barranco:2002te}, leading to\nstringent limits on neutrino magnetic transition\nmoments~\\cite{Miranda:2003yh}.\nKamLAND has also played a key role in identifying oscillations as\n``the'' solution to the solar neutrino problem~\\cite{pakvasa:2003zv}\nand also in pinning down the LMA parameter region among previous wide\nrange of oscillation\nsolutions~\\cite{Gonzalez-garcia:2000sq,fogli:2001vr}.\n\nHowever, there is still some fragility in the interpretation of the\ndata in the presence of sub-weak strength ($\\sim \\varepsilon G_F$)\nnon-standard neutrino interactions (NSI) (Fig.~\\ref{fig:nuNSI}).\nIndeed, most neutrino mass generation mechanisms imply the existence\nof such dimension-6 operators. They can be of two types:\nflavour-changing (FC) and non-universal (NU). Their presence leads to\nthe possibility of resonant neutrino conversions even in the absence\nof neutrino masses~\\cite{valle:1987gv}.\n\\begin{figure}[t] \\centering\n    \\includegraphics[height=3cm,width=.55\\linewidth]{diagram-fc.eps}\n    \\caption{\\label{fig:nuNSI} %\n      Non-standard neutrino interactions arise, e.~g., from the\n      non-unitary structure of charged current weak interactions\n      characterizing seesaw-type schemes~\\cite{schechter:1980gr}.}\n\\end{figure}\nWhile model-dependent, the expected NSI magnitudes may well\nfall within the range that will be tested in future precision\nstudies~\\cite{Huber:2004ug}.\nFor example, in the inverse seesaw model~\\cite{Deppisch:2004fa} the\nnon-unitary piece of the lepton mixing matrix can be sizeable, hence\nthe induced non-standard interactions.  Relatively sizeable NSI\nstrengths may also be induced in supersymmetric unified\nmodels~\\cite{hall:1986dx} and models with radiatively induced neutrino\nmasses~\\cite{zee:1980ai,babu:1988ki}.\n\nThe determination of atmospheric neutrino parameters $\\Dma$ and\n$\\sin^2\\theta_\\Atm$ is hardly affected by the presence of NSI on\ndown-type quarks~\\cite{fornengo:2001pm}. \n\nIn contrast, the determination of solar neutrino parameters is not\nquite robust against the existence of NSI~\\cite{Miranda:2004nb}, even\nif reactor data are included. \nThe issue can only be resolved by future low and intermediate energy\nsolar neutrino data mentioned by Raghavan, with enough precision to\nsort out the detailed profile of the solar neutrino conversion\nprobability.\n \n\\subsection{Future prospects}\n\\label{sec:future-prospects}\n\nUpcoming reactor and accelerator long baseline experiments aim at\nimproving the sensitivity on $\\sin^2\\theta_{13}$~\\cite{Huber:2004ug}.\nThe value of $\\theta_{13}$ is a key input and the start of an\nambitious long-term effort towards probing CP violation in neutrino\noscillations~\\cite{Alsharoa:2002wu,apollonio:2002en,albright:2000xi}.\nProspects of accelerator and reactor neutrino oscillation experiments\nfor the coming years have been extensively discussed in the literature\nand there have been several talks at this conference, for example\nthose of Declais, Huber, Kajita, Kato, Kopp, Lindner, Nunokawa and\nSchwetz. Here I simply illustrate in Fig.~\\ref{fig:t13fut} the\nanticipated evolution of the $\\theta_{13}$ discovery reach for the\nglobal neutrino program, see details in Ref.~\\cite{Albrow:2005kw}.\n\\begin{figure}[t] \\centering\n\\includegraphics[height=6cm,width=.7\\linewidth]{disclimitbandnova.eps}\n\\caption{\\label{fig:t13fut} The hunt for $\\theta_{13}$: artist's view\n  of anticipated sensitivities on $\\theta_{13}$ given in\n  Ref.~\\cite{Albrow:2005kw}.}\n\\end{figure}\n                                    \nOne important comment is that even a small residual non-standard\ninteraction in this ``solar'' (e-tau) channel has dramatic\nconsequences for the sensitivity loss for $\\theta_{13}$ at a neutrino\nfactory~\\cite{huber:2001de}.\nTo make these experiments meaningful it is a must to have a near\ndetector capable of sorting out for NSI with high sensitivity.\n\nIn contrast, future neutrino factories will probe flavor changing\nnon-standard neutrino-matter interactions in the ``atmospheric''\n(mu-tau) channel with sensitities which are substantially improved\nwith respect to current ones~\\cite{huber:2001zw}.  For example, a 100\nGeV NUFACT can probe these at the level of $|\\epsilon| < \\mathrm{few}\n\\times 10^{-4}$ at 99 \\% C.L.\n \nImproving the sensitivities on NSI constitutes at a near detector is a\nnecessary pre-requisit. In short, probing for NSI is an important item\nand a window of opportunity for neutrino physics in the precision age.\n\nTo close this section let me mention that day\/night effect studies in\nlarge water Cerenkov solar neutrino experiments such as UNO, Hyper-K\nor LENA has also been suggested as an alternative way to probe\n$\\theta_{13}$~\\cite{Akhmedov:2004rq}.\n\n\\section{Lepton flavour violation}\n\\label{sec:lept-flav-viol}\n\nThe discovery of neutrino oscillations demonstrates that lepton\nflavour conservation is not a fundamental symmetry of nature. It is\ntherefore natural to expect that it may show up elsewhere, for example\n\\(\\mu\\to e\\gamma\\) or nuclear $\\mu^-- e^-$ conversion, as seen in\nFig.~\\ref{fig:Diagrams}.\n\\begin{figure}[h]\n\\centering\n\\includegraphics[clip,height=4.5cm,width=0.8\\linewidth]{mue-conv.eps}\n\\caption{Contributions to the nuclear $\\mu^-- e^-$ conversion: (a)\n  long-distance and (b) short-distance. For numerical results see\n  Ref.~\\cite{Deppisch:2004fa,Deppisch:2005zm}}\n     \\label{fig:Diagrams}\n\\end{figure}\nIndeed, in seesaw-type schemes of neutrino mass, lepton flavour violation  is induced\neither from neutral heavy lepton\nexchange~\\cite{Bernabeu:1987gr,Ilakovac:1994kj} as discussed here by\nVogel, or through the exchange of charginos (neutralinos) and\nsneutrinos (charged sleptons) as discussed here by Masiero\n~\\cite{Hall:1985dx,borzumati:1986qx,casas:2001sr,Antusch:2006vw} (see\ntop panel in Fig.~\\ref{fig:mueg}).\n\\begin{figure}[h] \\centering\n\\includegraphics[height=4cm,width=.8\\linewidth]{mueg.ps}\n\\includegraphics[width=\\linewidth,height=5cm]{mue-muegamma.eps} \n\\vglue -1cm\n\\caption{\\label{fig:mueg} Supersymmetric Feynman diagrams for lepton flavour violation \n  and correlation between mu-e conversion and \\(Br(\\mu\\to e\\gamma)\\)\n  in the supersymmetric inverse seesaw model of\n  Ref.~\\cite{Deppisch:2004fa}.}\n\\end{figure}\nAs illustrated in Fig.~\\ref{fig:mueg} the rates for both processes can\nbe sizeable and fall within the sensitivity of upcoming experiments.\nThe calculation in Fig.~\\ref{fig:mueg} is performed in the framework\nof the generalized supersymmetric seesaw model of\nRef.~\\cite{Deppisch:2004fa} to where I address you if you wish to\nunderstand the interplay of neutral heavy\nlepton~\\cite{Bernabeu:1987gr} and supersymmetric contributions.\nIf the neutral heavy leptons are in the TeV range (a situation not\nrealizable in the minimal seesaw mechanism), the \\(Br(\\mu\\to\ne\\gamma)\\) rate can be enhanced even in the {\\sl absence} of\nsupersymmetry. In this case the neutral heavy leptons that mediate\nlepton flavour violation  may be directly produced at accelerators~\\cite{Dittmar:1990yg}.\n\nFig.~\\ref{fig:mueg} also illustrates the correlation between nuclear\n$\\mu^--e^-$ conversion and $\\mu^-\\to e^-\\gamma$ decay in the inverse\nseesaw model for the nuclei Au, Ti, Al.  The shaded area and vertical\nline denote the current bound and future sensitivity (PSI) on\n$Br(\\mu\\to e\\gamma)$, respectively.  The horizontal lines denote the\ncurrent bound (Au\/SINDRUM II) and expected future sensitivities\n(Al\/MECO, Ti\/PRISM) on $R(\\mu^-N\\to eN)$. More details in\nRef.