diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzeni" "b/data_all_eng_slimpj/shuffled/split2/finalzeni"
new file mode 100644--- /dev/null
+++ "b/data_all_eng_slimpj/shuffled/split2/finalzeni"
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction and Motivation} \nIn the background of any investigation into learning algorithms are no-free-lunch phenomena: roughly, the observation that assumption-free statistical learning is infeasible in general (see, e.g., \\cite[Ch. 5]{MLbook} for a formal statement). Common wisdom is that learning algorithms and architectures must adequately reflect non-trivial features of the data-generating distribution to gain inductive purchase. \n\nFor many purposes we need to move beyond passive observation, focusing instead on what \\emph{would} happen were we to \\emph{act} upon a given system. Even further, we sometimes desire to \\emph{explain} the behavior of a system, raising questions about what \\emph{would have} occurred had some aspects of a situation been different. Such questions depend not just on the data distribution; they depend on deeper features of underlying data-generating \\emph{processes} or \\emph{mechanisms}. It is thus generally acknowledged that stronger assumptions are  required if we want to draw \\emph{causal} conclusions from data \\citep{Spirtes2001,Pearl2009,Imbens2015,Peters,9363924}. \n\n\nWhether implicit or explicit, any approach to causal inference involves a space of candidate causal models, viz. data-generating processes. Indeed, a blunt way of incorporating inductive bias is simply to \\emph{omit} some class of possible causal hypotheses from consideration. Many (im)possibility results in the literature can accordingly be understood as pertaining to all models within a class. \nFor instance, if we can restrict attention to \\emph{Markovian} models that satisfy \\emph{faithfulness}, then we can \\emph{always} identify the structure of a model from experimental data (e.g., \\cite{Eberhardt2005,Spirtes2001}). If we can restrict attention to Markovian (continuous) models with linear functions and \\emph{non}-Gaussian noise, then \\emph{every} model can be learned even from purely observational data \\citep{JMLR:v7:shimizu06a}. As a negative example, in the larger class of (not necessarily Markovian) models, \\emph{no} model can \\emph{ever} be determined from observational data alone \\citep{Spirtes2001,BCII2020}.\n\nAt the same time, in many settings it is sensible to aim for results with ``nearly universal'' force. It is natural to ask, e.g., within the class of all Markovian models, how ``typical'' are those in which the faithfulness condition is violated? This might tell us, for instance, how typically we could expect failure of a method that depended on these assumptions. A well-known result shows that, fixing any particular causal dependence graph, such violations have \\emph{measure zero} for any smooth (e.g., Lebesgue) measure on the parameter space of distributions consistent with that graph \\citep{Meek1995}. In fact, the standard notion of statistical consistency itself, which underlies many possibility results in causal inference, requires omission of some purportedly ``negligible'' set of possible data streams \\citep{DiaconisFreedman,Spirtes2001}.\n\nThere are two standard mathematical approaches to making concepts like ``typical'' and ``negligible'' rigorous: \\emph{measure-theoretic} and \\emph{topological}. While the two approaches often agree, they capture slightly different intuitions \\citep{Oxtoby}.  One virtue of the measure-theoretic approach is its natural  probabilistic interpretation: intuitively, we are exceedingly \\emph{unlikely} to hit upon a set with measure zero. At the same time, the measure-theoretic approach is sometimes criticized in statistical settings for its alleged  dependence on a measure, and this has been argued to favor topological approaches (see, e.g.,  \\cite{BELOT2020159} on no-free-lunch theorems). The latter of course in turn demands an appropriate topology.\n\nIn the present work we show how to define a sequence of meaningful topologies on the space of causal models, each corresponding to a progressively coarser level of the so called \\emph{causal hierarchy} (\\cite{pearl2018book,BCII2020}; see Fig. \\ref{fig:hierarchy} for an abbreviated pictorial summary).\nWe aim to demonstrate that topologizing causal models in this way helps clarify the scope and limits of causal inference under different assumptions, as well as the potential empirical status of those very assumptions, in a highly general setting. \n\nOur starting point is a canonical topology on the space of Borel probability distributions called the \\emph{weak topology}. The weak topology is grounded in the fundamental notion of \\emph{weak convergence} of probability distributions \\citep{Billingsley} and is thereby closely related to problems of statistical inference (see, e.g., \\cite{DemboPeres}). Recent work has sharpened this correspondence, showing that open sets in the weak topology correspond exactly to the statistical hypotheses that can be naturally deemed  \\emph{verifiable} \\citep{Genin2018,GeninKelly}.\nWe extend the correspondence to higher levels of the causal hierarchy, including the most refined and expansive ``top'' level consisting of all (well-founded) causal models. Lower levels and  natural subspaces (e.g., corresponding to prominent causal assumption classes) emerge as coarsenings and projections of this largest space. As an illustration of the general approach, we prove a topological version of the causal hierarchy theorem from \\cite{BCII2020}. Rather than showing that collapse happens only in a measure zero set  as in \\cite{BCII2020}, our Theorem \\ref{thm:hierarchyFormal} show that collapse is \\emph{topologically meager}. Conceptually, this highlights a  different (but complementary) intuition:\nnot only is collapse  exceedingly unlikely in the sense of measure, meagerness implies that collapse could never be \\emph{statistically verified}. Correlatively, this implies that any causal assumption that would generally allow us to infer counterfactual probabilities from experimental (or ``interventional'') probabilities must itself be statistically unverifiable (Corollary \\ref{cor:hierarchy}). \n\nTo derive such a result we actually show something slightly stronger (see Lem.~\\ref{lem:separation}): even with respect to the subspace of models consistent with a fixed \\emph{temporal order} on variables, the causal hierarchy theorem holds. Merely knowing the temporal order of the variables is not enough to render collapse of the hierarchy a statistically verifiable proposition.\nFurthermore, we show that the \\emph{witness to collapse} can be taken as any of the well-known counterfactual ``probabilities of causation'' (see, e.g., \\cite{Pearl1999}): probabilities of necessity, sufficiency, necessity \\emph{and} sufficiency, enablement, or disablement. That is, none of these important quantities are fully determined by experimental data except in a meager set. \n\n\n\nIn \\S\\ref{scms} we give background on causal models, and in \\S\\ref{sec:causalhierarchy} we present a model-theoretic characterization of the causal hierarchy as a sequence of spaces. Topology is introduced in \\S\\ref{sec:weaktopology}, and the main results about collapse appear in \\S\\ref{sec:main}.\nFor the technical results, we include proof sketches in the main text to provide the core intuitions, relegating some of the details to an exhaustive technical appendix, which also includes additional supplementary material.\n\n\n\\section{Structural Causal Models} \\label{scms}\nA fundamental building block in the theory of causality is the \\emph{structural causal model} \\citep{Pearl1995,Spirtes2001,Pearl2009} or SCM, which formalizes the notion of a data-generating process. In addition to specifying data-generating distributions, these models also specify the generative mechanisms that produce them. For the purpose of causal inference and learning, SCMs provide a broad, fine-grained hypothesis space.\n\nThe notions in this section have their usual definition following, e.g., \\cite{Pearl2009}, but we have\nrecast them in the standard language of Borel probability spaces so as to handle the case of infinitely many variables rigorously.\nWe start with notation, basic assumptions, and some probability theory\n\n\n\\begin{notation*}\nThe signature (or range) of a variable $V$ is denoted $\\chi_V$. \nWhere $\\*S$ is a set of variables, let $\\chi_{\\*S} = \\bigtimes_{S \\in \\*S} \\chi_S$.\nGiven an indexed family of sets $\\{S_\\beta\\}_{\\beta \\in B}$ and elements $s_\\beta \\in S_\\beta$, let $(s_\\beta)_\\beta$ denote the tuple whose element at index $\\beta$ is $s_\\beta$, for all $\\beta$.\nFor $B' \\subset B$ write $\\pi_{B'} : \\bigtimes_{\\beta \\in B} S_\\beta \\to \\bigtimes_{\\beta \\in B'} S_{\\beta}$ for the \\emph{projection map} sending each $(s_\\beta)_{\\beta \\in B} \\mapsto (s_{\\beta'})_{\\beta' \\in B'}$; abbreviate $\\pi_{\\beta'} = \\pi_{\\{\\beta'\\}}$, where $\\beta' \\in B$. \n\\end{notation*}\n\nThe reader is referred to standard texts \\cite{johnkelley1975,Bogachev2007} for elaboration on the concepts used below.\n\n\\begin{definition}[Topology]\\label{def:basictopology}\nFor discrete spaces (like $\\chi_S$, for a single categorical variable $S$) we use the discrete topology and for product spaces (like $\\chi_{\\*S}$ for a set of variables $\\*S$) we use the product topology.\nNote that the so-called \\emph{cylinder sets} of the form $\\pi^{-1}_{\\*Y}(\\{\\*y\\})$ for finite subsets $\\*Y \\subset \\*S$ and $\\*y \\in \\chi_{\\*Y}$\nform a basis for the product topology on $\\chi_{\\*S}$. This cylinder set is a subset of $\\chi_{\\*S}$, and contains exactly those valuations agreeing with the value $\\pi_Y(\\*y)$ specified in $\\*y$ for $Y$, for every $Y \\in \\*Y$. Following standard statistical notation this cylinder is abbreviated as simply $\\*y$.\n\\end{definition}\n\\begin{definition}[Probability]\n\\label{def:probability}\nWhere $\\vartheta$ is a topological space write $\\mathcal{B}(\\vartheta)$ for its Borel $\\sigma$-algebra of measurable subsets.\nLet $\\mathfrak{P}(\\vartheta)$ be the set of probability measures on $\\mathcal{B}(\\vartheta)$.\nSpecifically, elements of $\\mathfrak{P}(\\vartheta)$ are functions $\\mu: \\mathcal{B}(\\vartheta) \\to [0, 1]$ assigning a probability to each measurable set such that $\\mu(\\vartheta) = 1$ and $\\mu\\big(\\bigcup_{i=1}^{\\infty}(S_i)\\big) = \\sum_{i=1}^{\\infty} \\mu(S_i)$ for each sequence $S_1, S_2, \\dots$ of pairwise disjoint sets from $\\mathcal{B}(\\vartheta)$.\nA map $f: \\vartheta_1 \\to \\vartheta_2$ is said to be \\emph{measurable} if $f^{-1}(S_2) \\in \\mathcal{B}(\\vartheta_1)$ for every $S_2 \\in \\mathcal{B}(\\vartheta_2)$.\n\\end{definition}\n\n\\begin{fact}[Lemma~1.9.4 \\cite{Bogachev2007}]\nA Borel probability measure is determined by its values on a basis\n\\end{fact}\n\n\\subsection{SCMs, Observational Distributions}\\label{ss:scml1}\n\nLet $\\*V$ be a set of \\emph{endogenous variables}.\nWe assume for simplicity every variable $V \\in \\*V$ is dichotomous\nwith $\\chi_V = \\{0, 1\\}$, although the results here generalize to any larger countable range.\nInfluences among endogenous variables are the main phenomena our formalism aims to capture.\nA well-founded\\footnote{See Appendix~\\ref{app:causalmodels} for additional background on orders and relations.} \\emph{direct influence} relation $\\rightarrow$ on $\\*V$ encapsulates the notion of one endogenous variable possibly influencing another. For each $V \\in \\*V$, we call $\\{V' \\in \\*V : V' \\rightarrow V\\} = \\mathbf{Pa}(V)$ the \\emph{parents} of $\\*V$.\nWe assume every set $\\mathbf{Pa}(V)$ is finite; this condition is called \\emph{local finiteness}. These two assumptions (well-foundedness and local finiteness) generalize the common \\emph{recursiveness} assumption to the infinitary setting, and have an alternative characterization in terms of ``temporal'' orderings:\n\\begin{fact}\\label{fact:omegalike}\nSay that a total order $\\prec$ on $\\*V$ is \\emph{$\\omega$-like} if every node has finitely many predecessors: for each $V \\in \\*V$, the set $\\{V' : V' \\prec V\\}$ is finite.\nThen the influence relation $\\rightarrow$ is extendible to an $\\omega$-like order iff $\\rightarrow$ is well-founded and locally finite.\n\\end{fact}\nIn addition to endogenous variables, causal models have \\emph{exogenous variables} $\\*U$.\nEach endogenous $V$ depends on a subset $\\*U(V)\\subset \\*U$ of ``exogenous parents'' and\nuncertainty enters via exogenous noise, that is, a distribution from $\\mathfrak{P}(\\chi_{\\*U})$.\nA \\emph{structural function} (or \\emph{mechanism}) for $V \\in \\*V$ is a measurable $f_{V} : \\chi_{\\mathbf{Pa}(V)} \\times \\chi_{\\*U(V)} \\to \\chi_{V}$\nmapping parental endogenous and exogenous valuations to values.\n\\begin{definition}\\label{def:scm:lit}\nA \\emph{structural causal model} is a tuple $\\mathcal{M} = \\langle \\*U, \\*V, \\{f_V\\}_{V \\in \\*V}, P \\rangle$ where $\\*U$ is a collection of exogenous variables, $\\*V$ is a collection of endogenous variables, $f_V$ is a structural function for each $V \\in \\*V$, and $P \\in \\mathfrak{P}(\\chi_{\\*U})$ is a probability measure on (the Borel $\\sigma$-algebra of) $\\chi_{\\*U}$.\n\\end{definition}\n\nAs is well known, recursiveness implies that each $\\*u \\in \\chi_{\\*U}$ induces a unique $\\*v \\in \\chi_{\\*V}$ that solves the simultaneous system of structural equations $\\{V = f_V\\}_V$:\n\\begin{proposition}\nAny SCM ${\\mathcal{M}}$ with well-founded, locally finite parent relation $\\rightarrow$ induces a unique measurable $m^{{\\mathcal{M}}} : \\chi_{\\*U} \\to \\chi_{\\*V}$ such that $f_V\\big( \\pi_{\\mathbf{Pa}(V)}(m^{{\\mathcal{M}}}(\\*u)), \\pi_{\\*U(V)}(\\*u) \\big) = \\pi_{V}\\big(m^{{\\mathcal{M}}}(\\*u)\\big)$ for all $\\*u \\in \\chi_{\\*U}$ and $V \\in \\*V$.\n\\end{proposition}\nMeasurability then entails that the exogenous noise $P$ induces a distribution on joint valuations of $\\*V$, called the \\emph{observational distribution},\nwhich characterizes passive observations of the system.\n\\begin{definition}\nThe observational distribution $p^{{\\mathcal{M}}} \\in \\mathfrak{P}(\\chi_{\\*V})$ is defined on open sets by $p^{{\\mathcal{M}}}(\\*y) = P\\big((m^{{\\mathcal{M}}})^{-1}(\\*y)\\big)$. Here recall that $\\*y$ represents a cylinder subset (Definition~\\ref{def:basictopology}) of $\\chi_{\\*V}$.\n\\end{definition}\n\n\n\n\n\n\n\\subsection{Interventions}\\label{ss:scml2}\nWhat makes SCMs distinctively causal is the way they accommodate statements about possible manipulations of a causal setup capturing, e.g., observations resulting from a controlled experimental trial. This is formalized in the following definition.\n\\begin{definition}\nAn \\emph{intervention} is a choice of a finite subset of variables $\\*W \\subset \\*V$ and $\\*w \\in \\chi_{\\*W}$. This intervention is written $\\*W \\coloneqq \\*w$, and we let $A$ be the set of all interventions.\nUnder this intervention, each $W \\in \\*W$ is held fixed to its value $\\pi_{W}(\\*w) \\in \\chi_W$ in $\\*w$ while the mechanism for any $V \\in \\*V \\setminus \\*W$ is left unchanged.\nSpecifically, where ${\\mathcal{M}}$ is as in Definition \\ref{def:scm:lit}, the manipulated model for $\\*W \\coloneqq \\*w$ is the model ${\\mathcal{M}}_{\\*W \\coloneqq \\*w} = \\langle \\*U, \\*V, \\{f^{\\*W \\coloneqq \\*w}_V\\}_{V \\in \\*V}, P\\rangle$ where\n\\begin{align*}\nf^{\\*W \\coloneqq \\*w}_V = \\begin{cases}\nf_V, & V \\notin \\*W\\\\\n\\text{constant func. mapping to } \\pi_V(\\*w), & V \\in \\*W.\n\\end{cases}\n\\end{align*}\n\\end{definition}\nThe \\emph{interventional} or \\emph{experimental distribution} $p^{\\mathcal{M}_{\\*W \\coloneqq \\*w}} \\in \\mathfrak{P}(\\chi_{\\*V})$ is just the observational distribution for the manipulated model ${\\mathcal{M}}_{\\*W \\coloneqq \\*w}$, and it encodes the probabilities for an experiment in which the variables $\\*W$ are fixed to the values $\\*w$.\n\\begin{remark}\nEmpty interventions $\\varnothing \\coloneqq ()$ are just passive observations, i.e., $p^{{\\mathcal{M}}_{\\varnothing \\coloneqq ()}} = p^{{\\mathcal{M}}}$.\n\\end{remark}\n\n\n\\subsection{Counterfactuals}\\label{ss:scml3}\nBy permitting multiple manipulated settings to share exogenous noise, not only the distribution arising from a single manipulation, but also joint distributions over multiple can be considered. These are often called \\emph{counterfactuals}. The set $\\mathfrak{P}(\\chi_{A \\times \\*V})$ encompasses the combined joint distributions over $\\*V$ for any combination of interventions from $A$. A basis for the space $\\chi_{A \\times \\*V}$ are the cylinder sets of the following form, for some sequence $(\\*X \\coloneqq \\*x, \\*Y), \\dots, (\\*W \\coloneqq \\*w, \\*Z)$ of pairs, where $ \\*Y, \\dots, \\*Z \\subset \\*V$ are finite, and $\\*X \\coloneqq \\*x, \\dots, \\*W \\coloneqq \\*w \\in A$ are interventions:\n\\begin{align*}\n    \\pi^{-1}_{\\{\\*X \\coloneqq \\*x\\}\\times \\*Y}(\\{\\*y\\}) \\cap \\dots \\cap \\pi^{-1}_{\\{\\*W \\coloneqq \\*w\\}\\times \\*Z}(\\{\\*z\\}).\n\\end{align*}\nWe will abbreviate this open set as ${\\*y}_{\\*x}, \\dots, {\\*z}_{\\*w}$, writing, e.g. simply $\\*x$ for the intervention $\\*X = \\*x$.\n\\begin{definition}\nGiven ${\\mathcal{M}}$, define a counterfactual distribution $p_{\\text{cf}}^{{\\mathcal{M}}} \\in \\mathfrak{P}(\\chi_{A \\times \\*V})$ on a basis as follows:\n\\begin{equation*}\n    p_{\\text{cf}}^{{\\mathcal{M}}}( \\*y_{\\*x}, \\dots, {\\*z}_{\\*w} ) = P\\big((m^{{\\mathcal{M}}_{\\*X \\coloneqq \\*x}})^{-1}(\\*y) \\cap  \\dots \\cap (m^{{\\mathcal{M}}_{\\*W \\coloneqq \\*w}})^{-1}(\\*z)\\big).\n\\end{equation*}\nHere, the letters $\\*y, \\dots, \\*z$ on the right-hand side abbreviate the respective cylinder sets (Definition~\\ref{def:basictopology}) $\\pi^{-1}_{\\*Y}(\\{\\*y\\}), \\dots, \\pi^{-1}_{\\*Z}(\\{\\*z\\})$.\n\\end{definition}\n\\begin{remark}\nMarginalizing $p^{{\\mathcal{M}}}_{\\text{cf}}$ to any single intervention $\\*W \\coloneqq \\*w$ yields $p^{{\\mathcal{M}}_{\\*W \\coloneqq \\*w}}$. If $\\chi_{\\*U}$ is finite, we obtain a familiar \\cite{Galles1998} sum formula\n$p_{\\text{cf}}^{{\\mathcal{M}}}(\\*y_{\\*x}, \\dots, {\\*z}_{\\*w}) = \\sum_{\\{\\*u \\mid m^{{\\mathcal{M}}_{\\*X \\coloneqq \\*x}}(\\*u) \\in \\*y, \\dots, m^{{\\mathcal{M}}_{\\*W \\coloneqq \\*w}}(\\*u) \\in \\*z\\}} P(\\*u)$.\n\\end{remark}\n\n\\begin{example} As a very simple example (drawn from \\cite{Pearl2009,BCII2020}), just to illustrate the previous definitions and notation, consider a scenario with two binary exogenous variables $\\mathbf{U} = \\{U_1,U_2\\}$ and two binary endogenous variables $\\mathbf{V} = \\{X,Y\\}$. Let $U_1,U_2$ both be uniformly distributed, and define $f_X:\\chi_{U_1} \\rightarrow \\chi_X$ to be the identity, and $f_Y:\\chi_X \\times \\chi_{U_2} \\rightarrow \\chi_Y$ by $f_Y(u,x) = ux + (1-u)(1-x)$. This fully defines an SCM ${\\mathcal{M}}$ with influence $X \\rightarrow Y$, and produces an observational distribution $p^{\\mathcal{M}}$ such that $p^{\\mathcal{M}}(x,y) = \\nicefrac{1}{4}$ for all four settings $X=x,Y=y$. \n\nThe space $A$ of interventions in this example includes the empty intervention and all combinations of $X:=x$ and $Y:=y$, with $x,y \\in \\{0,1\\}$. Notably, all interventional distributions here collapse to observational distributions, e.g., $p^{{\\mathcal{M}}_{X:= x}}(X,Y) = p^{\\mathcal{M}}(X,Y)$, for both values of $x$. Thus, ``experimental'' manipulations of this system reveal little interesting causal structure. The counterfactual distribution $p^{\\mathcal{M}}_{\\mathrm{cf}}$, however, does not trivialize. For instance, $p^{\\mathcal{M}}_{\\mathrm{cf}}((X:=1, Y=1),(X:=0,Y=0)) = \\nicefrac{1}{2}$. This term is known as the \\emph{probability of necessity and sufficiency} \\cite{Pearl1999}, which we can abbreviate by $p^{\\mathcal{M}}_{\\mathrm{cf}}(y_x,y'_{x'})$. Note that $p^{\\mathcal{M}}_{\\mathrm{cf}}(y_x,y'_{x'}) \\neq  p^{\\mathcal{M}}_{\\mathrm{cf}}(y_x)p^{\\mathcal{M}}_{\\mathrm{cf}}(y'_{x'}) = \\nicefrac{1}{4}$. Similarly, $p^{\\mathcal{M}}_{\\mathrm{cf}}(y'_x,y_{x'}) = \\nicefrac{1}{2}$.\n\n\\end{example}\n\n\n\n\n\\subsection{SCM classes}\nWe now define several subclasses of SCMs that we will use throughout the paper.\nNotably, we do not require their endogenous variable sets $\\*V$ to be finite. It is infinite in many applications, for instance, in time series models, or generative models defined by probabilistic programs (see, e.g., \\citep{II2019,Tavares}). Because the proofs call for slightly different methods, we deal with the infinite and finite cases separately. We make one additional assumption in the infinite case.\n\\begin{definition}\n$\\mu \\in \\mathfrak{P}(\\vartheta)$ is \\emph{atomless} if $\\mu(\\{t\\}) = 0$ for each $t \\in \\vartheta$;\n ${\\mathcal{M}}$ is atomless if $p^{{\\mathcal{M}}}_{\\text{cf}}$ is atomless.\n\\end{definition}\nIntuitively, an atomless distribution is one in which weight is always ``smeared'' out continuously and there are no point masses; infinitely many fair coin flips, for example, generate an atomless distribution as the probability of obtaining any given infinite sequence is zero.\n\\begin{definition}\nFor the remainder of the paper, fix a countable endogenous variable set $\\*V$. Define the following classes of SCMs:\n\\begin{align*}\n    \\mathfrak{M}_\\prec &= \\text{SCMs over } \\*V \\text{ whose influence relation is extendible to the } \\omega\\text{-like order } \\prec;\\\\\n    \\mathfrak{M}_X &= \\text{SCMs over } \\*V \\text{ in which the variable } X \\text{ has no parents: } \\mathbf{Pa}(X) = \\varnothing;\\\\\n    \\mathfrak{M} &= \\text{all SCMs over } \\*V = \\bigcup_{\\prec} \\mathfrak{M}_\\prec = \\bigcup_X \\mathfrak{M}_X.\n\\end{align*}\nIf $\\*V$ is infinite then all SCMs in the classes above are assumed to be atomless.\n\\end{definition}\n\n\\section{The Causal Hierarchy} \\label{sec:causalhierarchy}\n\nImplicit in \\S{}\\ref{scms}, and indeed in much of the literature on causal inference, is a hierarchy of causal expressivity. Following the metaphor offered in \\cite{pearl2018book}, it is natural to characterize three levels of the hierarchy as the \\emph{observational}, \\emph{interventional} (experimental), and \\emph{counterfactual} (explanatory). Drawing on recent work \\citep{BCII2020,ibelingicard2020} we make this characterization explicit. The levels will be defined in descending order of causal expressivity (the reverse of \\S\\ref{scms}). Fig.~\\ref{fig:hierarchy}(a) summarizes our definitions.\n\nHigher levels determine lower levels---counterfactuals determine interventionals, and the observational is just an (empty) interventional.\nThus movement ``downward'' in the causal hierarchy corresponds to a kind of projection.\nFor indexed $\\{S_\\beta\\}_{\\beta \\in B}$\nand $B' \\subset B$\nlet $\\varsigma_{B'} : \\mathfrak{P}(\\bigtimes_{\\beta \\in B} S_\\beta) \\to \\mathfrak{P}(\\bigtimes_{\\beta \\in B'} S_\\beta)$\nbe the \\emph{marginalization} map taking a joint distribution to its marginal on $B'$.\n\\begin{definition}\nDefine three composable \\emph{causal projections} $\\{\\varpi_i\\}_{1 \\le i \\le 3}$\nwith signatures and definitions\n\\begin{gather*}\n\\varpi_3 : \\mathfrak{M} \\to \\mathfrak{P}(\\chi_{A \\times \\*V}), \\quad \\varpi_2: \\mathfrak{P}(\\chi_{A \\times \\*V}) \\to  \\bigtimes_{\\alpha \\in A} \\mathfrak{P}(\\chi_{\\*V}), \\quad \\varpi_1: \\bigtimes_{\\alpha \\in A} \\mathfrak{P}(\\chi_{\\*V}) \\to \\mathfrak{P}(\\chi_{\\*V});\\\\\n    \\varpi_3: {\\mathcal{M}} \\mapsto p^{{\\mathcal{M}}}_{\\mathrm{cf}}, \\quad \\varpi_2: \\mu_3 \\mapsto \\big(\\varsigma_{\\{\\alpha\\} \\times \\*V}(\\mu_3)\\big)_{\\alpha \\in A}, \\quad \\varpi_1: (\\mu_\\alpha)_{\\alpha \\in A} \\mapsto  \\mu_{\\varnothing \\coloneqq ()} = \\pi_{\\varnothing \\coloneqq ()}\\big((\\mu_\\alpha)_\\alpha\\big).\n\\end{gather*}\nThe \\emph{causal hierarchy} consists of three sets $\\{\\mathfrak{S}_i\\}_{1 \\le i \\le 3}$ defined as images or projections of $\\mathfrak{M}$:\n\\begin{equation*}\n    \\mathfrak{S}_3 = \\varpi_3(\\mathfrak{M}), \\quad \\mathfrak{S}_2 = \\varpi_2(\\mathfrak{S}_3), \\quad \\mathfrak{S}_1 = \\varpi_1(\\mathfrak{S}_2).\n\\end{equation*}\n\\end{definition}\nThese are the three \\emph{Levels} of the hierarchy. The definitions cohere with those of \\S{}\\ref{scms} (and, e.g., \\cite{Pearl2009,BCII2020}):\n\\begin{fact}\nLet ${\\mathcal{M}} \\in \\mathfrak{M}$.\nThen\n$\\mu_3 = \\varpi_3({\\mathcal{M}}) \\in \\mathfrak{S}_3$ trivially coincides with its counterfactual distribution as defined in \\S{}\\ref{ss:scml3}, while\n$(\\mu_\\alpha)_\\alpha = \\varpi_2(\\mu_3) \\in \\mathfrak{S}_2$ coincides with the indexed family of all its interventional distributions (\\S{}\\ref{ss:scml2}), i.e., $\\pi_{\\*W \\coloneqq \\*w} \\big((\\mu_\\alpha)_\\alpha\\big) = p^{{\\mathcal{M}}_{\\*W \\coloneqq \\*w}}$ for each $\\*W \\coloneqq \\*w \\in A$.\nFinally $\\mu = \\varpi_1\\big((\\mu_\\alpha)_\\alpha\\big) \\in \\mathfrak{S}_1$ coincides with its observational distribution (\\S{}\\ref{ss:scml1}).\n\\end{fact}\nThus, e.g., $\\mathfrak{S}_3$ is the set of counterfactual distributions that are consistent with at least some SCM from $\\mathfrak{M}$. It is a fact that $\\mathfrak{S}_3 \\subsetneq \\mathfrak{P}(\\chi_{A \\times \\*V})$ and similarly not every interventional family belongs to $\\mathfrak{S}_2$; see\nAppendix \\ref{app:causalhierarchy} for explicit characterizations.\nAt the observational level, this is simple:\n\\begin{fact}\n$\\mathfrak{S}_1 = \\mathfrak{P}(\\chi_{\\*V})$ in the finite case. In the infinite case, $\\mathfrak{S}_1 = \\{ \\mu \\in \\mathfrak{P}(\\chi_{\\*V}): \\mu \\text{ is atomless}\\}$. \\label{prop:alternative}\n\\end{fact}\nWe will also use the subsets $\\{\\mathfrak{S}^\\prec_i\\}_{i}$ and $\\{\\mathfrak{S}^X_i\\}_{i}$, which are defined analogously but via projection from $\\mathfrak{M}_\\prec$ and $\\mathfrak{M}_X$ respectively.\n\n\n\n\n\n\\subsection{Problems of Causal Inference} \\label{subsection:probs}\nAs elucidated in \\cite{pearl2018book,BCII2020}, the causal hierarchy helps characterize many standard problems of causal inference, in as far as these problems typically involve ascending levels of the hierarchy. Some examples include: \n\\begin{enumerate}\n    \\item Classical identifiability: given observational data about some variables in $\\*V$, estimate a \\emph{causal effect} of setting variables $\\*X$ to values $\\*x$ \\citep{Pearl1995,Spirtes2001}. In the notation here, given information about $p^{\\mathcal{M}}(\\*V)$, can we determine $p^{\\mathcal{M}_{\\*X \\coloneqq \\*x}}(\\*Y)$?\n    \\item General identifiability: given a mix of observational data and limited  experimental data---that is, information about $p^{\\mathcal{M}}(\\*V)$ as well as some experimental distributions of the form $p^{\\mathcal{M}_{\\*W \\coloneqq \\*w}}(\\*V)$---determine $p^{\\mathcal{M}_{\\*X \\coloneqq \\*x}}(\\*Y)$ \\citep{tian-2002,lee2019general}.\n    \\item Structure learning: given observational data, and perhaps experimental data, infer properties of the underlying causal influence relation $\\rightarrow$ \\citep{Spirtes2001,Peters}.\n    \\item Counterfactual estimation: given a combination of observational and experimental data, infer a counterfactual quantity, such as probability of necessity  \\citep{robins-greenland}, or probability of necessity and sufficiency \\citep{Pearl1999,Tian2000} (see also \\S\\ref{section:pns} below). \n        \\item Global identifiability: given observational data drawn from $p^{\\mathcal{M}}(\\*V)$ infer the full counterfactual distribution $p_{\\text{cf}}^{{\\mathcal{M}}}(A \\times \\*V)$ \\citep{JMLR:v7:shimizu06a,drton2011}.\n\\end{enumerate} This is not an exhaustive list, and these problems are not all independent of one another. They are also all unsolvable in general. Problems 1, 2, and 3 involve ascending to Level 2 given information at Level 1 (and perhaps partial information at Level 2); problems 4 and 5 ask us to ascend to Level 3 given only Level 1 (and perhaps also Level 2) information. The upshot of the causal hierarchy theorem from \\cite{BCII2020} is that these steps are impossible without assumptions, formalizing the common wisdom, ``no causes in, no causes out'' \\cite{Cartwright}. \nTo understand the statement of the causal hierarchy theorem---and our topological version of it---we explain what it means for the hierarchy to collapse. \n\n\\subsection{Collapse of the Hierarchy}\\label{ss:collapse}\n\nIn the present setting a \\emph{collapse} of the hierarchy can be understood in terms of injectivity of the functions $\\varpi_i$.