~\\cite{Deppisch:2005zm}.\n\nNote also that since lepton flavour violation  and CP violation can occur in the massless\nneutrino limit, hence the allowed rates need not be suppressed by the\nsmallness of neutrino\nmasses~\\cite{Bernabeu:1987gr,Ilakovac:1994kj,branco:1989bn,rius:1990gk}.\n\n\\section{Absolute scale of neutrino mass}\n\\label{sec:neutr-double-beta}\n\nNeutrino oscillations are insensitive to absolute masses and can not\nprobe whether neutrinos are Dirac or Majorana. Current data can not\ndetermine whether the spectrum is normal or inverted, as illustrated\nin Fig.~\\ref{fig:Which}.\n\\begin{figure}[h]\n  \\centering\n\\includegraphics[clip,width=.46\\linewidth,height=3.5cm]{nuschemes.eps}\n\\includegraphics[height=3.5cm,width=.51\\linewidth]{plot-mi-m3.eps}\n\\caption{Which spectrum?}\n \\label{fig:Which}\n\\end{figure}\nTo settle the issue one needs kinematical tests, such as beta decay\nstudies~\\cite{Drexlin:2005zt}, as discussed here by Sisti and\nWeinheimer. The upcoming high precision neutrino mass experiment\nKATRIN scales up both the size \\& source intensity, aiming at a\nsensitivity one order of better than that of the current Mainz-Troitsk\nexperiments.\n\nNeutrinoless double beta decay and other lepton number violation  processes, such as\nneutrino transition electromagnetic\nmoments~\\cite{schechter:1981hw,Wolfenstein:1981rk}\n\\cite{pal:1982rm,kayser:1982br} can probe the basic nature of\nneutrinos.\n\nThe significance of neutrinoless double beta decay stems from the\nfact that, in a gauge theory, irrespective of the mechanism that\ninduces $0\\nu\\beta\\beta$ , it necessarily implies a Majorana neutrino\nmass~\\cite{Schechter:1982bd}, as illustrated in Fig.  \\ref{fig:bbox}.\n\\begin{figure}[h]\n  \\centering\n\\includegraphics[width=6cm,height=3.2cm]{blackbox.eps}\n\\caption{Neutrinoless double beta decay and Majorana mass are\n  equivalent~\\cite{Schechter:1982bd}.}\n \\label{fig:bbox}\n\\end{figure}\nThus it is a basic issue. Quantitative implications of the\n``black-box'' argument are model-dependent, but the theorem itself\nholds in any ``natural'' gauge theory.  \n\n\nNow that oscillations are experimentally confirmed we know that $0\\nu\\beta\\beta$ \nmust be induced by the exchange of light Majorana neutrinos, the\nso-called \"mass-mechanism\". The $0\\nu\\beta\\beta$  amplitude depends on the 3\nmasses, 2 mixing angles, and 2 CP phases. Hence it involves the\nabsolute scale of neutrino mass, as well as the Majorana\nphase~\\cite{schechter:1980gr}, neither of which can be probed in\noscillations~\\cite{bilenky:1980cx,Schechter:1981gk,doi:1981yb}.\nThe phenomenological situation was described here by Avignone,\nBettini, Fiorini, Pavan, Simkovik, Vala and Vogel. It clearly\ndistinguishes between normal and inverted hierarchical spectra, as\nseen in Fig.~\\ref{fig:nbbfut}: in the former hierarchy case there is\nin general no lower bound on the $0\\nu\\beta\\beta$  rate, since there can be a\ndestructive interference amongst the neutrino amplitudes. In contrast,\nthe inverted neutrino mass hierarchy implies a ``lower'' bound for the\n$0\\nu\\beta\\beta$  amplitude.\n\\begin{figure}[h]\n \\centering\n\\includegraphics[clip,width=.8\\linewidth,height=5.4cm]{nh_paper.eps}\n\\includegraphics[clip,width=.8\\linewidth,height=5.4cm]{ih_paper.eps}\n \\caption{$0\\nu\\beta\\beta$  amplitude versus current oscillation data,\n   from Ref.~\\cite{Bilenky:2004wn}.}\n\\label{fig:nbbfut}\n\\end{figure}\nThe best current limit on $\\langle m_{\\nu} \\rangle$ comes from the Heidelberg-Moscow\nexperiment. The current claim~\\cite{Klapdor-Kleingrothaus:2004wj} (see\nalso Ref.~\\cite{Aalseth:2002dt}) and the sensitivities of the upcoming\nexperiments are indicated in the compilation, courtesy of Simkovik,\ndisplayed in Fig. \\ref{fig:nbbfut}. It shows the estimated average\nmass parameter characterizing the neutrino exchange contribution to\n$0\\nu\\beta\\beta$  versus the lightest and heaviest neutrino masses.  The\ncalculation takes into account the current neutrino oscillation\nparameters in \\cite{Maltoni:2004ei} and state-of-the-art nuclear\nmatrix elements~\\cite{Bilenky:2004wn}.\nThe upper (lower) panel corresponds to the cases of normal (inverted)\nneutrino mass spectra. In these plots the ``diagonals'' correspond to\nthe case of quasi-degenerate\nneutrinos~\\cite{babu:2002dz}~\\cite{caldwell:1993kn}~\\cite{ioannisian:1994nx}.\n\n\\begin{figure}[h] \\centering\n    \\includegraphics[width=.6\\linewidth,height=3.5cm]{plotRmaxcomplexc1.eps}\n    \\caption{\\label{fig:bbn-a4} %\n      Lower bound on $|\\vev{m_{ee}}|\/\\Dma$ vs $|\\mathrm{cos}\n      (\\phi_1)|$ where $\\phi_1$ is a Majorana phase. The lines in dark\n      (red) and grey (green) correspond to normal and inverse\n      hierarchy. Model of Ref.~\\cite{Hirsch:2005mc}.}\n\\end{figure}\nWe now give two examples of model $0\\nu\\beta\\beta$  expectations. First,\nRef.~\\cite{Hirsch:2005mc} proposes a specific normal hierarchy model\nfor which a lower bound on $0\\nu\\beta\\beta$  can be placed, as a function of the\nvalue of the Majorana violating phase $\\phi_1$, as indicated in\nFig.~\\ref{fig:bbn-a4}.  Second, the $A_4$ model~\\cite{babu:2002dz}\ngives a lower bound on the absolute Majorana neutrino mass\n$m_{\\nu}\\raise0.3ex\\hbox{$\\;>$\\kern-0.75em\\raise-1.1ex\\hbox{$\\sim\\;$}} 0.3$ eV and may therefore be tested in $0\\nu\\beta\\beta$  searches.\n\n\nThe absolute scale of neutrino masses will be tested by its effect on\nthe cosmic microwave background and the large scale structure of the\nUniverse, as discussed here by Elgaroy, Palazzo, Pastor and\nViel~\\cite{Lesgourgues:2006nd,Hannestad:2006zg,Fogli:2006yq}, and\nillustrated in Fig.~\\ref{fig:sergio}.\n\\begin{figure}[h] \\centering\n\\includegraphics[height=4cm,width=.45\\textwidth]{plot-Sergio.eps}\n\\caption{\\label{fig:sergio} %\n  Sensitivity of cosmology to neutrino mass, see\n  Ref.~\\cite{Lesgourgues:2006nd}.}\n\\end{figure}\n\n\n\n\\section{Neutrinos as probes }\n\\label{sec:neutrinos-as-probes}\n\nNot only neutrino properties can be probed via cosmology and\nastrophysics, but also, once well-determined, they can be used as\nastro-probes (Sun, Supernovae, pulsars, etc), cosmo-probes even\ngeo-probes. Here there were many related talks by Elgaroy, Mangano,\nPastor, Villante, and Viel. Neutrinos affect nucleosynthesis, \nlarge scale structure, the CMB and possibly the generation of \nthe matter anti-matter asymmetry.\n\nLike photons, cosmic rays and gravitational waves, neutrinos are one\nof the basic messengers of the Big Bang capable of probing early\nstages of its evolution. In the leptogenesis scenarios (discussed here\nby Akhmedov, di Bari, Ma and Petcov) neutrinos could probe the\nUniverse even down to epochs prior to the electroweak phase\ntransition.\n\nNeutrinos are also basic probes is astrophysics. Having only weak\ninteractions, they are ideal to monitor the interior of stars, such as\nthe the Sun.\n\nSupernova neutrinos were discussed by Cardall, Fleurot, Lunardini and\nVagins. The measurement of a large number of neutrinos from a future\ngalactic supernova neutrino signal will give us important information\non supernova parameters. Here I give a simplified plot (small\n$\\theta_{13}$ approximation) taken from Ref.~\\cite{Minakata:2001cd} as\nan illustration of the potential to probe astrophysics from a precise\nknowledge of neutrino properties.\n\\begin{figure}[h] \\centering\n\\includegraphics[height=4cm,width=.48\\linewidth]{sn87a.eps}\n\\includegraphics[width=.49\\linewidth,height=4.2cm]{EavEb.eps} \n\\vglue -.5cm     \n\\caption{\\label{fig:sn} Improved supernova parameter determination\n  attainable from the neutrino signal from 10 kpc galactic supernova\n  within future neutrino telescopes Hyper-K versus Super-K.}\n\\end{figure}\n\nLet me also mention that new effects in neutrino conversions at the\ncore of supernovae (neutron-rich regime) are expected when neutrinos\nhave non-standard\ninteractions~\\cite{valle:1987gv,Nunokawa:1996tg,Nunokawa:1996ve}.\nThese could induce new inner resonant conversions, over and above\nthose that arise from oscillations.\n\nBefore closing this section, let me mention that neutrinos are ideal\nprobes of the high energy Universe, discussed here by Billoir, Karle,\nFlaminio, Stanev and Sigl. For example, one expects high energy\nneutrinos from AGNs and GRBs. Typically the accelerated primaries make\npions, leading to comparable fluxes of neutrinos and gammas due to\nisospin. In contrast to gamma-rays, the neutrino spectrum is\nessentially unmodified. One set of observables to monitor are the\nflavor ratios, which are sensitive both to neutrino oscillations as\nwell as neutrino non-standard interactions.\n\nHere I give a very useful roadmap-plot presented by Sigl (see his talk\nat these proceedings for details). This plot makes extrapolations as\nto what high energy neutrino fluxes could be on the basis of the\ngamma-flux constraints (e.~g. by EGRET) at lower energies.\n\\begin{figure}[h] \\centering\n\\includegraphics[height=6.5cm,width=\\linewidth]{roadmap-plot.ps}\n\\vglue -.8cm     \n  \\caption{\\label{fig:road}\nSigl's map to high energy astrophysics.}\n\\end{figure}\n\n\n\\section{Origin of neutrino mass}\n\\label{sec:origin-neutrino-mass}\n\nThis is one of the most well-kept secrets of nature.  Gauge theories\nprefer Majorana neutrinos (see historical talk by\nEsposito)~\\cite{Weinberg:1980bf,schechter:1980gr} irrespective the\ndetailed mechanism of neutrino mass generation. While possible, the\nemergence of Dirac neutrinos would constitute a surprise, indicating\nthe existence of an accidental lepton number symmetry whose\nfundamental origin should be understood.  There are some ideas for\ngenerating light Dirac neutrinos. For example, theories involving\nlarge extra dimensions offer a novel scenario to account for small\nDirac neutrino masses. Within this picture, right-handed neutrinos\npropagate in the bulk, while left-handed neutrinos, being a part of\nthe lepton doublet, live only on the Standard Model (SM) branes. In\nthis picture neutrinos get naturally small Dirac masses via mixing\nwith a bulk fermion.\n\nHowever neutrinos  are more likely Majorana.\n\nIn contrast to SM charged fermions, neutrinos do not get masses after\nthe electroweak symmetry breaks through by the nonzero vacuum\nexpectation value (vev) of the Higgs scalar doublet, since they come\nin just one chiral species. There is, however, an effective lepton\nnumber violating dimension-five operator $LL\\Phi\\Phi$ (where L denotes\nany of the lepton doublets and $\\Phi$ the Higgs) which can be added to\nthe SM~\\cite{Weinberg:1980bf}.  After the Higgs mechanism this\noperator induces Majorana neutrino masses, thus providing a natural\nway to account for their smallness, irrespective of their specific\norigin. Little more can be said from first principles about the\nmechanism giving rise to this operator, its associated mass scale or\nits flavour structure. Its strength may be suppressed by a large scale\nin the denominator (top-down) scenario, as in seesaw\nschemes~\\cite{Valle:2006vb}.  Alternatively, the strength may be\nsuppressed by small parameters (e.g. scales, Yukawa couplings) and\/or\nloop-factors (bottom-up scenario) with no need for a large scale, as\nalso reviewed in Ref.~\\cite{Valle:2006vb}.\n\n\n\\def1{1}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\chapter{Introduction}\n\n\\noindent\nMatrix models\nprovide not only a novel formulation of low dimensional\nstring\ntheory but one which is integrable and exactly\nsolvable.\nThey lead to exact string equations in $D<1$ and a wealth\nof results for the free energy and correlation functions [1].\nThe largest model understood so far is the two dimensional\nstring theory,\ndescribed by a\nsimple dynamics of a (matrix) harmonic oscillator [2-8].\n\nWhile the numerical results follow straightforwardly,\nthe physical picture encoded\nin the matrix model is however not seen directly. It is exhibited once\nappropriate physical observables (collective fields)\n[3] are identified.\nFor the tachyon one has the bosonic collective field defining\nperturbative states. While the matrix model\nis linear, the collective field\nexhibits a nonlinear interaction which leads\nto nontrivial physical scattering processes\n[4].\nA fermi liquid\ndescription can be used to give a semiclassical\npicture of the scattering\n[5].\nThe field theory is integrable: it exhibits an infinite\nsequence of conserved charges and an even larger symmetry of\n$W_{\\infty}$\ngenerators\n[6].\n\nString theory is however most naturally described in terms of the\nworld sheet string coordinates and\nassociated conformal vertex operators [9].\nThese indeed exhibit similar symmetries [10] and can be seen to\ngive the\nsame correlation functions.  Except for the coincidence of various\nresults a closer connection between the matrix model description and\nthe string language is still lacking.\n\nIt is the purpose of this paper to address this problem and give\na more direct relationship between linear states of the matrix model\nand nonlinear scattering states of string theory.\nOne has the matrix harmonic\noscillator\n\n$$\n           L = {1 \\over 2} {\\rm Tr} \\bigl( {\\dot M}^2 + M(t)^\n2 \\bigr),\n           \\eqno\\eq\n$$\n\n\\noindent\nwith\n\n$$     \\eqalign {\n               A_{\\pm} &\\equiv P \\pm M = \\dot M \\pm M, \\cr\n               A_{\\pm}(t) &= A_{\\pm}(0)\\, e^\n{\\pm t}  \\cr\n}\n$$\n\n\\noindent\nbeing standard creation-annihilation operators. In terms of\nthese one easily\n writes down the eigenstates of the hamiltonian\n\n$$\n                 H = {1 \\over 2}\\, {\\rm Tr\\,} \\bigl( (P+M)(P-M) \\bigr) =\n                     {1 \\over 2}\\, {\\rm Tr\\,} A_{+} A_{-}.\n$$\n\n\n\\noindent\nFor example, the one-parameter set\n\n$$\n        A_{n}^{\\pm} = {\\rm Tr\\,} (P\\pm M)^\nn\n$$\n\n\\noindent\ngives imaginary eigenvalues with energies\n$\\epsilon_n = \\mp i\\,n$.\nReal energy\nstates are obtained by analytic continuation $n=ik$:\n\n$$\n        B_{k}^{(\\pm )} = {\\rm Tr\\,} (P\\pm M)^\n{ik}.   \\eqn\\continuedstates\n$$\n\nThe question is then how this simple set of exact matrix model\nstates translates into nontrivial string scattering states.\nContinuing on the constructions begun in [6], we\nshall explain a correspondence in section 2 and\ndescribe a simple\nderivation of general string scattering amplitudes using the\nintegrable states.\nAs such we exhibit how the\nnonlinear string dynamics follows from the linear and\nintegrable matrix dynamics.\n\nIn section 3 we discuss the symmetry algebra of the theory.\nWe demonstrate there a close connection between the matrix\n$W_\\infty$ generators and those of the conformal string theory.\nIn particular we shall see that the collective (tachyon) field\nrepresentation of these operators is nothing but the representation\ndefined in the conformal approach by Klebanov in [11].