\n\nFor $i = 1, 2$ let $\\mathfrak{C}_i \\subset \\mathfrak{S}_i$ be the injective fibers of $\\varpi_i$, i.e.,\n$\\mathfrak{C}_i = \\{\\mu_i \\in \\mathfrak{S}_i : \\mu_{i+1} = \\mu_{i+1}' \\text{ whenever } \\varpi_i(\\mu_{i+1}) = \\varpi_i(\\mu'_{i+1}) = \\mu_i \\}$.\nEvery element $\\mu \\in \\mathfrak{C}_i$ is a witness to (global) collapse of the hierarchy: knowing $\\mu$ would be sufficient to determine the Level $i+1$ facts completely.\n\n\n\\begin{figure} \\centering \n\\subfigure [Causal Hierarchy] {\n   \\begin{tikzpicture}[framed]\n  \\node (m) at (0,0) {$\\mathfrak{M}$}; \n  \\node (ss3) at (1.5,0) {$\\mathfrak{S}_3$};\n  \\node (ss2) at (3,0) {$\\mathfrak{S}_2$};\n  \\node (ss1) at (4.5,0) {$\\mathfrak{S}_1$};\n  \n  \\path (m) edge[->]  (ss3);\n  \\node (l1) at (.7,.15) {\\small{}$\\varpi_3$};\n  \\path (ss3) edge[->]  (ss2);\n    \\node (l2) at (2.25,.15) {\\small{}$\\varpi_2$};\n  \\path (ss2) edge[->]  (ss1);\n    \\node (l3) at (3.75,.15) {\\small{}$\\varpi_1$};\n  \n  \\node (b1) at (-.5,0) {};\n  \\node (b2) at (5.5,0) {}; \n  \n  \\node (ns1) at (1.4,-1.25) {$\\mathfrak{S}^{X \\to Y}_{2}$};\n  \\node (b1) at (2.8,-1.25) {$\\dots$};\n  \\node (ns2) at (4.4,-1.25) {$\\mathfrak{S}^{X' \\to Y'}_{2}$};\n  \n    \\path (ss2) edge[->]  (ns1);\n   \\node (l3) at (1.6,-.55) {\\small{}$\\varpi^{X \\to Y}_{2}$};\n  \\path (ss2) edge[->]  (ns2);\n  \\node (l4) at (4.5,-.55) {\\small{}$\\varpi^{X' \\to Y'}_{2}$};\n   \\end{tikzpicture}\n   }\n\\subfigure [Collapse Set $\\mathfrak{C}_2$] {\n   \\begin{tikzpicture}\n\\draw (.7,1.2) ellipse (2.5cm and .5cm);\n\\draw [gray!60,fill=gray!20] (.15,0) ellipse (1.1cm and .25cm);\n\\draw (.7,0) ellipse (2cm and .4cm);\n\\node (s0) at (1.5,0) {$\\textcolor{blue}{\\bullet}$}; \n\\node (t0) at (1.2,1.2) {$\\textcolor{blue}{\\bullet}$};\n\\node (t1) at (1.55,1.2) {$\\textcolor{blue}{\\bullet}$};\n\\node (t2) at (1.9,1.2) {$\\textcolor{blue}{\\bullet}$};\n\\path (t0) edge[->] (s0);\n\\path (t1) edge[->] (s0);\n\\path (t2) edge[->] (s0);\n\n\\node (s5) at (2.15,0) {$\\textcolor{violet}{\\bullet}$}; \n\\node (t6) at (2.3,1.2) {$\\textcolor{violet}{\\bullet}$};\n\\node (t7) at (2.75,1.2) {$\\textcolor{violet}{\\bullet}$};\n\\path (t6) edge[->] (s5);\n\\path (t7) edge[->] (s5);\n\n\n\\node (s1) at (.15,0) {$\\textcolor{orange}{\\bullet}$};\n\\node (s2) at (-.3,0) {$\\textcolor{green}{\\bullet}$};\n\\node (t3) at (.15,1.2) {$\\textcolor{orange}{\\bullet}$};\n\\node (t4) at (-.3,1.2) {$\\textcolor{green}{\\bullet}$};\n\\node (t5) at (-1,1.2) {$\\textcolor{red}{\\bullet}$};\n\\path (t3) edge[->] (s1);\n\\path (t4) edge[->] (s2);\n\\path (t5) edge[->,color=gray] (s2);\n\\node (x) at (-.675,.6) {$\\textcolor{red}{\\mathsf{X}}$};\n\\node (C2) at (.65,0) {$\\mathfrak{C}_2$};\n\n\\node (ss3) at (3.6,1.2) {$\\mathfrak{S}_3$};\n\\node (ss2) at (3.6,0) {$\\mathfrak{S}_2$};\n\n\\path (ss3) edge[->] (ss2);\n\n\\node (f) at (3.3,.6) {$\\varpi_2$};\n\n\\end{tikzpicture} \n}\n    \\caption{(a) $\\mathfrak{S}_3$ can be seen as a coarsening of $\\mathfrak{M}$, abstracting from irrelevant ``intensional'' details. $\\mathfrak{S}_2$ is obtained from $\\mathfrak{S}_3$ by marginalization (also a coarsening), while $\\mathfrak{S}_1$ is a projection of $\\mathfrak{S}_2$ via the ``empty'' intervention. Each map $\\varpi_i$, $i=1,2$, is continuous in the respective weak topology (Prop. \\ref{prop:causalprojectioncontinuous}). The projections $\\varpi^{X \\to Y}_{2}$ from $\\mathfrak{S}_2$ to the 2VE-spaces are likewise continuous and also open (Prop. \\ref{prop:causalprojectioncontinuous}). \\\\\n    (b) The shaded region, $\\mathfrak{C}_2 \\subset \\mathfrak{S}_2$, is the collapse set in which Level 2 facts determine all Level 3 facts: those points in $\\mathfrak{S}_2$ whose $\\varpi_2$-preimage in $\\mathfrak{S}_3$ is a singleton set. The main result of this paper is that $\\mathfrak{C}_2$ is \\emph{meager} in weak topology on $\\mathfrak{S}_2$ (Thm. \\ref{thm:hierarchyFormal}). This means $\\mathfrak{C}_2$ contains no open subset, which by Thm. \\ref{thm:l2learning} implies no part of $\\mathfrak{C}_2$ is statistically verifiable, even with infinitely many ideal experiments.}\\label{fig:hierarchy}\n\\end{figure}\n\n\\begin{comment} \n\\begin{figure}\\begin{center}\n   \\begin{tikzpicture}\n\\draw (.7,1.2) ellipse (2.5cm and .5cm);\n\\draw [gray!60,fill=gray!20] (.15,0) ellipse (1.1cm and .25cm);\n\\draw (.7,0) ellipse (2cm and .4cm);\n\\node (s0) at (1.5,0) {$\\textcolor{blue}{\\bullet}$}; \n\\node (t0) at (1.2,1.2) {$\\textcolor{blue}{\\bullet}$};\n\\node (t1) at (1.55,1.2) {$\\textcolor{blue}{\\bullet}$};\n\\node (t2) at (1.9,1.2) {$\\textcolor{blue}{\\bullet}$};\n\\path (t0) edge[->] (s0);\n\\path (t1) edge[->] (s0);\n\\path (t2) edge[->] (s0);\n\n\\node (s5) at (2.15,0) {$\\textcolor{violet}{\\bullet}$}; \n\\node (t6) at (2.3,1.2) {$\\textcolor{violet}{\\bullet}$};\n\\node (t7) at (2.75,1.2) {$\\textcolor{violet}{\\bullet}$};\n\\path (t6) edge[->] (s5);\n\\path (t7) edge[->] (s5);\n\n\n\\node (s1) at (.15,0) {$\\textcolor{orange}{\\bullet}$};\n\\node (s2) at (-.3,0) {$\\textcolor{green}{\\bullet}$};\n\\node (t3) at (.15,1.2) {$\\textcolor{orange}{\\bullet}$};\n\\node (t4) at (-.3,1.2) {$\\textcolor{green}{\\bullet}$};\n\\node (t5) at (-1,1.2) {$\\textcolor{red}{\\bullet}$};\n\\path (t3) edge[->] (s1);\n\\path (t4) edge[->] (s2);\n\\path (t5) edge[->,color=gray] (s2);\n\\node (x) at (-.65,.6) {$\\textcolor{red}{\\times}$};\n\\node (C2) at (.65,0) {$\\mathfrak{C}_2$};\n\n\\node (ss3) at (3.5,1.2) {$\\mathfrak{S}_3$};\n\\node (ss2) at (3.5,0) {$\\mathfrak{S}_2$};\n\n\\path (ss3) edge[->] (ss2);\n\n\\node (f) at (3.2,.6) {$\\varpi_2$};\n\n\\end{tikzpicture} \\end{center}\n    \\caption{Collapse of the hierarchy}\n    \\label{fig:my_label}\n\\end{figure}\n\\end{comment}\n\n\nA first observation is that $\\varpi_1$ is \\emph{never} injective. In other words, the distribution $p^{\\mathcal{M}}(\\*V)$ never determines all the interventional distributions $p^{\\mathcal{M}_{\\*X \\coloneqq \\*x}}(\\*Y)$. This is essentially a way of stating that correlation never implies causation absent assumptions. (See also \\cite[Thm. 1]{BCII2020}.) \n\\begin{proposition}\n$\\mathfrak{C}_1 = \\varnothing$. That is, Level 2 \\emph{never} collapses to Level 1 without assumptions. \\label{prop:collapse1}\n\\end{proposition}\n\nTo overcome this formidable inferential barrier, researchers often assume we are not working in the ``full'' space $\\mathfrak{M}$ of all causal models, but rather some proper subset embodying a range of causal assumptions. This may effectively eliminate counterexamples to collapse (cf. Fig. \\ref{fig:hierarchy}(b)). For problems of type 1 or 2 (from the list above in \\S\\ref{subsection:probs}) it is common to assume we are only dealing with models whose graph (direct influence relation) $\\rightarrow$ satisfies a fixed set of properties. For problems of type 3 it is common to assume that $p^\\mathcal{M}$ and $\\rightarrow$ relate in some way (for instance, through an assumption like \\emph{faithfulness} or \\emph{minimality} \\cite{Spirtes2001}). All of these problems become solvable with sufficiently strong assumptions about the form of the functions $\\{f_V\\}_V$ or the probability space $P$.\n\nIn some cases, the relevant causal assumptions are justified by appeal to background or expert knowledge. In other cases, however, an assumption will be justified by the fact that it rules out only a ``small'' or ``negligible'' or ``measure zero'' part of the full set $\\mathfrak{M}$ of possibilities. As emphasized by a number of authors \\cite{Freedman1997,Uhler,pmlr-v117-lin20a}, not all ``small'' subsets are the same, and it seems reasonable to demand further justification for eliminating one over another. We believe that the framework presented here can contribute to this positive project, but our immediate interest is in solidifying and clarifying limitative results about what cannot be done.\n\n\n\nThe issue of collapse becomes especially delicate when we turn to $\\mathfrak{C}_2$. When do interventional distributions fully determine counterfactual distributions? In contrast to Prop. \\ref{prop:collapse1} we have:\n\\begin{proposition} $\\mathfrak{C}_2 \\neq \\varnothing$. That is, there exists an SCM in which Level 3 collapses to Level 2. \n\\end{proposition}\n\\begin{proof}[Proof sketch] As a very simple example in the finite case, any fully deterministic SCM will result in collapse. This is because, if $(\\mu_{\\alpha})_{\\alpha \\in A}$ are all binary-valued then the measure $\\mu_3 \\in \\mathfrak{P}\\big(\\bigtimes_{\\alpha \\in A} \\chi_{\\*V}\\big)$ that produces the marginals $\\mu_{\\alpha}$ is completely determined: each $\\mu_{\\alpha}$ specifies an element of $\\chi_{\\*V}$, so $\\mu_3$ must assign unit probability to the tuple that matches $\\mu_{\\alpha}$ at the $\\alpha$ projection. \nIn the infinite case, any example must be non-deterministic by atomlessness, but collapse is still possible; see Example \\ref{example:collapse} in Appendix \\ref{app:causalhierarchy}.\n\\end{proof}\n\n\\subsection{Probabilities of Causation} \\label{section:pns}\n\nA handful of counterfactual quantitites over two given variables, collected below, have been particularly prominent in the literature (e.g., \\cite{Pearl1999}). Our main result will show that \\emph{any} of these six quantities (for any two fixed variables) is robust against collapse.\nBelow, fix two distinct variables $Y \\neq X \\in \\*V$ and distinct values $x \\neq x' \\in \\chi_X$, $y \\neq y' \\in \\chi_Y$.\n\\begin{definition}\\label{def:probcaus}\nThe \\emph{probabilities of causation} are the following quantities:\n\\begin{align*}\n    P(y_x, y'_{x'}): & \\text{ probability of necessity and sufficiency}\\\\\n    P(y'_x, y_{x'}): & \\text{ converse prob. of necessity and sufficiency}\\\\\n    P(y'_{x'} \\mid x, y): & \\text{ prob. of necessity} \\qquad P(y_x \\mid x', y'): \\text{ prob. of sufficiency}\\\\\n    P(y'_{x'} \\mid y): & \\text{ prob. of disablement} \\qquad P(y_x \\mid y'): \\text{ prob. of enablement}\n\\end{align*}\n\\end{definition}\nConsider, for example, the probability of necessity and sufficiency (PNS), which is the joint probability that $Y$ would take on value $y$ if $X$ is set by intervention to $x$, and $y'$ if $X$ is set to $x'$.\nPNS has been thoroughly studied \\citep{Pearl1999,Tian2000,avin:etal05}, in part due to its widespread relevance: from medical treatment to online advertising, we would like to assess which interventions are likely to be both \\emph{necessary} and \\emph{sufficient} for a given outcome.\nUsing the notation from \\S{}\\ref{ss:scml3}, PNS concerns the measure of sets\n$y_{x}, y'_{x'} = \\pi^{-1}_{(X \\coloneqq x, Y)}(\\{y\\}) \\cap \\pi^{-1}_{(X \\coloneqq x', Y)}(\\{y'\\})$.\n\n\nThe probabilities of causation are paradigmatically Level 3, and we will be interested in their manifestations at Level 2. In that direction we introduce a small part of $\\mathfrak{S}_2$, just enough to witness the behavior of $Y$ (and $X$) under the empty intervention and the two possible interventions on $X$:\n\\begin{definition}\nLet $A_X =\\{\\varnothing \\coloneqq (), X \\coloneqq 0, X \\coloneqq 1\\}$.\nDefine a small subspace $\\mathfrak{S}^{X \\to Y}_{2}\\subset \\bigtimes_{\\alpha \\in A_X} \\mathfrak{P}(\\chi_{\\{X,Y\\}})$ as the image of the map $\\varpi^{X\\to Y}_{2} = \\big(\\varsigma_{\\{X, Y\\}} \\times \\varsigma_{\\{X, Y\\}} \\times \\varsigma_{\\{X, Y\\}}\\big) \\circ \\pi_{A_X}$ (see Fig. \\ref{fig:hierarchy}(a)). Call $\\mathfrak{S}^{X \\to Y}_{2}$ a \\emph{two-variable effect} (2VE) \\emph{space}; fixing $X$, we have a 2VE-space for each $Y$.\n\\end{definition}\n\nIt is known in the literature that the probabilities of causation are not identifiable from the data $p(X,Y)$, $p(Y_{x})$, and $p(Y_{x'})$ (see, e.g., \\cite{avin:etal05} for PNS). As part of our proof of Theorem \\ref{thm:hierarchyFormal} below, we will strengthen this considerably to show them all to be \\emph{generically} unidentifiable, in a topological sense to be made precise.\n\n\n\n\n\n\n\n\\section{The Weak Topology} \\label{sec:weaktopology}\n\nWe now demonstrate how $\\mathfrak{S}_1,\\mathfrak{S}_2$ and $\\mathfrak{S}_3$ can be topologized.  \nIn general, given a space $\\vartheta$ and the set $\\mathfrak{S} = \\mathfrak{P}(\\vartheta)$ of Borel probability measures on $\\vartheta$, a natural topology on $\\mathfrak{S}$ can be defined as follows:\n\\begin{definition}\nFor a sequence $(\\mu_n)_n$ of measures in $\\mathfrak{S}$,\nwrite $(\\mu_n)_n \\Rightarrow \\mu$ and say it \\emph{converges weakly} \\citep[p.~7]{Billingsley} to $\\mu$ if $\\int_{\\vartheta} f \\, \\mathrm{d}\\mu_n \\to \\int_{\\vartheta} f \\, \\mathrm{d}\\mu$ for all bounded, continuous $f : \\vartheta \\to \\mathbb{R}$.\nThen the \\emph{weak topology} $\\tau^{\\mathrm{w}}$ on $\\mathfrak{S}$ is that with the following closed sets: $E \\subset \\mathfrak{S}$ is closed in $\\tau^{\\mathrm{w}}$ iff for any weakly convergent sequence $(\\mu_n)_n \\Rightarrow \\mu$ in which every $\\mu_n \\in E$, the limit point $\\mu$ is in $E$.\n\\end{definition}\nThere are several alternative characterizations of $\\tau^{\\mathrm{w}}$, which hold under very general conditions. For instance, it coincides with the topology induced by the so called L\\'{e}vy-Prohorov metric \\citep{Billingsley}. \nThe most useful characterization for our purposes is that it \ncan be generated by subbasic open sets of the form \\begin{equation}  \\{\\mu: \\mu(X)>r\\}\n\\label{subbasis-weak}\n\\end{equation} with $X$ ranging over basic clopens in $\\vartheta$ and $r$ over rationals (see, e.g., \\cite[Lemma A.5]{GeninKelly}).\n\nConceptually, the explication of $\\tau^{\\mathrm{w}}$ in terms of weak convergence strongly suggests a connection with statistical learning. We now make this connection precise, building on existing work \\cite{DemboPeres,Genin2018,GeninKelly}. \n\n\\subsection{Connection to Learning Theory}\nRoughly speaking, we will say a hypothesis $H \\subseteq \\mathfrak{S}$ is \\emph{statistically verifiable} if there is some error bound $\\epsilon$ and a sequence of statistical tests that converge on $H$ with error at most $\\epsilon$, when data are generated from $H$. More formally, a \\emph{test} is a function $\\lambda: \\vartheta^n \\rightarrow \\{\\mathsf{accept},\\mathsf{reject}\\}$, where $\\vartheta^n$ is the $n$-fold product of $\\vartheta$, viz. finite data streams from $\\vartheta$. The interest is in whether a ``null'' hypothesis can be rejected given data observed thus far. The \\emph{boundary} of a set $A\\subseteq \\vartheta$, written $\\mathsf{bd}(A)$, is the difference of its closure and its interior. Intuitively, a learner will not be able to decide whether to accept or reject on the boundary. Consequently it is assumed that $\\lambda$ is \\emph{feasible} in the sense that the boundary of its acceptance zone (in the product topology on $\\vartheta^n$) always has measure 0, i.e., $\\mu^n[\\mathsf{bd}(\\lambda^{-1}(\\mathsf{accept}))] = 0$  for every $\\mu \\in \\mathfrak{S}$, where $\\mu^n$ is the $n$-fold product measure of $\\mu$.\n\nSay a hypothesis $H \\subseteq \\mathfrak{S}$ is \\emph{verifiable} \\cite{Genin2018} if there is $\\epsilon>0$ and a sequence $(\\lambda_n)_{n\\in\\mathbb{N}}$ of feasible tests (of the complement of $H$ in $\\mathfrak{S}$, i.e., the ``null hypothesis'') such that \\begin{enumerate}\n   \\item $\\mu^n[\\lambda_n^{-1}(\\mathsf{reject})] \\leq \\epsilon$ for all $n$, whenever $\\mu \\notin H$;\n   \\item $\\underset{n\\rightarrow\\infty}{\\mbox{lim}}\\;\\mu^n[\\lambda_n^{-1}(\\mathsf{reject})] = 1$, whenever $\\mu \\in H$.\n\\end{enumerate} That is, to be verifiable we only require a sequence of tests that converges in probability to the true hypothesis in the limit of infinite data (requirement 2), while incurring (type 1) error only up to a given bound at finite stages (requirement 1). As an illustrative example, \\emph{conditional dependence} is verifiable \\cite{Genin2018}. This is a relatively lax notion of verifiability.\nFor instance, the hypothesis need not also be \\emph{refutable} (and thus ``decidable''). For our purposes this generality is a virtue: we want to show that certain hypotheses are not statistically verifiable by any method, even in this wide sense. The fundamental link between verifiability and the weak topology is the following, due to \\cite{Genin2018,GeninKelly}:\n\\begin{theorem} \\label{thm:genin} A set $H \\subseteq \\mathfrak{S}$ is verifiable if and only if it is open in the weak topology.\n\\end{theorem}\n\n\\subsection{Topologizing Causal Models}\nWe now reinterpret $\\tau^{\\mathrm{w}}$ at each level of the causal hierarchy: \n\\begin{definition}\nThe \\emph{weak causal topology} $\\tau^{\\mathrm{w}}_i$, $1 \\le i \\le 3$, is the subspace topology on $\\mathfrak{S}_i$, induced by\n\\begin{align*}\n\\text{if } i=3: \\tau^{\\mathrm{w}} \\text{ on } \\mathfrak{P}(\\chi_{A \\times \\*V}); \\quad\n\\text{if } i=2: \\text{product of }\\tau^{\\mathrm{w}} \\text{ on } \\bigtimes_{\\alpha \\in A} \\mathfrak{P}(\\chi_{\\*V}); \\quad\n\\text{if } i=1: \\tau^{\\mathrm{w}} \\text{ on } \\mathfrak{P}(\\chi_{\\*V}).\n\\end{align*}\n\\end{definition}\n\\begin{proposition}\\label{prop:causalprojectioncontinuous}\nIn the weak causal topologies,\n$\\{\\varpi_i\\}_{i = 1, 2}$ are continuous and all projections\n$\\varpi^{X \\to Y}_{2}$ are continuous and open.\n\\end{proposition}\nA significant observation is that the learning theoretic interpretation, originally intended for $\\tau^{\\mathrm{w}}_1$, naturally extends to $\\tau^{\\mathrm{w}}_2$. While data streams at Level 1 amount to passive observations of $\\*V$, data streams at Level 2 can be seen as sequences of experimental results, i.e., observations of ``potential outcomes'' $\\*Y_{\\*x}$. To make verifiability as easy as possible we assume a learner can observe a sample from all conceivable experiments at each step.  A learner is thus a function $\\lambda:\\mathcal{E}^n \\rightarrow \\{\\mathsf{accept},\\mathsf{reject}\\}$, where $\\mathcal{E}^n = ((\\chi_{\\*V})^n)_\\alpha$  is the set of potential experimental observations over $n$ trials (with $\\alpha$ indexing the experiments). Construing $\\mathcal{E}^n$ as a product space we can again speak of \\emph{feasibility} of $\\lambda$.  \n\nRecall that elements of $\\mathfrak{S}_2$ are tuples $(\\mu_\\alpha)_{\\alpha \\in A}$ of measures. Say  a hypothesis $H\\subseteq \\mathfrak{S}_2$ is \\emph{experimentally verifiable} if there is $\\epsilon>0$ and a sequence $(\\lambda_n)_{n \\in \\mathbb{N}}$ of feasible tests such that 1 and 2 above hold, replacing $\\mu^n[\\lambda_n^{-1}(\\mathsf{reject})]$ with $\\prod_{\\alpha} \\mu_\\alpha^n[(\\lambda_n^{-1}(\\mathsf{reject}))_\\alpha]$. That is, when experimental data are drawn from the interventional distributions $(\\mu_\\alpha)_{\\alpha \\in A} \\in H$, we require that the learner eventually converge on $H$ with bounded error at finite stages. We can then show (see Appendix \\ref{app:empirical}): \n\\begin{theorem} A set $H \\subseteq \\mathfrak{S}_2$ is experimentally verifiable if and only if it is open in $\\tau^{\\mathrm{w}}_2$. \\label{thm:l2learning}\n\\end{theorem}\nA similar result can be given for $(\\mathfrak{S}_3,\\tau^{\\mathrm{w}}_3)$, although it is less clear what the empirical content of this result would be.\nNote also that $\\tau^{\\mathrm{w}}_1,\\tau^{\\mathrm{w}}_2,\\tau^{\\mathrm{w}}_3$ give a sequence of increasingly fine topologies on the set of actual SCMs $\\mathfrak{M}$ by simply pulling back the projections. The point is that $\\tau^{\\mathrm{w}}_2$ is the finest that has clear empirical significance, while $\\tau^{\\mathrm{w}}_3$ is the finest in terms of relevance to the causal hierarchy.\n\n\n\n\\section{Collapse is Meager} \\label{sec:main}\nRecall that a set $X \\subseteq \\vartheta$ is \\emph{nowhere dense} if every open set contains an open  $Y$ with $X \\cap Y = \\varnothing$. A countable union of nowhere dense sets is said to be \\emph{meager} (or \\emph{of first category}). The complement of a meager set is \\emph{comeager}. Intuitively, a meager set is one that can be ``approximated'' by sets ``perforated with holes'' \\citep{Oxtoby}. Meagerness is notably preserved when taking the preimage under a map that is both continuous and open.\n\nAs discussed above, one intuition highlighted by the weak topology $\\tau^{\\mathrm{w}}$ is that open sets are the kinds of probabilistic propositions that could, in the limit of infinite data, be verified (Thms. \\ref{thm:genin},  \\ref{thm:l2learning}). Correlatively, meager sets in $\\tau^{\\mathrm{w}}$ are so negligible as to be unverifiable: as a meager set contains no non-empty open subsets (by the Baire Category Theorem \\cite{Oxtoby}), it is statistically unverifiable.\nWe will now show that the injective collapse set $\\mathfrak{C}_2$ from \\S{}\\ref{ss:collapse} is topologically meager.\n\nThe crux is to identify a ``good'' comeager 2VE-subspace where collapse \\emph{never} occurs (with separation witnessed by probabilities of causation).\nIn this subspace, the constraints circumscribing Level 3 have sufficient slack to make a tweak without thereby disturbing Level 2 (cf. Figure \\ref{fig:example:separation}).\nWe define the good set as the locus of a set of strict inequalities:\n\\begin{definition}\nA family $(\\mu_\\alpha)_{\\alpha \\in A_X} \\in \\mathfrak{S}^{X\\to Y}_{2}$ is \\emph{$Y$-good}\nif we have the following, abbreviating the members of $A_X$ as $(), x, x'$:\n\\begin{gather}\n0 < \\mu_x(y') - \\mu_{()}(x, y')\n < \\mu_{()}(x'),\\label{cns:ineq:1}\\\\\n0 < \\mu_{()}(x', y') < \\mu_{()}(x')\\label{cns:ineq:2}.\n\\end{gather}\n\\end{definition}\n\\begin{lemma}\n\\label{lem:goodcomeager}\nThe subspace of $Y$-good families is comeager in $\\mathfrak{S}^{X \\to Y}_{2}$.\n\\end{lemma}\n\\begin{proof}[Proof sketch]\nThe non-strict versions of \\eqref{cns:ineq:1}, \\eqref{cns:ineq:2} hold universally, so the complement of the good set is defined by equalities. This is closed and contains no nonempty open by the weak subbasis \\eqref{subbasis-weak}. \\end{proof}\nFigure~\\ref{fig:example:separation} presents the construction in a small, two-variable case, and Lemma~\\ref{lem:separation} below is proven by generalizing it to arbitrary $\\*V$.\nGuaranteeing agreement on every interventional distribution in the general case is subtle (Appendix~\\ref{app:main}):\nit has been observed that enlarging $\\*V$ can enable additional inferences (e.g., \\cite{9363924}), though the next result reflects a dependence on further assumptions.\n\n\\begin{lemma}\\label{lem:separation}\nSuppose $\\prec$ is an order in which $X$ comes first and $(\\mu_\\alpha)_{\\alpha \\in A} \\in \\mathfrak{S}^{\\prec}_2$ is such that $\\varpi^{X\\to Y}_{2}\\big((\\mu_\\alpha)_{\\alpha}\\big)$ is $Y$-good, and\nlet $\\varphi$ be PNS, the converse PNS, the probability of sufficiency, or the probability of enablement (Definition~\\ref{def:probcaus}).\nThen for any $\\mu_3 \\in \\mathfrak{S}^{\\prec}_3$ such that $\\varpi_2(\\mu_3) = (\\mu_\\alpha)_{\\alpha}$, there exists a $\\mu'_3 \\in \\mathfrak{S}^{\\prec}_3$ such that $\\mu_3$ and $\\mu'_3$ disagree on $\\varphi$.\n\\end{lemma}\n\\begin{figure} \\centering \n\\subfigure [$Y$-good Model] {\n\\begin{tabular}{ lllll } \n& & ${\\mathcal{M}}$ & & \\\\\n\\toprule\n$u$ & $P(u)$ & $X_u$ & $Y_{x, u}$ & $Y_{x', u}$\\\\\n\\midrule\n$u_0$ & $\\nicefrac{1}{2}$ & $x'$ & $y$ & $y$\\\\\n$u_1$ & $\\nicefrac{1}{2}$ & $x'$ & $y'$ & $y'$\\\\\n\\bottomrule\n\\end{tabular}\n}\n\\subfigure [Example Separating Levels 2 and 3] {\n\\begin{tabular}{ lllll } \n& & ${\\mathcal{M}}'$ & & \\\\\n\\toprule\n$u$ & $P(u)$ & $X_u$ & $Y_{x, u}$ & $Y_{x', u}$\\\\\n\\midrule\n$u_0$ & $\\nicefrac{1}{2}- \\varepsilon$ & $x'$ & $y$ & $y$\\\\\n$u_1$ & $\\varepsilon$ & $x'$ & $y$ & $y'$\\\\\n$u_2$ & $\\nicefrac{1}{2}- \\varepsilon$ & $x'$ & $y'$ & $y'$\\\\\n$u_3$ & $\\varepsilon$ & $x'$ & $y'$ & $y$\\\\\n\\bottomrule\n\\end{tabular}\n}\n\\caption{(a): the structural functions and exogenous noise for a model ${\\mathcal{M}}$ with direct influence $X \\rightarrow Y$. This ${\\mathcal{M}}$ meets \\eqref{cns:ineq:1} and \\eqref{cns:ineq:2}, so we may apply Lemma~\\ref{lem:separation}, constructing the model $\\mathcal{M}'$ in (b), where $0 < \\varepsilon < \\nicefrac{1}{2}$. Note that $p^{{\\mathcal{M}}}_{\\text{cf}}(y_x, y'_{x'}) = 0$ while $p^{{\\mathcal{M}}'}_{\\text{cf}}(y_x, y'_{x'}) = \\varepsilon$, so that the two models disagree on a Level 3 PNS quantity; on the other hand, it is easy to check agreement on all of Level 2. Similarly, ${\\mathcal{M}}$ and ${\\mathcal{M}}'$ disagree on the converse PNS, probability of sufficiency, and probability of enablement (Definition~\\ref{def:probcaus}).}\n  \\label{fig:example:separation}\n\\end{figure}\nNote that by reversing the roles of $x$ and $x'$, we may obtain the same for the probability of necessity and probability of disablement.\nThe main theorem and its important learning-theoretic corollary are now straightforward.\n\\begin{theorem}[Topological Hierarchy] The set $\\mathfrak{C}_2$ of points where all Level 3 facts are identifiable from Level 2 is meager in $(\\mathfrak{S}_2,\\tau^{\\mathrm{w}}_2)$. \\label{thm:hierarchyFormal}\n\\end{theorem}\n\\begin{proof}\nLet $\\mathfrak{D}^{X, Y}_2 \\subset \\mathfrak{S}_2^X$ be the preimage under $\\varpi^{X \\to Y}_{2}$ of the set of $Y$-good tuples in $\\mathfrak{S}^{X \\to Y}_{2}$.\nLemma~\\ref{lem:separation} implies that $\\mathfrak{C}_2 \\cap \\mathfrak{S}_2^{X}$ is contained in $\\mathfrak{S}_2^{X} \\setminus \\mathfrak{D}^{X, Y}_2$, for \\emph{any} $Y \\neq X$.\nMeanwhile, since $\\varpi^{X \\to Y}_{2}$ is continuous and open, Lemma~\\ref{lem:goodcomeager} implies that $\\mathfrak{S}_2^{X} \\setminus \\mathfrak{D}^{X, Y}_2$ is meager in $\\mathfrak{S}_2^{X}$, and thereby also in $\\mathfrak{S}_2$.\nThus $\\mathfrak{C}_2 = \\bigcup_{X \\in \\*V} \\mathfrak{C}_2 \\cap \\mathfrak{S}_2^{X}$ is a countable union of meager sets, and hence meager.\n\\end{proof}\n\\begin{corollary} \\label{cor:hierarchy} No causal hypothesis licensing arbitrary counterfactual inferences (and specifically those of the probabilities of causation) from observational and experimental data is itself statistically (even experimentally) verifiable.\n\\end{corollary}\n\n\\section{Conclusion}\\label{section:conclusion}\n\nWe introduced a general framework for topologizing spaces of causal models, including the space of all (discrete, well-founded) causal models. As an illustration of the framework we characterized levels of the causal hierarchy  topologically, and proved a topological version of the causal hierarchy theorem from \\cite{BCII2020}. While the latter shows that collapse of the hierarchy (specifically of Level 3 to Level 2) is \\emph{exceedingly unlikely} in the sense of (Lebesgue) measure, we offer a complementary result: any condition guaranteeing that we could infer arbitrary Level 3 information from purely Level 2 information must be \\emph{statistically  unverifiable}, even by experimental means. Both results capture an important sense in which collapse is ``negligible'' in the space of all possible models. As an added benefit, the topological approach extends seamlessly to the setting of infinitely many variables.\n\nThere are many natural extensions of these results. For instance, we have begun work on a version for continuous endogenous variables. Also of interest are subspaces embodying familiar causal assumptions or other well-studied coarsenings of SCMs (see, e.g., \\cite{pmlr-v117-lin20a} on Bayesian networks, or \\cite{GeninMayoWilson,Genin2021} on linear non-Gaussian models), which often render important inference problems solvable, though sometimes only ``generically'' so.\nIn the opposite direction, we expect analogous hierarchy theorems to hold for extensions of the SCM concept, e.g., that dropping the well-foundedness or recursiveness requirements \\cite{bongers2021foundations}.\nAs emphasized by \\cite{BCII2020}, a causal hierarchy theorem should not be construed as a purely limitative result, but rather as further motivation for understanding the whole range of causal-inductive assumptions, how they relate, and what they afford. We submit that the topological constructions presented here can help clarify and systematize this broader landscape.