\n\n\n\\chapter{From States to Scattering}\n\n\\noindent\nStrings in two dimensions are described by the coordinates\n$X^\\mu \\equiv (X, \\phi)$, where $X$ is (usually) taken\nas spacelike and $\\phi$ is the nontrivial Liouville coordinate [9].\nOne has translation invariance in the $X$ direction (this is the\ntime coordinate of the matrix model, \\ie $X = it$) and only\nasymptotic translation invariance in $\\phi$ due to an\nexponential wall $\\mu\\, e^{-\\sqrt{2}\\phi}$.  The vertex operators\nof the lowest string modes (massless tachyons) are\n\n$$\n\\eqalign{\n  V_{\\pm} &= e^{ipX + \\beta_{\\pm} \\phi},   \\cr\n  \\beta_{\\pm} &= -\\sqrt 2 \\pm |k|.\n}  \\eqno\\eq\n$$\n\n\\noindent\nOnly the $+$ branch describes physical scattering states.  The\n$-$ operators grow at $\\phi\\to -\\infty$ and are termed \\lq\\lq wrongly\ndressed\". For scattering one has left movers (as initial states)\nand right movers (as final states) respectively denoted by\n\n$$\n  T^{(\\pm)}_k = e^{ikX + (-\\sqrt 2 + |k|)\\phi},\n  \\eqno\\eq\n$$\n\n\\noindent\nwhere $\\pm = {\\rm sign\\,} k$.\n\nStates of the matrix model can be seen to be in close correspondence.\nIn particular, of \\continuedstates\\ half of the states have a\nscattering interpretation as\n\n$$ \\eqalign {\n       B_{-k}^{(-)} |0\\rangle &= {\\rm Tr\\,} (P-M)^\n{-ik} = |k\\,; {\\rm in}\\rangle, \\cr\n       B_{k}^{(+)} |0\\rangle  &= {\\rm Tr\\,} (P+M)^\n{ik} = |k\\,; {\\rm out}\\rangle, \\cr\n} \\eqno\\eq\n$$\n\n\\noindent\nThis physical interpretation will arise once the spatial\n(Liouville) coordinate is identified.\nThis was understood to be related to the eigenvalue index\nof the matrix variable.  The\nphysical world is the positive real axis with a barrier at the\norigin, and so one only considers an in state that is left moving and\nan out state that is right moving.\n\nThe identification of physical states and of the extra Liouville\nmomentum is seen in a\ntransition to the collective field theory language [3].\nThis transition can be summarized [3-6] by the\nfollowing set of replacement rules:\n\n$$ \\eqalign {\n                       M &\\to x,      \\cr\n                       P &\\to \\alpha (x,t),   \\cr\n                     {\\rm Tr} &\\to {\\int {{dx} \\over {2\\,\\pi}}\\,} \\da.  \\cr\n}  \\eqno\\eq\n$$\n\n\\noindent\nThe matrix hamiltonian then becomes\n\n$$\n    H  = {1 \\over 6} {\\int {{dx} \\over {2\\,\\pi}}\\,} \\bigl( {\\alpha_+}^3 - {\\alpha_-}^\n3 \\bigr)\n       - {1 \\over 2} {\\int {{dx} \\over {2\\,\\pi}}\\,} x^\n2\\, \\bigl( {\\alpha_+} -  {\\alpha_-} \\bigr),\n    \\eqno\\eq\n$$\n\n\\noindent\ndescribing a scalar field $\\phi(x,t)$ and its conjugate $\\Pi (x,t)$,\nwith\n$\n       {\\alpha_{\\pm}} = \\partial_x \\Pi\\,\\pm\\,\\pi\\phi.\n$\n\nThe collective representation exhibits in addition to the time $t$ a\nspatial dimension $x$. One has a classical background field\n$  {\\pi\\phi_0} = \\sqrt{x^\n2 - 2\\mu} $, which induces a reparametrization of the\nnew spatial coordinate to\n$\n            \\tau = \\int{dx \\over {\\pi\\phi_0}(x)},\n$\nor\n\n$$  \\eqalign {\n          \\quad x(\\tau)  &= \\sqrt{2\\mu}\\, \\cosh(\\tau),\\cr\n          {\\pi\\phi_0} (\\tau)     &= \\sqrt{2\\mu}\\, \\sinh(\\tau).\\cr\n}                 \\eqn\\Trajectory\n$$\n\n\\noindent\nAsymptotic translations in $\\tau$ are scale transformations of\n$x$ since\n\n$$\n                 x(\\tau) \\sim \\sqrt{{\\mu \\over 2}}\\,e^\n{\\tau}.\n    \\eqn\\AsTrajectory\n$$\n\n\\noindent\nIndeed, in\naddition to time translation the collective\nLagrangian transforms covariantly under\nscale transformations\n\n$$  \\eqalign {\n                    x &\\to \\lambda\\,x, \\cr\n    \\alpha (x,t) &\\to {1 \\over \\lambda}\\, \\alpha (\\lambda\\,x,t), \\cr\n                    H &\\to {1 \\over \\lambda\n^4}\\, H. \\cr\n} \\eqno\\eq\n$$\n\n\\noindent\nThis symmetry is the origin of a second (spatial momentum)\nquantum number $p_{\\tau}$.\n\nIn linearized approximation with\n\n$$ \\eqalign {\n   &\\phi(x) = \\phi_0(x) + \\partial_x\\,\\psi(x), \\qquad\n              p(x) = - \\partial_x\\Pi(x), \\cr\n   &\\ \\psi(\\tau) = \\psi(x), \\qquad \\qquad p(\\tau) = {\\pi\\phi_0}\\, p(x), \\cr\n            & {\\alpha_{\\pm}}(x) = \\pm\\,{\\pi\\phi_0} + {1 \\over {\\pi\\phi_0}}\\, {\\at_{\\pm}}(\\tau). \\cr\n} \\eqn\\ChangeVarE\n$$\n\n\\noindent\none has right-left moving massless modes (tachyons)\n\n$$ \\eqalign {\n {\\at_{\\pm}}(\\tau,t)\\,&= f(t\\,\\mp \\,\\tau) =\n   \\pm\\int_{-\\infty}^{\\infty}dk\\,\\alpha^{\\pm}_k\\, e^\n{-ik(t\\mp \\tau)},\\cr\n}    \\eqn\\LeftRight\n$$\n\n\\noindent\nsatisfying\n\n$$\n         (\\partial_t \\pm \\partial_{\\tau}) {\\at_{\\pm}}  = 0.\n         \\eqno\\eq\n$$\n\n\\noindent\nwith the energy momentum values\n\n$$\n    \\alpha^\n{\\pm}_{-k} : \\quad p_0 = k, \\quad p_{\\tau} = \\pm k.\n    \\eqno\\eq\n$$\n\nThe exact states of the matrix model are directly translated into the\nfield theoretic representation. We have as exact tachyon eigenstates\n\n$$\n            T^{(\\pm)}_n = {\\int {{dx} \\over {2\\,\\pi}}\\,} \\int d\\alpha\\, (\\alpha \\pm x)^\n{n} =\n            {\\int {{dx} \\over {2\\,\\pi}}\\,} { (\\alpha \\pm x)^\n{n+1} \\over n+1},\n            \\eqno\\eq\n$$\n\n\\noindent\nintroduced by Avan and one of the authors in [6].\nUsing the Poisson brackets $\\{\\alpha (x),\\alpha (y)\\} = 2\\pi\\, \\delta '(x-y)$\none easily shows\n\n$$\n             \\{H,T^{(\\pm)}_n\\} = \\pm n\\,T^\n{(\\pm)}_n\n             \\eqno\\eq\n$$\n\n\\noindent\nand one has eigenstates with $ip_0 = \\pm n$. Defining\n\n$$\n      p_{\\tau} = {\\it scale\\  dimension\\\/} - 4, \\eqno\\eq\n$$\n\n\\noindent\none has\n\n$$\n                        p_{\\tau} = - 2 + n.\n$$\n\n\\noindent\nThese states stand in comparison with\nthe vertex operators of conformal field\ntheory\n\n$$\n\\eqalign{\n              T_p^{(\\pm)} &\\equiv e^\n{ i\\,p\\,X + (-\\sqrt 2 + |p|)\\varphi}  \\cr\n  &\\leftrightarrow {\\int {{dx} \\over {2\\,\\pi}}\\,} {(\\alpha \\pm x)^{n+1} \\over n+1}  \\cr\n  &\\leftrightarrow {\\rm Tr \\,} (P\\pm M)^n.  \\cr\n}  \\eqno\\eq\n$$\n\n\\noindent\nThe tachyon vertex operators with opposite (Liouville)\ndressing correspond to singular\noperators in the matrix model\n\n$$\n\\eqalign{\n  e^\n{ i\\,p\\,X + (-2 - |p|)\\,\\varphi} &\\leftrightarrow\n  {\\int {{dx} \\over {2\\,\\pi}}\\,} { (\\alpha \\pm x)^\n{1-n} \\over 1-n} \\cr\n   &\\leftrightarrow   {\\rm Tr\\,} (P\\pm M)^{-n}. \\cr\n} \\eqno\\eq\n$$\n\nWe have now described a one to one correspondence between the matrix\nmodel states and string states.  Scattering amplitudes can be\nderived immediately once this correspondence is understood.\n\nWe note that the collective field theory seemingly introduces a\ndegeneracy. For each state of the matrix model one can define\ntwo states\nin collective field theory since we can replace $P \\to {\\alpha_{\\pm}}\n(x,t)$. Each\nof the separate fields ${\\alpha_+}$ or ${\\alpha_-}$\ncan be used to define states with the\nabove quantum numbers. In particular\n\n$$\n    {\\int {{dx} \\over {2\\,\\pi}}\\,} { ({\\alpha_+} \\pm x)^\n{1 \\pm ik} \\over {1 \\pm ik} }\n$$\n\n\\noindent\nand\n\n$$\n    {\\int {{dx} \\over {2\\,\\pi}}\\,} { ({\\alpha_-} \\pm x)^\n{1 \\pm ik} \\over {1 \\pm ik} }\n$$\n\n\\noindent\nboth have the same quantum numbers\n\n$$\n           p_0 = k, \\qquad p_{\\tau} = -2 \\pm ik.\n$$\n\n\\noindent\nThese have to be identified, up to a phase factor.\nIt can be\nshown (below) that boundary conditions fix the phase factor\nto be $-1$. So one has\n\n$$\n    {\\int {{dx} \\over {2\\,\\pi}}\\,} { ({\\alpha_{\\mp}} \\pm x)^\n{1 \\mp ik} \\over {1 \\mp ik} } =\n    - {\\int {{dx} \\over {2\\,\\pi}}\\,} { ({\\alpha_{\\pm}} \\pm x)^\n{1 \\mp ik} \\over {1 \\mp ik} },\n    \\eqn\\Identification\n$$\n\n\\noindent\nimplying a nonlinear relation between left and right movers.\nThis equation, which\nfollows from simple kinematical reasoning, determines the complete\ntree level scattering amplitude. Expanding\n\n$$\n   {\\alpha_{\\pm}}(x) = \\pm\\,x \\mp {1 \\over x}\n             \\bigl(\\mu \\mp \\hat\\alpha_{\\pm}(\\tau) \\bigr) +\n{1 \\over x^\n2}\n              {\\rm \\ terms},\n              \\eqn\\asymptotic\n$$\n\n\\noindent\nwe shall find the relation\n\n$$\n\\eqalign {\n   \\int_{-\\infty}^\\infty d\\tau\\, e^{\\pm ik\\tau}\\,\n        {{\\hat\\alpha}_{\\pm}\\over\\mu}\n       &=\n       - \\int_{-\\infty}^{\\infty} {d\\tau \\over ik\\pm 1}\\, e^\n{\\mp ik\\tau}\n       \\Biggl[ \\Bigl[ 1 + {\\hat\\alpha_{\\mp} \\over \\mu} \\Bigr]^\n{ik\\pm 1} - 1\n             \\Biggr].\\cr\n} \\eqn\\Moore\n$$\n\n\\noindent\nThis functional equation relating left and right moving waves\nof the collective\nfield was shown to represent a solution to the\nscattering problem in\n[8].  Here we exhibited how this nonlinear scattering equation\nemerges directly from the\nexact oscillator states.\nThe fact that the left and the right hand side of the equation\nare interpreted as eigenstates of collective field theory\nimplies also the following:  a complete quantization procedure was\ngiven [4] for the field theory Hamiltonian, involving normal\nordering and the subtraction of counterterms.  The same procedure\ncan be applied to the states and will lead to a fully quantum\nversion of the scattering equation.\n\nThe main ingredient in obtaining the scattering equation are\nthe proper boundary conditions.\nLet us now elaborate on this question.\nThe issue of boundary conditions is of paramount\nimportance in a correct treatment of\nthe spectrum within the collective\napproach. In QCD-like unitary matrix models,\nit is well known that as the system moves from\na strong coupling regime to a weak coupling regime where the classical\ndensity of states $\\phi_0$ has only finite support, Dirichlet boundary\nconditions must be imposed on the shifted field $\\psi(\\tau)$. This is\nessentially due to the fact that $\\phi_0(\\tau=0)=\\phi_0(\\tau=L\n\\to\\infty)=0$,\nand in this way\nthe time independence of the original constraint condition $\\int\\,dx\\,\n \\phi=N$\nis preserved\n[3]. For $c=1$ strings,\nthis \\lq\\lq constraint\" equation determines the value of the\ncosmological constant. Therefore, apart from problems of consistency,\na choice other than Dirichlet boundary conditions would result in a\ntime\ndependent cosmological constant.\nNotice that this implies that in a density\nvariable description of \\lq\\lq wall\"\nscattering, the \\lq\\lq wall\" at $\\tau=0$ is\nrigid.\nA creation-annihilation basis that automatically enforces Dirichlet\nboundary conditions on the scalar field $\\psi$ is defined by the\nexpansion\n\n$$ \\eqalign{\n  &{\\at_{\\pm}}(\\tau)\\,= \\pm\\,\\int_{-\\infty}^\n{\\infty}{dk \\over \\sqrt{|k|}}\\,\n  e^\n{\\pm\\,i\\,k\\tau} a_k , \\quad a_{-k}\\equiv \\adag,\\cr\n  &\n[a_k,\\adag] = \\delta (k-k').\\cr\n}   \\eqn\\StandExp\n$$\n\n\\noindent\nWe could equally well have chosen the \\lq\\lq left-right\" basis\n\\LeftRight.\nOnce one expresses a scalar theory with fields satisfying boundary\nconditions in a left-right basis, there is a standard problem, also\npresent in the critical open string: the functions $e^\n{ik\\tau}$ are not\northogonal over the half line, and therefore the computation of\nFourier\ncoefficients require some modification. To this standard problem\nthere\nis a standard solution\n[12]: one notices that\nthe definitions of all the fields in \\LeftRight\\ naturally extend to\nnegative values of $\\tau$. Therefore we extend the definition of\nthe fields from $ 0 \\le \\tau < \\infty$ to $-\\infty < \\tau < \\infty$\nby requiring\n\n$$   \\eqalign {\n            \\psi(-\\tau) &= - \\,\\psi(\\tau),\\cr\n            {\\at_{\\pm}}(-\\tau) &= - \\,{\\at_{\\mp}}(\\tau).\\cr\n}      \\eqn\\Involut\n$$\n\n\\noindent\nIn other words, the fields of interest to us are the fields defined\non the\nfull line which are odd (in coordinate free form) under the involution\n$\\tau \\to -\\tau$. This point of view has been extensively used in works\nrelating critical open string amplitudes to those of the closed string\n[13].\nOne can then compute Fourier coefficients of ${\\tilde\\alpha_+}$, say:\n\n$$\n    \\int_0^{\\infty} {d\\tau \\over 2\\pi}\\, e^\n{\\mp\\,ik\\tau}\\, {\\at_{\\pm}}(\\tau)-\n    \\int_0^{\\infty} {d\\tau \\over 2\\pi}\\, e^\n{\\pm\\,ik\\tau}\\, {\\at_{\\mp}}(\\tau) =\n    \\pm \\alpha_k.\n    \\eqn\\Extend\n$$\n\n\\noindent\nWe can now reformulate the problem as follows:\nsuppose we introduce the arbitrary left-right expansion \\LeftRight.\nEquation \\Involut\\ is then equivalent to\n\n$$\n            \\alpha^-_k = - \\alpha^\n+_k.     \\eqn\\BigEq\n$$\n\n\\noindent\nPhysically, this simply means that in order to preserve the boundary\nconditions of the system,\nif a right mover is created then a left mover must also be created\nwith\namplitude minus one, and similarly for annihilation operators.\nThis means that the Dirichlet boundary conditions cause the\n left and right\nmovers to combine into standing waves, which are perturbative tachyon\nstates in the matrix model.\n\nNow, in terms of the matrix variables  \\StandExp\\\ndescribed above, this condition\nis immediately built into the expansion of the fields. However, for\nasymptotic incoming and outgoing states,\nwhich are naturally defined on the full line,\nthe analogue of condition \\BigEq, imposed on the\nthe exact states of the system, leads to the nonlinear scattering\nmatrix.\n\nWe now concentrate on $T^\n{(+)}_{ik}$ and introduce the following notation\nto represent the two degenerate states described previously\n\n$$ \\eqalign {\n      T^{(+)}_{ik} &= {\\int {{dx} \\over {2\\,\\pi}}\\,} \\da\\,(\\alpha+x)^\n{ik}\n                   \\equiv T^{+}_{ik\\,(+)} - T^\n{+}_{ik\\,(-)} \\cr\n     &= {\\int {{dx} \\over {2\\,\\pi}}\\,} { ({\\alpha_{\\mp}} + x)^\n{ik + 1} \\over {ik + 1} } -\n        {\\int {{dx} \\over {2\\,\\pi}}\\,} { ({\\alpha_{\\pm}} + x)^\n{ik + 1} \\over {ik + 1} }.\\cr\n}   \\eqn\\ExactDefX\n$$\n\n\\noindent\nThe equation relating left and right moving fields reads\n\n$$  \\eqalign {\n     \\Tp &= {1 \\over 2} \\int_{-\\infty}^\n{\\infty} {d\\tau \\over 2\\pi}\\,\n           {{\\pi\\phi_0} \\over ik+1} \\Biggl[\n           \\Bigl( ({\\pi\\phi_0}+x) + {{\\alpha_+}(\\tau) \\over {\\pi\\phi_0}} \\Bigr)^\n{ik+1}\n             - x^\n{ik+1} \\Biggr] \\cr\n         &= - {1 \\over 2} \\int_{-\\infty}^\n{\\infty} {d\\tau \\over 2\\pi}\\,\n           {{\\pi\\phi_0} \\over ik+1} \\Biggl[\n           \\Bigl( (-{\\pi\\phi_0}+x) + {{\\alpha_-}(\\tau) \\over {\\pi\\phi_0}} \\Bigr)^\n{ik+1}\n             - x^\n{ik+1} \\Biggr] \\cr\n         &= - \\Tm.\\cr\n}   \\eqn\\IdentificationB\n$$\n\n\\noindent\nThe range of integration has been extended as described above equation\n\\Extend. This is a restatement of equation \\Identification.\nSince the c-number contributions $C_{\\pm}^\n{(+)}$ to the above operators are\nare the same, we rewrite this condition as\n\n$$\n       \\Tp - C_{+}^{(+)} = - \\bigl( \\Tm - C_{-}^\n{(+)} \\bigr).\n       \\eqn\\TheCondition\n$$\n\n\\noindent\nThis equality is a necessary consequence of Dirichlet boundary\nconditions.\nTo linear order, it is straightforward to show that\nequation \\TheCondition\\ is equivalent to equation \\BigEq .