\n\n\n\\subsubsection*{Acknowledgments} \nThis material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-16565. We are very grateful to the five anonymous NeurIPS reviewers for insightful and detailed comments and questions that led to significant improvements in the paper. We would also like to thank Jimmy Koppel, Krzysztof Mierzewski, Francesca Zaffora Blando, and especially Kasey Genin for helpful feedback on earlier versions. Finally, we are indebted to Saminul Haque for identifying a gap in the published version of the paper, which has been corrected in the present arXiv version (see in particular the augmented statement of Prop.~\\ref{prop:causalprojectioncontinuous}). \n\n\n\\medskip\n\n{\n\\small\n\n\\bibliographystyle{abbrvnat}\n\n\\section{Structural Causal Models (\\S\\ref{scms})} \\label{app:causalmodels}\n\\subsection{Background on Relations and Orders}\n\n\\begin{definition}\nLet $C$ be a set.\nThen a subset $R \\subset C \\times C$ is called a \\emph{binary relation} on $C$. We write $cRc'$ if $(c, c') \\in R$.\nThe binary relation $R$ is \\emph{well-founded} if every nonempty subset $D \\subset C$ has a minimal element with respect to $R$, i.e., if for every nonempty $D \\subset C$, there is some $d \\in D$, such that there is no $d' \\in D$ such that $d' R d$.\nThe binary relation $\\left.\\prec\\right. \\subset C \\times C$ is a (strict) \\emph{total order} if it is irreflexive, transitive, and \\emph{connected}: either $c \\prec c'$ or $c' \\prec c$ for all $c \\neq c' \\in C$.\n\\end{definition}\n\\begin{example}\nThe edges of a dag form a well-founded binary relation on its nodes. If $\\*V = \\{V_n\\}_{n \\ge 0}$, then the binary relation $\\rightarrow$ defined by $V_m \\rightarrow V_n$ iff either $0 < m < n$ or $n = 0 < m$ is well-founded but not extendible to an $\\omega$-like total order (see Fact~\\ref{fact:omegalike}) and not locally finite: $V_0$ has infinitely many predecessors $V_1, V_2, \\dots$\n\\end{example}\n\n\\subsection{Proofs}\n\n\\begin{proof}[Proof of Proposition~1.]\nWe assume without loss that $\\*U(V) = \\*U$ for every $V \\in \\*V$.\nFor each $\\*u \\in \\chi_{\\*U}$,\nwell-founded induction along $\\rightarrow$ shows unique existence of a $m^{{\\mathcal{M}}}(\\*u) \\in \\chi_{\\*V}$\nsolving $f_{V}\\big(\\pi_{\\mathbf{Pa}(V)}(m^{{\\mathcal{M}}}(\\*u)), \\*u\\big) = \\pi_V(m^{{\\mathcal{M}}}(\\*u))$ for each $V$.\nWe claim the resulting function $m^{{\\mathcal{M}}}$ is measurable.\nOne has a clopen basis of cylinders, so it suffices to show each preimage $(m^{{\\mathcal{M}}})^{-1}(v)$ is measurable. Recall that here $v$ denotes the cylinder set $\\pi^{-1}_V(\\{v\\}) \\in \\mathcal{B}(\\chi_{\\*V})$, for $v \\in \\chi_V$.\nOnce again this can be established inductively. Note that\n\\begin{align*}\n(m^{{\\mathcal{M}}})^{-1}(v) =\n\\bigcup_{\\mathbf{p} \\in \\chi_{\\mathbf{Pa}(V)}}\\Big[ (m^{{\\mathcal{M}}})^{-1}(\\*p) \\cap \\pi_{\\*U}\\big(f_V^{-1}(\\{v\\}) \\cap (\\{\\*p\\} \\times \\chi_{\\*U})\\big) \\Big].\n\\end{align*}\nwhich is a finite union (by local finiteness) of measurable sets (by the inductive hypothesis) and therefore measurable.\nThus for any ${\\mathcal{M}}$\nthe pushforward $p^{{\\mathcal{M}}} = m^{{\\mathcal{M}}}_*(P)$ is a measure on $\\mathcal{B}(\\chi_{\\*V})$ and gives the observational distribution (Definition~4).\n\\end{proof}\n\n\\begin{proof}[Remark on Definition~6]\nTo see that $p^{{\\mathcal{M}}}_{\\mathrm{cf}}$ thus defined is a measure,\nnote that $p^{{\\mathcal{M}}}_{\\mathrm{cf}} = p^{{\\mathcal{M}}_A}$ and apply Proposition~1, where the model ${\\mathcal{M}}_A$ is defined in Definition~\\ref{def:counterfactualmodel}. This is similar in spirit to the construction of ``twinned networks'' \\citep{BalkePearl} or ``single-world intervention graphs'' \\citep{Richardson}.\n\\end{proof}\n\n\\begin{definition}\\label{def:counterfactualmodel}\nGiven ${\\mathcal{M}}$ as in Def.~\\ref{def:scm:lit}\nand a collection of interventions $A$\nform the following \\emph{counterfactual model} ${\\mathcal{M}}_A = \\langle \\*U, A \\times \\*V, \\{f_{(\\alpha, V)}\\}_{(\\alpha, V)}, P\\rangle$, over endogenous variables $A \\times \\*V$. The counterfactual model has the influence relation $\\rightarrow'$, defined as follows.\nWhere $\\alpha', \\alpha \\in A$ let $(\\alpha', V') \\rightarrow' (\\alpha, V)$ iff $\\alpha' = \\alpha$ and $V' \\rightarrow V$.\nThe exogenous space $\\*U$ and noise distribution $P$ of $\\mathcal{M}_A$ are the same as those of $\\mathcal{M}$,\nthe exogenous parents sets $\\{\\*U(V)\\}_V$ are also identical,\nand the functions are $\\{f_{(\\alpha, V)}\\}_{(\\alpha, V)}$ defined as follows.\nFor any $\\*W \\coloneqq \\*w \\in A$, $V \\in \\*V$, $\\mathbf{p} \\in \\chi_{\\mathbf{Pa}(V)}$, and $\\*u \\in \\chi_{\\*U(V)}$ let\n\\begin{align*}\nf_{(\\*W \\coloneqq \\*w, V)}\\big((\\*W \\coloneqq \\*w, \\mathbf{p}), \\*u\\big) = \\begin{cases}\n\\pi_V(\\*w), & V \\in \\*W\\\\\nf_V(\\mathbf{p}, \\*u), & V \\notin \\*W\n\\end{cases}.\n\\end{align*}\n\\end{definition}\n\n\n\\section{Proofs from \\S\\ref{sec:causalhierarchy}} \\label{app:causalhierarchy}\n\\begin{proof}[Remark on exact characterizations of $\\mathfrak{S}_3$, $\\mathfrak{S}_2$]\nRich probabilistic languages interpreted over $\\mathfrak{S}_3$ and $\\mathfrak{S}_2$ were axiomatized in \\cite{ibelingicard2020}.\nThis axiomatization, along with the atomless restriction, gives an exact characterization for the hierarchy sets.\nStandard form, defined below, gives an alternative characterization exhibiting each $\\mathfrak{S}_3^\\prec$ as a particular atomless probability space (Corollary~\\ref{cor:standardform}).\nFor $\\mathfrak{S}^{X \\to Y}_{2}$ (or $\\mathfrak{S}_2$ in the two-variable case) we need the characterization for the proof of the hierarchy separation result, so it is given explicitly as Lemma~\\ref{prop:p2:characterization:binary} in the section below on 2VE-spaces.\n\\end{proof}\n\\subsection{Standard Form}\\label{ss:app:standardform}\nFix $\\prec$. Note that the map $\\varpi_3$ restricted to $\\mathfrak{M}_\\prec$ does \\emph{not} inject into $\\mathfrak{S}_3^\\prec$, as any trivial reparametrizations of exogenous noise are distinguished in $\\mathfrak{M}_\\prec$.\nIt is therefore useful to identify a ``standard'' subclass $\\mathfrak{M}_\\prec^{\\mathrm{std}}$ on which $\\varpi_3$ is injective with image $\\mathfrak{S}_3^\\prec$, and in which we lose no expressivity.\n\\begin{notation*}\nLet $\\mathbf{Pred}(V) = \\{V' : V' \\prec V\\}$ and denote a \\emph{deterministic} mechanism for $V$ mapping a valuation of its predecessors to a value as $\\texttt{f}_V \\in \\chi_{\\mathbf{Pred}(V)} \\to \\chi_V$. Write an entire collection of such mechanisms, one for each variable, as $\\texttt{\\textbf{f}} = \\{\\texttt{f}_V\\}_{V}$.\nA set $\\*B \\subset \\*V$ is \\emph{ancestrally closed} if $\\mathbf{B} = \\bigcup_{V \\in \\*B} \\mathbf{Pred}(V)$.\nFor any ancestrally closed $\\*B$\nlet $\\xi(\\*B) = \\big\\{(V, \\*p): V \\in \\*B, \\*p \\in \\chi_{\\mathbf{Pred}(V)}\\big\\}$. Note that $\\texttt{\\textbf{F}}(\\*B) = \\bigtimes_{(V, \\mathbf{p}) \\in \\xi(\\*B)} \\chi_V$ encodes the set of all possible such collections of deterministic mechanisms, and we write, e.g., $\\texttt{f} \\in \\texttt{F}(\\*B)$.\nAbbreviate $\\xi(\\*V)$, $\\texttt{F}(\\*V)$ for the entire endogenous variable set $\\*V$ as $\\xi$, $\\texttt{F}$ respectively. We also use $\\texttt{f}$ to abbreviate the set\n\\begin{align}\n\\label{eq:probabilityofmechanisms}\n\\bigcap_{\\substack{V \\in \\*B\\\\ \\*p \\in \\chi_{\\mathbf{Pred}(V)}}} \\pi^{-1}_{(\\mathbf{Pred}(V) \\coloneqq \\*p, V)}(\\{\\texttt{f}(\\*p)\\}) \\in \\mathcal{B}(\\chi_{A \\times \\*V})\n\\end{align}\nso we can write, e.g., $p^{{\\mathcal{M}}}_{\\mathrm{cf}}(\\texttt{f})$ for the probability  in ${\\mathcal{M}}$ that the effective mechanisms $\\texttt{f}$ have been selected (by exogenous factors) for the variables $\\*B$.\n\\end{notation*}\n\\begin{definition}\\label{def:standardform}\n\nThe SCM ${\\mathcal{M}} = \\langle \\*U, \\*V, \\{f_V\\}_V, P\\rangle$ of Def.~\\ref{def:scm:lit} is in \\emph{standard form} over $\\prec$, and we write ${\\mathcal{M}} \\in \\mathfrak{M}_\\prec^{\\mathrm{std}}$, if we have that $\\left.\\rightarrow\\right. = \\left.\\prec\\right.$ for its influence relation, $\\*U = \\{U\\}$ for a single exogenous variable $U$ with $\\chi_U = \\texttt{{F}}$, $P \\in \\mathfrak{P}(\\texttt{F})$ for its exogenous noise space, and for every $V$, we have that $\\*U(V) = \\*U = \\{U\\}$ and the mechanism $f_V$ takes $\\mathbf{p}, (\\{ \\texttt{f}_V\\}_V) \\mapsto \\texttt{f}_V(\\mathbf{p})$ for each $\\*p \\in \\chi_{\\mathbf{Pred}(V)}$ and joint collection of deterministic functions $\\{ \\texttt{f}_V\\}_V \\in \\texttt{F} = \\chi_{U}$.\n\\end{definition}\nEach unit $\\*u$ in a standard form model amounts to a collection $\\{ \\texttt{f}_V\\}_V$ of deterministic mechanisms, and each variable is determined by a mechanism specified by the ``selector'' endogenous variable $U$.\n\\begin{lemma}\nLet ${\\mathcal{M}} \\in \\mathfrak{M}_\\prec$. Then there exists ${\\mathcal{M}}^{\\mathrm{std}} \\in \\mathfrak{M}_\\prec^{\\mathrm{std}}$ such that $\\varpi_3({\\mathcal{M}}) = \\varpi_3({\\mathcal{M}}^{\\mathrm{std}})$.\n\\end{lemma}\n\\begin{proof}\nTo give ${\\mathcal{M}}^{\\mathrm{std}}$ define a measure $P \\in \\mathfrak{P}(\\texttt{F})$ as in Def.~\\ref{def:standardform} on a basis of cylinder sets by the counterfactual in ${\\mathcal{M}}$\n\\begin{multline}\nP\\big(\\pi^{-1}_{(V_1, \\*p_1)}(\\{v_1\\}) \\cap \\dots \\cap \\pi^{-1}_{(V_n, \\*p_n)}(\\{v_n\\})\\big)\\\\\n= p_{\\mathrm{cf}}^{{\\mathcal{M}}}\\big( \\pi^{-1}_{(\\mathbf{Pred}(V_1) \\coloneqq \\*p_1, V_1)}(\\{v_1\\}) \\cap \\dots \\cap \\pi^{-1}_{(\\mathbf{Pred}(V_n) \\coloneqq \\*p_n, V_n)}(\\{v_n\\}) \\big). \\label{eq:standardformdefinition}\n\\end{multline}\nTo show that $\\varpi_3({\\mathcal{M}}) = \\varpi_3({\\mathcal{M}}^{\\mathrm{std}})$ it suffices to show that any two models agreeing on all counterfactuals of the form \\eqref{eq:standardformdefinition} must agree on all counterfactuals in $A$.\nSuppose $\\alpha_i \\in A$, $V_i \\in \\*V$, $v_i \\in \\chi_{V_i}$ for $i = 1, \\dots, n$.\nLet $\\*{B} = \\bigcup_{i} \\mathbf{Pred}(V_i)$ and given $\\texttt{f} = \\{\\texttt{f}_V\\}_V$,\ndefine $\\texttt{f}^{\\*W \\coloneqq \\*w}_V$ to be a constant function mapping to $\\pi_V(\\*w)$ if $V \\in \\*W$ and $\\texttt{f}^{\\*W \\coloneqq \\*w}_V = \\texttt{f}_V$ otherwise.\nWrite $\\texttt{f} \\models V = v$ if $\\pi_V(\\*v) = v$ for that $\\*v \\in \\chi_{\\*V}$ such that $\\texttt{f}_{V}\\big(\\pi_{\\mathbf{Pred}(V)}(\\*v)\\big) = \\pi_{V}(\\*v)$ for all $V$.\nFinally, \nnote that\n\\begin{align*}\n \\bigcap_{i=1}^n \\pi^{-1}_{(\\alpha_i, V_i)}(\\{v_i\\})  = \\bigsqcup_{\\substack{\\{\\texttt{\\textbf{f}}_{V}\\}_{V \\in \\*B} \\in \\texttt{\\textbf{F}}(\\*B) \\\\\\{\\texttt{\\textbf{f}}^{\\alpha_i}_{V}\\}_{V \\in \\*B} \\models V_i = v_i \\\\ \\text{for each } i }} \\{\\texttt{f}_V\\}_{V \\in \\*B}\n\\end{align*}\nwhere each set in the finite disjoint union is of the form \\eqref{eq:probabilityofmechanisms}.\nThus the measure of the left-hand side can be written as a sum of measures of such sets, which use only counterfactuals of the form \\eqref{eq:standardformdefinition},\nshowing agreement of the measures (by Fact~1).\n\\end{proof}\n\\begin{corollary}\\label{cor:standardform}\n$\\mathfrak{S}_3^\\prec$ bijects with the set of atomless measures in $\\mathfrak{P}(\\texttt{F})$, which we denote $\\mathfrak{S}^\\prec_{\\mathrm{std}}$.\nWe write the map as $\\varpi^\\prec_{\\mathrm{std}} : \\mathfrak{S}_3^\\prec \\to \\mathfrak{S}^\\prec_{\\mathrm{std}}$.\n\\qed\n\\end{corollary}\nWhere the order $\\prec$ is clear, the above result permits us to abuse notation, using e.g. $\\mu$ to denote either an element of $\\mathfrak{S}_3^\\prec$ or its associated point $\\varpi^\\prec_{\\mathrm{std}}(\\mu)$ in $\\mathfrak{S}^\\prec_{\\mathrm{std}}$.\nWe will henceforth indulge in such abuse.\n\n\\begin{proof}[Proof of Fact~\\ref{prop:alternative}]\nThe follows easily from Lem.~\\ref{lem:acausals1canonical} below, adapted from \\citet[Thm.~1]{suppes:zan81}.\nThis shows that every atomless distribution is generated by some SCM; furthermore, it can  chosen so as to exhibit no causal effects whatsoever.\n\\end{proof}\n\\begin{definition}\nSay that $\\nu \\in \\mathfrak{P}\\big(\\texttt{\\textbf{F}}(\\*V)\\big)$ is \\emph{acausal} if $\\nu(\\pi^{-1}_{(V, \\*p)}(\\{v_1\\}) \\cap \\pi^{-1}_{(V, {\\*p}')}(\\{v_2\\})\\big) = 0$\nfor every $(V, \\*p), (V, \\*p') \\in \\xi$ and $v_1 \\neq v_2 \\in \\chi_V$.\n\n\\end{definition}\n\\begin{lemma}\\label{lem:acausals1canonical}\nLet $\\mu \\in \\mathfrak{P}(\\chi_{\\*V})$ be atomless. Then there is a ${\\mathcal{M}} \\in \\mathfrak{M}^{\\mathrm{std}}_\\prec$ (see Def.~\\ref{def:standardform}) with an acausal noise distribution such that $\\mu = (\\varpi_1 \\circ \\varpi_2 \\circ \\varpi_3)({\\mathcal{M}})$. \n\\end{lemma}\n\\begin{proof}\nConsider $\\nu \\in \\mathfrak{P}\\big(\\texttt{\\textbf{F}}(\\*V)\\big) = \\mathfrak{P}\\big(\\bigtimes_{(V, \\*p)} \\chi_V \\big)$ determined on a basis as follows:\n$\\nu\\big( \\pi^{-1}_{(V_1, \\mathbf{p}_1)}(\\{v_1\\})\\cap \\dots \\cap \\pi^{-1}_{(V_n, \\*p_n)}(\\{v_n\\}) \\big) = \\mu\\big( \\pi^{-1}_{V_1}(\\{v_1\\}) \\cap \\dots \\cap \\pi^{-1}_{V_n}(\\{v_n\\}) \\big)$.\nThis is clearly acausal and atomless.\n\\end{proof}\n\n\\subsection{Proofs from \\S{}{3.2}}\n\n\\begin{proof}[Proof of Prop. \\ref{prop:collapse1} (Collapse set $\\mathfrak{C}_1$ is empty)]\nLet $\\mu \\in \\mathfrak{S}_1$ and $\\nu \\in \\mathfrak{S}^\\prec_{\\mathrm{std}}$ with $(\\varpi_1 \\circ \\varpi_2 \\circ \\varpi_{\\mathrm{std}}^{-1})(\\nu) = \\mu$.\nBy Lemma \\ref{lem:acausals1canonical} we may assume $\\nu$ is acausal.\nLet $X$ be the first, and $Y$ the second variable with respect to $\\prec$.\nNote there are $x^*$, $y^*$ such that $\\mu(\\pi^{-1}_X(\\{x^*\\}) \\cap \\pi^{-1}_Y(\\{y^*\\})) > 0$;\nlet $x^\\dagger \\neq x^*$, $y^\\dagger \\neq y^*$.\nConsider $\\nu'$ defined as follows where $\\digamma_3$ stands for any set of the form\n$\\pi^{-1}_{(V_1, {\\*p}_1)}(\\{v_1\\}) \\cap \\dots \\cap \\pi^{-1}_{(V_n, {\\*p}_n)}(\\{v_n\\}) \\subset \\texttt{\\textbf{F}}(\\*V)$, for $V_i \\in \\*V$, $\\*p_i \\in \\chi_{\\mathbf{P}(V_i)}$, $v_i \\in \\chi_{V_i}$,\nand $\\digamma_1$ is the corresponding $\\pi^{-1}_{V_1}(\\{v_1\\}) \\cap \\dots \\cap \\pi^{-1}_{V_n}(\\{v_n\\}) \\subset \\chi_{\\*V}$.\n\\begin{multline*}\n \\nu'\\big( \\pi^{-1}_{(X, ())}(\\{x\\}) \\cap \\pi^{-1}_{(Y, (x^*))}(\\{y_*\\}) \\cap \\pi^{-1}_{(Y, (x^{\\dagger}))}(\\{y_\\dagger\\}) \\cap \\digamma_3 \\big) =\\\\\n \\begin{cases}\n \\mu\\big(\\pi^{-1}_X(\\{x^*\\}) \\cap \\pi^{-1}_Y(\\{y^*\\}) \\cap \\digamma_1 \\big), & x = x^*, y_* = y^* \\neq y_\\dagger\\\\% = y^{\\dagger}\\\\\n 0, & x = x^*, y_* = y^{\\dagger} \\neq y_\\dagger\\\\% = y^*\\\\\n 0, & x = x^*, y_* = y_{\\dagger} = y^*\\\\\n \\mu\\big(\\pi^{-1}_X(\\{x^*\\}) \\cap \\pi^{-1}_Y(\\{y^\\dagger\\}) \\cap \\digamma_1 \\big), & x = x^*, y_* = y_{\\dagger} = y^\\dagger\\\\\n \\mu\\big(\\pi^{-1}_X(\\{x^\\dagger\\}) \\cap \\pi^{-1}_Y(\\{y\\}) \\cap \\digamma_1 \\big), & x = x^\\dagger\n \\end{cases}\n\n\n\\end{multline*}\nWe claim that $\\mu = \\mu'$ where $\\mu' = (\\varpi_1 \\circ \\varpi_2 )(\\nu')$; it suffices to show agreement on sets of the form $\\pi^{-1}_X(\\{x\\}) \\cap \\pi^{-1}_Y(\\{y\\}) \\cap \\digamma_1$. If $x = x^\\dagger$ then the last case above occurs; if $x = x^*$ and $y = y^\\dagger$ then we are in the fourth case; if $x = x^*$ and $y = y^*$ then exclusively the first case applies. In all cases the measures agree.\nLet $(\\nu_\\alpha)_\\alpha = \\varpi_2(\\nu)$ and $(\\nu'_\\alpha)_\\alpha = \\varpi_2(\\nu')$\nbe the Level 2 projections of $\\nu$, $\\nu'$ respectively.\nNote that $\\nu_{X \\coloneqq x^\\dagger}(y^\\dagger) < \\nu'_{X \\coloneqq x^\\dagger}(y^\\dagger)$.\nThis shows that the standard-form measures $\\nu$, $\\nu'$ project down to different points in $\\mathfrak{S}_2$ (in particular differing on the $Y$-marginal at the index corresponding to the intervention $X \\coloneqq x^\\dagger$) while projecting to the same point in $\\mathfrak{S}_1$.\nThus $\\mu \\notin \\mathfrak{C}_1$ and since $\\mu$ was arbitrary, $\\mathfrak{C}_1 = \\varnothing$.\n\\end{proof}\n\n\n\n\\begin{example}[Collapse set $\\mathfrak{C}_2$ is nonempty]\nWe present a $\\mu \\in \\mathfrak{S}_{\\mathrm{std}}^{\\prec}$ for which $\\varpi_2(\\mu) \\in \\mathfrak{C}_2$.\nLet ${\\*S}_n \\subset \\*V$ be the ancestrally closed (\\S\\ref{ss:app:standardform}) set of the $n$ least variables with respect to $\\prec$ and $X$ be the first variable with respect to $\\prec$; thus, e.g., ${\\*S}_1 = \\{X\\}$.\nWhere $\\texttt{\\textbf{f}} = \\{\\texttt{f}_V\\}_{V \\in {\\*S}_n} \\in \\texttt{F}(\\textbf{S}_n)$, define $\\mu(\\texttt{\\textbf{f}}) = 0$ if there is any $V \\in {\\*S}_n \\setminus \\{X\\}$, $\\*p \\neq (0, \\dots, 0) \\in \\chi_{\\mathbf{Pred}(V)}$ such that $\\texttt{f}_V(\\*p) = 0$, and otherwise define $\\mu(\\texttt{\\textbf{f}}) = 1\/2^n$.\nNote that this example is \\emph{monotonic} in the sense of \\cite{Angrist,Pearl1999}.\n\nWe claim $\\mu' = \\mu$ for any $\\mu' \\in \\mathfrak{S}_{\\mathrm{std}}^\\prec$ projecting to the same Level 2, i.e., such that $\\varpi_2(\\mu') = \\varpi_2(\\mu)$; note that it suffices to consider only candidate counterexamples with order $\\prec$ since $\\varpi_2(\\mu) \\notin \\mathfrak{S}_2^{\\prec'}$ for any $\\left.\\prec'\\right. \\neq \\left.\\prec\\right.$.\nIt suffices to show that $\\mu(\\texttt{f}) = \\mu'(\\texttt{f})$ for any $n$ and $\\texttt{\\textbf{f}} = \\{\\texttt{f}_V\\}_{V \\in {\\*S}_n}$; recall that in the measures, $\\texttt{f}$ denotes a set of the form \\eqref{eq:probabilityofmechanisms}.\nLet $(\\mu_\\alpha)_\\alpha = \\varpi_2(\\mu) \\in \\mathfrak{S}_2^\\prec$ and $(\\mu'_\\alpha)_\\alpha = \\varpi_2(\\mu')$, with $(\\mu_\\alpha)_\\alpha = (\\mu'_\\alpha)_\\alpha$.\nSince $\\mu'_{\\mathbf{Pred}(V) \\coloneqq \\*p}(\\pi^{-1}_V(\\{1\\})) = 1$ for any $V \\in {\\*S}_n \\setminus \\{X\\}$, $\\*p \\neq (0, \\dots, 0)$, probability bounds show $\\mu'(\\texttt{f})$ vanishes unless $\\texttt{f}_V(\\*p) = 1$ for each such $\\*p$, in which case\n\\begin{equation}\\label{eq:toreducetol2}\n\\mu'(\\texttt{f})=\n\\mu'\\Big(\n\\bigcap_{i=1}^n \\pi^{-1}_{(V_i, \\{V_1, \\dots, V_{i-1}\\} \\coloneqq (0, \\dots, 0))}(\\{v_i\\})\\Big)\n\\end{equation}\nfor some $v_i \\in \\chi_{V_i}$,\nwhere we have labeled the elements of ${\\*S}_n$ as $V_1, \\dots, V_n$,\nwith $V_1 \\prec \\dots \\prec V_n$.\nWe claim this is reducible---again using probabilistic reasoning alone---to a linear combination of quantities fixed by $(\\mu'_\\alpha)_\\alpha$, the Level 2 projection of $\\mu'$, which is the same as the projection $(\\mu_\\alpha)_\\alpha$ of $\\mu$.\nThis can be seen by an induction on the number $m = \\left|M\\right|$ where $M = \\{i : v_i = 1 \\}$: note\n\\eqref{eq:toreducetol2} becomes\n\\begin{multline*\n\\mu'\\Big(\n\\bigcap_{\\substack{i \\notin M}} \\pi^{-1}_{(V_i, \\{V_1, \\dots, V_{i-1}\\} \\coloneqq (0, \\dots, 0))}(\\{0\\})\\Big)\\\\\n- \\sum_{M' \\subsetneq M}\\mu'\\Big(\n\\bigcap_{\\substack{i \\notin M'}} \\pi^{-1}_{(V_i, \\{V_1, \\dots, V_{i-1}\\} \\coloneqq (0, \\dots, 0))}(\\{0\\})\n\\cap\n\\bigcap_{\\substack{i \\in M'}} \\pi^{-1}_{(V_i, \\{V_1, \\dots, V_{i-1}\\} \\coloneqq (0, \\dots, 0))}(\\{1\\})\n\\Big)\n\\end{multline*}\nand the inductive hypothesis implies each summand  can be written in the sought form\nwhile the first term becomes\n$\\mu'\\big(\\bigcap_{i\\notin M} \\pi^{-1}_{(V_i, ())}(\\{0\\})\\big) = \\mu'_{()}\\big(\\bigcap_{i \\notin M}\\pi^{-1}_{V_1}(\\{0\\})\\big) = \\mu_{()}\\big(\\bigcap_{i \\notin M}\\pi^{-1}_{V_1}(\\{0\\})\\big)$.\nHere $()$ abbreviates the empty intervention $\\varnothing \\coloneqq ()$.\nThus any Level 3 quantity reduces to Level 2, on which the two measures agree by hypothesis.\n\\label{example:collapse}\n\\end{example}\n\n\\subsection{Remarks on \\S{}{3.3}}\\label{ss:app:3.3}\n\n\\begin{lemma}\n\\label{prop:p2:characterization:binary}\n  Let $(\\mu_{\\alpha})_{\\alpha} \\in \\bigtimes_{\\alpha \\in A^{X \\to Y}_{2}} \\mathfrak{P}(\\chi_{X, Y})$.\n  Then $(\\mu_\\alpha)_{\\alpha} \\in \\mathfrak{S}^{X \\to Y}_{2}$ iff\n    \\begin{align}\\label{eq:snsr}\n    \\mu_{X \\coloneqq x}( x ) = 1\n    \\end{align}\n    for every $x \\in \\chi_X$\n    and \\begin{align}\\label{eq:snscm}\n    \\mu_{X \\coloneqq x}(y)\n    \\ge \\mu_{()}(x, y)\n    \\end{align}\n    for every $x \\in \\chi_{X}$, $y \\in \\chi_Y$. Here $x, y$ abbreviates the basic set $\\pi_X^{-1}(\\{x\\}) \\cap \\pi_Y^{-1}(\\{y\\})$.\n\\end{lemma}\n\\begin{proof\nIt is easy to see that \\eqref{eq:snsr}, \\eqref{eq:snscm} hold for any $(\\mu_\\alpha)_\\alpha$.\nFor the converse,\nconsider the two-variable model over endogenous $\\*Z = \\{X, Y\\}$ with $X \\prec Y$; note that $|\\texttt{\\textbf{F}}(\\*Z)| = 8$.\nA result of\n\\citet{tian:etal06} gives that this model is characterized exactly by \\eqref{eq:snsr}, \\eqref{eq:snscm} so for any such $(\\mu_\\alpha)_\\alpha$ there is a distribution on $\\texttt{\\textbf{F}}(\\*Z)$ such that this model induces $(\\mu_\\alpha)_\\alpha$.\nIt is straightforward to extend this distribution to an atomless measure on $\\texttt{\\textbf{F}}(\\*V)$.\n\\end{proof}\n\n\n\\section{Proofs from \\S\\ref{sec:weaktopology}} \\label{app:empirical}\n\n\\begin{proof}[Proof of Prop.~\\ref{prop:causalprojectioncontinuous}]\nThe continuity of any of the maps amounts to the continuity of projections in product spaces and marginalizations in weak convergence spaces. The latter follows easily from results in \\S{}3.1.3 of \\cite{Genin2018} or \\cite{Billingsley}.\n\nAs for the openness of any $\\varpi_2^{X \\to Y}$, note we can write any $S \\subset \\mathfrak{S}_2$ as $S = \\bigcup_{\\prec} S \\cap \\mathfrak{S}_2^\\prec$ where $\\prec$ in the union ranges over all total orders of $\\*V$. It thus suffices to show that for any $\\prec$ the image of any open $S \\subset \\mathfrak{S}_2^\\prec$ is open.\nDefine the map $\\varpi_2^{\\prec}: \\mathfrak{S}_{\\mathrm{std}}^\\prec \\to \\mathfrak{S}_2^\\prec$ and the map $\\varpi_2^{X \\to Y, \\prec}: \\mathfrak{S}_2^\\prec \\to \\mathfrak{S}_2^{X \\to Y}$ as restrictions of $\\varpi_2$ and $\\varpi_2^{X \\to Y}$ respectively. Evidently, $\\varpi_2^{\\prec}$ is continuous so it suffices to show that $\\varpi_2^{X \\to Y, \\prec} \\circ \\varpi_2^{\\prec}$ is open.\n\nFor any $n \\ge 1$ let $\\*S_{n, \\prec}$ be the initial segment of the first $n$ variables in $\\*V$ when ordered according to $\\prec$, as in Ex.~\\ref{example:collapse}, and define sets $\\mathfrak{S}_{\\mathrm{std}}^{n, \\prec}$, $\\mathfrak{S}_2^{n, \\prec}$ analogously to $\\mathfrak{S}_{\\mathrm{std}}^\\prec$, $\\mathfrak{S}_2^{\\prec}$ but over the set of variables $\\*{S}_{n, \\prec}$.\nDefine maps $\\varpi_{2}^{n, \\prec} : \\mathfrak{S}_{\\mathrm{std}}^{n, \\prec} \\to \\mathfrak{S}_2^{n, \\prec}$ and $\\varpi_2^{X \\to Y, n, \\prec} :\\mathfrak{S}_2^{n, \\prec} \\to \\mathfrak{S}_2^{X \\to Y}$, where $X, Y \\in \\*{S}_{n, \\prec}$, in a similar fashion.\nDefine also a map $\\varpi_{\\mathrm{std}}^{n, \\prec}: \\mathfrak{S}_{\\mathrm{std}}^{\\prec} \\to \\mathfrak{S}_{\\mathrm{std}}^{n, \\prec}$ as a marginalization taking a distribution over mechanisms (recall \\S{}\\ref{ss:app:standardform}) determining all of $\\*V$ to a distribution over deterministic mechanisms for $\\*S_{n, \\prec}$.\nLet $S \\subset \\mathfrak{S}_{\\mathrm{std}}^\\prec$ be an arbitrary basic open set and let $n$ be least such that $\\*{S}_{n, \\prec}$ contains $X$, $Y$, and every variable whose structural equation appears as a cylinder in the finite intersection defining $S$.\nThen note that $\\big(\\varpi_2^{X \\to Y, \\prec} \\circ \\varpi_2^{\\prec}\\big)(S) = \\big(\\varpi_2^{X \\to Y, n, \\prec} \\circ \\varpi_2^{n, \\prec} \\circ \\varpi_{\\mathrm{std}}^{n, \\prec}\\big)(S)$ and $\\varpi_{\\mathrm{std}}^{n, \\prec}(S)$ is certainly open, so it suffices to show that $\\varpi_2^{X \\to Y, n, \\prec} \\circ \\varpi_2^{n, \\prec} :  \\mathfrak{S}_{\\mathrm{std}}^{n, \\prec} \\to \\mathfrak{S}_2^{X \\to Y}$ is open for any $n$.\n\nTo see this, note that $\\mathfrak{S}_2^{X \\to Y}$ and $\\mathfrak{S}_{\\mathrm{std}}^{n, \\prec}$ in the weak topology are homeomorphic to (subsets of products of) probability simplices in appropriate Euclidean spaces $\\mathbb{R}^m$ with the standard topology, as they are distributions over a discrete space.\nThe latter in fact is exactly a probability simplex while $\\mathfrak{S}_2^{X \\to Y}$ is polyhedral by Lemma~\\ref{prop:p2:characterization:binary}, and $\\varpi_2^{X \\to Y, n, \\prec} \\circ \\varpi_2^{n, \\prec}$ is the restriction of a surjective linear mapping under the aforementioned homeomorphism.\nThis must be open by \\citet[Corollary~8]{MIDOLO20091186}.\n\\end{proof}\n\n\\begin{proof}[Proof of Thm. \\ref{thm:l2learning}]\nWe show how Theorem~3.2.1 of \\cite{Genin2018} can be applied to derive the result. Specifically, let $\\Omega = \\bigtimes_\\alpha \\chi_{\\*V}$. Let $\\mathcal{I}$ be the usual clopen basis, and let $W$ be the set of Borel measures $\\mu \\in \\mathfrak{P}(\\Omega)$ that factor as a product $\\mu = \\times_\\alpha \\mu_\\alpha$ where each $\\mu_\\alpha \\in \\mathfrak{S}_1$ and $(\\mu_\\alpha)_\\alpha \\in \\mathfrak{S}_2$. This choice of $W$ corresponds exactly to our notion of experimental verifiability.\n\nIt remains to check that a set is open in $W$ iff the associated set is open in $\\mathfrak{S}_2$ (homeomorphism).\nIt suffices to show their convergence notions agree. Suppose $(\\nu_n)_n$ is a sequence, each $\\nu_n \\in W$, converging to $\\nu = \\times_\\alpha \\mu_\\alpha \\in W$. We have for each $n$ that $\\nu_n = \\times_\\alpha \\mu_{n, \\alpha}$ such that $(\\mu_{n,\\alpha})_\\alpha \\in \\mathfrak{S}_2$. By Theorem~3.1.4 in \\cite{Genin2018}, which is straightforwardly generalized to the infinite product, for each fixed $\\alpha$ we have $(\\mu_{n, \\alpha})_n \\Rightarrow \\mu_\\alpha$. This is exactly pointwise convergence in the product space $\\mathfrak{S}_2$, and the same argument in reverse works for the converse.\n\\end{proof}\n\n\\section{Proofs from \\S\\ref{sec:main}} \\label{app:main}\n\nWe will use the following result to categorize sets in the weak topology.\n\\begin{lemma}\\label{prop:probiscontinuous}\n  If $X \\subset \\vartheta$ is a basic clopen,\n  the map $p_X : (\\mathfrak{S}, \\tau^{\\mathrm{w}}) \\to ([0, 1], \\tau)$ sending $\\mu \\mapsto \\mu(X)$ is continuous and open (in its image), where $\\tau$ is as usual on $[0, 1] \\subset \\mathbb{R}$.\n\\end{lemma}\n\\begin{proof}\nContinuous: the preimage of the basic open $(r_1, r_2) \\cap p_X(\\mathfrak{S})$ where $r_1, r_2 \\in \\mathbb{Q}$ is $\\{ \\mu: \\mu(X) > r_1 \\} \\cap \\{ \\mu : \\mu(X) < r_2 \\} = \\{ \\mu: \\mu(X) > r_1 \\} \\cap \\{ \\mu : \\mu(\\vartheta \\setminus X) > 1- r_2 \\}$, a finite intersection of the subbasic sets \\eqref{subbasis-weak} from \\S{}\\ref{sec:weaktopology}.\nSee also \\citet[Corollary~17.21]{Kechris1995}.\n\nOpen: if $X = \\varnothing$ or $\\vartheta$, then $p_X(\\mathfrak{S}) = \\{0\\}$ or $\\{1\\}$ resp., both open in themselves.\nElse $p_X(\\mathfrak{S}) = [0, 1]$;\nwe show any $Z = p_X\\big(\\bigcap_{i = 1}^n \\{\\mu: \\mu(X_i)>r_i\\} \\big)$ is open.\nConsider a mutually disjoint, covering $\\mathcal{D} = \\big\\{ \\bigcap_{i=0}^n Y_i : Y_0 \\in \\{X, \\vartheta \\setminus X\\},\\text{ each } Y_i \\in \\{X_i, \\vartheta \\setminus X_i\\}\\big\\}$ \nand space $\\Delta = \\{(\\mu(D))_{D \\in \\mathcal{D}} : \\mu \\in \\mathfrak{S}\\} \\subset \\mathbb{R}^{2^{n+1}}$.\nJust as in the Lemma, we have $\\textsf{p}_{S} : \\Delta \\to [0, 1]$, for each $S \\subset \\mathcal{D}$ taking $(\\mu(D))_D \\mapsto \\sum_{D \\in S} \\mu(D)$.\nNote $Z = \\textsf{p}_{\\{D: D\\cap X \\neq \\varnothing\\}}\\big( \\bigcap_{i=1}^n\\textsf{p}^{-1}_{\\{D: D\\cap X_i \\neq \\varnothing\\}}((r_i, 1])  \\big)$ so it suffices to show $\\textsf{p}_S$ is continuous and open; this is straightforward (see the end of the proof of Prop.~\\ref{prop:causalprojectioncontinuous}).\n\\end{proof}\n\n\\begin{proof}[Full proof of Lem.~\\ref{lem:goodcomeager}]\nWe show a stronger result, namely  that the complement of the good set is nowhere dense.\nBy rearrangement and laws of probability we find that the second inequality in \\eqref{cns:ineq:1} is equivalent to\n\\begin{align*}\n\\mu_x(y') \n &< \\mu_{()}(x') + \\mu_{()}(x, y')\\\\\n 1- \\mu_x(y) &< \\underbrace{\\mu_{()}(x') + \\mu_{()}(x)}_1 - \\mu_{()}(x, y)\\\\\n \\mu_x(y) &> \\mu_{()}(x, y).\n\\end{align*}\nLemma~\\ref{prop:p2:characterization:binary} then entails the non-strict analogues of all four inequalities in \\eqref{cns:ineq:1}, \\eqref{cns:ineq:2} are met for any $(\\mu_\\alpha)_\\alpha \\in \\mathfrak{S}^{X \\to Y}_{2}$, so we show that converting each to an equality yields a nowhere dense set, whose finite union is also nowhere dense.