\n\nAs $x \\gg \\sqrt{2\\mu}$ we wish to express this condition in terms\nof the asymptotic fields $\\hat \\alpha_\\pm$ defined in equation\n\\asymptotic, using the asymptotic behaviour\n\\AsTrajectory.\nWe remind ourselves that\n$\\hat\\alpha_{-}(t+\\tau)$ is an incoming left-moving wave\nand $\\hat\\alpha_{+}(t-\\tau)$ is the outgoing, right-moving wall\nscattered\nwave.\nIn\nthe asymptotic description the fact\nthat, from the collective field theory point of view, the \\lq\\lq wall\"\nat\n$\\tau=0$ is rigid, is not immediately built into the definition of\nthe fields.\nThis condition has to be imposed on the exact states\nof the system, i.e., equation\n\\TheCondition\\  must be satisfied.  Expressing the exact states in\nterms of the variables \\asymptotic\\ we get\n\n$$  \\eqalign {\n    \\Tp - C_{+}^\n{(+)}\n        &= {1\\over ik+1}\\,{\\int {{dx} \\over {2\\,\\pi}}\\,} {({\\alpha_+}+x)^{ik+1} - ({\\pi\\phi_0}+x)^\n{ik+1}}\\cr\n        &= { \\sqrt{2\\mu}^{\\,ik+1}\\over ik+1} \\int_{-\\infty}^\n{\\infty}\n{d\\tau \\over 8\\pi}\\,\n           {e^{ik\\tau}}\\, e^\n{2\\tau} \\cr\n   &\\ \\Biggl[\n                \\sum_{j=0}^\n\\infty\n                {\\Gamma(ik+2) \\over  \\Gamma(ik + 2 - j)\\,j!}\n                \\,(-)^j\\,e^\n{-2j\\tau}\n      \\left\\{\\left(1-{\\hat\\alpha_+\\over\\mu}\\right)^\nj-1\\right\\}\n             \\Biggr],  \\cr\n}  \\eqno\\eq\n$$\n$$\n\\eqalign{\n   - \\bigl( \\Tm - C_{-}^\n{(+)} \\bigr)\n        &= - {1\\over ik+1}{\\int {{dx} \\over {2\\,\\pi}}\\,} ({\\alpha_-}+x)^{ik+1} - (-{\\pi\\phi_0}+x)^\n{ik+1} \\cr\n        &= - {\\sqrt{2\\mu}^{\\,ik+1}\\over ik+1} \\int_{-\\infty}^\n{\\infty}\n              {d\\tau \\over 8\\pi} \\,\n           e^{-ik\\tau} \\sum_{p=1}^\n\\infty\n                {\\Gamma(ik+2) \\over  \\Gamma(ik + 2 - p)\\,p!}\n                \\left( {\\hat\\alpha_- \\over \\mu} \\right)^\np.  \\cr\n} \\eqno\\eq\n$$\n\n\\noindent\nEquating these expressions as required by the condition\n\\TheCondition, and applying partial integrations and a Fourier\ntransform, we obtain\n\n$$  \\eqalign {\n        &\n                \\sum_{j=0}^\\infty e^\n{-2(j-1)\\tau}\\,\n                {\\Gamma(-\\partial+2) \\over  \\Gamma(-\\partial + 2 -\nj)\\,j!}\n                \\,(-)^j\\,e^\n{-2j\\tau}\n  \\left\\{\\left(1-{\\hat\\alpha_+(\\tau)\\over\\mu}\\right)^\nj-1\\right\\} \\cr\n  &\\ =\n        -\n           \\sum_{p=1}^\n\\infty\n                {\\Gamma(\\partial+2) \\over  \\Gamma(\\partial + 2 -\np)\\,p!}\n                \\left( {\\hat\\alpha_-(-\\tau) \\over \\mu} \\right)^\np.  \\cr\n} \\eqno\\eq\n$$\n\n\\noindent\nWe can now extract the asymptotic limit by letting $\\tau\\to\\infty$.\nWe find that on the left hand side only the $j=1$ term contributes\n(lower order terms would correspond, on the right hand side, to\nterms dropped in the asymptotic definition \\AsTrajectory).\nAs $\\tau\\to \\infty$\n\n$$  \\eqalign {\n        &           (-\\partial+1)\n            {\\hat\\alpha_+(\\tau)\\over\\mu}\n\\cr\n  &\\ =\n        -\n           \\sum_{p=1}^\n\\infty\n                {\\Gamma(-\\partial+2) \\over  \\Gamma(-\\partial + 2 -\np)\\,p!}\n                \\left( {\\bar\\alpha_-(\\tau) \\over \\mu} \\right)^\np,  \\cr\n} \\eqno\\eq\n$$\n\n\\noindent\nwhere $\\bar\\alpha_-(\\tau) \\equiv \\hat\\alpha_-(-\\tau)$.  It follows\nthat $$\n\\eqalign {\n        &            {\\alpha_+(\\tau)}\n   =     -\n           \\sum_{p=1}^\n\\infty\n                {\\Gamma(-\\partial+1) \\over  \\Gamma(-\\partial + 2 -\np)\\,p!} \\, \\left({1\\over\\mu}\\right)^\n{p-1}\\,\n                \\bar\\alpha_-(\\tau)^\np.  \\cr\n} \\eqno\\eq\n$$\n\nThis relation expressing left moving fields in terms of right moving\nones is the result \\Moore\\ for the scattering problem [8].\n\n\n\\chapter{Symmetries}\n\n\\noindent\nThe spacetime field theory given by the collective field exhibits a\nlarge ($W_\\infty$) spacetime symmetry of 2-dimensional string theory\n[6].\nThe generators of this symmetry can be directly found or simply\ninduced from the matrix model.  There one has the invariant\noperators\n\n$$\n  {\\rm Tr\\,} (P^r M^s),       \\eqno\\eq\n$$\n\n\\noindent\nwhich are closed under commutation.  The field theory operators\nread\n\n$$\n  H_m^n =      {\\int {{dx} \\over {2\\,\\pi}}\\,} {\\alpha_+^{m-n}\\over m-n}\\,x^{m-1}  \\eqno\\eq\n$$\n\n\\noindent\nand can be shown to satisfy the $w_\\infty$ algebra\n\n$$\n  [H^{n_1}_{m_1}, H^{n_2}_{m_2}]\n   = i\\, [(m_2-1)\\,n_1 - (m_1-1)\\,n_2]\\, H^{n_1+n_2}_{m_1+m_2-2}.\n\\eqno\\eq\n$$\n\n\\noindent\nOf particular relevance to us are the spectrum generating operators\n\n$$\n  O_{JM} \\equiv {\\rm Tr\\,} (P+M)^{J-M}(P-M)^{J+M},   \\eqno\\eq\n$$\n\n\\noindent\nwhich become, in the collective field theory representation\n\n$$\n  O_{JM} =     {\\int {{dx} \\over {2\\,\\pi}}\\,}\\int_{\\alpha_-}^{\\alpha_+}\n   d\\alpha\\, (\\alpha+x)^{J+M+1}\\,(\\alpha-x)^{J-M+1}.\n   \\eqn \\specgen\n$$\n\n\\noindent\nOne sees that these are linear combinations of the basic\n$w_\\infty$ operators\n\n$$\n  O_{JM} = H_1^{-2J-2} + 2M\\, H_2^{-2j} + (2M^2-J-1)\\, H_3^{2J+2}\n   + \\dots.\n\\eqno\\eq\n$$\n\n\\noindent\nand it follows that the spectrum generating algebra is precisely a\n$w_\\infty$, \\ie,\n\n$$\n  [\\,O_{J_1,M_1}, O_{J_2,M_2}\\,]\n   = i\\, [(M_2-1)\\,J_1 - (M_1-1)\\,J_2]\\, O_{J_1+J_2,M_1+M_2}.\n  \\eqn\\algebra\n$$\n\n\\noindent\nThis can also be shown directly from \\specgen\\ by doing partial\nintegrations [6].\n\nThere is a close connection between these $w_\\infty$ operators\nand the operators describing exact tachyon states of the\nfield theory.  Recall\nthe one parameter family of operators\n\n$$\n  T^{(\\pm)}_n =     {\\int {{dx} \\over {2\\,\\pi}}\\,}\\int_{\\alpha_-}^{\\alpha_+}\n   d\\alpha\\, (\\alpha\\pm x)^n.\n\\eqno\\eq\n$$\n\n\\noindent\nTheir commutators give the generators of the $w_\\infty$ algebra,\n\\ie, one can show that\n\n$$\n  O_{JM} = {1\\over 2i\\,(J-M+2)\\,(J+M+2)}\\, [\\,T^+_{J+M+2},\n    T^-_{J-M+2}\\,].\n\\eqno\\eq\n$$\n\n\\noindent\nThe symmetry generators were also written down in the\nconformal field theory approach [10].\nThere is a close parallel with all of\nthe matrix model relationships\nand the commutators are\nsimply replaced by operator products.  The above implies for example\nthat the\n$w_\\infty$ generators are obtained as operator products of basic\ntachyon vertex operators.\nA closer correspondence is seen by comparing the above field\ntheory forms with  the representations deduced for the action of the\nsymmetry generators on the tachyon module [11].\n\n\n\\section {Fourier Expansion}\n\n\\noindent\nWe now\nconsider the spectrum generating operators $O_{JM}$ of \\specgen\\\nin more detail.\nWe shall see that the correspondence with the conformal field theory\nresults\nof [11] will then follow.   To expand in terms of\ncreation-annihilation\noperators we make the substitutions\n\n$$\n\\eqalign{\n  \\alpha_+ &= x+ \\bar\\alpha_+,  \\cr\n  \\alpha_- &= -x + \\bar\\alpha_- \\cr\n} \\eqn\\defA\n$$\n\n\\noindent\nin the spectrum generating operators \\specgen.\nApplying partial integration to \\specgen\\\nand inserting the limits \\defA, one finds\n\n$$\n\\eqalign{\n  O_{JM} =      {\\int {{dx} \\over {2\\,\\pi}}\\,}\n         \\sum_{k=0}^{J+M+1} (-)^k\\,\n          & {(J-M+1)!\\,(J+M+1)! \\over (J-M+2+k)! \\,(J+M+1-k)!