\nNote that we have a continuous, open (again, refer to the end of the proof of Prop.~\\ref{prop:causalprojectioncontinuous}), and surjective observational projection $\\pi_{()}: \\mathfrak{S}^{X \\to Y}_{2} \\to \\mathfrak{P}\\big(\\chi_{\\{X, Y\\}}\\big)$,\nand the first inequality in \\eqref{cns:ineq:2} is met iff $(\\mu_\\alpha)_\\alpha \\in \\big(p_{x', y'} \\circ \\pi_{()}\\big)^{-1}(\\{0\\})$ where $p_{x', y'}$ is the map from Lemma~\\ref{prop:probiscontinuous} and $x', y'$ denotes the set $\\pi_X^{-1}(\\{x'\\}) \\cap \\pi_Y^{-1}(\\{y'\\}) \\subset \\chi_{\\{X, Y\\}}$.\nThis is nowhere dense as it is the preimage of the nowhere dense set $\\{0\\}\\subset [0, 1]$ under a map which is continuous and open by Lemma~\\ref{prop:probiscontinuous}. The second inequality of \\eqref{cns:ineq:2} is wholly analogous after rearrangement.\n\nAs for \\eqref{cns:ineq:1},\ndefine a function $d: \\mathfrak{S}^{X \\to Y}_2 \\to [0, 1]$ taking $(\\mu_\\alpha)_\\alpha \\mapsto \\mu_{X \\coloneqq x}(y') - \\mu_{()}(x, y')$; this function $d$ is continuous by Lemma~\\ref{prop:probiscontinuous} and the continuity of addition and projection, and is once again open. Note\nthat the first inequality of \\eqref{cns:ineq:1} holds iff $d((\\mu_\\alpha)_\\alpha) = 0$.\nFor any $\\mu \\in \\mathfrak{S}_3^{X}$ such that $(\\varpi^{X \\to Y}_{2}\\circ \\varpi_2)(\\mu) = (\\mu_\\alpha)_\\alpha$, note that $d((\\mu_\\alpha)_\\alpha) = \\mu\\big(x', y'_x\\big)$ where $x', y'_x$ abbreviates the basic set $\\pi_{((), X)}^{-1}(\\{x'\\}) \\cap \\pi_{(X \\coloneqq x, Y)}^{-1}(\\{y'\\}) \\in \\mathcal{B}(\\chi_{A \\times \\*V})$.\nThus $d$ is surjective, so that $d^{-1}(\\{0\\})$ is nowhere dense since $\\{0\\} \\subset [0, 1]$ is nowhere dense.\nThe second inequality in \\eqref{cns:ineq:1} is again totally analogous.\n\\end{proof}\n\n\\begin{proof}[Proof of Lem.~\\ref{lem:separation}]\nAbbreviate $\\mu_3$ as $\\mu$, and without loss take $\\mu \\in \\mathfrak{S}^\\prec_{\\mathrm{std}}$.\nNote that \\eqref{cns:ineq:1}, \\eqref{cns:ineq:2} entail\n\\begin{equation*}\n0 < \\mu(x', y'_x) < \\mu(x'), \\quad 0 < \\mu(x', y'_{x'}) < \\mu(x').\n\\end{equation*}\nand therefore\n\\begin{align*}\n0 < \\mu\\big(\\pi^{-1}_{((), X)}(\\{x'\\}) \\cap \\pi^{-1}_{(x^*, Y)}(\\{1\\})\\big) < \\mu\\big(\\pi^{-1}_{((), X)}(\\{x'\\})\\big)\n\\end{align*}\nfor each $x^* \\in \\chi_X = \\{0, 1\\}$.\nIn turn this \nentails that there are some values $y_{0}, y_{1} \\in \\{ 0, 1 \\}$ such that\n$\\mu(\\Omega_1) > 0$, $\\mu(\\Omega_2) > 0$\nwhere the disjoint sets $\\{\\Omega_i\\}_i$ are defined as\n\\begin{align*}\n\\Omega_1 &= \\pi^{-1}_{((), X)}(\\{x'\\}) \\cap \\pi^{-1}_{(X \\coloneqq 0, Y)} (\\{y_0\\}) \\cap \\pi^{-1}_{(X \\coloneqq 1, Y)} (\\{y_1\\})\\\\\n\\Omega_2 &= \\pi^{-1}_{((), X)}(\\{x'\\}) \\cap \\pi^{-1}_{(X \\coloneqq 0, Y)} (\\{y^\\dagger_0\\}) \\cap \\pi^{-1}_{(X \\coloneqq 1, Y)} (\\{y^\\dagger_1\\})\n\\end{align*}\nwhere in the second line, $y_0^\\dagger = 1-y_0$ and $y_1^\\dagger = 1 - y_1$.\nNote that for $i = 1, 2$ we have conditional measures $\\mu_i(S_i) = \\frac{\\mu(S_i)}{\\mu(\\Omega_i)}$ for $S_i \\in \\mathcal{B}(\\Omega_i)$; further, $\\Omega_i$ is Polish, since each is clopen.\nThis implies $\\Omega_i$ is a standard atomless (since $\\mu$ is) probability space under $\\mu_i$.\nBy \\citet[Thm.~17.41]{Kechris1995}, there are Borel isomorphisms $f_i : \\Omega_i \\hookdoubleheadrightarrow [0, 1]$ pushing $\\mu_i$ forward to Lebesgue measure $\\lambda$, i.e., $\\mu_i(f_i^{-1}(B)) = \\lambda(B)$ for $B \\in \\mathcal{B}([0, 1])$.\nThus $g = f_2^{-1} \\circ f_1 : \\Omega_1  \\hookdoubleheadrightarrow \\Omega_2$ is $\\mu_i$-preserving: for $X_1 \\in \\mathcal{B}(\\Omega_1)$,\n\\begin{align}\n\\mu(g(X_1)) = \\frac{\\mu(\\Omega_2)}{\\mu(\\Omega_1)} \\mu(X_1).\\label{eqn:NOCBKNantKPlTpiViI0=}\n\\end{align}\n\nConsider $\\mu' = \\varpi_3(\\mathcal{M}')$ for a new $\\mathcal{M}' \\in \\mathfrak{M}_\\prec$, given as follows. Its exogenous valuation space is $\\chi_{\\*U} = \\Omega'$ where we define the sample space $\\Omega' = \\texttt{\\textbf{F}}(\\*V) \\times \\{ \\mathrm{T}, \\mathrm{H} \\}$; that is, a new exogenous variable representing a coin flip is added to some representation of the choice of deterministic standard form mechanisms.\nFix constants $\\varepsilon_1, \\varepsilon_2 \\in (0, 1)$ with $\\varepsilon_1 \\cdot \\mu(\\Omega_1) = \\varepsilon_2 \\cdot \\mu(\\Omega_2)$ and define its exogenous noise distribution $P$ by\n\\begin{equation}\\label{eq:modprobz}\nP(X \\times \\{\\mathrm{S}\\}) = \n\\begin{cases}\n(1-\\varepsilon_1)\\cdot \\mu(X), & X \\subset \\Omega_1, \\mathrm{S} = \\mathrm{T}\\\\\n\\varepsilon_1 \\cdot \\mu(X), & X \\subset \\Omega_1, \\mathrm{S} = \\mathrm{H}\\\\\n(1-\\varepsilon_2)\\cdot\\mu(X), & X \\subset \\Omega_2, \\mathrm{S} = \\mathrm{T}\\\\\n\\varepsilon_2 \\cdot \\mu(X), & X \\subset \\Omega_2, \\mathrm{S} = \\mathrm{H}\\\\\n\\mu(X), & X \\subset \\texttt{\\textbf{F}}(\\*V) \\setminus (\\Omega_1 \\cup \\Omega_2), \\mathrm{S} = \\mathrm{T}\\\\\n0, & X \\subset \\texttt{\\textbf{F}}(\\*V) \\setminus (\\Omega_1 \\cup \\Omega_2), \\mathrm{S} = \\mathrm{H}\n\\end{cases}.\n\\end{equation}\nWhere $\\texttt{\\textbf{f}}  \\in \\texttt{\\textbf{F}}(\\*V)$ and $V \\in \\*V$ write $\\texttt{f}_V$ for the deterministic mechanism (of signature $\\chi_{\\mathbf{Pred}(V)} \\to \\chi_V$) for $V$ in $\\texttt{f}$. (Note that each $\\texttt{f}$ is just an indexed collection of such mechanisms $\\texttt{f}_V$.)\nThe function $f'_V$ in $\\mathcal{M}'$ is defined at the initial variable $X$ as\n$f'_X(\\texttt{\\textbf{f}}, \\mathrm{S}) = \\texttt{\\textbf{f}}_X$ for both values of $\\mathrm{S}$,\nand for $V \\neq X$ is defined as follows, where $\\*p \\in \\mathbf{Pred}(V)$:\n\\begin{equation} \\label{eqn:modseqz}\n{f}'_V\\big(\\*p, (\\texttt{\\textbf{f}}, \\mathrm{S})\\big) =\n \\begin{cases}\n      (g(\\texttt{\\textbf{f}}))_V(\\*p), & \\texttt{\\textbf{f}}  \\in\\Omega_1, \\mathrm{S} = \\mathrm{H},\\, \\pi_X(\\*p) = x\\\\\n      (g^{-1}(\\texttt{\\textbf{f}}))_V(\\*p), & \\texttt{\\textbf{f}}  \\in\\Omega_2, \\mathrm{S} = \\mathrm{H},\\, \\pi_X(\\*p) = x\\\\\n      \\texttt{\\textbf{f}}_V(\\*p), &\\textnormal{otherwise}\n                                                 \\end{cases}.\n                                                 \\end{equation}\n\nWe claim that $\\varpi_2(\\mu') = \\varpi_2(\\mu)$.\nIt suffices to show for any $\\*Z \\coloneqq \\*z \\in A$ and ${\\mathbf{w}} \\in \\chi_{{\\*W}}$, $\\*W$ finite,\nwe have\n\\begin{equation}\n    \\mu(\\theta) = \\mu'(\\theta), \\text{ where }\\theta = \\bigcap_{W \\in {\\*W}}\\pi^{-1}_{(\\*Z \\coloneqq \\*z, W)}(\\{\\pi_W(\\*w)\\}).\\label{eqn:IOXCVOIXCJVSOIJDklfsjdlfksdj}\n\\end{equation}\nAssume $\\pi_Z(\\*w) = \\pi_Z(\\*z)$ for every $Z \\in \\*Z \\cap {\\*W}$, since both sides of \\eqref{eqn:IOXCVOIXCJVSOIJDklfsjdlfksdj} trivially vanish otherwise.\nWhere $\\texttt{\\textbf{f}} \\in \\texttt{\\textbf{F}}(\\*V)$\nwrite, e.g., $\\texttt{\\textbf{f}} \\models \\theta$ if $m^{{\\mathcal{M}}_A}(\\texttt{\\textbf{f}}) \\in \\theta$, where ${\\mathcal{M}}$ is a standard form model (Def.~\\ref{def:standardform}); for\n$\\omega' \\in \\Omega'$ write $\\omega' \\models' \\theta$\nif $m^{{\\mathcal{M}}'_A}(\\omega') \\in \\theta$. By the last two cases of \\eqref{eqn:modseqz} we have\n\\begin{align}\n\\mu'(\\theta)\n&= \\sum_{\\mathrm{S} =  \\mathrm{T}, \\mathrm{H}}P\\big( \\{ (\\texttt{\\textbf{f}}, \\mathrm{S}) \\in \\Omega' : (\\texttt{\\textbf{f}}, \\mathrm{S}) \\models' \\theta \\} \\big)\\nonumber\\\\\n&= \\mu\\big( \\{ \\texttt{\\textbf{f}} \\in \\texttt{\\textbf{F}}(\\*V) \\setminus (\\Omega_1 \\cup \\Omega_2) : \\texttt{\\textbf{f}} \\models \\theta \\}\\big) +\n\\sum_{\\substack{\\mathrm{S} = \\mathrm{T}, \\mathrm{H}\\\\ i = 1, 2}} P\\big( \\{ (\\texttt{\\textbf{f}}, \\mathrm{S}) \\in \\Omega' : \\texttt{\\textbf{f}} \\in \\Omega_i, (\\texttt{\\textbf{f}}, \\mathrm{S}) \\models' \\theta \\} \\big).\\label{eqn:CXJSDFJKSDKFJKSDVC}\n\\end{align}\nApplying the first four cases of \\eqref{eq:modprobz} and the third case of \\eqref{eqn:modseqz}, the second term of \\eqref{eqn:CXJSDFJKSDKFJKSDVC} becomes\n\\begin{equation}\n\\sum_i \\Big[\\varepsilon_i \\cdot \\mu\\big( \\{ \\texttt{\\textbf{f}} \\in \\Omega_i : (\\texttt{\\textbf{f}}, \\mathrm{H}) \\models' \\theta \\} \\big) +\n \\left(1- \\varepsilon_i\\right) \\cdot\\mu\\big( \\{ \\texttt{\\textbf{f}} \\in \\Omega_i : \\texttt{\\textbf{f}} \\models \\theta \\} \\big)\\Big] .\\label{eqn:ckxljvXIOCJVOX}\n\\end{equation}\nEither $X \\in \\*Z$ and $\\pi_X(\\*z) = x$, or not.\nIn the former case: defining $X_i = \\{\\texttt{\\textbf{f}} \\in \\Omega_i : \\texttt{\\textbf{f}} \\models \\theta\\}$ for each $i = 1, 2$,\nthe first two cases of \\eqref{eqn:modseqz} yield that\n\\begin{align}\n\\{ \\texttt{\\textbf{f}} \\in \\Omega_1 : (\\texttt{\\textbf{f}}, \\mathrm{H}) \\models' \\theta \\} &= \\{\\texttt{\\textbf{f}} \\in \\Omega_1 : g(\\texttt{\\textbf{f}}) \\models \\theta\\}\n= g^{-1}(X_2 )\\nonumber\\\\\n\\{ \\texttt{\\textbf{f}} \\in \\Omega_2 : (\\texttt{\\textbf{f}}, \\mathrm{H}) \\models' \\theta \\} &= \\{\\texttt{\\textbf{f}} \\in \\Omega_2 : g^{-1}(\\texttt{\\textbf{f}}) \\models \\theta\\}\n= g(X_1).\\label{eqn:yDuRPR\/5r83+sSdJ2Iw=}\n\\end{align}\nApplying \\eqref{eqn:yDuRPR\/5r83+sSdJ2Iw=} and \\eqref{eqn:NOCBKNantKPlTpiViI0=}, \\eqref{eqn:ckxljvXIOCJVOX} becomes\n\\begin{gather}\n\\varepsilon_1 \\cdot \\frac{\\mu(\\Omega_1)}{\\mu(\\Omega_2)} \\cdot \\mu\\big( X_2 \\big)\n + \\left(1- \\varepsilon_1\\right) \\cdot\\mu\\big( X_1 \\big)\n  + \\varepsilon_2 \\cdot\\frac{\\mu(\\Omega_2)}{\\mu(\\Omega_1)} \\cdot \\mu\\big( X_1\\big)\n + \\left(1- \\varepsilon_2\\right) \\cdot\\mu\\big( X_2 \\big)\\nonumber\\\\\n = \\mu(X_1) + \\mu(X_2),\\label{eqn:CxoFLyrcYfY8on7xY\/A=}\n\\end{gather}\nthe final cancellation by choice of $\\varepsilon_1, \\varepsilon_2$.\nIn the latter case: since $m^{{\\mathcal{M}}}(\\texttt{\\textbf{f}}) \\in \\pi^{-1}_X(\\{x'\\})$ for any $\\texttt{\\textbf{f}} \\in \\Omega_1 \\cup \\Omega_2$, the third case of \\eqref{eqn:modseqz} gives $\\{ \\texttt{\\textbf{f}} \\in \\Omega_i : (\\texttt{\\textbf{f}}, \\mathrm{H}) \\models'\\theta\\} = X_i$. Thus \\eqref{eqn:ckxljvXIOCJVOX} becomes \\eqref{eqn:CxoFLyrcYfY8on7xY\/A=} in either case.\nPutting in \\eqref{eqn:CxoFLyrcYfY8on7xY\/A=} as the second term in \\eqref{eqn:CXJSDFJKSDKFJKSDVC}, we find $\\mu(\\theta) = \\mu'(\\theta)$.\n\nNow we claim $\\mu(\\zeta) \\neq \\mu'(\\zeta)$\nfor $\\zeta = \\zeta_0 \\cap \\zeta_1$ where\n$\\zeta_1 = \\pi^{-1}_{(X \\coloneqq 1, Y)}(\\{y_1\\})$ and $\\zeta_0 = \\pi^{-1}_{(X \\coloneqq 0, Y)}(\\{y_0\\})$.\nWe have\n\\begin{align}\n\\mu'(\\zeta) = &\\mu\\big( \\{ \\texttt{\\textbf{f}} \\in \\Omega \\setminus (\\Omega_1 \\cup \\Omega_2) : \\texttt{\\textbf{f}} \\models \\zeta \\} \\big)\\nonumber\\\\\n&+ \\sum_{i=1,2} \\Big[\\varepsilon_i \\cdot \\mu\\big( \\{ \\texttt{\\textbf{f}} \\in \\Omega_i : (\\texttt{\\textbf{f}}, \\mathrm{H}) \\models' \\zeta \\} \\big) +\n\\left(1- \\varepsilon_i\\right) \\cdot\\mu\\big( \\{ \\texttt{\\textbf{f}} \\in \\Omega_i : \\texttt{\\textbf{f}} \\models \\zeta \\} \\big)\\Big].\\label{eqn:XT26+J4OsmYVQhi6+dY=}\n\\end{align}\nFirst suppose that $x = 0$.\nIf $\\texttt{\\textbf{f}} \\in \\Omega_1$, then\nnote that $(\\texttt{\\textbf{f}}, \\mathrm{H}) \\models' \\zeta_0$ iff $g(\\texttt{\\textbf{f}}) \\models \\zeta_0$, but this is never so, since $g(\\texttt{\\textbf{f}}) \\in \\Omega_2$.\nIf $\\texttt{\\textbf{f}} \\in \\Omega_2$,\nthen\n$(\\texttt{\\textbf{f}}, \\mathrm{H}) \\models' \\zeta_1$ iff $\\texttt{\\textbf{f}} \\models \\zeta_1$, which is never so again by choice of $\\Omega_2$.\nIf $x = 1$ then we find that $(\\texttt{\\textbf{f}}, \\mathrm{H}) \\not\\models \\zeta_1$ (if $\\texttt{\\textbf{f}} \\in \\Omega_1$) and $(\\texttt{\\textbf{f}}, \\mathrm{H}) \\not\\models \\zeta_0$ (if $\\texttt{\\textbf{f}} \\in \\Omega_2$).\nThus $(\\texttt{\\textbf{f}}, \\mathrm{H}) \\not\\vDash ' \\zeta$ for any $\\texttt{\\textbf{f}} \\in \\Omega_1 \\cup \\Omega_2$ and \\eqref{eqn:XT26+J4OsmYVQhi6+dY=} becomes\n\\begin{equation*}\n \\mu\\big( \\{ \\texttt{\\textbf{f}} \\in \\Omega : \\texttt{\\textbf{f}} \\models \\zeta \\} \\big)\n - \\sum_{i=1,2} \\varepsilon_i \\cdot \\mu\\big( \\{ \\texttt{\\textbf{f}} \\in \\Omega_i : \\texttt{\\textbf{f}} \\models \\zeta \\} \\big)\n = \\mu\\big( \\{ \\texttt{\\textbf{f}} \\in \\Omega : \\texttt{\\textbf{f}} \\models \\zeta \\} \\big) - \\varepsilon_1 \\cdot\\mu(\\Omega_1) < \\mu(\\zeta).\n\\end{equation*}\n\nIt is straightforward to check (via casework on the values $y_0$, $y_1$) that $\\mu$ and $\\mu'$ disagree also on the PNS: $\\mu(y_x, y'_{x'}) \\neq \\mu'(y_x, y'_{x'})$ as well as its converse.\nAs for the probability of sufficiency (Definition \\ref{def:probcaus}), note that\n\\begin{align*}\n    P(y_x \\mid x', y') = \\frac{P(y_x, x', y'_{x'}) + \\overbrace{P(y_x, y'_x, x', x)}^0}{P(x', y')}\n\\end{align*}\nand it is again easily seen (given the definition of the $\\Omega_i$) that $\\mu(y_x, x', y'_{x'}) \\neq \\mu'(y_x, x', y'_{x'})$ while the two measures agree on the denominator; similar reasoning shows disagreement on the probability of enablement, since\n\\begin{equation*}\n    P(y_x \\mid y') = \\frac{P(y_x, y'_{x'}, x') + \\overbrace{P(y_x, y'_{x}, x)}^0}{P(y')}. \\qedhere\n\\end{equation*}\n   \n   \n\\end{proof}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\nNon-equilibrium quantum many-body systems are nowadays routinely probed in experiments with ultracold atoms with unprecedented control over their parameters such as particle number, interaction strength, and external potentials. A plethora of non-equilibrium systems has been realized to address a wide variety of physics questions, including Anderson localization of Bose-Einstein condensates (BECs) \\cite{clement_suppression_2005, fort_effect_2005, schulte_routes_2005, billy_direct_2008, roati_anderson_2008}, many-body localization in disordered lattices \\cite{schreiber_observation_2015, lukin_probing_2019}, pre-thermalization of one-dimensional (1D) BECs \\cite{gring_relaxation_2012}, and quench-dynamics of spin-model systems \\cite{bernien_probing_2017}, to name just a few.\\\\ \nAt the same time, the theoretical understanding of non-equilibrium quantum many-body systems still lags behind that obtained for stationary or equilibrium systems. In particular, the role of ``classical'' chaos on non-equilibrium quantum many-body systems is currently subject of intense scrutiny, see e.g.~\\cite{ho_periodic_2019, hallam_lyapunov_2019, lewis-swan_unifying_2019, xu_does_2020}. For a bosonic quantum many-body system, the mean-field limit can be viewed as the classical limit in which the particle creation and annihilation operators lose their quantum properties and start to act as classical fields. The mean-field limit, typically involves non-linear (partial) differential equations, and can exhibit chaos with exponential separation in Hilbert space characterized by a positive Lyapunov exponent $\\lambda$ \\cite{brezinova_wave_2011, cassidy_threshold_2009}. One of the open questions is, whether wave chaos, more specifically the positive $\\lambda$, imprints itself onto the dynamics of a quantum many-body system even at finite particle numbers and how such an imprint could be measured. Recently, out-of-time-order correlators have been suggested as suitable probes for a positive $\\lambda$ (see e.g.~\\cite{Kitaev2014,sekino_fast_2008,shenker_black_2014,maldacena_bound_2016,lewis-swan_unifying_2019, xu_does_2020}). \\\\\nIn this paper, we find an imprint of chaos on a different observable within a paradigmatic bosonic system: A quasi 1D BEC initially trapped harmonically and then released to expand in a shallow disordered or periodic potential. We show that the fraction of non-condensed particles increases exponentially over time and that the associated rate is given by the Lyapunov exponent $\\lambda$ obtained from mean-field chaos. The depletion occurs on time scales during which most of the initial interaction energy is converted into kinetic energy, and comes to a halt at times close to the so-called scrambling time (or Ehrenfest time) \\cite{rammensee_many-body_2018, tomsovic_post-ehrenfest_2018, maldacena_bound_2016}. We observe chaos-induced depletion both in shallow disordered as well as periodic potentials showing that the effect is quite general and does not rely on the intrinsic randomness of a disordered landscape.\\\\\nFinally, we demonstrate that the condensate depletion and thus the Lyapunov exponent $\\lambda$ is accessible experimentally through the analysis of fluctuations of the total particle density in momentum space. While the condensed part is coherent and leads to interference fringes in the total density, the non-condensed part is incoherent and piles up over time as a non-fluctuating background. Analyzing the interference fringes after time of flight would thus allow to extract experimentally the fraction of non-condensed particles as a function of time and compare to the theoretically obtained $\\lambda$. Condensate depletion thus offers itself as an experimentally accessible probe to investigate the role of (mean-field or classical) chaos in non-equilibrium quantum matter.\\\\\nWhile our findings are generally applicable to bosonic systems that exhibit mean-field chaos, we pick one specific system already realized experimentally \\cite{billy_direct_2008} to obtain numerical results. The initially harmonically trapped quasi 1D BEC of $N$ $^{87}$Rb atoms is released at $t=0$ to expand in a shallow potential. As units we use $\\hbar=m=\\omega_0=1$, with $\\omega_0$ being the frequency of the initial longitudinal harmonic trap which amounts to a time unit of $t_0\\approx30$ms and a space unit of $x_0\\approx4.6\\mu$m. For the number of atoms, we take $N=1.2\\times 10^4$ following \\cite{billy_direct_2008}, as well as larger values, i.e.~$N=1.2\\times10^5$ and $N=1.2\\times10^6$ to investigate the effect of varying $N$.\\\\\nDescribing the quasi-1D system on a mean-field level the Gross-Pitaevskii equation (GPE) takes the form\n\\begin{equation}\ni\\frac{\\partial \\psi(x,t)}{\\partial t} = \n\\left(-\\frac{1}{2}\\frac{\\partial^2}{\\partial x^2}\n+V(x)+g|\\psi(x,t)|^2\\right)\\psi(x,t),\n\\label{eq:gpe}\n\\end{equation}\nwhere the nonlinearity is $g\\approx400$ with the above parameters and the normalization $\\int dx|\\psi(x,t)|^2=1$. The potential $V(x)$ corresponds to the harmonic potential at $t=0$, and to the periodic or disordered speckle potential at $t>0$ with amplitude much smaller than the mean energy per particle $e$. This system exhibits chaos on the mean-field level \\cite{brezinova_wave_2011}: Two wave functions, $\\psi_a(x,0)$ and $\\psi_{b}(x,0)$, respectively, initially very close to each other in Hilbert space as measured by a distance norm, separate exponentially in time until quasi-orthogonality is reached, see Fig.~\\ref{fig:N6}.\n\\begin{figure}[t]\n\t\\includegraphics[width=\\columnwidth]{N6_grid_bog_comp_log.pdf}\n\t\\caption{$d^{(2)}$ (Eq.~\\ref{eq:d2}) for two initial conditions obtained by linear distortion of the mean-field ground state (black dashed-dotted line), distance function $\\bar d^{(2)}$ (Eq.~\\ref{eq:bar_d2}) averaged over the stochastic ensemble (triangles), and fraction of incoherent particles $n_\\text{incoh}$ (squares). Data shown at integer values of $t$ (except for $d^{(2)}$), the lines serve as guides for the eye. The red dashed lines mark an exponential increase with $\\lambda=1.44t_0^{-1}$. Number of particles is $N=1.2\\times 10^6$, periodic potential $V(x)=V_P\\cos{(k_Px)}$ used with $V_P=0.3e$, $k_P=\\pi\/3\\xi$ and $\\xi$ the healing length.}\n\t\\label{fig:N6}\n\\end{figure}\nThe rate of the exponential growth is given by the Lyapunov exponent $\\lambda$. For the distance norm we take\n\\begin{align}\nd^{(2)}_{a,b}(t) &= \\frac{1}{2}\\int dx |\\psi_a(x,t)-\\psi_{b}(x,t)|^2 .\n\\label{eq:d2}\n\\end{align}\nThe Lyapunov exponent $\\lambda$ shows systematic trends as a function of the parameters of the system: It vanishes in absence of inter-particle interactions for arbitrary potentials, as well as in presence of inter-particle interactions in free space (i.e.~without any potential). At fixed period of the periodic potential $k_P$, or fixed correlation length $\\sigma$ of the speckle potential, it increases both with nonlinearity and the potential amplitude \\cite{brezinova_wave_2011}, see the supplemental material (SM).\\\\\nTo find imprints of chaos on measurable observables of the quantum many-body system with finite $N$ a theory beyond mean-field has to be applied. The multi-configurational time-dependent Hartree method for bosons (see, e.g.~\\cite{alon_multiconfigurational_2008,lode_colloquium_2020}), while being in principle exact for a sufficient number of orbitals, suffers from the exponentially growing configuration space. For the particle numbers considered, only two orbitals can be afforded numerically \\cite{brezinova_wave_2012}. As more than two orbitals are populated during the propagation, the MCTHB method entails a large and not easy to quantify error. We, therefore, resort to the truncated Wigner approximation (TWA), see e.g.~\\cite{Steel1998,Sinatra2002,blakie_dynamics_2008,dujardin_breakdown_2016} which employs the Wigner representation $W$ for (in general) a many-body density matrix $\\hat \\rho$\n\\begin{align}\nW(&\\psi_1,\\ldots,\\psi_M,\\psi_1^* \\ldots,\\psi_M^*) = \\frac{1}{\\pi^{2M}}\\nonumber \\\\\n&\\times \\int dz^{2M}\\Tr\\left[\\hat \\rho e^{i\\sum_j\\left( z_j^*\\hat\\psi_j^\\dagger\n+iz_j\\hat{\\psi_j}\\right)}\n\\right]\ne^{-i\\sum_j\\left(z_j^*\\psi_j^*-iz_j\\psi_j\\right)}.\n\\end{align}\n$W$ can be viewed as a phase-space representation of the quantum many-body state. $M$ is the total number of modes in which particles can be created or annihilated and $\\hat \\psi_j^\\dagger$ and $\\hat \\psi_j$ are the corresponding creation and annihilation operators, respectively. In general, particles can be created or annihilated in an arbitrary single particle mode denoted by $j$. We choose $j$ to represent a specific point in space assuming for simplicity an equidistant spatial discretization. We have made sure, however, that the spatial grid is fine enough, i.e.~the distance between grid points $dx<1\/k_\\text{max}$ with $k_\\text{max}$ being the largest relevant momentum in the system, such that we are still in the continuum limit.\\\\ \nHaving $W$ as a function of time at disposal would allow to evaluate all expectation values of symmetrized products of creation and annihilation operators. The exact equation of motion for $W$ can be obtained using von Neumann's equation of motion for $\\hat\\rho$. However, it proves to be intractable, such that approximations have to be invoked. Within the TWA \\cite{Steel1998, Sinatra2002, blakie_dynamics_2008}, the \ntime evolution of $W$ is sampled stochastically with an ensemble of trajectories obeying the GPE, Eq.~\\ref{eq:gpe}. (The only modification comes from the fact that we have to discretize space such that the second derivative in Eq.~\\ref{eq:gpe} has to be replaced by its second-order finite difference approximation.) It has been shown \\cite{schlagheck_enhancement_2019, tomsovic_post-ehrenfest_2018, dujardin_describing_2015} that this approximation amounts to neglecting non-classical trajectories as well as interferences between distinct trajectories in many-body Hilbert space. The question then arises at which times do these neglected effects start to play a role and become non-negligible. For single- or few-particle systems, sampling the time evolution with classical trajectories is accurate up to the point where an initially maximally localized state has spread over the whole system. This time is called the Ehrenfest time $\\tau_E$ \\cite{ehrenfest_bemerkung_1927} which is, in presence of classical chaos, inversely proportional to $\\lambda$ and grows logarithmically with $1\/\\hbar$. This concept can be extended into the many-body regime for bosonic systems with $\\hbar$ being replaced by the effective Planck constant $\\hbar_\\text{eff}\\simeq 1\/N$. Following the lines of \\cite{rammensee_many-body_2018, tomsovic_post-ehrenfest_2018} we thus assume that our results are accurate up to the time $\\tau_E=\\frac{1}{\\lambda}\\log{N}$.\\\\\nThe initial conditions within the stochastic ensemble of trajectories are constructed such as to correctly sample the phase-space distribution of the underlying initial quantum state, which in our case is a BEC at zero temperature. The stochasticity of the ensemble comes solely from the sampling of this initial state since Eq.~\\ref{eq:gpe} is completely deterministic. We follow \\cite{Sinatra2002, blakie_dynamics_2008, Steel1998} and construct the initial wave functions by adding to the mean-field ground state in the harmonic trap vacuum fluctuations in form of Gaussian noise (see the SM).\\\\\nThe most relevant observables in our case will be the coherent part of the particle density given by $\\rho_\\text{coh}(x_j,t) = |\\langle \\hat \\psi_j(t) \\rangle|^2$, as well as the one-particle reduced density matrix (1RDM) $D_{ij}(t)=\\langle \\hat\\psi_i^\\dagger(t)\\hat\\psi_j(t)\\rangle$. The term ``coherent\" in defining $\\rho_\\text{coh}(x_j,t)$ points to the fact that only a macroscopically occupied state with a spatially non-random phase will survive the averaging. $\\rho_\\text{coh}(x_j,t)$ can therefore be associated with the density of condensed particles. Alternatively \\cite{penrose_bose-einstein_1956,leggett_bose-einstein_2001}, the condensate state is defined through a macroscopic occupation of one eigenstate of the 1RDM. We show in the SM that these two definitions of the condensate give practically identical results for the depletion over time such that we use throughout the remainder of the paper the term coherent synonymously to condensed.\\\\\nWithin the stochastic ensemble of trajectories, expectation values can be calculated as\n$\\langle \\hat\\psi_j(t)\\rangle = \\frac{1}{N_s}\\sum_{s=1}^{N_s}\\psi_s(x_j,t)\n$,\nwith $N_s$ ($\\gg1$) being the number of Gross-Pitaevskii trajectories $\\psi_{s}(x_j,t)$ within the ensemble. To calculate the 1RDM, one has to rewrite $\\langle \\hat\\psi_i^\\dagger\\hat\\psi_j + \\hat \\psi_j\\hat\\psi_i^\\dagger\\rangle = \n2\\langle \\hat\\psi_i^\\dagger\\hat\\psi_j \\rangle + \\frac{1}{Ndx}\\delta_{ij}$ using the commutator relation $[\\hat\\psi_j,\\hat\\psi_i^\\dagger] = \\frac{1}{Ndx}\\delta_{ij}$.\nThe term $\\delta_{ij}\/dx$ is the discrete version of the $\\delta$-function for a continuous system, and \nthe factor $1\/N$ comes from our normalization of the wave functions of Eq.~\\ref{eq:gpe} to one, or equivalently, the creation and annihilation operators to $1\/N$.\nThe 1RDM is then given by\n$D_{ij}(t) = \\frac{1}{N_s}\\sum_{s=1}^{N_s}\\psi^{*}_s(x_i,t)\\psi_s(x_j,t) - \\frac{1}{2Ndx}\\delta_{ij}\n$,\nand the total particle density is $\\rho_\\text{total}(x_j,t) = D_{jj}(t)$. The fraction of coherent particles is determined by \n$n_\\text{coh}(t)=\\sum_j dx\\rho_\\text{coh}(x_j,t)$. Accordingly, the fraction of incoherent particles is $n_\\text{incoh}(t) = 1-n_\\text{coh}(t)$. The crucial observation now is that\n\\begin{align}\nn_\\text{incoh}(t) = \\frac{1}{N_s^2}\\sum_{s,r}d^{(2)}_{s,r}(t)\n-\\frac{L}{2Ndx} = \\bar d^{(2)}(t),\n\\label{eq:bar_d2}\n\\end{align}\nwhich we obtain using Eq.~\\ref{eq:d2}, taking into account that the norm of the wave functions within the ensemble is $1\/N_s\\sum_s\\sum_j dx |\\psi_s(x_j)|^2 = 1 + L\/2Ndx$ with $L$ the length of the system. (Note that the term $L\/dx$ counts the number of single-particle modes to which vacuum fluctuations have been added.) For a detailed derivation, see the SM. The right-hand side of Eq.~\\ref{eq:bar_d2} is (apart from a constant term) the arithmetic mean over the distance function between all pairs of mean-field trajectories, and we denote it with $\\bar d^{(2)}(t)$.\\\\\nWe have explicitly verified the equality of Eq.~\\ref{eq:bar_d2} numerically by independently calculating the arithmetic mean of the distance function, $\\bar d^{(2)}(t)$, and comparing it to $n_\\text{incoh}(t)$, see Fig.~\\ref{fig:N6}.\n\\begin{figure}[t]\n\t\\includegraphics[width=\\columnwidth]{N_number_comp_combined.pdf}\n\t\\caption{Left column (a) and (b) for the periodic potential with $V_P=0.3e$ and $k_P=\\pi\/3\\xi$: (a) Fraction of incoherent particles $n_\\text{incoh}$ for different particle numbers $N$ ($N$ being $1.2$ times the number near each curve), and $d^{(2)}$ for the linearly distorted initial conditions. Red dashed lines correspond to an exponential increase with $\\lambda=1.44 t_0^{-1}$. (b) Linear plot of (a) including $n^\\text{total}_\\text{low-env}$ extracted from the total density only, see Fig.