}\n            \\times\\cr\n         \\times\\bigl\\{\n           & {\\bar\\alpha_+}^{J-M+2+k}\\,\n           (\\bar\\alpha_+ + 2x)^{J+M+1-k}   \\cr\n          -\n         &\n           (\\bar\\alpha_- - 2x)^{J-M+1+k}\\,\n           {\\bar\\alpha_-}^{J+M+2-k}\n           \\bigr\\}. \\cr\n}  \\eqn  \\partialA\n$$\n\n\\noindent\nThe leading term in $\\bar\\alpha_+$ is of order\n${\\bar\\alpha_+}^{J-M+2}$.  The leading term in $\\bar\\alpha_-$\nseems to be linear in $\\bar\\alpha_-$.  However, this is not true, as\ncareful consideration shows that there are two terms linear\nin $\\bar\\alpha_-$ which cancel.  One might expect that in general\nsomething\nsimilar happens also for higher order terms in $\\bar\\alpha_-$.\nThis indeed turns out to be the case.  The easiest way to see this,\nis to do the partial integration of \\specgen\\ in the other\n\\lq\\lq direction\".  One finds\n\n$$\n\\eqalign{\n  O_{JM} =      {\\int {{dx} \\over {2\\,\\pi}}\\,}\n          \\sum_{k=0}^{J-M+1} (-)^k\\,\n           &{(J+M+1)!\\,(J-M+1)! \\over (J+M+2+k)!\\, (J-M+1-k)!}\n           \\times\\cr\n         \\times\\bigl\\{\n          &{\\bar\\alpha_+}^{J-M+1-k}\\,\n           (\\bar\\alpha_+ + 2x)^{J+M+2+k}   \\cr\n          -\n          &(\\bar\\alpha_- - 2x)^{J-M+1-k}\\,\n           {\\bar\\alpha_-}^{J+M+2+k}\n           \\bigr\\}. \\cr\n}  \\eqn  \\partialB\n$$\n\n\\noindent\nThe terms in $\\bar\\alpha_+$ and $\\bar\\alpha_-$ in \\partialA\\\nand \\partialB\\ must separately be equal, up to c-number terms\nof the form ${\\int {{dx} \\over {2\\,\\pi}}\\,} x^{2J+3}$.  Substituting the\nchange of variables \\AsTrajectory, this becomes\n$\\sim \\int_{-\\infty}^\\infty {d\\tau\\over 2\\pi}\\, e^{(2J+3)\\tau}$,\nwhich can in general be argued to vanish after an analytic\ncontinuation $\\tau\\to i\\tau$ (see below).  It therefore follows\nthat we can write the expansion\n\n$$\n\\eqalign{\n  O_{JM} =\n      &{\\int {{dx} \\over {2\\,\\pi}}\\,}\n         \\sum_{k=0}^{J+M+1} (-)^k\\,\n           {(J-M+1)!\\,(J+M+1)! \\over (J-M+2+k)!\\, (J+M+1-k)!}\\,\n           {\\bar\\alpha_+}^{J-M+2+k} \\times  \\cr\n     & \\qquad\\qquad\\qquad\\qquad \\times\n           (\\bar\\alpha_+ + 2x)^{J+M+1-k}   \\cr\n           &- {\\int {{dx} \\over {2\\,\\pi}}\\,}\n          \\sum_{k=0}^{J-M+1} (-)^k\\,\n           {(J+M+1)!\\,(J-M+1)! \\over (J+M+2+k)!\\, (J-M+1-k)!}\\,\n           {\\bar\\alpha_-}^{J+M+2+k} \\times  \\cr\n     & \\qquad\\qquad\\qquad\\qquad \\times\n           (\\bar\\alpha_- - 2x)^{J-M+1-k}.  \\cr\n}  \\eqn  \\fullCharge\n$$\n\n\\noindent\nThus to lowest order in the fields, one finds\n\n$$\n\\eqalign{\n  O_{JM} &= {1\\over J-M+2}     {\\int {{dx} \\over {2\\,\\pi}}\\,} (2x)^{J+M+1} \\,\n            {\\bar\\alpha_+}^{J-M+2}  \\cr\n       &\\ - {1\\over J+M+2}     {\\int {{dx} \\over {2\\,\\pi}}\\,} (-2x)^{J+M+1} \\,\n            {\\bar\\alpha_-}^{J+M+2}. \\cr\n}  \\eqno\\eq\n$$\n\nNow, applying the change of variables \\AsTrajectory, \\ie,\n\n$$\n\\eqalign{\n  x &= \\sqrt {\\mu\\over 2}\\, e^\\tau,      \\cr\n  \\bar\\alpha_\\pm &\\to {d\\tau\\over dx} \\bar\\alpha_\\pm,  \\cr\n}  \\eqno\\eq\n$$\n\n\\noindent\none finds that the leading order expression for the charges is\ngiven by\n\n$$\n\\eqalign{\n  O_{JM} &= {{2}^{J+1}{\\mu}^{M}\n  \\over J-M+2}    \\, {\\int {{d\\tau} \\over {2\\,\\pi}}\\,}\n  e^{2M\\tau}\\,\n            {\\bar\\alpha_+}^{J-M+2}  \\cr\n       &\\ - (-)^{J-M+1}\\,\n       {{2}^{J+1}\\, {\\mu}^{-M}\n       \\over J+M+2}\\,\n           {\\int {{d\\tau} \\over {2\\,\\pi}}\\,} e^{-2M\\tau}\\,\n            {\\bar\\alpha_-}^{J+M+2}. \\cr\n}  \\eqno\\eq\n$$\n\n\\noindent\nExpanding in right and left moving modes\n\n$$\n\\eqalign{\n  \\bar\\alpha_+ &= \\int_{-\\infty}^\\infty\n       dk\\, \\bar\\alpha(k)\\, e^{-ik(t-\\tau)},  \\cr\n  \\bar\\alpha_- &= \\int_{-\\infty}^\\infty\n       dk\\, \\bar\\beta (k)\\, e^{-ik(t+\\tau)}  \\cr\n}    \\eqno\\eq\n$$\n\n\\noindent\nand applying the rotation $\\tau\\to i\\tau$, $k\\to -ik$,\nwe find that in terms of the analytically continued oscillators\n\n$$\n\\eqalign{\n   \\alpha(k) &\\equiv \\bar\\alpha(-ik),  \\cr\n   \\beta(k) &\\equiv \\bar\\beta(-ik)   \\cr\n} \\eqn\\defB\n$$\n\n\\noindent\nthe charges have the form\n\n$$\n\\eqalign{\n  O_{JM} = &{{2}^{J+1}{\\mu}^{M}\n  \\over J-M+2}\\, i   \\int dk_1 \\dots dk_{J-M+2}\\, \\times \\cr\n    &\\qquad \\times\\alpha (k_1) \\dots \\alpha (k_{J-M+2}) \\,\n      \\delta\\left(\\sum k_i + 2M\\right)  \\cr\n       &\\ - (-)^{J-M+1}\\,\n       {{2}^{J+1} {\\mu}^{-M}\n       \\over J+M+2}\\,i  \\int dp_1 \\dots dp_{J+M+2}\\,\\times  \\cr\n    &\\qquad \\times\\beta  (p_1) \\dots \\beta  (p_{J+M+2}) \\,\n      \\delta\\left(\\sum p_i + 2M\\right).  \\cr\n}  \\eqn\\fullrep\n$$\n\nWe emphasize that this is the expression\nfor the charges to lowest order in the fields, which corresponds to the\nleading order in $\\mu$. The full expression \\fullCharge\\ has\ncorrections in $1\/\\mu$ that are higher order polynomials in the\nfields.  In the remainder of the discussion we do not consider\nthese corrections.\n\nDefining\n\n$$\n\\eqalign{\n  a(k) &\\equiv \\alpha(k), \\qquad b(p) \\equiv \\beta(p),  \\cr\n  a^\\dagger(k) &\\equiv \\alpha (-k)\/k,\n  \\qquad b^\\dagger(p) \\equiv \\beta(-p)\/p  \\cr\n}  \\eqno\\eq\n$$\n\n\\noindent\nsatisfying $[a(k), a^\\dagger(k')] = \\delta (k - k')$,\n         $[b(p), b^\\dagger(p')] = \\delta (p-p')$,\nwe have the expressions of [11] (up to an inessential difference\nin normalization),\nplus additional contributions.  To see\nthese, note that in addition to the term\n\n$$\n\\eqalign{\n             {2}^{J+1}{\\mu}^{M}\\,\n               i  & \\int_0^\\infty dk\n               \\int_0^\\infty dk_1 \\dots dk_{J-M+1}\\times  \\cr\n  &\\quad \\times k\\,a^\\dagger(k)\\,\n    a (k_1) \\dots a (k_{J-M+1}) \\,\n      \\delta\\left(\\sum k_i - k + 2M\\right)  \\cr\n}\n\\eqn \\termA\n$$\n\n\\noindent\nfound in [11], we in general also have terms of higher\norder in the creation operators.  The next term would be, for\nexample\n\n$$\n\\eqalign{\n             {2}^{J+1}{\\mu}^{M} (J-M  &  +1)\\,\n               i  \\int_0^\\infty dk\\,dk'\n               \\int_0^\\infty dk_1 \\dots dk_{J-M  }\\,\\times  \\cr\n   &\\times  kk'\\,a^\\dagger(k)\\, a^\\dagger(k')\\,\n    a (k_1) \\dots a (k_{J-M }) \\,\n      \\delta\\left(\\sum k_i - k - k'+ 2M\\right).  \\cr\n}\n\\eqn\\termB\n$$\n\n\\noindent\nIf $M<0$, we also get an additional contribution of the form\n\n$$\n  \\eqalign{\n            {{2}^{J+1}{\\mu}^{M}\\over J-M+2}\\,\n               i\n               &\\int_0^\\infty dk_1 \\dots dk_{J-M+2}\\,\\times  \\cr\n    &\\ \\times a (k_1) \\dots a (k_{J-M+2}) \\,\n      \\delta\\left(\\sum k_i + 2M\\right).  \\cr\n}\n\\eqn \\termC\n$$\n\nThese additional contributions have to be included in order\nto obtain a representation of the algebra  \\algebra.  The reason\nfor this is that terms of the type \\termB, commuted\nwith terms of the type \\termC, give additional contributions of\nthe type \\termA, which are needed to again obtain a member of the\nalgebra on the right hand side.  This effect cannot be produced\nby only using terms of the type \\termA.  To see where the\nrepresentation \\termA\\ fails, one has to take careful account of\nthe regions of momentum integration.  For example, if one were\nto use only terms of the type \\termA\\ one would find for the\ncommutator\n\n$$\n\\eqalign{\n \\left[ O_{MM}, O_{{1 \\over 2},-{1 \\over 2}} \\right]\n       =  &\\int_0^\\infty dk_1 dk_2\\,\n          (-2k_1 - 2k_2 - 4M)\\, (k_1+k_2+M-{1 \\over 2}) \\times \\cr\n    &\\qquad\\qquad\\times a^\\dagger (k_1+k_2+M-{1 \\over 2})\\,a(k_1)\\,a(k_2)\n\\cr\n         &+\\int_{0,k_1+k_2>{1 \\over 2}}^\\infty dk_1 dk_2\\,\n          (2k_1 + 2k_2 - 1)\\, (k_1+k_2+M-{1 \\over 2}) \\times  \\cr\n  &\\qquad\\qquad\\times a^\\dagger (k_1+k_2+M-{1 \\over 2})\\,a(k_1)\\,a(k_2).\n\\cr\n}\n\\eqno\\eq\n$$\n\n\\noindent\nwhich would give $(-4M-1)\\, O_{M+{1 \\over 2},M-{1 \\over 2}}$, were it not for\nthe fact that the regions of integration do not match.  It is now\nnot difficult to see how to fix the representation \\termA.  Simply\nremove the restrictions on the ranges of integration, \\ie, take\nthem to be $\\int_{-\\infty}^\\infty dk$ instead of\n$\\int_{0}^\\infty dk$.  This solves the problem on a formal level,\nand imposing the reality conditions $a_{-n} = n a^\\dagger_n\n\\equiv \\alpha_{-n}$, we recover our full representation \\fullrep.\n\nOne can now ask whether the Ward identities derived in [11] for\nthe tachyon scattering amplitudes will be affected by these\ncorrections.  As we will show in the next section, they will\nnot be affected.\n\n\n\\section {Ward Identities}\n\n\\noindent\nOne can now identify the spectrum generating operators\nas we did for the tachyons by comparing quantum numbers as\nin \\Identification, or alternatively, by imposing Dirichlet\nboundary conditions as in \\IdentificationB.\nOne simply requires\n\n$$\n  O_{JM,+} = - O_{JM,-},   \\eqno\\eq\n$$\n\n\\noindent\nwhich implies that to leading order\n\n$$\n\\eqalign{\n  O_{JM} &= 2i\\,{{2}^{J+1}{\\mu}^{M}\n  \\over J-M+2}\\,   \\int dk_1 \\dots dk_{J-M+2}\\times  \\cr\n    &\\qquad \\times \\alpha (k_1) \\dots \\alpha (k_{J-M+2}) \\,\n      \\delta\\left(\\sum k_i + 2M\\right)  \\cr\n       &= 2i\\,(-)^{J-M+1}\\,\n       {{2}^{J+1} {\\mu}^{-M}\n       \\over J+M+2}\\,  \\int dp_1 \\dots dp_{J+M+2} \\times \\cr\n    &\\qquad \\times\\beta  (p_1) \\dots \\beta  (p_{J+M+2}) \\,\n      \\delta\\left(\\sum p_i + 2M\\right).  \\cr\n}  \\eqn\\specgenIdent\n$$\n\n\\noindent\nThis identification\nwill, in practice, be very useful in\nexplicit calculations of\nWard identities, as will be seen below.\n\nThe \\lq\\lq bulk\" scattering amplitudes only involve\nfixed, discrete values of the outgoing momenta.  This can be\ninterpreted in our formalism as follows:  Imposing the above\nidentification of quantum numbers, one has\n\n$$\n\\eqalign{\n  T^{(-)}_{2M} \\equiv O_{M-1,M}\n    &= 2i\\, (-)^{2M+1} \\left( {2\\over\\mu}\\right)^M \\beta(2M)  \\cr\n    &= 2i\\, (2\\mu)^M \\int dk_1\\dots dk_{2M+1}\\,\n        \\alpha(k_1) \\dots \\alpha (k_{2M+1})\n         \\,\\delta (\\sum k_i - 2M).   \\cr\n}  \\eqn\\Tident\n$$\n\nThus, in terms of the oscillators $\\alpha(k)$ and $\\beta(p)$\ndefined in\n\\defA\\ and \\defB, we find an S-matrix that is different from the\none we previously calculated in terms of the \\lq\\lq asymptotic\"\nvariables \\AsTrajectory.  In particular, to leading order\nan out state $\\langle 0|\\beta (2M)$ is $(2M+1)$-linear in\n$\\alpha$, so that a correlation function\n$\\langle\\, \\beta(p)\\alpha(k_1)\\dots\\alpha(k_N)\\,\\rangle$ can only\nbe nonvanishing to this order if\n\n$$\n  p = N-1.   \\eqn \\sumrule\n$$\n\n\\noindent\nExcept for an overall factor of ${1 \\over 2}$, due to different normalization\nof the momentum, this agrees with the \\lq\\lq sum rule\" stated in [11].\n\nNow, to see how the Ward identities can be derived in our formalism,\nnote that if one has an operator $O$ that annihilates the vacuum from\nthe left and the right, \\ie,\n\n$$\n  \\langle 0| O = 0 = O |0\\rangle,   \\eqno\\eq\n$$\n\n\\noindent\nthen, starting from the expectation value\n\n$$\n  \\langle \\,\\beta(p)\\, O \\,\\alpha(k_1) \\dots \\alpha(k_N)\\,\\rangle\n   \\eqn\\expectation\n$$\n\n\\noindent\none can write, commuting $O$ respectively to the left and to\nthe right\n\n$$\n  \\langle\\, [\\beta(p),O] \\,\\alpha(k_1) \\dots \\alpha(k_N)\\,\\rangle\n    =\n  \\langle\\, \\beta(p)\\, [O, \\alpha(k_1) \\dots \\alpha(k_N)]\\,\\rangle.\n \\eqn  \\wardId\n$$\n\n\\noindent\nThis equation expresses the Ward identities, and for suitable\nchoices of the charges $O$, can be used to derive recursion relations\nrelating scattering amplitudes.\n\nIn general, however, our representation \\fullrep\\ of the\ncharges $O_{JM}$ have terms of the type $aa\\dots a$ or $a^\\dagger\na^\\dagger\\dots a^\\dagger$, and therefore would fail to annihilate the\nvacuum from either the left or the right.  Also, in addition to the\nterms \\termA, which were considered in the analysis of [11], one\nmight expect corrections to the Ward identities derived in [11]\ndue to our extra terms such as \\termB\\ and \\termC.  However, counting\nnumbers of creation and annihilation operators, one sees that indeed\nonly terms of the type \\termA\\ contribute to the Ward identities.\nFor example, consider the Ward identity relating $N+1\\to 1$\namplitudes to $N\\to 1$ amplitudes.  The relevant charge is\n$O_{{1 \\over 2},-{1 \\over 2}} \\sim aaa + a^\\dagger a a + a^\\dagger a^\\dagger a$,\nand we should take the momentum of the out state to be $p=N-1$.\nThe out state is then, from our previous discussion, of order\n\n$$\n\\langle 0|\\,\\beta(p) \\sim \\langle 0|a^N,  \\eqno\\eq\n$$\n\n\\noindent\nso that one can write \\expectation\\ as\n\n$$\n  \\eqalign{\n  &\\langle \\,\\beta(p)\\, O_{{1 \\over 2}, -{1 \\over 2}}\\,\n  \\alpha(k_1) \\dots \\alpha(k_{N+1})\\,\\rangle  \\cr\n  &\\ \\sim \\langle a^N \\,( aaa + a^\\dagger aa + a^\\dagger\n       a^\\dagger a )\\, (a^\\dagger)^{N+1}\\rangle. \\cr\n}  \\eqno \\eq\n$$\n\n\\noindent\nIt immediately follows that only the term linear in $a^\\dagger$\ncontributes.  This argument generalizes to the identity for\nexpressing $N\\to 1$ amplitudes directly in terms of $2\\to1$\namplitudes, where the relevant charge is $Q_{N\/2-1, -N\/2-1}$.\nThe conclusion is therefore that for the purpose of deriving\nthese Ward identities, it\nis sufficient to consider only the terms \\termA\\ linear in\nthe creation operators, as was done in [11].\n\nFinally,\nas an example, we calculate the Ward identity relating\n$3\\to 1$ amplitudes to $2\\to 1$ amplitudes.  Using the\nidentification \\specgenIdent, we have\n\n$$\n\\eqalign{\n  Q_{{1 \\over 2},-{1 \\over 2}}\n   &= {4\\sqrt2\\, i\\over \\sqrt\\mu}\n      \\int_0^\\infty dk_1 dk_2 dk_3\\,\\, k_1 \\,a^\\dagger (k_1)\\,\n         a(k_2)\\, a(k_3)\\, \\delta(-k_1+k_2+k_3-1)  \\cr\n   &= {4\\sqrt{2\\mu}\\, i}\n      \\int_0^\\infty dp_1 dp_2 \\,\\, p_1 \\,b^\\dagger (p_1)\\,\n         b(p_2)\\, \\delta(-p_1+p_2-1).  \\cr\n}  \\eqno\\eq\n$$\n\n\\noindent\nup to terms that we have argued to be irrelevant.\nInserting this into the general formula \\wardId, for $p=1$ and\n$k_1+k_2+k_3 = 2$, we obtain the Ward identity\n\n$$\n  \\langle\\, b(2)\\,a^\\dagger(k_1)\\,a^\\dagger(k_2)\\,a^\\dagger(k_3)\n  \\,\\rangle\n  = {1\\over\\mu}\\, (k_1+k_2-1)\\,\n  \\langle\\, b(1)\\,a^\\dagger(k_1+k_2-1)\\, a^\\dagger(k_3)\\,\\rangle\n   + {\\rm cyclic}.\n$$\n\nIt is also possible to derive recursion relations in our formalism\nusing methods similar to those used in [11].  The argument\nroughly goes as follows:  The operators\n\n$$\n  O_{NN} \\equiv \\int dx \\int d\\alpha\\, (\\alpha+x)^{2N+1} (\\alpha-x)\n$$\n\n\\noindent\nhave quantum numbers\n$\n  p_x = N = p_\\tau,\n$\nwhile $T_k$ has $p_x = k$, $p_\\tau = -1 + k$.\nAdding these, it follows that the commutator $[\\,O_{NN}, T_k\\,]$\nhas quantum numbers $p_x = N+k$, $p_\\tau = -1 + (n+k)$, and should\ntherefore be identified with $T_{k+N}$, \\ie,\n\n$$\n  [\\,O_{NN}, T_k\\,] \\sim T_{k+n}.  \\eqno\\eq\n$$\n\n\\noindent\nSimilarly, the charge $O_{N+M,M}$ has quantum numbers\n$p_x = M$, $p_\\tau = N+M$.  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