~\\ref{fig:incoh_den}. The Ehrenfest time $\\tau_E=1\/\\lambda \\ln{N}$ is marked for each curve. Right column (c) and (d) same as left column but for one realization of speckle disorder with $V_D=0.3e$ and correlation length $\\sigma =0.57\\xi$. In (c) the Lyapunov exponent is $\\lambda =1.43 t_0^{-1}$. Data is shown at integer values of $t$ (except for $d^{(2)}$), the lines serve as guides for the eye.}\n\t\\label{fig:N_numb}\n\\end{figure}\n\\begin{figure}[t]\n\t\\includegraphics[width=\\columnwidth]{incoherent_density_k_2.pdf}\n\t\\caption{Total density $\\tilde\\rho_\\text{total}(k,t)$ (solid), as well as coherent $\\tilde\\rho_\\text{coh}(k,t)$ (filled), and incoherent part $\\tilde\\rho_\\text{incoh}(k,t)$ (dashed) for $N=1.2\\times10^6$ at (a) $t=5t_0$ and (b) $t=9t_0$. The orange dots mark the lower envelope of the strong fluctuations of $\\tilde\\rho_\\text{total}(k,t)$, and can be used as an accurate estimate for $\\tilde\\rho_\\text{incoh}(k,t)$. Same potential as in Fig.~\\ref{fig:N6}.}\n\t\\label{fig:incoh_den}\n\\end{figure}\nThe observed rate of exponential growth does not depend on the specific choice of the two close initial conditions such that we can clearly associate it with a Lyapunov exponent $\\lambda$. The equality between $\\bar d^{(2)}(t)$ and $n_\\text{incoh}(t)$ proves that, if the mean-field limit is chaotic, the fraction of incoherent particles will grow exponentially with a rate given exactly by the mean-field $\\lambda$. The exponentially fast depletion is thus chaos-induced, or seen from another perspective, measures mean-field chaos. Importantly, the exponential increase happens on shorter time scales than $\\tau_E$, i.e.~before effects neglected within the TWA start to play a role.\\\\\nWhile Fig.~\\ref{fig:N6} depicts the exponential increase for $N=1.2\\times10^6$ particles, we see the same exponential increase, i.e.~the same $\\lambda$, also for smaller particle numbers, see Fig.~\\ref{fig:N_numb}. We have varied $N$ while keeping the nonlinearity $g\\propto a_sN$ constant, which amounts to increasing the scattering length $a_s$ by the same factor $N$ is decreased, which preserves the classical phase space. Indeed, with decreasing $N$ and increasing $a_s$, the BEC naturally shows larger initial depletion, but upon expansion in the periodic potential, the same $\\lambda$ emerges.\\\\ \nWe now turn to the question of how the present chaos-induced depletion could be observed in an experiment. We analyze the total particle density in momentum space $\\tilde\\rho_\\text{total}(k,t)$ which is accessible in experiments through time-of-flight measurements, see e.g.~\\cite{gericke_high-resolution_2008, erne_universal_2018}. During the expansion of the BEC, matter waves start to scatter at the potential landscape preserving initially their phase coherence. This scattering creates fluctuations in momentum space with increasingly higher frequencies as waves originating from points increasingly farther apart in real space coherently interfere. Ultimately, the density exhibits strong fluctuations reaching down to almost zero density, provided that inelastic scattering has been negligible up until this point in time, Fig.~\\ref{fig:incoh_den} (a). During inelastic scattering particles lose energy, phase information, and with it, the ability to create interference fringes. These particles constitute the incoherent part of the density which piles up in form of an almost non-fluctuating background. Using a simple algorithm that determines the lower envelope of the fluctuations in the total density, we obtain a functional form very close to $\\tilde\\rho_\\text{incoh}(k,t)$, see Fig.~\\ref{fig:incoh_den} (b). Interpolating between the points of the lower envelope and integrating, we obtain $n^\\text{total}_\\text{low-env}$, which follows $n_\\text{incoh}$ closely, see Fig.~\\ref{fig:N_numb} (b). We emphasize that $n^\\text{total}_\\text{low-env}$ is extracted from the total density only. From Fig.~\\ref{fig:N_numb} (b) it is obvious that the extraction mechanism will work best for high particle numbers with a small scattering length (e.g., for $N=1.2\\times 10^6$ two orders of magnitude of exponential growth can be resolved). For smaller $N$ and correspondingly larger scattering lengths $a_s$ the incoherent density starts to pile up before coherent scattering produces sufficiently strong fluctuations in the coherent part of the density. Therefore, the close association of a non-fluctuating density with $\\tilde\\rho_\\text{incoh}(k,t)$ is broken initially. It becomes, however, more and more accurate over time such that, in the experiment, one could observe the behavior of $\\tilde\\rho_\\text{incoh}(k,t)$ also beyond $\\tau_E$, where interferences of many-body trajectories not included within the TWA become relevant.\\\\\nIn order to measure the incoherent fraction of the total density in an experiment it is pivotal to resolve the deep minima of the fluctuations. Half of the distance between two minima is $\\Delta k\\gtrsim 0.04 x_0^{-1}$. Assuming a linear pixel size of a CCD camera of $2\\mu$m the fluctuations could be resolved after about $300$ms time of flight. The peak amplitude of the fluctuations is $\\tilde\\rho_\\text{total}(k)\\gtrsim0.03 x_0$ leading to $\\tilde\\rho_\\text{total}(k)\\Delta k = 1.2\\times10^{-3}$ such that the number of particles within each hump is greater than $100$ for $N=1.2\\times10^5$ and $N=1.2\\times10^6$. Despite the large time of flight necessary, we believe that the here proposed extraction could be realized in state-of-the-art BEC experiments.\\\\\nFor the disorder potential, we mostly see the same behavior as for the periodic potential, see Fig.~\\ref{fig:N_numb} (c) and (d): \n\\begin{figure}[t]\n\t\\includegraphics[width=\\columnwidth]{incoherent_density_k_disorder.pdf}\n\t\\caption{Particle density in momentum space at $t=8t_0$ for the speckle disorder with $V_D=0.3e$ and $\\sigma =0.57\\xi$ averaged over $10$ realizations (additional smoothing of the curves has been applied). The vertical lines mark the Landau velocity approximated by $k_L=\\sqrt{\\mu(t)}$ with $ \\mu(t)$ the chemical potential at time $t$.}\n\t\\label{fig:incoh_den_dis}\n\\end{figure}\n$n_\\text{incoh}$ grows exponentially with $\\lambda$ independent of the particle number $N$. Note that we did not perform any averages over disorder realizations here. As to the extraction of the incoherent part of the density from $\\tilde \\rho_\\text{total}(k,t)$ there is one point worth mentioning. Due to the broad spectrum of frequencies the speckle disorder offers, within few time steps, slow particles start to be scattered coherently and intertwine with particles that have lost their coherence through inelastic (i.e.~incoherent) scattering near $k=0$. The result is a local maximum in $\\tilde \\rho_\\text{total}(k,t)$ near $k=0$, and local minima near the Landau velocity $\\pm k_L$ due to inelastic scattering out of this momentum, see Fig.~\\ref{fig:incoh_den_dis}. Since, however, slow particles scatter from positions in space close to each other, this scattering produces fluctuations with low frequencies as compared to the fluctuations observed for larger $k$. It is, therefore, impossible to identify $\\tilde \\rho_\\text{incoh}(k,t)$ near $k=0$ based on the fluctuations of the total density, initially. At later times the local maximum near $k=0$ consists of incoherent particles only such that $n^\\text{total}_\\text{low-env}$ again accurately predicts the value of $n_\\text{incoh}$, see Fig.~\\ref{fig:N_numb}. For $N=1.2\\times10^6$ the agreement between $n^\\text{total}_\\text{low-env}$ and $n_\\text{incoh}$ is accurate only after $t\\gtrsim12t_0$ such that we refrained from plotting it.\\\\\nIn conclusion, we have shown that a BEC expanding in a shallow periodic or disordered potential is subject to an exponentially growing depletion, and that the depletion is characterized by the ``classical\" (mean-field) Lyapunov exponent $\\lambda$. We have thus found a new observable that allows to identify the finger-print of classical chaos on the non-equilibrium many-body dynamics of a quantum system with a finite number of particles. In addition, we have shown how our results could be measured in an experiment by analyzing the visibility of the fluctuations of the particle density after time of flight. This opens up the possibility to verify our predictions experimentally for a real many-body system.\\\\\n\nWe thank Joachim Burgd\u00f6rfer, David Gu\u00e9ry-Odelin, Dana Orsolits, Thorsten Schumm, and Juan-Diego Urbina for helpful discussions. This work has been supported by the WWTF grant MA14-002. S. D. acknowledges support by the International Max Plank Research School of Advanced Photon Science (IMPRS-APS). Calculations\nwere performed on the Vienna Scientific Cluster (VSC3). \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{Introduction}\n\nMost of efficient cetacean localisation systems are based on the Time Delay Of Arrival (TDOA) estimation from detected\\footnote{As click\/whistles detector, matching filter is often prefered} animal's click\/whistles signals \\cite{nosalGT,fredericbenard_herveglotin_2009}. Long-base hydrophones'array is involving several fixed, efficient but expensive hydrophones \\cite{3Dtracking} while short-base version is requiring a precise array's self-localization to deliver accurate results. Recently (see \\cite{IFA_DCL}), based on Leroy's attenuation model versus frequencies \\cite{Leroy}, a range estimator have been proposed. This approach is working on the detected most powerful pulse inside the click signal and is delivering a rough range' estimate robust to head orientation variation of the animal. Our purpose is to use i) these hydrophone' array measurements recorded in diversified sea conditions and ii) the associated ground-truth trajectories of spermwhale (obtained by precise TDAO and\/or Dtag systems) to regress both position and azimuth of the animal from a third-party hydrophone\\footnote{We assume that the velocity vector is colinear with the head's angle.} (typically onboard, standalone and cheap model).\n\nWe claim, as in computer-vision field, that BoF approach can be successfully applied to extract a global and invariant representation of click's signals. Basically, the pipeline of BoF approach is composed of three parts: i) a local features extractor, ii) a local feature encoder (given a dictionary pre-trained on data) and iii) a pooler aggregating local representations into a more robust global one. Several choice for encoding local patches have been developed in recent years: from hard-assignment to the closest dictionary basis (trained for example by $K$means algorithm) to a sparse local patch reconstruction (involving for example Orthognal Maching Pursuit (OMP) or LASSO algorithms).\n\n\\section{Global feature extraction by spare coding}\n\n\n\\subsection{Local patch extraction}\n\nLet's denote by $\\g{C}\\triangleq\\{\\g{C}^j\\}$, $j=1,\\ldots,H$ the collection of detected clicks associated with the $j^{th}$ hydrophone of the array composed by $H$ hydrophones. Each matrix $\\g{C}^j$ is defined by $\\g{C}^j\\triangleq\\{\\g{c}_i^j\\}$, $i=1,\\ldots,N^j$ where $\\g{c}_i^j\\in\\mathds{R}^n$ is the $i^{th}$ click of the $j^{th}$ hydrophone. For our \\textit{Bahamas2} dataset \\cite{3Dtracking}, we choose typically $n=2000$ samples surrounding the detected click. The total number of available clicks is equal to $N=\\sum\\limits_{i=1}^{H}N^j$.\n\nAs local features, we extract simply some local signal patches of $p\\leq n$ samples (typically $p=128$) and denoted by $\\g{z}_{i,l}^j\\in\\mathbb{R}^p$. Furthermore all $\\g{z}_{i,l}^j$ are $\\ell_2$ normalized. For each $\\g{c}_{i}^j$, a total of $L$ local patches $\\g{Z}_{i}^j\\triangleq\\{\\g{z}_{i,l}^j\\}$, $l=1,\\ldots,L$ equally spaced of $\\lceil\\frac{n}{L}\\rceil$ samples are retrieved (see Fig.~\\ref{patch_extraction}). All local patches associated with the $j^{th}$ hydrophone is denoted by $\\g{Z}^{j}\\triangleq\\{\\g{Z}_i^{j}\\}$, $i=1,\\ldots,N^j$ while $\\g{Z}\\triangleq\\{\\g{Z}^j\\}$ is denoting all the local patches matrix for all hydrophones. A final post-processing consists in uncorrelate local features by PCA training and projection with $p'\\leq p$ dimensions.\n\n\\begin{figure*}[!ht]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[height=5cm,width=7.5cm]{ex_click.pdf} &\n\\includegraphics[height=5cm,width=7.5cm]{local_features.pdf}\n\\end{tabular}\n\\caption{Left: Example of detected click with $n=2000$. Right: extracted local features with $p=128$, $L=1000$ (one local feature per column).}\n\\label{patch_extraction}\n\\end{center}\n\\end{figure*}\n\n\\subsection{Local feature encoding by sparse coding}\n\nIn order to obtain a global robust representation of $\\g{c}\\subset\\g{C}$, each associated local patch $\\g{z}\\subset\\g{Z}$ are first linearly encoded \\textit{via} the vector $\\g{\\alpha}\\in\\mathbb{R}^k$ such as $\\g{z}\\approx\\g{D}\\g{\\alpha}$ where $\\g{D}\\triangleq [\\g{d}_1,\\ldots,\\g{d}_k]\\in\\mathbb{R}^{p\\times k}$ is a pre-trained dictionary matrix whose column vectors respect the constraint $\\g{d}_j^T\\g{d}_j=1$. In a first attempt to solve this linear problem, $\\g{\\alpha}$ can be the solution of the Ordinary Least Square (OLS) problem:\n\\begin{equation}\\label{1}\nl_{OLS}(\\g{\\alpha}|\\g{z};\\g{D}) \\triangleq \\min_{\\g{\\alpha} \\in \\mathbb{R}^k}\\left\\lbrace\\frac{1}{2}\\Vert \\g{z} - \\g{D}\\g{\\alpha}\\Vert_2^2 \\right\\rbrace.\n\\end{equation}\nOLS formulation can be extended to include regularization term avoiding overfitting. We obtain the ridge regression (RID) formulation:\n\\begin{equation}\nl_{RID}(\\g{\\alpha}|\\mathbf{z};\\mathbf{D}) \\triangleq \\min_{\\g{\\alpha} \\in \\mathbb{R}^k}\\left\\lbrace\\frac{1}{2}\\Vert \\g{z} - \\g{D}\\g{\\alpha}\\Vert_2^2 + \\beta \\Vert \\g{\\alpha} \\Vert_2^2 \\right\\rbrace.\n\\end{equation}\nThis problem have an analytic  solution $\\g{\\alpha} = (\\g{D}^T\\g{D} + \\beta\\g{I}_k)^{-1}\\g{D}^T\\g{z}$. Thanks to semi-positivity of $\\g{D}^T\\g{D} + \\beta\\g{I}_k$, we can use a cholesky factor on this matrix to solve efficiently this linear system. In order to decrease reconstruction error and to have a sparse solution, this problem can be reformuled as a constrained Quadratic Problem (QP):\n\\begin{equation}\nl_{SC}(\\g{\\alpha}|\\g{z};\\g{D}) \\triangleq \\min_{\\g{\\alpha} \\in \\mathbb{R}^k} \\frac{1}{2}\\Vert \\g{z} - \\g{D} \\g{\\alpha} \\Vert_2^2 \\ s.t. \\ \\ \\Vert\\boldsymbol{\\alpha}\\Vert_1 = 1.\n\\end{equation}\nTo solve this problem, we can use a QP solver involving high combinatorial computation to find the solution. Under RIP assumptions \\cite{Tibshirani94regressionshrinkage}, a greedy approach can be used efficiently to solve and eq. 3 and this latter can be rewritten as:\n\\begin{equation}\\label{2}\nl_{SC}(\\g{\\alpha}|\\g{z};\\g{D}) \\triangleq \\min_{\\g{\\alpha} \\in \\mathbb{R}^k} \\frac{1}{2}\\Vert \\g{z} - \\g{D} \\g{\\alpha} \\Vert_2^2 + \\lambda\\Vert\\g{\\alpha}\\Vert_1,\n\\end{equation}\nwhere $\\lambda$ is a regularization parameter which controls the level of sparsity. This problem is also known as basis pursuit \\cite{Chen98atomicdecomposition} or the Lasso \\cite{Tibshirani94regressionshrinkage}. To solve this problem, we can use the popular Least angle regression (LARS) algorithm.\n\n\\subsection{Pooling local codes}\n\nThe objective of pooling \\cite{Boureau10atheoretical,Feng11} is to transform the joint feature representation into a new, more usable one that preserves important information while discarding irrelevant detail.\nFor each click signal, we usually compute $L$ codes denoted $\\g{V} \\triangleq \\left\\lbrace\\g{\\alpha}_i\\right\\rbrace$, $i = 1,\\ldots,L$.\nLet define $\\g{v}^{j}\\in\\mathbb{R}^L$, $j=1,\\ldots,k$ as the $j^{th}$ row vector of $\\g{V}$. It is essential to use feature pooling to map the response vector $\\g{v}^{j}$ into a statistic value $f(\\g{v}^{j})$ from some spatial pooling operation $f$. We use $\\g{v}^{j}$, the response vector, to summarize the joint distribution of the $j^{th}$ compounds of local features over the region of interest (ROI). We will consider the $\\ell_{\\mu}$-norm pooling and defined by:\n\\begin{equation}\nf_n(\\g{v};\\mu) = \\left(\\sum_{m=1}^L |v_m|^{\\mu}\\right)^{\\frac{1}{\\mu}} \\ \\ s.t. \\ \\mu \\neq 0.\n\\end{equation}\nThe parameter $\\mu$ determines the selection policy for locations. When $\\mu = 1$, $\\ell_{\\mu}$-norm pooling is equivalent to sum-pooling and aggregates the responses over the entire region uniformly. When $\\mu$ increases, $\\ell_{\\mu}$-norm pooling approaches max-pooling. We can note the value of $\\mu$ tunes the pooling operation to transit from sum-pooling to max-pooling.\n\n\n\\subsection{Pooling codes over a temporal pyramid}\nIn computer vision, Spatial Pyramid Matching (SPM) is a technic (introduced by \\cite{Lazebnik2006}) which improves classification accuracy by performing a more robust local analysis. We will adopt the same strategy in order to pool sparse codes over a temporal pyramid (TP) dividing each click signal into ROI of different sizes and locations. Our TP is defined by the matrix $\\g{\\Lambda}$ of size $(P \\times 3)$ \\cite{sebastienparis_xanaduhalkias_herveglotin_2013}:\n\\begin{equation}\n\\g{\\Lambda} = [\\g{a}, \\g{b}, \\g{\\Omega}],\n\\end{equation}\nwhere $\\g{a}$, $\\g{b}$, $\\g{\\Omega}$ are 3 $(P \\times 1)$ vectors representing subdivision ratio, overlapping ratio and weights respectively. $P$ designs the number of layers in the pyramid. Each row of $\\g{\\Lambda}$ represents a temporal layer of the pyramid, \\textit{i.e.} indicates how do divide the entire signal into sub-regions possibly overlapping. For the $i^{th}$ layer, the click signal is divided into $D_i=\\lfloor\\frac{1-a_i}{b_i}+1\\rfloor$ ROIs where $a_i$, $b_i$ are the $i^{th}$ elements of vector $\\g{a}$, $\\g{b}$ respectively. For the entiere TP, we obtain a total of $D=\\sum\\limits_{i=1}^{P}D_i$ ROIs. Each click signal $\\g{c}$ $(n \\times 1)$ is divided into temporal ROI $\\g{R}_{i,j}$, $i=1,\\ldots,P$, $j=1,\\ldots,D_i$ of size $(\\lfloor a_i.n\\rfloor \\times 1)$. All ROIs of the $i^{th}$ layer have the same weight $\\Omega_i$. For the $i^{th}$ layer, ROIs are shifted by $\\lfloor b_i.n\\rfloor$ samples. A TP with $\\g{\\Lambda} = \\left[\\begin{array}{ccc}\n1 & 1 & 1  \\\\ \\frac{1}{2} & \\frac{1}{4} & 1  \\end{array}\\right]$ is designing a 2-layers pyramid with $D=1+4$ ROIs, the entiere signal for the first layer and $4$ half-windows of $\\frac{n}{2}$ samples with $25\\%$ of overlapping for the second layer.\nAt the end of pooling stage over $\\g{\\Lambda}$, the global feature $\\g{x}\\in\\mathbb{R}^d$, $d=D.k$ is defined by the weighted concatenation (by factor $\\Omega_i)$ of $L$ pooled codes associated with $\\g{c}$.\n\n\n\n\\subsection{Dictionary learning}\n\nTo encode each local features by sparse coding (see eq.~\\ref{2}), a dictionary $\\g{D}$ is trained offline with an important collection of $M\\leq N.L$ local features as input. One would minimize the regularized empirical risk $\\mathcal{R}_M$:\n\\begin{equation}\n\\begin{array}{c}\n\\mathcal{R}_M(\\g{V},\\g{D}) \\triangleq \\displaystyle\\frac{1}{M}\\sum\\limits_{i=1}^M \\frac{1}{2}\\Vert \\g{z}_i - \\g{D} \\g{\\alpha}_i \\Vert_2^2 + \\lambda\\Vert\\g{\\alpha}_i\\Vert_1\n\\\\\n\\\\\n\\ s.t. \\ \\g{d}_j^T\\g{d}_j=1.\n\\end{array}\n\\end{equation}\nUnfortunatly, this problem is not jointly convex but can be optimized by alternating method:\n\\begin{equation}\n\\mathcal{R}_M(\\g{V}|\\g{\\hat{D}}) \\triangleq \\frac{1}{M}\\sum\\limits_{i=1}^M \\frac{1}{2}\\Vert \\g{z}_i - \\g{\\hat{D}} \\g{\\alpha}_i \\Vert_2^2 + \\lambda\\Vert\\g{\\alpha}_i\\Vert_1,\n\\end{equation}\nwhich can be solved in parallel by LASSO\/LARS and then:\n\\begin{equation}\n\\mathcal{R}_M(\\g{D}|\\g{\\hat{V}}) \\triangleq \\frac{1}{M}\\sum\\limits_{i=1}^M \\frac{1}{2}\\Vert \\g{z}_i - \\g{D} \\g{\\hat{\\alpha}}_i \\Vert_2^2 \\ \\ s.t. \\ \\g{d}_j^T\\g{d}_j=1.\\label{eq_dico}\n\\end{equation}\nEq.~\\ref{eq_dico} have an analytic solution involving a large matrix $(k\\times k)$ inversion and a large memory occupation for storing the matrix $\\g{V}$ $(k\\times M)$. Since $M$ is potentially very large (up to 1 million), an online method to update dictionary learning is prefered \\cite{Mairal_2009}. Figure \\ref{baha_click_range_dico} depicts 3 dictionary basis vectors learned \\textit{via} sparse coding. As depicted, some elements reprensents more impulsive responses while some more harmonic responses.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[height=5.5cm,width=7.5cm]{baha_click_range_dico.pdf}\n\\caption{Example of trained dictionary basis with sparse coding.}\n\\label{baha_click_range_dico}\n\\end{center}\n\\end{figure}\n\n\n\\section{Range and azimuth logistic regression from global features}\n\nAfter the pooling stage, we extracted unsupervisly $N$ global features $\\g{X}\\triangleq\\{x_i\\}\\in\\mathbb{R}^{d\\times N}$. We propose to regress \\textit{via} logistic regression both range $r$ and azimuth $az$ (in $x-y$ plan, when animal reach surface to breath) from the animal trajectory groundtruth denoted $\\g{y}$. For the current train\/test splitsets of the data, such as $\\g{X}=\\g{X}_{train}\\bigcup\\g{X}_{test}$, $\\g{y}=\\g{y}_{train}\\bigcup\\g{y}_{test}$ and $N=N_{train}+N_{test}$, $\\forall$ $\\{\\g{x}_i,y_i\\}\\in\\g{X}_{train}\\times \\g{y}_{train}$, we minimize:\n\\begin{equation}\n\\widehat{\\g{w}}_{\\theta}=\\arg\\min\\limits_{\\g{w}_{\\theta}}\\left\\{\\frac{1}{2}\\g{w}_{\\theta}^T\\g{w}_{\\theta} + C\\sum\\limits_{i=1}^{N_{train}}\\log(1+e^{-y_i\\g{w}_{\\theta}^T\\g{x}_i})\\right\\},\\label{logistic_regression}\n\\end{equation}\nwhere $y_i$ denotes $r_i$ and $az_i$ for $\\theta=r$ and $\\theta=az$ respectively. Eq.~\\ref{logistic_regression} can be efficiently solved for example with Liblinear software \\cite{liblinear2008}. In the test part, range and azimuth for any $\\g{x}_i\\in\\g{X}_{test}$ are recontructed linearly by $\\widehat{r}_i=\\widehat{\\g{w}}_r^T\\g{x}_i$ and by $\\widehat{az}_i=\\widehat{\\g{w}}_{az}^T\\g{x}_i$ respectively.\n\n\\section{Experimental results}\n\n\\subsection{bahamas2 dataset}\n\nThis dataset \\cite{3Dtracking} contains a total of $N=6134$ detected clicks for $H=5$ different hydrophones (named $H^7$, $H^8$, $H^9$, $H^{10}$ and $H^{11}$ and with $N^7=1205$, $N^8=1238$, $N^9=1241$, $N^{10}=1261$ and $N^{11}=1189$ respectively).\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[height=5.5cm,width=7.5cm]{baha_traj_and_hydros.pdf}\n\\caption{The 2D trajectory (in $x\u2212y$ plan) of the single sperm whale observed during $25$ min and corresponding hydrophone's positions.}\n\\label{true_trajectory}\n\\end{center}\n\\end{figure}\nTo extract local features, we chose $n=2000$, $p=128$ and $L=1000$ (tuned by model selection). For both the dictionary learning and the local features encoding, we chose $\\lambda=0.2$ and fixed $15$ iterations to train dictionary on a subset of $M=400.000$ local features drawn uniformaly. We performed\n$K=10$ cross-validation where training sets reprensented $70\\%$ of the total of extracted global features, the rest for the testing sets. Logistic regression parameter $C$ is tuned by model selection. We compute the average root mean square error (ARMSE) of range\/azimuth estimates per hydrophone: $ARMSE(l)=\\frac{1}{K}\\sum\\limits_{i=1}^{K}\\sqrt{\\sum\\limits_{j=1}^{N_{test}^l}(y_{i,j}^l-\\widehat{y}_{i,j}^l)^2}$ where $y_{i,j}^l$, $\\widehat{y}_{i,j}^l$\n and $N_{test}^l$ represent the ground truth, the estimate and the number of test samples for the $l^{th}$ hydrophone respectively. The global ARMSE is then calculated by $\\overline{ARMSE}=\\frac{1}{H}\\sum\\limits_{l=1}^{H}ARMSE(l)$.\n\n \\subsection{$\\ell_{\\mu}$-norm pooling case study}\n\n For prilimary results, we investigate the influence of the $\\mu$ parameter during the pooling stage. We fix the number of dictionary basis to $k=128$ and the temporal pyramid equal to $\\g{\\Lambda}_1=\\left[1,1,1\\right]$, \\textit{i.e.} we pool sparse codes on whole the temporal click signal.\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[height=5.5cm,width=7.5cm]{baha_click_range_nu_pooling.pdf}\n\\caption{$\\overline{ARMSE}$ vs. $\\mu$ for range estimation.}\n\\label{baha_click_range_nu_pooling}\n\\end{center}\n\\end{figure}\nA value of $\\mu=\\{3,4\\}$ seems to be a good choice for this pooling procedure. For $\\mu\\geq20$, results are similar to those obtained by max-pooling. For azimuth, we observe also the same range of $\\mu$ values.\n\n \\subsection{Range and azimuth regression results}\n\n Here, we fixed the value of $\\mu=3$ and we varied the number of dictionary basis $k$ from $128$ to $4096$ elements. We also investigated the influence of the temporal pyramid and we give results for two particulary choices: $\\g{\\Lambda}_1=\\left[1,1,1\\right]$ and $\\g{\\Lambda}_2=\\small\\left[\\begin{array}{ccc}1 & 1 &1\\\\ \\frac{1}{3} & \\frac{1}{3}&1\\end{array}\\right]$. For $\\g{\\Lambda}_2$, the sparse are first pooled over all the signal then pooled over 3 non-overlapping windows for a total of $1+3=4$ ROIs.\n In order to compare results of our presented method, we also give results for an hand-craft feature \\cite{IFA_DCL} specialized for spermwhales and based on the spectrum of the most energetic pulse d\u00e9tected inside the click. This specialized feature, denoted \\textit{Spectrum feature}, is a 128 points vector.\n\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[height=5.5cm,width=7.5cm]{baha_click_range.pdf}\n\\caption{$\\overline{ARMSE}$ vs. $k$ for range estimation with $\\mu = 3$.}\n\\label{baha_click_range}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[height=5.5cm,width=7.5cm]{baha_click_azimuth.pdf}\n\\caption{$\\overline{ARMSE}$ vs. $k$ for azimuth estimation with $\\mu = 3$.}\n\\label{baha_click_azimuth}\n\\end{center}\n\\end{figure}\n\nFor both range and azimuth estimate, from $k=2048$, our method outperforms results of the \\textit{Spectrum feature} and particulary for azimuth estimate. Using a temporal pyramid for pooling permits also to improve slightly results.\n\n\\section{Conclusions and perspectives}\n\nWe introduced in the paper, for spermwhale localization, a BoF approach \\textit{via} sparse coding delivering rough estimates of range and azimuth of the animal, specificaly towarded for mono-hydrophone configuration. Our proposed method works directly on the click signal without any prior pulses detection\/analysis while being robust to signal transformation issue by the propagation. Coupled with non-linear filtering such as particle filtering \\cite{Arulampalam02}, accurate animal position estimation could be perform even in mono-hydrophone configuration. Applications for anti-collision system and whale whatching are targeted with this work.\n\nAs perspective, we plan to investigate other local features such as spectral features, MFCC \\cite{Davis80,Rabiner93}, Scattering transform features \\cite{AndenM11}. These latter can be considered as a hand-craft first layer of a deep learning architecture with 2 layers.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\n\n\nThis note is a development based on a recent paper \\cite{Kallosh:2021ors} where we have performed a covariant (Lagrangian) quantization of gravity in a black hole background in the Regge-Wheeler  set up \\cite{Regge:1957td,Zerilli:1971wd,Martel:2005ir}.  The gauge-fixing condition in  \\cite{Kallosh:2021ors} includes the Regge-Wheeler gauge for ${\\ell }\\geq 2$ modes, and a certain background covariant gauge  for ${\\ell }<2$  modes, where Regge-Wheeler gauge is not valid. We will refer to the  gauge in  \\cite{Kallosh:2021ors} covering all ${\\ell }$ modes, as a `generalized  Regge-Wheeler gauge'.\n\n\n\n\n\n\nThe Feynman path integral for gravity, viewed as quantum field theory (QFT), is  defined by De Witt-Faddeev-Popov  \\cite{DeWitt:1967ub,Faddeev:1967fc} and takes a form, in absence of sources \n\\begin{equation}\n \\int  D h\n \\,J_\\chi ( g,h)  \\delta \\big (\\chi_\\alpha ( g,h)\\big ) e^{\\mathrm{i} S ( g+h) } \\, .\n\\label{pi}\\end{equation}\nHere we integrate over the perturbations $h$ in the background metric $g$. The gauge-fixing conditions are $\\chi_\\alpha(g, h)=0$.  The Jacobian designed to make this path integral independent on the choice of the gauge-fixing conditions   can be presented with the help of the Faddeev-Popov (FP) ghosts \\cite{Faddeev:1967fc} \\begin{equation}\nJ_\\chi = \\int D\\bar C^\\alpha DC_\\beta \\, e^{\\mathrm{i} \\int \\mathrm{d}^4 x \\, \\bar C^\\alpha(x)  Q_{\\alpha} {}^{ \\beta}  (g, h)C_\\beta(x) }\\, .\n\\end{equation}\nThe differential operator in the ghost action is defined by the gauge variation of the gauge-fixing finctions\n$\n\\delta \\chi_\\alpha = Q_{\\alpha}{}^{\\beta} \\xi_\\beta\n$. For the  choice of the gauge-fixing functions made in \\cite{Kallosh:2021ors}, which in addition to Regge-Wheeler gauge for $l\\geq 2$ modes includes gauges for $l<2$ modes, a generalized  Regge-Wheeler gauge, we have found that the ghost actions do not have time derivatives  in  Schwarzschild coordinates. We therefore predicted that in the a generalized  Regge-Wheeler gauge  \\cite{Kallosh:2021ors} the canonical Hamiltonian according to the rules for gauge theories \n\\cite{Faddeev:1969su,Fradkin:1970pn,Faddeev:1973zb} is expected to be unitary.\n\nIn this note we will  present the quadratic in gravitational perturbations $h$ part of the gravity Hamiltonian in the spherical harmonic basis. Before doing this we will perform the standard counting of physical degrees of freedom in this case. The structure of the Hamiltonian will confirm this counting.\n\nThe standard counting of physical degrees of freedom in gauge theories in the QFT context of the Feynman path integral is the same in either Lagrangian or Hamiltonian quantization, and it is also gauge independent, if performed correctly. \nThe general counting formula is formulated for the number of gauge field components equal to $n+k$ in case of $k$ gauge symmetries.  The total number of physical degrees of freedom is\n\\begin{equation}  n-k\n\\label{number}\\end{equation}\nThis final counting formula in QFT is valid for any choice of  gauge-fixing, but  the procedure is different for unitary and pseudo-unitary gauges. \nFor example in 4D the metric has $n+k=10$   components and there are $k=4$ gauge symmetries, the counting is $n-k=(10- 4)-4=2$.\n\n\n\nIn QFT\nin the class of unitary gauges Hamiltonians have manifestly ghost-free underlying Hilbert spaces. There are $(p^*, q^*)$ variables in Faddeev's theorem \\cite{Faddeev:1969su} as described in \\cite{Kallosh:2021ors}.\nThis means that all $n-k$ physical states have positive definite metric. The S-matrix is unitary.\n\\begin{equation}\n\\# \\, {\\rm degrees \\, of\\, freedom}_{\\rm unitary \\, H} \\,  =  n-k\n\\label{numberU}\\end{equation}\nMeanwhile, \nin other gauges, for example, 4D Lorentz covariant gauges in gravity,  the Hamiltonians are ``pseudo-unitary'' with underlying state spaces with negative-norm ghost degrees of freedom \\cite{Fradkin:1977hw,Batalin:1977pb}. In such case the counting goes as follows: there are $n+k$ states with positive norm and $2k$ states with negative norm presented by FP anti-commuting ghosts, so the total counting, with account of negative norm states,  is the same as in unitary gauges\n\\begin{equation}\n\\# \\, {\\rm degrees \\, of\\, freedom}_{\\rm pseudo-unitary \\, H}  =  n+k -2k  \\Rightarrow n-k\n\\end{equation}\n The S-matrix is pseudo-unitary in a space of states with the indefinite metric.\n\n\n\n\n\n \nWe will see that the quadratic in $h$ part of the gravity Hamiltonian in spherical harmonic basis does support  this counting. In the class of gauges used in  \\cite{Kallosh:2021ors} the canonical Hamiltonian is unitary, as predicted there.\n\nThe complete form of the Hamiltonian to all orders of $h$  is beyond the scope of this paper. However, in \\cite{Kallosh:2021ors} we have argued that the non-linear couplings of ghosts to all orders in $h$ are free of time derivatives on ghosts. Therefore one would expect that the non-linear in $h$ terms in $H$ will be consistent with the unitarity of the Hamiltonian which will be deduced in this note  at the level quadratic in $h$.\n\n The corresponding part of the action $S( g+h)$, quadratic in $h$, is of the form\n\\begin{equation}\nS= {1\\over 2} \\int h_{\\mu\\nu} S^{ \\mu\\nu  \\lambda \\delta} (g) h_{\\lambda \\delta}\n\\label{quad} \\end{equation}\nHere $S^{ \\mu\\nu  \\lambda \\delta} (g)$ is a differential operator depending on the background metric $g$. The left hand side of equations of motion ${\\delta S\\over \\delta h_{\\mu\\nu} }=0$ linear in $h$ takes the form\n\\begin{equation}\nQ^{\\mu\\nu} \\equiv {\\delta S\\over \\delta h_{\\mu\\nu} }= S^{ \\mu\\nu  \\lambda \\delta} (g) h_{\\lambda \\delta}\n\\label{EOM}\\end{equation}\nOne can restore the action in eq. \\rf{quad} from the information available in eq. \\rf{EOM}.\n\nIn the spherical harmonic basis the 4D spacetime  is split into $\\mathcal{M} = \\mathcal{M}_2\\times\\mathbb{S}^2$  with coordinates $(x^{a},\\theta^A)$, $a=1,2$.  The 4D perturbations $h_{\\mu\\nu} $ are  represented by  2D fields for each $({\\ell }, m)$ \\cite{Regge:1957td,Zerilli:1971wd,Martel:2005ir}. \nThe corresponding equations are known and we will use them as derived in \\cite{Martel:2005ir} in Schwarzschild coordinates. Once the quadratic Lagrangian is known, it is possible to derive the relevant quadratic in $h$ Hamiltonian.\nFor ${\\ell }\\geq 2$ modes the corresponding quadratic Hamiltonian was constructed by Moncrief in \\cite{Moncrief:1974am} where also the relevant Regge-Wheeler \\cite{Regge:1957td} and Zerilli \\cite{Zerilli:1971wd} equations were re-derived in the form of Hamiltonian equations of motion. The Hamiltonian was derived in \\cite{Moncrief:1974am} in absence of source terms.\nFor ${\\ell }<2$ the Hamiltonian was not studied, to the best of our   knowledge. In \\cite{Moncrief:1974am} it was explained that  the  attention was restricted to modes with $l\\geq 2$ since the modes with $l<2$ are nonradiative and require a special treatment.\n\n\nHere we will use the known field equations \\rf{EOM} in the form given in  \\cite{Martel:2005ir} in Schwarzschild coordinates, which allow to derive the Lagrangian in \\rf{quad}. From the quadratic Lagrangian we derive  a canonical  quadratic part of the Hamiltonian, with account of the algebraic constraints in our gauges.  We will conclude that there are no physical degrees of freedom suitable for quantization  at ${\\ell }<2$. Our definition of quantized degrees of freedom involves the QFT quantization conditions in 2D space of the form\n\\begin{equation}\n[q(r, t) , p(r', t) ]= i \\delta (r-r')\n\\label{quant}\\end{equation}\nThe classical field equations for low multipoles  in presence of sources are known to have non-trivial solutions. For example for the monopoles ${\\ell }=m=0$   there are solutions  like \n$\nh^{00}_{tt} \\sim {\\delta M\\over r}\n$, \nthey are known to affect  the  the black hole mass. However,  there are no solutions of the constraint equations compatible with the quantization condition \\rf{quant} for ${\\ell }<2$.\n\nAll our results are valid for any mass $M$ of the Schwarzschild black hole, and the limit to $M = 0$ is continuous. This means that they apply not only to the quantization in the black hole background, but also to the unitary quantization of the gravitational field in the Minkowski space background in spherical coordinates.\n\n\n\n\n\\section{ Counting Gravity  Physical Degrees of Freedom in the Spherical Harmonic Basis}\nThe ansatz of Regge-Wheeler for the metric perturbations $h_{\\mu\\nu}$  with  spherical harmonics of definite parity  is given in  \\cite{Regge:1957td,Zerilli:1971wd,Martel:2005ir}. In our \n recent paper \\cite{Kallosh:2021ors} it was adapted for the purpose of quantization following  the formalism and notations in \\cite{Martel:2005ir}. In particular, we have presented the gauge symmetry transformations to all orders in $h$. The background metric in Schwarzschild coordinates is\n \\begin{equation}\ng_{\\mu\\nu}dx^\\mu dx^\\nu=\t-f(r)\\,\\mathrm{d}t^2 + \\frac{\\mathrm{d}r^2}{f(r)} + r^2(x)\\,\\mathrm{d}\\Omega^2_2\\, ,  \\qquad f(r)  = 1-\\frac{2GM}{r}\n\t\\label{SchldBackgroundGauge}\n\\end{equation} \n\n\\\n\n The 2D fields representing all components of $h_{\\mu\\nu}$ in 4D include the following  \n\\begin{eqnarray} \\label{RWA}\n&& h_{ab}^{\\ell m(+)}, \\quad j_{a}^{\\ell m(+)}, \\quad K^{\\ell m(+)}, \\quad G^{\\ell m(+)}\\,  \\hskip 3 cm {\\ell } >1  , \\quad  {\\rm even}\\\\\n&&  h_{a}^{\\ell m(-)}, \\quad h_2^{\\ell m(-)}\\,    \\hskip 6.5 cm {\\ell } > 1 , \\quad  {\\rm odd} \\\\\n&&h_{ab}^{1 m(+)}, \\quad  j_{a}^{1 m(+)},  \\quad K^{1m(+)}   \\qquad \\hskip 4 cm {\\ell }=1, \\quad  {\\rm even}\n\\label{ansatzDe} \\\\\n&&h_{a}^{1 m(-)}  \\hskip 8.2 cm  {\\ell }=1, \\quad   {\\rm odd}\\\\\n&&h_{ab}^{0 0(+)},  \\quad K^{00(+)}  \\hskip 6.6 cm  {\\ell }=0,  \\quad  {\\rm even}\n\\label{ansatzM} \n \\end{eqnarray}\nThe gauge symmetries are also expanded in spherical harmonics. In the form   given in our recent paper \\cite{Kallosh:2021ors} these are\n \\begin{eqnarray}\n\\xi^{{\\ell } > 1 }  \\quad  {\\rm even}  \\qquad   &&\\Rightarrow \\qquad \\{ \\xi_{a}^{\\ell m(+)}, \\xi^{\\ell m(+)} \\}\n\\label{sym}\\\\\n\\xi^{{\\ell } > 1}  \\quad  {\\rm odd}   \\qquad   &&\\Rightarrow \\qquad \\{  \\xi^{\\ell m(-)} \\}\n\\label{sym}\\\\\n\\xi^{{\\ell }=1}  \\quad  {\\rm even}  \\qquad   &&\\Rightarrow \\qquad  \\{ \\xi_{a}^{1 m(+)}, \\xi^{1 m(+)} \\} \n\\label{symDe} \\\\\n\\xi^{{\\ell }=1}  \\quad  {\\rm odd}  \\qquad  && \\Rightarrow \\qquad  \\{  \\xi^{1 m(-)}  \\} \\\\\n\\xi^{{\\ell }=0}   \\quad  {\\rm even}   \\qquad   &&\\Rightarrow \\qquad  \\{  \\xi_{a}^{0 0(+)} \\}\n\\label{symM} \n \\end{eqnarray}\nThe gauge symmetry parameters $\\xi_{a}^{\\ell m (+)}$, $\\xi^{\\ell m (+)}$, $\\xi^{\\ell m (-)}$  \ncan be regarded as scalar and vector fields on $\\mathcal{M}_2$.\n\nThe counting of physical degrees of freedom in these 5 sectors is\n\\begin{enumerate}\n  \\item ${\\ell } >1   \\quad  {\\rm even}:  \\quad  \\, n+k = 7, \\quad k= 3\\, \\quad \\Rightarrow \\quad n+k -2k = 7- 2\\cdot 3=1$\n  \\item ${\\ell } >1   \\, \\, \\quad  {\\rm odd}:  \\quad  \\, n+k = 3, \\quad k= 1\\, \\quad \\Rightarrow \\quad n+k -2k= 3- 2\\cdot 1=1$\n  \\item ${\\ell } =1    \\quad  {\\rm even}:  \\quad  \\, n+k = 6, \\quad k= 3\\, \\quad \\Rightarrow \\quad n+k -2k= 6- 2\\cdot 3=0$\n  \\item ${\\ell } =1   \\, \\,  \\quad  {\\rm odd}:  \\quad  \\, n+k = 2, \\quad k= 1\\, \\quad \\Rightarrow \\quad n+k -2k= 2- 2\\cdot 1=0$ \n  \\item ${\\ell } =0    \\quad  {\\rm even}:  \\quad  \\, n+k = 4, \\quad k= 2\\, \\quad \\Rightarrow \\quad n+k -2k= 4- 2\\cdot 2=0$\n\\end{enumerate}\nThus we find that in $l\\geq 2$ sector there is one even and one odd physical degree of freedom for each $({\\ell },m)$. There are no degrees of freedom for any of ${\\ell }<2$.\n\n\\section{Quadratic Lagrangian\/Hamiltonian  for ${\\ell }\\geq 2 $ Modes}\n\\subsection{${\\ell }\\geq 2 $ even} \n There are 7 fields here, $h_{ab}^{\\ell m(+)}, \\quad j_{a}^{\\ell m(+)}, \\quad K^{\\ell m(+)}, \\quad G^{\\ell m(+)}$.\n There are 7 equations of motion for these fields.   Now we can add the 3 Regge-Wheeler gauge-fixing conditions\n \\begin{equation}\n G=j_a=0\n \\end{equation} \nThe remaining 4 fields are $h_{ab}^{\\ell m(+)},  K^{\\ell m(+)}$. We expect to identify 3 constraints which will leave us with just one canonical degree of freedom. These equations are according to \\cite{Martel:2005ir}\n \\begin{eqnarray} \\label{MarP}\nQ^{tt} &=& -\\frac{\\partial^2}{\\partial r^2} {K} \n- \\frac{3r-5M}{r^2 f} \\frac{\\partial}{\\partial r} {K} \n+ \\frac{f}{r} \\frac{\\partial}{\\partial r} {h}_{rr}\n+ \\frac{(\\lambda+2)r + 4M}{2r^3} {h}_{rr} \n+ \\frac{\\mu}{2r^2 f} {K}, \\\\ \nQ^{tr} &=& \\frac{\\partial^2}{\\partial t \\partial r} {K} \n+ \\frac{r-3M}{r^2 f} \\frac{\\partial}{\\partial t} {K} \n- \\frac{f}{r} \\frac{\\partial}{\\partial t} {h}_{rr} \n- \\frac{\\lambda}{2r^2} {h}_{tr}, \\cr\nQ^{rr} &=& -\\frac{\\partial^2}{\\partial t^2} {K} \n+ \\frac{(r-M)f}{r^2} \\frac{\\partial}{\\partial r} {K} \n+ \\frac{2f}{r} \\frac{\\partial}{\\partial t} {h}_{tr} \n- \\frac{f}{r} \\frac{\\partial}{\\partial r} {h}_{tt}  \n+ \\frac{\\lambda r + 4M}{2r^3} {h}_{tt} \n- \\frac{f^2}{r^2} {h}_{rr} \n- \\frac{\\mu f}{2r^2} {K}, \\cr \nQ^\\flat &=& -\\frac{\\partial^2}{\\partial t^2} {h}_{rr} \n+ 2 \\frac{\\partial^2}{\\partial t \\partial r} {h}_{tr} \n- \\frac{\\partial^2}{\\partial r^2} \\tilde{h}_{tt} \n- \\frac{1}{f} \\frac{\\partial^2}{\\partial t^2} {K} \n+ f \\frac{\\partial^2}{\\partial r^2} \\tilde{K} \n+ \\frac{2(r-M)}{r^2 f} \\frac{\\partial}{\\partial t} {h}_{tr} \n- \\frac{r-3M}{r^2 f} \\frac{\\partial}{\\partial r} {h}_{tt} \n\\cr & & \\mbox{} \n- \\frac{(r-M)f}{r^2} \\frac{\\partial}{\\partial r} {h}_{rr} \n+ \\frac{2(r-M)}{r^2} \\frac{\\partial}{\\partial r} {K} \n+ \\frac{\\lambda r^2-2(2+\\lambda)Mr+4M^2}{2r^4 f^2}{h}_{tt}   \n- \\frac{\\lambda r^2-2\\mu Mr-4M^2}{2r^4} {h}_{rr}\\nonumber \n\\end{eqnarray} \nHere \n\\begin{equation}\n\\lambda = {\\ell }({\\ell }+1) \\qquad \\mu = ({\\ell }-1)({\\ell }+2)\n\\end{equation}\nThe quadratic in $h$ Lagrangian can be restored from these  equations  as explained in eqs. \\rf{quad}, \\rf{EOM}. One can proceed by defining for each of the 4  fields their canonical momenta. For example, there is no time derivative on $h_{tt}$ in the action, therefore $p_{tt}=0$, the other 3 coordinates in the form of ${\\cal L} (q, \\dot q)$ do have time derivatives, however, two more combinations of $q$'s and $p$'s are constrained. Only one independent canonical degree of freedom out of 4  is left.\n\nThe Hamiltonian of the related system starting with the  Arnowitt, Deser, Misner construction  was derived in \\cite{Moncrief:1974am}. We skip the details of the derivation here starting with the field equations \n\\rf{MarP} since the answer for the corresponding Lagrangian can be also reconstructed from the \n Zerilli-Moncrief function \\cite{Zerilli:1971wd,Moncrief:1974am} which in Regge-Wheeler gauge is\n\\[\n\\Psi_{\\rm even} = \\frac{2r}{{\\ell }({\\ell }+1)} \\biggl[ {K}   \n+ \\frac{2f}{\\Lambda} \\biggl( f {h}_{rr} \n- r  {K}_{,r} \\biggr) \\biggr], \\qquad l\\geq 2\n\\]\nwhere $\\Lambda = ({\\ell }-1)({\\ell }+2)  + 6M\/r$. The equation of motion in the form of the Zerilli-Moncrief function $\\Psi^{lm}_{even}$ as given in \\cite{Martel:2005ir} is\n\\begin{equation}\n(\\Box - V_{\\rm even})  \\Psi_{\\rm even}=S_{\\rm even}\n\\label{ZM}\\end{equation}\nwhere $\\Box= g^{ab} {\\cal D}_a {\\cal D}_b$ is the Laplacian operator on ${\\cal M}_2$, $V_{\\rm even}$ depends on $r$ as well as on  $M$ and on $l$,   and $S_{\\rm even}$ is the contribution from sources.  We refer to details given in \\cite{Martel:2005ir}, where also the relation between Zerilli-Moncrief function and the  original Regge-Wheeler function is explained.\nEquation  \\rf{ZM} can be derived from the Lagrangian of the form \\rf{quad}\n\\begin{equation}\n{\\cal L} = \\sum_{{\\ell } \\geq 2 ,  m} \\Big [  {1\\over 2} \\Psi_{\\rm even}( \\Box - V_{\\rm even})\\Psi_{\\rm even}  - \n \\Psi_{\\rm even} S_{\\rm even}\\Big ]\\end{equation}\n This can be rewritten in the form producing a quadratic part of the Hamiltonian. With $\\Psi_{\\rm even}\\equiv Q_{\\rm even}$ and its canonically conjugate $P_{\\rm even}$ and,  in absence of sources\n\\begin{equation}\n H_{{\\ell } \\geq 2, \\rm even}= {1\\over 2} \\sum_{{\\ell }\\geq 2, m} \n \\int \\Big [dr  f(  P^{{\\ell }, m } )^{2 }_{\\rm even}+  f (Q_{, r}^{{\\ell }, m })_{\\rm even}^2 +   V_{\\rm even}\n (Q^{{\\ell }, m })_{\\rm even}^2\\Big ]\n\\label{Heven} \\end{equation}\nwhere\n\\begin{equation} \nV_{\\rm even} = \\frac{1}{\\Lambda^2} \\biggl[ \\mu^2 \\biggl(\n  \\frac{\\mu+2}{r^2} + \\frac{6M}{r^3} \\biggr) \n+ \\frac{36M^2}{r^4} \\biggl(\\mu + \\frac{2M}{r} \\biggr) \\biggr] \n\\label{4.26}\n\\end{equation}\nThis is an example of the Faddeev's theorem  \\cite{Faddeev:1969su}, which we described in   \\cite{Kallosh:2021ors}, where starting from the original constrained  variables $(p_i, q^i)$  with constraints $\\phi^\\alpha(p,q)$ one can perform a canonical transformation with $p'_\\alpha =\\chi_\\alpha (p,q) =0$ and  $ q^{'\\alpha} = q^{'\\alpha} (p^*, q^*)$ \nso that the independent set of canonical variables is $(p^*, q^*)$. In this particular case we find just one set of $(p^*, q^*)$, which are the \n Zerilli-Moncrief function $\\Psi$ of the original variables,  and its canonical conjugate.\n\n \n\\subsection{${\\ell }\\geq 2 $ odd} \n\nThere are 3 fields in this sector:  $h_{a}^{\\ell m(-)}, \\quad h_2^{\\ell m(-)}$. In the RW gauge\n\\begin{equation}\nh_2^{\\ell m(-)}=0\\, .\n\\end{equation}\nEquations of motion for the remaining two fields are\n\\begin{eqnarray*} \nP^t &=& - \\frac{\\partial^2}{\\partial t \\partial r} {h}_r \n+ \\frac{\\partial^2}{\\partial r^2} {h}_t \n- \\frac{2}{r} \\frac{\\partial}{\\partial t} {h}_r \\\n- \\frac{\\lambda r - 4M}{r^3 f} {h}_t, \\\\ \nP^r &=& \\frac{\\partial^2}{\\partial t^2} {h}_r \n- \\frac{\\partial^2}{\\partial t \\partial r} {h}_t \n+ \\frac{2}{r} \\frac{\\partial}{\\partial t} {h}_t \n+ \\frac{\\mu f}{r^2} {h}_r,\n\\end{eqnarray*} \nRestoring the quadratic Lagrangian and using partial integration one can identify one  field which enters into Lagrangian without a time derivative, this is $h_t$.  \n\\begin{equation} \n{\\cal L} = h_t \\Big (- \\frac{\\partial^2}{\\partial t \\partial r} {h}_r \n+ {1\\over 2} \\frac{\\partial^2}{\\partial r^2} {h}_t \n- \\frac{2}{r} \\frac{\\partial}{\\partial t} {h}_r \\\n- {1\\over 2} \\frac{\\lambda r - 4M}{r^3 f} {h}_t \\Big ) + {1\\over 2} h_r \\Big (\\frac{\\partial^2}{\\partial t^2} {h}_r \n+ \\frac{\\mu f}{r^2} {h}_r\\Big ) \n\\end{equation}\nThus we find\n\\begin{eqnarray}\n&&p_t=0\\\\\n&&p_r= h_{t, r} - h_{r, t} -{2\\over r} h_t\n\\end{eqnarray}\nand there is a constraint for $h_t$ algebraically related to $p_r$  \n\\begin{equation}\n \\Big ( \\partial_r + \\frac{2}{r}  \\Big ) \\Big (p_r + h_{t, r} -{2\\over r} h_t\n\\Big ) \n-  {h}_{t,rr} \n+  \\frac{\\lambda r - 4M}{r^3 f} {h}_t =0\n\\end{equation}\nTherefore there is one independent degree of freedom  $(h_r, p_r)$. These are Faddeev's $(p^*, q^*)$ variables, exactly one set in agreement with the counting give above. One can write the corresponding Hamiltonian $H(h_r, p_r)$ and the field equations.\n\nOn the other hand, the Hamiltonian for this system was already derived in \\cite{Moncrief:1974am} in  the framework of the Arnowitt, Deser, Misner construction.  The field equations were derived in \\cite{Cunningham:1978zfa}, where the corresponding Cunningham-Price-Moncrief function was introduced. In notation of  \\cite{Martel:2005ir}, this function is\n\\begin{equation}\n\\Psi_{\\rm odd}^{lm}  = \\frac{2r}{({\\ell }-1) ({\\ell }+2) } \\biggl( \n {h}_{t , r} ^{{\\ell }m} - \n {h}_{r,t}^{{\\ell }m}  - \\frac{2}{r} {h}_t^{{\\ell }m}  \n\\biggr). \n\\end{equation}\nThis function in terms of canonical variables above depends on $(p_r, h_r)$.\nAs in the even case  discussed  above we are lead to a single field   equation for the Cunningham-Price-Moncrief  function\n\\begin{equation}\n(\\Box - V_{\\rm odd})  \\Psi_{\\rm odd}=S_{\\rm odd}\n\\label{odd}\\end{equation}\nHere the expressions for $V_{\\rm odd}$ and $  S_{\\rm odd}$ are given in \\cite{Martel:2005ir}, where also \nthe relation between Cunningham-Price-Moncrief  function  and the original Regge-Wheeler function is explained.\nWith $\\Psi_{\\rm odd}\\equiv Q_{\\rm odd}$ the Hamiltonian is\n\n \\begin{equation}\n H_{{\\ell } \\geq 2, \\rm odd}= {1\\over 2} \\sum_{{\\ell }\\geq 2, m} \n \\int \\Big [dr  f(  P^{{\\ell }, m } )^{2 }_{\\rm odd}+  f (Q_{, r}^{{\\ell }, m })_{\\rm odd}^2 +  \\Big ( \\frac{{\\ell }({\\ell }+1) }{r^2} - {6M \\over r^3}\\Big ) \n (Q^{{\\ell }, m })_{\\rm odd}^2\\Big ]\n \\label{Hodd}\\end{equation}\n  \n\\section{Quadratic Lagrangian\/Hamiltonian  for ${\\ell }<2$ Modes}\n\\subsection{${\\ell }=1$ even} \nOur 6 fields are $h_{ab}^{1 m(+)},   j_{a}^{1 m(+)},   K^{1m(+)}$.  We take a gauge-fixing condition \\cite{Kallosh:2021ors}\n\\begin{equation}\nj_{a}^{1 m(+)}= K^{1m(+)}=0\n\\end{equation}\nThe remaining filelds $h_{ab}^{1 m(+)}$ in this gauge satisfy the field equations\n\\begin{eqnarray*} \nQ^{tt} &=& \n \\frac{f}{r} \\frac{\\partial}{\\partial r}{h}_{rr}\n+ \\frac{2(r + M)}{r^3} {h}_{rr}, \\\\ \nQ^{tr} &=& \n- \\frac{f}{r} \\frac{\\partial}{\\partial t} {h}_{rr} \n- \\frac{1}{r^2} {h}_{tr}, \\\\\nQ^{rr} &=& \n \\frac{2f}{r} \\frac{\\partial}{\\partial t} {h}_{tr} \n- \\frac{f}{r} \\frac{\\partial}{\\partial r}{h}_{tt}  \n+ \\frac{ r + 2M}{r^3} {h}_{tt} \n- \\frac{f^2}{r^2}{h}_{rr} \n\\end{eqnarray*} \nWe can therefore reconstruct the Lagrangian of the form \\rf{quad} which will produce these equations.\n\\begin{equation}\n{\\cal L} = h_{tt} Q^{tt} +  \\Big (\\frac{\\partial}{\\partial t} h_{tr}\\Big)   \\frac{ 2f}{r}  {h}_{rr} \n-   h_{tr} \\frac{1}{2r^2} {h}_{tr} -h_{rr} \\frac{f^2}{ 2r^2}{h}_{rr} \n\\end{equation}\nWe now define $q\\equiv  h_{tr}, \\, p \\equiv  \\frac{ 2f}{r}  {h}_{rr}$ and $h_{tt} \\equiv \\lambda$\n\\begin{equation}\n{\\cal L} =  \\dot q p + \\lambda Q^{tt} (p, \\partial_r p)  \n-   q^2 \\frac{1}{2r^2} -{1\\over 8} p^2\n\\end{equation}\nWe integrate out the Lagrange multiplier and find\n\\begin{equation}\n{\\cal L} =  \\dot q p \n-   q^2 \\frac{1}{ 2r^2} -{1\\over 8} p^2\n\\end{equation}\nwhere \n\\begin{equation}\n \\frac{f}{r} \\frac{\\partial}{\\partial r}{rp\\over 2f}\n+ \\frac{2(r + M)}{r^3} {rp\\over 2f}=0  \\qquad \\Rightarrow \\qquad  p_{,r} + F(r) p=0\n\\label{Cp}\\end{equation}\nThe algebraic constraint which $p$ has to satisfy contradicts the  commutation relation which have to be imposed for quantization, as shown in eq. \\rf{quant}.\nThere is no solution of the algebraic constraint \\rf{Cp} for the canonical momentum $p(t,r)$ which would be consistent with the quantization condition, only $p=0$ is a consistent one. We conclude there that there are no physical degrees of freedom left in this sector,\n\\begin{equation}\nH_{{\\ell }=1, \\rm  even} =0\n\\end{equation}\n This is in agreement with the counting we presented above.\n\n\n\\subsection{${\\ell }=1$ odd} \n\nThere are 2 fields: $h_{a}^{1 m(-)}$. We take a gauge-fixing condition $h_{r}^{1 m(-)}=0$ \\cite{Kallosh:2021ors}. In this gauge the remaining field equation is\n\\begin{equation}\nP^t =  \\frac{\\partial^2}{\\partial r^2} {h}_t \n- \\frac{2}{r^2 } {h}_t\n\\end{equation}\nThe Lagrangian which will generate this equation is\n\\begin{equation}\n{\\cal L}={1\\over 2} h_t \n\\Big (\\frac{\\partial^2}{\\partial r^2} {h}_t \n- \\frac{2}{r^2} {h}_t \\Big  )\n\\end{equation}\nThere is one field here where the Lagrangian $ {\\cal L}(q)$ does not have time derivative of this field, therefore $p={\\delta {\\cal L}\\over \\dot h_t}=0 $. There are no canonical variables here and the Hamiltonian vanishes \n\\begin{equation}\nH_{{\\ell }=1, \\rm  odd} =0\n\\end{equation}\nThis is in agreement with the counting we presented above.\n\n\n\\subsection{${\\ell }=0$ even} \n\nThere are 4 fields here: $h_{ab}^{0 0(+)},  \\quad K^{00(+)}$. We take a gauge-fixing conditions $K=h_{tr}=0$ \\cite{Kallosh:2021ors}.\nThe remaining field equations are\n\\begin{eqnarray*} \nQ^{tt} &=&  \\frac{f}{r} \\frac{\\partial}{\\partial r} {h}_{rr}\n+ \\frac{r + 2M}{r^3} {h}_{rr} \n, \\\\ \nQ^{rr} &=&  \n- \\frac{f}{r} \\frac{\\partial}{\\partial r} {h}_{tt}  \n+ \\frac{ 2M}{r^3} {h}_{tt} \n- \\frac{f^2}{r^2} {h}_{rr}\n\\end{eqnarray*} \nThe Lagrangian which will generate these equations is\n\\begin{equation}\n{\\cal L}=h_{tt} \\Big (\\frac{f}{r} \\frac{\\partial}{\\partial r} {h}_{rr}\n+ \\frac{(r + 2M)}{r^3} {h}_{rr} \\Big ) -{f^2\\over 2r^2} h_{rr}^2\n\\end{equation}\nThere are 2 fields, $q^1, q^2$, but there are no time derivatives in the Lagrangian, $p_1=p_2=0$, no canonical variables and the Hamiltonian vanishes\n\\begin{equation}\nH_{{\\ell }=0} =0\n\\end{equation}\n This is again in agreement with the counting we presented above.\n\n\n\n\\section{A special role of ${\\ell }=0,1$ in quantization of gravity}\n\nIs there any relation between the well known fact about the absence of radiation from monopoles and dipoles in gravity and the fact we observed here, that there are no quantum physical degrees of freedom in monopoles and dipoles when gravity is quantized in spherical harmonics basis? The answer is yes, and it has to do with the tensor nature of gravity, so that radiation starts with quadrupoles ${\\ell }\\geq 2$. \n\nRegge-Wheeler ansatz for ${\\ell }\\geq 2$ has 10 functions depending on coordinates of ${\\cal M}_2$ listed in eqs. \\rf{RWA}-\\rf{ansatzM}. Here we show them in the matrix form contracted with spherical functions.\n\\begin{equation}\nh_{\\mu\\nu}^{{\\ell }>1} =\\begin{pmatrix}\n h^{{\\ell }m}_{ab} Y^{{\\ell }m}   &   &  &  &  {\\color {blue}j^{{\\ell }m}_a Y_B^{{\\ell }m}} \\\\\n\\cr  \n {\\color {blue} j^{{\\ell }m}_a Y_B^{{\\ell }m} } &   &  &  & r^2 K^{{\\ell }m} \\Omega_{AB} Y^{{\\ell }m} + {\\color {red} G^{{\\ell }m} Y_{AB}^{{\\ell }m}}\\end{pmatrix}^{(+)} + \\begin{pmatrix}\n0   &   &  &  &  {\\color {blue} h^{{\\ell }m}_a X_B^{{\\ell }m}} \\\\\n\\cr  \n {\\color {blue} h^{{\\ell }m}_a X_B^{{\\ell }m}}  &   &  &  & {\\color {red} h^{{\\ell }m}_2 X_{AB}^{{\\ell }m}} \\end{pmatrix}^{(-)} \\, .\n\\end{equation}\nThe number gauge symmetries  in all cases with ${\\ell } >0$ is the same since $\\xi_\\mu$ is a vector\n\\begin{equation}\n\\xi_{\\mu}^{{\\ell }>0} =\\begin{pmatrix}\n\\xi^{{\\ell }m}_{a} Y^{{\\ell }m}   \\\\\n\\cr  \n {\\color {blue}\\xi^{{\\ell }m} Y_A^{{\\ell }m}}  \\end{pmatrix}^{(+)}  \\, + \\begin{pmatrix}\n0    \\\\\n\\cr  \n {\\color {blue}\\xi^{{\\ell }m}_a X_B^{{\\ell }m} }  \\end{pmatrix}^{(-)} \\, .\n\\end{equation}\nTherefore we find that  instead of 10 fields (even and odd) as for ${\\ell }\\geq 2$ we have 8 fields (even and odd)  for ${\\ell }=1$, no fields  in red\n\\begin{equation}\nh_{\\mu\\nu}^{{\\ell }=1} =\\begin{pmatrix}\n h^{{\\ell }m}_{ab} Y^{{\\ell }m}   &   &  &  & j^{{\\ell }m}_a Y_B^{{\\ell }m} \\\\\n\\cr  \n j^{{\\ell }m}_a Y_B^{{\\ell }m}  &   &  &  & r^2 K^{{\\ell }m} \\Omega_{AB} Y^{{\\ell }m} \\end{pmatrix}^{(+)} + \\begin{pmatrix}\n0   &   &  &  &h^{{\\ell }m}_a X_B^{{\\ell }m} \\\\\n\\cr  \nh^{{\\ell }m}_a X_B^{{\\ell }m}  &   &  &  & 0 \\end{pmatrix}^{(-)} \\, .\n\\end{equation}\nTherefore from 10-2 =8 states we subtract a double set of 4 symmetries, and find no degrees of freedom for ${\\ell }=1$ since  8-8=0.\n\nAt  ${\\ell}=0$  $Y_{AB}^{00}= X_{AB}^{00}=0$, the terms in red are absent, but also $Y_{A}^{00}= X_{A}^{00}=0$, all blue terms are absent. \n\\begin{equation}\nh_{\\mu\\nu}^{{\\ell }=0} =\\begin{pmatrix}\n h^{{\\ell }m}_{ab} Y^{{\\ell }m}   &   &  &  & 0 \\\\\n\\cr  \n0  &   &  &  & r^2 K^{{\\ell }m} \\Omega_{AB} Y^{{\\ell }m} \\end{pmatrix}^{(+)} + \\begin{pmatrix}\n0   &   &  &  &0 \\\\\n\\cr  \n0  &   &  &  & 0 \\end{pmatrix}^{(-)} \\, .\n\\end{equation}\n\\begin{equation}\n\\xi_{\\mu}^{{\\ell }=0} =\\begin{pmatrix}\n\\xi^{{\\ell }m}_{a} Y^{{\\ell }m}   \\\\\n\\cr  \n0  \\end{pmatrix}^{(+)}  \\, + \\begin{pmatrix}\n0    \\\\\n\\cr  \n0 \\end{pmatrix}^{(-)} \\, .\n\\end{equation}\nWe are left with 4 fields and  2 gauge symmetries, there are no degrees of freedom for ${\\ell }=0$: 4-4=0.\n\n\n\\section{Quantization of Gravity in Spherical Harmonics Basis in the Flat Background}\nThe procedure of Lagrangian quantization performed in \\cite{Kallosh:2021ors} as well as the values of the unitary quadratic Hamiltonians presented in this paper, have a smooth limit from the Schwarzschild background to a flat one. In Schwarzschild  coordinates this means that the limit $M \\rightarrow 0$ is regular.\n\n In particular,   Zerilli-Moncrief function for ${\\ell }\\geq 2$ in  Regge-Wheeler gauge in the limit  $M \\rightarrow 0$  is \n\\begin{equation}\n\\Psi_{\\rm even}^{{\\ell }m} = \\frac{2r}{{\\ell }({\\ell }+1)} \\biggl[ {K}   \n+ \\frac{2}{({\\ell }-1)({\\ell }+2)} \\biggl( {h}_{rr} \n- r  {K}_{,r} \\biggr) \\biggr], \\qquad {\\ell }\\geq 2\n\\end{equation}\nThe Cunningham-Price-Moncrief function is\n\\begin{equation}\n\\Psi_{\\rm odd}^{{\\ell } m}  = \\frac{2r}{({\\ell }-1) ({\\ell }+2) } \\biggl( \n {h}_{t , r} ^{{\\ell }m} - \n {h}_{r,t}^{{\\ell }m}  - \\frac{2}{r} {h}_t^{{\\ell }m}  \n\\biggr \n), \n \\qquad {\\ell }\\geq 2 \\end{equation}\n The quadrartic part of the Hamiltonian in both cases  is\n \\begin{equation}\n H_{\\rm even\/odd}= {1\\over 2} \\sum_{{\\ell }\\geq 2, m} \n \\int \\Big [dr ( P^{{\\ell }, m } )^{2 }_{\\rm even\/odd}+  (Q_{, r}^{{\\ell }, m })_{\\rm even\/odd}^2 +   \\frac{{\\ell }({\\ell }+1) }{r^2} \n (Q^{{\\ell }, m })_{\\rm even\/odd}^2\\Big ]\n\\label{HamFlat} \\end{equation}\nHere $Q_{\\rm even\/odd} = \\Psi _{\\rm even\/odd}$ and $P_{\\rm even\/odd}$ is the corresponding canonical conjugate. At the quadratic level these are the only 2 physical states which appear in the unitary Hamiltonian.\n\nThe higher order terms in the each of the quantized actions, at the black hole background and in the flat background still have to be constructed.\n\n\\section{A comment on Regge-Wheeler and  Teukolsky formalism and gravity waves}\n\n\nThe Cunningham-Price-Moncrief (CPM) master function and the Zerilli-Moncrief (ZM) master function, which were identified here as canonical variables  in the gravity,  appear to play some role also in a more interesting case of the Kerr black holes. Namely, as pointed out in a  review \\cite{Pound:2021qin}, there is a relation via Chandrasekhar transformation between these functions and Teukolsky radial function. Note that \nTeukolsky equations for the Weyl tensor components  use the expansion in terms of the spin-weighted spheroidal harmonics. Such and expansion for the metric starts with ${\\ell }=2$.\n\nThere is also an interesting relation between the metric perturbation far from the source and our canonical variables in the generalized Regge-Wheeler gauge. Namely, according to  \\cite{Pound:2021qin} the gravitational wave strain can be determined directly from CPM and ZM functions of the metric. Using the \n Chandrasekhar transformation between these functions and Teukolsky radial function, and some properties of $\\psi_4= C_{n\\bar m n\\bar m}$ the gravitational strain was given as\n\n\n\\begin{equation}\nr(h_+ -i h_x) =\\sum_{{\\ell }\\geq 2} \\sum_{ |m|  \\leq {\\ell }} {D\\over 2} \\Big (\\Psi_{\\rm even}^{{\\ell }m}- i \\Psi_{\\rm even}^{{\\ell }m}\\Big ) \\, \\hskip 1 mm   {}_{- 2}  Y_{{\\ell }, m } (\\theta, \\phi)\n\\end{equation}\nwhere ${}_{- 2}  Y_{{\\ell }, m } (\\theta, \\phi)$ is the the spin-weighted spheroidal harmonic.\nThat equality holds in the limit $r  \\rightarrow  \\infty $  (at fixed $u=t-r_*$).\nHere the constant\n\\begin{equation}\nD= \\sqrt {({\\ell }-1) ({\\ell }+1) ({\\ell }+1)}\n\\end{equation}\nis the Schwarzschild limit of the constant that appears in the Teukolsky-Starobinsky identities. Clearly, the cases ${\\ell }=0,1$ drop from the formula for the gravitational waves. This is in agreement with the  fact established in this paper that these modes have no physical degrees of freedom.\n\n\n\n\\section{Summary} \n\n\n\nIn this note we have counted the number of physical quantized degrees of freedom  of Einstein gravity in spherical harmonic basis using the standard formula:  this number is given by $n-k$,  where  $n+k$ is the number of components of gauge fields and the  gauge theory has $k$ gauge symmetries. For example, in 4D the graviton has  $n+k= 6+4= 10$ components and there are $k=4$  gauge symmetries. The  number of physical degrees of freedom is $n-k= (n+k) - 2k= 10-8=2$.\n\n\nIn spherical harmonic basis we have found that for each ${\\ell }, m$ in ${\\ell } \\geq 2$ sector there is one degree of freedom for even parity states and one degree of freedom for odd parity states. In ${\\ell }<2$ sector of gravity we have found that there are no physical degrees of freedom.\n\nTo construct the Hamiltonian we start with the Regge-Wheeler formulation \\cite{Regge:1957td,Zerilli:1971wd,Martel:2005ir} of   Einstein gravity in spherical harmonic basis in the background of a Schwarzschild black hole. The part of the action $S(g+h)$ quadratic in perturbations $h_{\\mu\\nu}$ in eq. \\rf{quad} can be presented in  spherical harmonic basis using the explicit form of equations of motion linear in perturbations, as shown in eq. \\rf{EOM}. We take these explicit expressions $Q^{\\mu\\nu} = {\\delta S (g, h) \\over \\delta h_{\\mu\\nu}} $, which are linear in $h_{\\mu\\nu}$, \nfrom \\cite{Martel:2005ir}, and reconstruct  the part of the action $S(g+h)$ quadratic in perturbations $h_{\\mu\\nu}$. We impose the generalized Regge-Wheeler gauge\n \\cite{Kallosh:2021ors}. The action quadratic in fields we take in Schwarzschild  coordinates and proceed with canonical quantization, defining canonical momenta and constraints. \n\nFor ${\\ell } \\geq 2$ fields the procedure leads to one independent degree of freedom for even and one for odd modes in each case with ${\\ell }, m$, in agreement with the counting of physical degrees of freedom. We conclude that up to a canonical transformation such a Hamiltonian is equivalent to the one presented in \\cite{Moncrief:1974am} where the corresponding canonical variables are Zerilli-Moncrief function \\cite{Zerilli:1971wd,Moncrief:1974am} for even modes and a  Cunningham-Price-Moncrief function \\cite{Cunningham:1978zfa} for odd modes. In \\cite{Moncrief:1974am} the modes with ${\\ell }<2$ were not studied.\n\nWe apply our method also for  ${\\ell }<2$ modes. In each sector for ${\\ell }=1$, even and odd case and for \n${\\ell }=0$ we first reproduce the action from the explicit expressions $Q^{\\mu\\nu} = {\\delta S (g, h) \\over \\delta h_{\\mu\\nu}} $  linear in $h_{\\mu\\nu}$. We use the gauge-fixing condition for low multipoles in  \\cite{Kallosh:2021ors} and identify the canonical variables and constraints. In each case the conclusion is that there are no independent unconstrained canonical variables suitable for the quantized Hamiltonian. This is again in agreement with the counting of degrees of freedom performed earlier. \n\n\nThe original goal of this investigation was to develop a consistent method of quantization of gravitational field in the background  of a Schwarzschild black hole  \\cite{Kallosh:2021ors}.  However, we found that in Schwarzschild  coordinates the limit $M \\rightarrow 0$ is regular, and therefore the quantization procedure is valid in the Minkowski background as well. In this paper we found the  Hamiltonian describing unitary evolution of gravitational perturbations in spherical coordinates, which equally well applies to quantization of gravity in Minkowski  background as well as in the Schwarzschild black hole background. The choice of the generalized Regge-Wheeler gauge in \\cite{Kallosh:2021ors} where the gravity Hamiltonian is unitary requires to use the spherical harmonic basis for the metric perturbations. This {\\it unitary gauge} is a Regge-Wheeler gauge $G^{\\ell m(+)}=j_a^{\\ell m(+)}=h_2^{\\ell m(-)}=0$ for ${\\ell } \\geq 2$. For ${\\ell } =1$ it is $j_{a}^{1 m(+)}= K^{1m(+)}=h_{r}^{1 m(-)}=0$ and for ${\\ell } =0$ it is $K^{00}=h_{tr}^{00}=0$.\n\nIn this generalized Regge-Wheeler gauge, the quadratic part of the Hamiltonian for ${\\ell }<2$ modes is vanishing, whereas for ${\\ell }\\geq 2$ it is given in eqs. \\rf{Heven}, \\rf{Hodd} in the black hole background and in eq. \\rf{HamFlat} in Minkowski background. \n\n\\section*{Acknowledgement}\nI am   grateful to  A. Barvinsky,  E. Coleman, A. Linde,  E. Poisson,  A. Rahman, P. Stamp,  A. Starobinsky, A. Vainshtein,  A. Van Proeyen  and I. Volovich  for stimulating and helpful discussions.  \nI am supported by the SITP, by the US National Science Foundation Grant PHY-2014215 and  by the  Simons Foundation Origins of the Universe program (Modern Inflationary Cosmology collaboration).  \n \n\n\n\\\n\n\\\n\n\n\n\n\\bibliographystyle{JHEP}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe \\LaTeXe{} document class {\\tt appolb.cls} should be used \nby starting the file with\\\\\n{\\tt\\verb|\\documentclass{appolb}|}\n\nOur main goal is to let the authors see how the text and equations\nfit to our page layout --- the text column size is 126 mm\n$\\times$ 190 mm. The style is very similar to the original Latex\n{\\tt article}, \\ie most of the commands are used in the same way although\nsome of them result in a different text formatting.\nThere are also some new commands, which are described below. \n\n\\section{Options}\nOptional parameters to the {\\tt appolb} class can be given, as usually, in square\nbrackets, \\eg\\\\\n{\\tt\\verb|\\documentclass[letterpaper,draft]{appolb}|}\\\\\nDefault options are: {\\tt a4paper,final}.\n\n{\\parindent=0pt\\obeylines\nAvailable options:\n{\\tt draft} or {\\tt final} --- show or hide the overfull rule\n{\\tt letterpaper} or {\\tt a4paper} --- select paper size\n}\n\n\\section{Commands}\n\\parindent=0pt\n{\\tt\\verb|\\eqsec|}\n\nCall this macro before the first {\\tt\\verb|\\section|}\ncommand if you want equations numbered as \n(SectionNumber.EqNumber).\nYou can uncomment line \\the\\eLiNe\\ of this file \n({\\tt\\jobname.tex}) to see the effect.\n\n\\subsection{Shortcuts}\n{\\obeylines\n{\\tt \\verb|\\ie|} gives: \\ie\n{\\tt \\verb|\\eg|} gives: \\eg\n{\\tt \\verb|\\cf|} gives: \\cf\n}\nThe macros provide appropriate spacing\nwithout the need for any curly braces \\{\\}.\n\n\\subsection{Math mode operators}\n{\\tt \\verb|\\Tr|} gives: $\\Tr$\n\n{\\tt \\verb|\\e|} gives: $\\e$ --- straight `e' in math mode.\n\n\\subsection{{\\tt eqletters} environment}\n\nEnumarate equations with\na number and a lower-case letter, \\eg\n\\begin{eqletters}\n\\label{myeq}\n\\begin{eqnarray}\nA_1 &=& F(1)\\,,\n\\label{me1}\n\\\\\nA_2 &=& F(2)\\,.\n\\label{me2}\n\\end{eqnarray}\nAs long as the {\\tt eqletters} environment is active all equations are\nnumbered with letters, \\eg\n\\begin{equation}\nL = \\Half a = \\half A\n\\end{equation}\n\\end{eqletters}\n\nEquations (\\ref{me1}) and (\\ref{me2}) can be referenced as Eqs. (\\ref{myeq}).\nThe {\\tt \\verb|\\label|} statement used to generate the latter reference\nmust be placed outside any {\\tt eqnarray} or {\\tt equation} environment.\n\n\\end{document}\n\n\n\\section{Introduction}\n\nThese lectures concern the properties of strongly interacting matter at high energy density.  Such matter occurs in a  number of contexts.  The high density partonic matter that controls the early stages of hadronic collisions at very high energies is largely made of very coherent gluonic fields. In a single hadron, such matter forms the small x part of a wavefunction, a Color Glass Condensate.  After a collision of two hadrons, this matter almost instantaneously is transformed into longitudinal color electric and color magnetic fields.  The ensemble of these fields in their early time evolution is called the Glasma. The decay products of these fields thermalize and form a high temperature gas of quarks and gluons, the Quark Gluon Plasma.  In collisions at lower energy, and perhaps in naturally occurring objects such as neutron stars, there is high baryon density matter at low temperature.  This is Quarkyonic matter.\n\nThere is a very well developed literature concerning these various forms of matter.  It is not the purpose of these lectures to provide a comprehensive review.  I will concentrate on motivating and describing such matter from simple intuitive physical pictures and from simple structural aspects of QCD.  I will attempt at various places to relate what is conjectured or understood about such matter to experimental results from accelerator experiments.\n\n\n\n\\section{Lecture I: The Color Glass Condensate and the Glasma}\n\nThe parton distributions of gluons, valence quarks and sea quarks can be measured for some momentum scale less than a resolution scale $Q$ as a function of their fractional momentum $x$ of a high energy hadron.  The lowest value of $x$ accessible for a fixed hadron energy $E$ is typically\n$x_{min} \\sim \\Lambda_{QCD}\/E_{hadron}$.  The small x limit is therefore the high energy limit.\n\nIt is remarkable that as $x$ is decreased, as we go to the high energy limit, that the gluon density dominates the constituents of a hadron for $x \\le 10^{-1}$.  The various distributions are shown as a function of $x$ in Fig. \\ref{gluondominance}.  The gluon density rises like a power of $x$ like $x^{-\\delta}$, $\\delta \\sim .2-.3$\nat accessible energies\n\\begin{figure}[!htb]\n\\begin{center}\n  \\mbox{{\\epsfig{figure=gluondominance.pdf, width=0.70\\textwidth}}}\n   \\end{center}\n\\caption[*]{ \\small  The parton distribution as a function of $x$.}\n     \\label{gluondominance}\n\\end{figure}\nThe area of a hadron grows slowly with energies.  Cross sections grow roughly as $ln^2(1\/x)$ for small x.\nThis means that the rapidly growing gluon distribution results in a high density system of gluons.  At high density, the gluons have small separation and by asymptotic freedom, the intrinsic strength of their interaction must be weak.\n\nA small intrinsic interaction strength does not mean that interactions are weak.  Consider gravity:  The interactions between single protons is very weak, but the force of gravity is long range, and the protons in the earth act coherently, that is always with the same sign.  This results in a large force of gravity.  This can also happen for the gluons inside a hadron, if their interactions are coherent.\n\nTo understand how this might happen, suppose we consider gluons of a fixed size $r_0 \\sim 1\/p_T$ where\n$p_T$ is its transverse momentum.  We assume that at high energy, the gluons have been Lorentz contracted into a thin sheet, so we need only consider the distribution of gluons in the transverse plane.  If\nwe start with a low density of gluons at some energy, and then evolve to higher energy, the density of gluons increases.  When the density is of order one gluon per size of the gluon, the interaction remains weak because of asymptotic freedom.  When the density is of order $1\/\\alpha_S$, the coherent interactions are strong, and adding another gluon to the system is resisted by a force of order $1$.  The gluons act as hard spheres. One can add no more gluons to the system of this size.  It is however possible to add in smaller gluons, in the space between the closely packed gluons of size $r_0$.  This is shown in Fig. \\ref{saturation}\n\\begin{figure}[!htb]\n\\begin{center}\n  \\mbox{{\\epsfig{figure=saturation.pdf, width=0.60\\textwidth}}}\n   \\end{center}\n\\caption[*]{ \\small  Increasing the gluon density in a saturated hadron when going to higher energy.}\n     \\label{saturation}\n\\end{figure}\n\nThe physical picture we derive means that below a certain momentum scale, the saturation scale $Q_{sat}$,\nthe gluon density is saturated and above this scale it is diffuse.  The saturation momentum scale grows with energy and need not itself saturate\\cite{Gribov:1984tu}-\\cite{McLerran:1993ka}.\n\nThe high phase space density of gluons, $dN\/dyd^2p_Td^2r_T \\sim 1\/\\alpha_S$ suggests that one can describe the gluons as a classical field.  A phase space density has a quantum mechanical interpretation\nas density of occupation of quantum mechanical states.  When the occupation number is large, one is in the classical limit.\n\nOne can imagine this high density gluon field generated from higher momentum partons.  We introduce the idea of sources corresponding to high $x$ partons and fields as low $x$ partons.  Because the high $x$ parton sources are fast moving, their evolution in time is Lorentz time dilated.  The gluon field produced by these sources is therefore static and evolves slowly compared to its natural time scale of evolution.  This ultimately means that the different configurations of sources are summed over incoherently, as in a spin glass. \n\nWe call this high energy density configuration of colored fields a Color Glass Condensate.  The word color is because the gluons that make it are colored.  The word condensate is used because the phase space density of gluons is large, and because this density is generated spontaneously.  The word glass is used because the typical time scale of evolution of the classical fields is short compared to the Lorentz time dilated scales associated with the sources of color.\n\nThere is an elaborate literature on the Color Glass Condensate and an excellent review is by Iancu and Venugopalan\\cite{Iancu:2003xm}.  Evolution of the CGC to small values of x is understood, as well as many relationships between deep inelastic scattering, deep inelastic diffraction and high energy nucleus-nucleus,\nproton-nucleus and proton-proton scattering.  The CGC is a universal form of matter in the high energy limit.  The theoretical ideas underlying the CGC are largely unchallenged as a description of the high energy limit of QCD, but the issue of when the approximation appropriate for the high energy limit are valid remains contentious. \n\nIn the description of high energy hadron hadron collisions, we consider the collision of two sheets of CGC as shown in Fig. \\ref{sheets}.  The color electric and color magnetic fields of the CGC are visualized as sheets of Lenard-Wiechart potentials.  These are classical gluon fields whose polarization and color are random, with an intensity distribution determined by the underlying theory of the CGC.\n\\begin{figure}[!htb]\n\\begin{center}\n  \\mbox{{\\epsfig{figure=sheets.pdf, width=0.70\\textwidth}}}\n   \\end{center}\n\\caption[*]{ \\small  The collision of two sheets of CGC.}\n     \\label{sheets}\n\\end{figure}\n\nUpon collision of these sheets, the sheets become charged with color magnetic and color electric charge distributions of equal magnitude but opposite sign locally in the transverse plane of the sheets\\cite{Kovner:1995ja}-\\cite{Lappi:2006fp}. In the high\nenergy limit sources of color electric and color magnetic field must be treated on an equal footing because of the self duality of QCD.  This induced charge density produces longitudinal color electric and color magnetic fields between the two sheets.  These fields are longitudinally boost invariant and therefore have the correct structure to account for Bjorken's initial conditions in heavy ion collisions\\cite{Bjorken:1982qr}.  The typical transverse length scale over which the flux tubes vary is $1\/Q_{sat}$. The initial density of produced gluons is on dimensional grounds\n\\begin{equation}\n  {1 \\over {\\pi R^2}} {{dN} \\over {dy}} \\sim {{Q_{sat}^2} \\over {\\alpha_S}}\n\\end{equation}\nBecause there are both color electric and color magnetic fields, there is a topological charge density of maximal strength induced $FF^D \\sim Q_{sat}^2\/\\alpha_S$  \n\\begin{figure}[!htb]\n\\begin{center}\n  \\mbox{{\\epsfig{figure=glasma.pdf, width=0.70\\textwidth}}}\n   \\end{center}\n\\caption[*]{ \\small  Glasma flux tube produces after the collision.}\n     \\label{glasma}\n\\end{figure}\n\nThe decay of products of the Glasma is what presumably makes a thermalized Quark Gluon Plasma.  It is not clear how this thermalization takes place.  It is quite likely that in the decay of these fields, a turbulent fluid arises, and perhaps this fluid can generate an expansion dynamics similar to that of a thermalized QGP for at least some time\\cite{Dusling:2010rm}.\n\n\\subsection{The CGC and Electron-Hadron Scattering}\n\nIf the only momentum scale that controls high energy scattering is the saturation momentum,\nthen there will be scaling\\cite{Stasto:2000er}.  In particular, the cross section for deep inelastic scattering will be a function\n\\begin{equation}\n\\sigma_{\\gamma^*p} \\sim F(Q^2\/Q_{sat}^2)\n\\end{equation}\nrather than a function of $Q^2$ and $x$ independently.  The x dependence of the saturation momentum may be determined empirically as $Q_{sat}^2 \\sim Q_0^2\/x^\\delta$ where $\\delta = 0.2-0.3$, which is consistent with analysis of evolution equations\\cite{Balitsky:1995ub}-\\cite{Mueller:2002zm}.\nThe scaling relationship can be derived from the classical theory for $Q^2 \\le Q_{sat}^2$.  It can further be shown to extend over a much larger range of $Q^2$\\cite{Iancu:2002tr}.  For large values of $Q^2$ this scaling is a consequence of the linear evolution equations, but the global structure is determined by the physics of saturation.  Such a simple scaling relationship describes deep inelastic scattering data for $ x \\le 10^{-2}$.\n\\begin{figure}[!htb]\n\\begin{center}\n  \\mbox{{\\epsfig{figure=geometric.pdf, width=0.70\\textwidth}}}\n   \\end{center}\n\\caption[*]{ \\small  The geometric scaling in deep inelastic scattering.}\n     \\label{geometric}\n\\end{figure}\n\nUsing evolution equations for the CGC including the effects of running coupling constant\\cite{Albacete:2007yr}-\\cite{Balitsky:2008zza}, one can compute\ndeep inelastic scattering structure functions at small x\\cite{Albacete:2009fh}.  This involves very few parameters, and provides comprehensive description of deep inelastic scattering data at $x \\le 10^{-2}$.\nThe description of $F_2$ in deep inelastic scattering is shown in Fig. \\ref{f2}.  It should be noted that in\nthe CGC description of deep inelastic scattering, the gluon distribution function is the Fock space distribution of gluons inside a hadron.  It can never become negative.  In the description of the $F_2$ data,\nthe gluon distribution function is not becoming small at small $Q^2$ as is the case in some linear evolution fits.  This is intuitively reasonable since we have no reason to expect that the Fock space distribtuion of gluons in a hadron should become small at small $Q^2$.\n\\begin{figure}[!htb]\n\\begin{center}\n  \\mbox{{\\epsfig{figure=f2.pdf, width=0.90\\textwidth}}}\n   \\end{center}\n\\caption[*]{ \\small  The CGC description of F2 data in deep inelastic scattering.}\n     \\label{f2}\n\\end{figure}\n\nThe Color Glass Condensate description may also be applied to diffractive deep inelastic scattering,\nand with the same parameters that describe deep inelastic scattering does an excellent job of describing\nthe data.  In addition, there are measurements of the longitudinal structure function, a quantity directly proportional to the gluon density.  Conventional descriptions that use linear DGLAP evolution equations are somewhat challenged by this data, but the CGC description naturally fits the data.  \n\nTo summarize, the CGC description of deep inelastic scattering at small x naturally describes $F_2$, $F_L$\nand diffractive data. It is a successful phenomenology  Why is the CGC therefore not accepted as the standard description? The problem is that the linear evolution DGLAP descriptions describe $F_2$ adequately, except in the region where the perturbative computations most probably breaks down.  They do not do a very good job on the low $Q^2$ $F_L$ data, but this is where there is a fair uncertainty in the data.\nThe diffractive data is naturally described in the CGC framework, but there are other successful models.\nUltimately, there is no consensus within the deep inelastic scattering community that the CGC is needed in order to describe the data.\n\n\\subsection{The CGC and Heavy Ion Collisions}\n\n\\subsubsection{Multiplcities in RHIC Nulcear Collsions}\n\nOne of the early successes of the CGC was the description of multiplicity distributions in deep inelastic scattering\\cite{Kharzeev:2000ph}-\\cite{Kharzeev:2001yq}.  Recall that the phase space distribution of gluons up to the saturation momentum is of order $Q_{sat}^2\/\\alpha_S(Q_{sat})$.  We will assume that the distribution of\ninitially produced gluons is proportional to this distribution of gluons in the hadron wavefunctions of the colliding nuclei and further,that the multiplicity of produced gluon is proportional to the final\nstate distribution of pions.  We get\n\\begin{equation}\n{1 \\over \\sigma}~  {{dN} \\over {dy}} \\sim  {1 \\over{ \\alpha_S(Q_{sat})}}Q_{sat}^2 \\sim A^{1\/3} x^{-\\delta}\n\\end{equation}  \nHere $\\sigma$ is the area of overlap of the two nuclei in the collision and A the number of nucleons that participate in the collision.  $\\sigma ~Q_{sat}^2 \\sim A$\nat low energies assumes no shadowing of nucleon parton distributions and is consistent with\ninformation concerning deep inelastic scattering on nuclear targets.  In the collisions of nuclei one can directly measure the number of nucleonic participants in the collisions, a number that varies with the centrality of the collision. One can then compare the central region multiplicity with the number of participants so determined.  Such a comparison is shown in\\cite{Adcox:2004mh}-\\cite{Back:2004je} Fig. \\ref{aa} .\nThe saturation description of Kharzeev and Nardi provides a good description of the centrality dependence of the collisions\\cite{Kharzeev:2000ph}.  It also does well with the energy dependence.  Refinements of this description can provide a good description of the rapidity distribution of produced particles\\cite{Kharzeev:2001yq}.\n\\begin{figure}[!htb]\n\\begin{center}\n  \\mbox{{\\epsfig{figure=aa.pdf, width=0.90\\textwidth}}}\n   \\end{center}\n\\caption[*]{ \\small  The multiplicity as a function of the number of nucleon participants in heavy ion collisions.}\n     \\label{aa}\n\\end{figure}\n\n\\subsubsection{Limiting Fragmentation}\n\nA general feature of high energy hadronic scattering is limiting fragmentation.  If one measures the distribution of particles as a function of rapidity up to some fixed rapidity from the rapidity of one of the colliding particles, then the distribution is independent of collision energy.  The region over which this scaling occurs\nincreases as the energy of the colliding particles increases.  Such scaling is shown in Fig. \\ref{limfrag}.\n\\begin{figure}[!htb]\n\\begin{center}\n  \\mbox{{\\epsfig{figure=limfrag.pdf, width=0.90\\textwidth}}}\n   \\end{center}\n\\caption[*]{ \\small  Limiting fragmentation in RHIC nuclear  collisions.}\n     \\label{limfrag}\n\\end{figure}\nSuch limiting fragmentation is natural in the CGC approach.  For example in Fig. \\ref{limfrag},\nwe see that the region of limiting fragmentation increases as beam energy increases\\cite{Back:2004je}.  If we think of the region where there is limiting fragmentation as sources for fields at small more central rapidities, then we see that going to higher energies corresponds to treating a larger region as sources.  In a renormalization group language, this simply means that one is integrating out fluctuations at less central rapidities, to generate an effective theory for the particles at more central rapidity.  A quantitative description of limiting fragmentation within the theory of the  CGC is found in Ref.   \\cite{Gelis:2006tb}.\n\n\\subsubsection{Single Particle Distributions in dAu Collisions}\n\nSome of the early predictions of the CGC were generic features of the single particle inclusive distributions\nseen in hadron-nucleus collisions.  There are two competing effects.  The first is multiple scattering of a hadron as it traverses a nucleus.  This effect is included n the CGC gluon distributions as an enhancment\nof the gluon distribution for  $p_T$ at transverse momentum of the order of the saturation momentum,\nwith a corresponding depletion at smaller momentum.  There is little effect at high $p_T$.  The other effect is that in the evolution of the gluon distribution to small $x$, the saturation momentum acts as a cutoff in the\nbremstrahlung like integrals that generate such small x gluons.  Nuclei have a larger saturation momentum\nthan do hadrons, so the small x gluon distribution for nuclei will be suppressed relative to that for hadrons.  Put another way, this effect will generate a suppression for more central collisions.  The sum of these effects is shown in Fig. \\ref{dA}\\cite{Baier:2003hr}-\\cite{Iancu:2004bx}. \nThe different curves correspond to different rapidities of the produced particle,\nbeginning with the top curve being near the fragmentation region of the nucleus.  As one evolve further in rapidity, the enhancement at intermediate transverse momentum disappears and is replaced by a smooth curve with an overall suppression of produced particles.\n\\begin{figure}[!htb]\n\\begin{center}\n  \\mbox{{\\epsfig{figure=dA.pdf, width=0.90\\textwidth}}}\n   \\end{center}\n\\caption[*]{ \\small  The ratio of particles emitted in dA  and AA collisions to that in proton due to CGC effects.}\n     \\label{dA}\n\\end{figure}\n\nThe pattern of suppression suggested by the Color Glass Condensate was first seen in dAu collisions\nin the Brahms collaboration\\cite{Arsene:2004fa}, and later confirmed by the other experiments \\cite{Adcox:2004mh},\\cite{Back:2004je},\\cite{Adams:2005dq}.  The Brahms experiments demonstrated\nthat in the nuclear target fragmentation region that at intermediate $p_T$ there was en enhancement in $R_{dA}$\nas a function of centrality, but in the deuteron fragmentation region, there was a depletion as a function of centrality.  The CGC provided the only model that predicted such an effect, and it remains the only \ntheory that can quantitatively explain the suppression seen in the deuteron fragmentation region.\n\n\\subsubsection{Heavy Quark and $J\\Psi$ Production}\n\nIf the saturation momentum is small compared to a quark mass, it can be treated as very heavy.  It should have perturbative incoherent production cross sections.  If the saturation momentum is large compared to a quark\nmass, the quarks should be thought of as light mass.  Cross sections for production should be coherent,\nand for example in $pA$ collisions, scale as $A^{2\/3}$.   In the deuteron fragmentation region of dAu collisions we would expect suppression of heavy quark and charmonium cross sections relative to the nuclear fragmentation region. \n\\begin{figure}[!htb]\n\\begin{center}\n  \\mbox{{\\epsfig{figure=jpsi.pdf, width=0.90\\textwidth}}}\n   \\end{center}\n\\caption[*]{ \\small  The $J\/\\Psi$ production cross section as a function of centrality and rapidity.}\n     \\label{jpsi}\n\\end{figure}\n In Fig. \\ref{jpsi}, the ratio of central to peripheral cross sections for $J\/\\Psi$ production is shown as a function of centrality and rapidity.  Note the strong suppression in the forward region for central collisions, as expected from the CGC.  Precise computations are difficult for the charm quark since its mass is close to the saturation momentum.  Such computations are in agreement with the data at forward rapidity\\cite{Kharzeev:2008nw}-\\cite{Kopeliovich:2010nw}.\n \n \\subsubsection{Two Particle Correlations}\n \n The Glasma flux tubes induced by the collision of two hadrons will generate long range correlations in rapidity.  In heavy ion collisions, this may be seen in forward backward correlations, as measured in STAR.  The correlation increases in strength with higher energy collisions or more central collisions.  This is expected in the CGC-Glasma description because for more central collisions the saturation momentum\n is bigger, so that the system is more correlated.  (The coupling becoming weaker means the system is more\n classical, and therefore the leading order contribution associated with Glasma flux tubes becomes\n relatively more important.)  Such forward-backward correlations are shown in Fig. \\ref{fb} as a function of rapidity and centrality\\cite{:2009dqa}-\\cite{Lappi:2009vb}.  The value of the correlation coefficient b\n can be shown to be bounded $b \\le 1\/2$.\n \\begin{figure}[!htb]\n\\begin{center}\n  \\mbox{{\\epsfig{figure=fb.pdf, width=0.90\\textwidth}}}\n   \\end{center}\n\\caption[*]{ \\small  The strength of forward backward correlations as a function of rapidity and centrality.}\n     \\label{fb}\n\\end{figure}\n\nSuch two particle correlations in the Glasma can generate ridge like structures seen in two particle correlation experiments in azimuthal angle and rapidity\\cite{Shuryak:2007fu}-\\cite{Dumitru:2008wn}.  The long range rapidity correlation is intrinsic to the Glasma.  The angular correlation might be generated by flow effects at later times in the collision, by opacity and trigger bias effects, or by\nintrinsic angular correlations associated with the decay of Glasma flux tubes\\cite{Gavin:2008ev}-\\cite{Dumitru:2010iy}.\n\n\\subsubsection{The Negative Binomial Distribution}\n\nThe decay of a single Glasma flux tube generates a negative binomial distribution of produced particles\\cite{Gelis:2009wh}.\nA sum of negative binomial distributions is again a negative binomial distribution.  Such oa form of the distribution describes the RHIC data well.  It is difficult with the heavy ion data to isolate those effects due to an intrinsic negative binomial distribution and those due to impact parameter.  It is possible to isolate the effects of impact parameter, but it demands a high statistics study.  \n\n\\subsubsection{Two Particle Azimuthal Angular Correlations in dA Collisions}\n\nThe CGC will de-correlate forward-backward angular correlations when the the transverse momentum of produced particles is of order the saturation momentum\\cite{Kharzeev:2004bw}-\\cite{Albacete:2010rh}.\nThis is because near the produced particles get momentum from the CGC and\ntherefore are not back-to-back correlated.  In dAu collisions such an effect will be largest at forward rapidities near the fragmentation region of the deuteron, since this corresponds to the smallest values of x for the nuclear target.  This kinematic region is least affected by multiple scattering on the nucleus.  This effect has been seen by the STAR and PHENIX collaborations\\cite{Braidot:2010ig}-\\cite{Meredith:2009fp}.  There is a good quantitative description by Tuchun and by Albacete and Marquet,\\cite{Tuchin:2009nf}-\\cite{Albacete:2010rh}  as shown in the figure \\ref{dafb}\n \\begin{figure}[!htb]\n\\begin{center}\n  \\mbox{{\\epsfig{figure=dafb.pdf, width=0.90\\textwidth}}}\n   \\end{center}\n\\caption[*]{ \\small Forward rapidity, forward backward angular correlations in dAu collisions\nas a function of centrality.}\n     \\label{dafb}\n\\end{figure}\n\n\\subsection{Concluding Comments on the CGC and the Glasma}\n\nThere is now a wide variety or experimental data largely consistent with the CGC and Glasma based description.  There is a well developed theoretical framework that provides a robust phenomenology\nof both electro-hadron scattering and hadron scattering,  There are new areas that are developing that I have not had time to discuss.  One is the possibility to see effects of topological charge change in heavy ion collisions, the Chiral Magnetic Effect\\cite{Kharzeev:2007jp}.  Another area is pp collisions at the LHC, where some work concerning recent experimental data was developed at this school\\cite{McLerran:2010ex}.  \n\n\n\\section{Lecture II: Matter at High Temperature: The Quark Gluon Plasma}\n\n\\section{Matter at Finite Temperature}\n\n\\subsection{Introdcution}\n\nIn this lecture I will describe the properties of matter at high temperature.  The discussion here will be theoretical.  There is a wide literature on the phenomenology of the Quark Gluon Plasma and its possible\ndescription of heavy ion collisions at RHIC energies.  The interested reader is referred to that literature.\nI will here develop the ideas of decofinement, chiral symmetry restoration based in part on a simple description using the large number of colors limit of QCD.\n\n\\subsection{Confinement}\n\nThe partition function is\n\\begin{equation}\n Z = Tr~e^{-\\beta H + \\beta \\mu_B N_B}\n\\end{equation}\nwhere the temperature is $T = 1\/\\beta$ and $N_B$ is the baryon number and $\\mu_B$ is the baryon number chemical potential.  Operator expectation values are\n\\begin{equation}\n <O> = {{Tr~ O ~e^{-\\beta H + \\beta \\mu_B N_B}} \\over Z}\n\\end{equation}\n Under the substitution $e^{-\\beta H} \\rightarrow e^{-itH}$, the partition function becomes the time\n evolution operator of QCD.  Therefore, if we change $t \\rightarrow it$,and redefine zeroth\n components of fields by  appropriate factors of i, and introduce Euclidean gamma matrices with anti-commutation relations\n \\begin{equation}\n   \\{ \\gamma^\\mu, \\gamma^\\nu \\} = -2 \\delta^{\\mu \\nu}\n \\end{equation}\n  then for QCD, the partition function has the path integral representation \n  \\begin{equation}\n  Z = \\int~[dA] [d\\overline \\psi ] [d\\psi] exp\\left\\{ -\\int_0^\\beta~ d^4x~\\left( {1 \\over 4 }F^2 \n  +\\overline \\psi \\left[ {1 \\over i} \\gamma \\cdot D + m+ i \\mu_Q \\gamma^0 \\psi \\right]\\right) \\right\\}\n  \\end{equation}\nHere the fermion field is a quark field so that the baryon number chemical potential is\n\\begin{equation}\n  \\mu_Q = {1 \\over N_c} \\mu_B\n\\end{equation}\nThis path integral is in Euclidean space and is computable using Monte Carlo methods when\nthe quark chemical potential vanishes.  If the quark chemical potential is non-zero, various contributions appear with different sign, and the Monte Carlo integrations are poorly convergent.  Boundary conditions\non the fields must be specified on account of the finite length of the integration in time.  They are periodic for Bosons and anti-periodic for Fermions, and follow from the trace in the definition of the partition function.\n\nA straightforward way to probe the confining properties of the QCD matter is to introduce a heavy\ntest quark.  If the free energy of the heavy test quark is infinite, then there is confinement,\nand if it is finite there is deconfinement.  We shall see below that the free energy of an quark added to the system is\n\\begin{equation}\n  e^{-\\beta F_q} = <L>\n  \\label{L}\n\\end{equation}\nwhere\n \\begin{equation}\n    L(\\vec{x}) = {1 \\over N_c} Tr~ P~ e^{i \\int ~dt~ A^0(\\vec{x},t)}\n \\end{equation}\nSo confinement means $<L> = 0$ and deconfinement means that $<L>$ is finite.  The path ordered phase integration which defines the line operator $L$ is shown in Fig. \\ref{line}.  Such a path ordered phase  is called a Polyakov loop or Wilson line.\n\\begin{figure}[ht]\n        \\begin{center}\n        \\includegraphics[width=0.50\\textwidth]{line.jpg}\n        \\end{center}\n        \\caption{The contour in the t plane which defines the Polyakov loop.  The space is closed in time because of the periodic boundary conditions imposed by the definition of the partition function.}\n\\label{line}\n\\end{figure}\n\nIt is possible to prove that the free energy of a heavy static quark added to the system is given by Eqn. \\ref{L}\nusing the effective action for a very heavy quark:\n\\begin{equation}\n S_{HQ} = \\int ~dt ~\\overline \\psi (\\vec{x},t) ~{1 \\over i}\\gamma^0 D^0~ \n\\psi (\\vec{x},t).\n\\end{equation}\nThe Yang-Mills action is invariant under gauge transformations that are periodic up to an element of the center of the gauge group.  The center of the gauge group is a set of diagonal matrices matrix $Z_p = e^{2\\pi i p\/N} \\overline I$ where $\\overline I$ is an identity matrix.\nThe quark contribution to the action is not invariant, and  $L \\rightarrow Z_p L$ under this transformation.  In a theory with only dynamical gluons, the energy of a system of $n$ quarks minus antiquarks is invariant under the center symmetry transformation only if $n$ is an integer multiple of $N$.  Therefore, when the center symmetry is realized, the only states of finite free energy\nare baryons plus color singlet mesons.\n\nThe realization of the center symmetry, $L \\rightarrow Z_p L$\nis equivalent to confinement.  This symmetry is like the global rotational symmetry of a spin system, and it may be either realized or broken.   At large separations, the correlation of a line and its adjoint, corresponding to a quark-antiquark pair is\n\\begin{equation}\n        lim_{r \\rightarrow \\infty} <L(r) L^\\dagger (0)> = Ce^{-\\kappa r} + <L(0)><L^\\dagger (0)>\n\\end{equation}        \nsince upon subtracting a mean field term, correlation functions should vanish exponentially.  Since\n\\begin{equation}\n   e^{-\\beta F_{q \\overline q} (r)} = <L(r) L^\\dagger (0)>\n\\end{equation}\nwe see that in the confined phase, where $<L> = 0$, the potential is linear, but in the unconfined phase,\nwhere $<L>$ is non-zero, the potential goes to a constant at large separations.\n\nThe analogy with a spin system is useful.  For the spin system corresponding to QCD\nwithout dynamical quarks,\nthe partition function  can be written as\n\\begin{equation}\n Z = \\int~ [dA]~e^{- {1\\over g^2} S[A]}\n\\end{equation}\nThe effective temperature of the spin system associated with the gluon fields is $T_{eff} \\sim g^2$.\nBy asymptotic freedom of the strong interactions, as real temperature gets larger, the effective temperature gets smaller.  So at large real temperature (small effective temperature) we expect an ordered system, where the $Z_N$ symmetry is broken, and there is deconfinement.  For small real temperature corresponding to large effective temperature, there is disorder or confinement.\n\nThe presence of dynamical fermions breaks the $Z_N$ symmetry.  This is analogous to placing a spin system in an external magnetic field.  There is no longer any symmetry associated with confinement, and the phase transition can disappear.  This is what is believed to happen in QCD for physical masses of quarks.  What was a first order phase transition for the theory in the absence of quarks becomes  a continuous change in the properties of the matter for the theory with quarks.\n\nAnother way to think about the confinement-decofinement transition is a change in the number of degrees of freedom.  At low temperatures, there are light meson degrees of freedom.  Since these\nare confined, the number of degrees of freedom is of order one in the number of colors.  In the unconfined world, there are $2(N_c^2-1)$ gluons, and $4N_cN_f$ fermions where $N_f$ is the number of light mass fermion families.  The energy density scaled by $T^4$ is a dimensionless number and directly proportional to the number of degrees of freedom.  We expect it to have the property shown in Fig. \\ref{et4} for pure QCD in the absence of quarks.  The discontinuity at the deconfinement temperature, $T_d$ is the latent heat of the phase transition.\n\\begin{figure}[ht]\n        \\begin{center}\n        \\includegraphics[width=0.60\\textwidth]{et4.jpg}\n        \\end{center}\n        \\caption{The energy density scaled by $T^4$ for QCD in the absence of dynamical quarks.}\n\\label{et4}\n\\end{figure}\n\nThe energy density can be computed using lattice Monte Carlo methods.  The result of such computation is shown in Fig. \\ref{et4lat}.  The discontinuity present for the theory with no quarks becomes a rapid cross over when dynamical quarks are present.\n\nThe large $N_c$ limit gives some insight into the properties of high temperature \nmatter\\cite{'tHooft:1973jz}-\\cite{Thorn:1980iv}. As $N_c \\rightarrow \\infty$, the energy density itself is an order parameter for the decofinement phase transition.  Viewed from the hadronic world, there is an amount of energy density $\\sim N_c^2$ which must be inserted \nto surpass the transition temperature.  At infinite $N_c$ this cannot happen, as this involves an infinite amount of energy.  There is a Hagedorn limiting temperature, which for finite $N_c$ would have been the deconfinement temperature.\n\\begin{figure}[ht]\n        \\begin{center}\n        \\includegraphics[width=0.60\\textwidth]{et4lat.jpg}\n        \\end{center}\n        \\caption{The energy density scaled by $T^4$ measured in QCD from lattice Monte Carlo simulation.  Here there are quarks with realistic masses.}\n\\label{et4lat}\n\\end{figure}\n\nThe Hagedorn limiting temperature can be understood from the viewpoint of the hadronic world as arising from an exponentially growing density of states.  In a few paragraphs, we will argue that mesons and glueballs are very weakly interacting in the limit of large $N_c$.  Therefore, the partition function is\n\\begin{equation}\n  Z = \\int~ dm~\\rho (m) e^{-m\/T}\n \\end{equation}\n Taking $\\rho(m) \\sim m^\\alpha e^{\\kappa m}$, so that \n \\begin{equation}\n   <m> \\sim {1 \\over {1\/T-\\kappa}}\n \\end{equation} \ndiverges when $T \\rightarrow 1\/\\kappa$\n\n\\subsection{A Brief Review of the Large $N_c$ Limit}\n\nThe large $N_c$ limit for an interacting theory takes $N_c \\rightarrow \\infty $ with the 't Hooft coupling\n$g^2_{'t Hooft} = g^2 N_c$ finite.  This approximation has the advantage that the interactions among quarks and gluons simplify.  For example, at finite temperature, the disappearance of confinement\nis associated with Debye screening by gluon loops, as shown in Fig. \\ref{loop}a.  This diagram generates a screening mass of order $M^2_{screening} \\sim g^2_{'t Hooft} T^2$.   On the other hand the quark loop contribution is smaller by a power of $N_c$ and vanishes in the large $N_c$ limit.\n\\begin{figure}[htbp]\n\\begin{center}\n\\begin{tabular} {l l l}\n\\includegraphics[width=0.50\\textwidth] {glueloop.jpg}  & &\n\\includegraphics[width=0.46\\textwidth] {quarkloop.jpg} \\\\\n& & \\\\\na & & b \\\\\n\\end{tabular}\n\\end{center}\n\\caption{a:  The gluon loop contribution to the heavy quark potential.  b:  The quark loop contribution to the potential}\n\\label{loop}\n\\end{figure}\n\nTo understand interactions, consider Fig. \\ref{int}a.  This corresponds to a mesonic current-current interaction through quarks.  In powers of $N_c$, it is of order $N_c$.  Gluon interactions will not change this overall factor.  The three current interaction is also of order $N_c$ as shown in Fig. \\ref{int}b.  The three meson vertex, $G$ which remains after amputating the external lines, is therefore of order $1\/\\sqrt{N_c}$.  A similar argument shows that the four meson interaction is of order $1\/N_c$.\nUsing the same arguments, one can show that the 3 glueball vertex is of order $1\/N_c$ and the four glueball interaction of order\n$1\/N_c^2$.\n\nThese arguments show that QCD at large $N_c$ becomes a theory of non-interacting mesons and glueballs.  There are an infinite number of such states because excitations can never decay.  In fact, the spectrum of mesons seen in nature does look to a fair approximation like non-interacting particles.  Widths of resonances are typically of order $200~ MeV$, for resonances with masses up to several $GeV$.\n\\begin{figure}[htbp]\n\\begin{center}\n\na  \\includegraphics[width=0.75\\textwidth ] {loop1.jpg}  \\\\\n ~ \\\\\n~~b ~~~ \\includegraphics[width=0.75\\textwidth ] {loop2.jpg} \\\\\n\n\\end{center}\n\\caption{a:  The quark loop corresponding to a current-current interaction.  b: A quark loop corresponding to a three current interaction.}\n\\label{int}\n\\end{figure}\n\\subsection{Mass Generation and Chiral Symmetry Breaking}\n\nQCD in the limit of zero quark masses has a $U(1) \\times SU_L(2) \\times SU_R(2)$ symmetry.  (The $U_5(1)$ symmetry is explicitly broken due to the axial anomaly.)  Since the pion field, $\\overline \\psi \\tau^a \\gamma_5 \\psi$ is generated by an $SU_{L-R}(2)$ transformation of the sigma field, $\\overline \\psi \\psi$, the energy (or potential) in the space of the pion-sigma field is degenerate under this transformation.\nIn nature, pions have anomalously low masses.  This is believed to be a consequence of chiral symmetry breaking, where the $\\sigma $ field acquires an expectation value, and the pion fields are Goldstone bosons associated with the degeneracy of the potential under the chiral rotations.\n\nSuch symmetry breaking can occur if the energy of a particle-antiparticle pair is less than zero, as shown in Fig. \\ref{hole}.  On the left of this figure is the naive vacuum where the negative energy states associated with quark are filled.  The right hand side of the figure corresponds to a particle hole excitation, corresponding to a sigma meson.  Remember that a hole in the negative energy sea corresponds to an antiparticle with the opposite momentum and energy. If the $\\sigma$ meson excitation has negative energy, the system is unstable with respect to forming a condensate of these mesons.\n\\begin{figure}[ht]\n        \\begin{center}\n        \\includegraphics[width=0.40\\textwidth]{hole.jpg}\n        \\end{center}\n        \\caption{The energy levels of the Dirac equation.  Unfilled states are open circles and filled states are solid circles.  For the free Dirac equation, negative energy states are filled and positive energy states are unoccupied, as shown on the left hand side.  A mesonic excitation corresponding to a particle hole pair is shown on the right hand side.}\n\\label{hole}\n\\end{figure}\n\nAt sufficiently high temperature, the chiral condensate might melt.  Indeed this occurs\\cite{Karsch:2001cy} .For QCD,\nthe chiral and deconfinement phase transition occur at the same temperature.  At a temperature of about $170 - 200~ MeV$, both the linear potential disappears and chiral symmetry is restored.  It is difficult to make a precise statement about the indentification of the chiral and deconfinement phase transitions,\nsince as argued above, for QCD with quarks, there is not a real phase transition associated with deconfinement\\cite{Bazavov:2009zn}-\\cite{Borsanyi:2010cj}.\nAlso, when quarks have finite masses, as they do in nature, chiral symmetry is not an exact symmetry, and there need be no strict phase transition associated with its restoration.  Nevertheless, the cross over is quite rapid, and there are rapid changes in the both the potential and the sigma condensate $<\\overline \\psi \\psi >$ at temperatures which are in a narrow range.\n\n\\section{Lecture III: Matter at High Baryon Number Density: Quarkyonic Matter}\n\nI now turn to a discussion of the phase diagram of QCD at finite baryon number density.\n\nIn the large $N_c$ limit of QCD, the nucleon mass is of order $N_c$\\cite{'tHooft:1973jz}-\\cite{Witten:1979kh}.  This means that in the confined phase of hadronic matter, for baryon chemical potential $\\mu_B \\le M_N$, the baryon number density is\nessentially zero:\n\\begin{equation}\n   <N_B> \\sim e^{(\\mu_B-M_N)\/T} \\sim e^{-N_c}\n\\end{equation}   \nFor temperatures above the de-confinement phase transition the baryon number is non-zero since there the baryon number density is controlled by $e^{-M_q\/T} \\sim 1$, and quark masses are independent of\n$N_c$.  For sufficiently large chemical potential the baryon number density can be nonzero also. The Hadronic Matter phase of QCD is characterized in large $N_c$ by zero baryon number density, but at higher density there is a new phase.\n \\begin{figure}\n\\centering\n\\includegraphics[scale=0.30]{phasediagram}\t       \n\\caption[]{The revised phase diagram of QCD}\n\\label{phasediagram}\n\\end{figure}\n\nIn the large $N_c$ limit, fermion loops are suppressed by a factor of $1\/N_c$.  Therefore the contribution to Debye screening from quarks cannot affect the quark potential until\n\\begin{equation}\n  M_{Debye}^2 \\sim \\alpha_{t'Hooft}~ \\mu_{quark}^2\/N_c \\sim \\Lambda^2_{QCD}\n\\end{equation}\nHere the quark chemical potential is $\\mu_B = N_c \\mu_{quark}$.  The relationship involving the Debye mass means there is a region parametrically large  chemical potential $M_N \\le \\mu_B \\le \\sqrt{N_c}M_N$ where matter is confined, and has finite baryon number.  This matter is different than either the Hadronic Matter or the De-Confined Phases.  It is called Quarkyonic because it exists at densities parametrically large compared to the QCD scale, where quark degrees of freedom are important,\nbut it is also confined so the degrees of freedom may be thought of also as those of confined \nbaryons\\cite{McLerran:2007qj}-\\cite{Hidaka:2008yy}.\n\nThe width of the transition region between the Hadronic phase and the Quarkyonic phase is estimated\nby requiring that the baryon number density become of order $N_B\/V \\sim k_{Fermi}^3 \\sim \\Lambda_{QCD}^3$.  Recall that the baryon chemical potential is $\\mu_B \\sim M_N + k_f^2\/2M_N$ for small $k_F$, so that the width of the transition in $\\mu_B$ is very narrow, of order $1\/N_c$. This is $\\delta \\mu_{qaurk} \\sim  1\/N_c^2$ when expressed in terms of $\\mu_{quark}$ which is the finite variable in the large $N_c$ limit.\n \\begin{figure}\n\\centering\n\\includegraphics[scale=0.60]{line_pbm}\t       \n\\caption[]{Chemical potentials and temperatures at decoupling.}\n\\label{line}\n\\end{figure}\n\n\nThe transition from Hadronic Matter to that of the Quark Gluon Plasma may be thought of as a change in the number of degrees of freedom of matter.  Hadronic Matter at low temperatures has  3 pion degrees of freedom.  The quark gluon plasma has of order $2(N_c^2-1)$ degrees of freedom corresponding to gluons and $4 N_c$ degrees of freedom for each light mass quark.  The change in degrees of freedom is of order $N_c^2$ in the large $N_c$ limit.  At very high baryon number densities, the quarks in the Fermi sea interact at short distances, and although strictly speaking are confined, behave like free quarks.  The number of degrees of freedom is therefore of order $N_c$.  Each phase has different numbers of degrees of freedom, and is presumably separated from the other by a rapid crossover.\n \\begin{figure}\n\\centering\n\\includegraphics[scale=0.60]{horn}\t       \n\\caption[]{Ratios of abundances of various particles .}\n\\label{horn}\n\\end{figure}\nQuarkyonic matter is confined and therefore thermal excitations such as mesons, glueballs, and Fermi surface excitations must be thought of as confined.  The quarks in the Fermi sea are effectively weakly interacting since their interactions take place at short distances.  So in some sense, the matter is ``de-confined\" quarks in the Fermi sea with confined glueball, mesons and Fermi surface excitations\\cite{Castorina:2010gy}.\n\nIn Hadronic Matter, chiral symmetry is broken and in Deconfined Matter it is broken.  In Quarkyonic Matter chiral symmetry is broken by the formation of charge density waves from binding of quark and quark hole excitations near the Fermi surface\\cite{Deryagin:1992rw}.  In order that the quark hole have small relative momentum to the quark, the quark hole must have momentum opposite to that of the quark.  This means the quark-quark hole excitation has total net momentum, and therefore the finite wavelength of the corresponding bound state leads to a breaking of translational invariance.  The chiral condensate turns out to be a chiral spiral where the chiral condensate rotates between different Goldstone boson\nas one moves through the condensate\\cite{Kojo:2009ha}.  Such condensation may lead to novel crystalline structures\\cite{tsvelik}.\n\nA figure of the hypothetical phase diagram of QCD is shown in Fig. \\ref{phasediagram} for $N_c = 3$.  Also shown is the weak liquid-gas phase transition, and the phase associated with color superconductivity.  Although the color superconducting phase cannot coexist with quarkyonic matter in infinite $N_c$, for finite $N_c$ there is such possibility.  The lines on this phase diagram might correspond to true phase transitions or rapid cross overs.  The confinement-deconfinement transition is known to be a cross over.  In the FPP-NJL model\\cite{Fukushima:2003fw}-\\cite{Pisarski:2000eq}, the Hadronic-Quarkyonic transition is first order\\cite{McLerran:2008ua},  but nothing is known from lattice computations.  If as we conjecture, there is region where chiral symmetry is broken by translationally non-invariant modes, then this region must be surrounded by a line of phase transitions.  I call this region Happy Island becuase it is an island of matter in the $\\mu_B-T$ plane.\n\nA remarkable feature of this plot is the triple point where the Hadronic Matter, Deconfined Matter and\nQuarkyonic Matter all meet\\cite{Andronic:2009gj}.  This triple point is reminiscent of the triple point for the liquid, gas and vapor phases of water. \n\nSince we expect a rapid change in the number of degrees of freedom across the transitions between\nthese forms of matter, an expanding system crossing such a transition will undergo much dilution would undergo much dilution at a fixed value of temperature or baryon chemical potential\\cite{BraunMunzinger:1994xr}-\\cite{Heinz:1999kb}.  One might expect in heavy ions to see decoupling of particle number changing processes at this transition, and the abundances of produced particles will be characteristic of the transition.  In Fig. \\ref{line}\n\nIn the Fig. \\ref{line}, the expectations for the confinement-deconfinement transition are shown with the dotted red line.  It is roughly constant with the baryon chemical potential, and the constant value of temperature is taken from lattice estimates.  The dark dashed curve represents $\\mu_B -T = cons \\times M_N$, corresponding to a simple model for the Quarrkyonic transition.  Such a very simple description does remarkably well.  \n\nA triple point is suggested at a baryon chemical potential near 400 MeV, and temperature near 160 MeV.  This corresponds to a center of mass energy for Pb-Pb collisions of 9-10 GeV.  This is near where there are anomalies in the abundances of rations of particles\\cite{Gazdzicki:1998vd}, as shown in Fig. \\ref{horn}.\nShown are fits using statistical models of abundances of particles using chemical potentials and temperature extracted from experimental data.  The sharp peak reflects the change in behavior as one proceeds along the dashed line of Fig. \\ref{line} corresponding to the Quarkyonic transition and joins to the dotted red line \nof the deconfinement transition\n\nIt is remarkable that the value of beam energy where this occurs corresponds to the hypothetical triple point of Fig. \\ref{line}, and that this is the density where the energy density stored in baryons becomes equal to that stored in mesons, Fig. \\ref{baryons},\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.60]{baryons}\t       \n\\caption[]{Energy density stored in baryons compared to that stored in mesons.}\n\\label{baryons}\n\\end{figure}\n\n\n\n\n\n\n\n\n\\section*{Acknowledgments}\n\nI gratefully acknowledge the organizers of the 50'th Crakow School of Theoretical Physics, in particular,\nMichal Praszalowicz, for making this wonderful and extraordinary meeting.\nThe research of  L. McLerran is supported under DOE Contract No. DE-AC02-98CH10886.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}