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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{moment map} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{symplectic_geometry}{}\paragraph*{{Symplectic geometry}}\label{symplectic_geometry} [[!include symplectic geometry - contents]] \hypertarget{the_moment_map}{}\section*{{The moment map}}\label{the_moment_map} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{Preliminaries}{Preliminaries}\dotfill \pageref*{Preliminaries} \linebreak \noindent\hyperlink{HamiltonianActionAndTheMomentMap}{Hamiltonian action and moment map}\dotfill \pageref*{HamiltonianActionAndTheMomentMap} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{AngularMomentum}{Angular moment}\dotfill \pageref*{AngularMomentum} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{relation_to_conserved_quantities}{Relation to conserved quantities}\dotfill \pageref*{relation_to_conserved_quantities} \linebreak \noindent\hyperlink{RelationToConstrainedMechanics}{Relation to constrained mechanics}\dotfill \pageref*{RelationToConstrainedMechanics} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak \noindent\hyperlink{relation_to_symplectic_reduction}{Relation to symplectic reduction}\dotfill \pageref*{relation_to_symplectic_reduction} \linebreak \noindent\hyperlink{relation_to_equivariant_cohomology}{Relation to equivariant cohomology}\dotfill \pageref*{relation_to_equivariant_cohomology} \linebreak \noindent\hyperlink{generalization_groupvalued_moment_maps}{Generalization: group-valued moment maps}\dotfill \pageref*{generalization_groupvalued_moment_maps} \linebreak \noindent\hyperlink{ReferencesInThermodynamics}{In thermodynamics}\dotfill \pageref*{ReferencesInThermodynamics} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} A \emph{moment map} is a dual incarnation of a [[Hamiltonian action]] of a [[Lie group]] (or [[Lie algebra]]) on a [[symplectic manifold]]. An [[action]] of a [[Lie group]] $G$ on a [[symplectic manifold]] $X$ by ([[Hamiltonian symplectomorphisms|Hamiltonian]]) [[symplectomorphisms]] corresponds [[infinitesimal space|infinitesimally]] to a Lie algebra [[homomorphism]] from the [[Lie algebra]] $\mathfrak{g}$ to the [[Hamiltonian vector fields]] on $X$. If this lifts to a coherent choice of [[Hamiltonians]], hence to a Lie algebra homomorphism $\mathfrak{g} \to (C^\infty(X), \{-,-\})$ to the [[Poisson bracket]], then, by [[dual vector space|dualization]], this is equivalently a [[Poisson manifold]] homomorphism of the form \begin{displaymath} \mu : X \to \mathfrak{g}^* \,. \end{displaymath} This $\mu$ is called the \emph{moment map} or \emph{momentum map} of the Hamiltonian action. The name derives from the special and historically first case of [[angular momentum]] in the [[rigid body dynamics|dynamics of rigid bodies]], see \emph{\hyperlink{AngularMomentum}{Examples - Angular momentum}} below. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} The \hyperlink{Preliminaries}{Preliminaries} below review some basics of [[Hamiltonian vector fields]]. The definition of the moment map itself is below in \emph{\hyperlink{HamiltonianActionAndTheMomentMap}{Hamiltonian action and the moment map}}. \hypertarget{Preliminaries}{}\subsubsection*{{Preliminaries}}\label{Preliminaries} This section briefly reviews the notion of [[Hamiltonian vector fields]] on a [[symplectic manifold]] The basic setup is the following: Let $(M,\omega)$ be a [[symplectic manifold]] with a [[Hamiltonian action]] of a [[Lie group]] $G$. In particular that means that there is an [[action]] $\nu\colon G \times M \to M$ via [[symplectomorphism]]s (diffeomorphisms $\nu_g$ such that $\nu_g^*(\omega) = \omega$). A vector field $X$ is [[symplectic vector field|symplectic]] if the corresponding flow preserves (again by pullbacks) $\omega$. The symplectic vector fields form a Lie subalgebra $\chi(M,\omega)$ of the Lie algebra of all smooth vector fields $\chi(M)$ on $M$ with respect to the Lie bracket. By the [[Cartan homotopy formula]] and closedness $d \omega = 0$ \begin{displaymath} \mathcal{L}_X \omega = d \iota_X \omega \end{displaymath} where $\mathcal{L}_X$ denotes the [[Lie derivative]]. Therefore a vector field $X$ is symplectic iff $\iota(X)\omega = d H$ for some function $H\in C^\infty(M)$, usually called Hamiltonian (function) for $X$. Here $X$ is determined by $H$ up to a locally constant function. Such $X = X_H$ is called the \textbf{Hamiltonian vector field} corresponding to $H$. The Poisson structure on $M$ is the bracket $\{,\}$ on functions may be given by \begin{displaymath} \{ f, g\} := [X_f,X_g] \end{displaymath} where there is a Lie bracket of vector fields on the right hand side. For $(M,\omega)$ a connected symplectic manifold, there is an exact sequence of [[Lie algebras]] \begin{displaymath} 0 \to \mathbf{R}\to (C^\infty(M), \{-,-\}) \to \chi(M,\omega) \to 0 \,. \end{displaymath} See at \emph{\href{Hamiltonian+vector+field#RelationToFunctions}{Hamiltonian vector field -- Relation to Poisson bracket}}. \hypertarget{HamiltonianActionAndTheMomentMap}{}\subsubsection*{{Hamiltonian action and moment map}}\label{HamiltonianActionAndTheMomentMap} Let $(X, \omega)$ be a [[symplectic manifold]] and let $\mathfrak{g}$ be a [[Lie algebra]]. Write $(C^\infty(X), \{-,-\})$ for the [[Poisson bracket]] Lie algebra underlying the corresponding [[Poisson algebra]]. \begin{defn} \label{MomentMap}\hypertarget{MomentMap}{} A \emph{[[Hamiltonian action]]} of $\mathfrak{g}$ on $(X, \omega)$ is a [[Lie algebra]] [[homomorphism]] \begin{displaymath} \tilde \mu \;\colon\; \mathfrak{g} \longrightarrow (C^\infty(X), \{-,-\}) \,. \end{displaymath} The corresponding function \begin{displaymath} \mu \;\colon\; X \longrightarrow \mathfrak{g}^* \end{displaymath} to the [[dual vector space]] of $\mathfrak{g}$, defined by \begin{displaymath} \mu \;\colon\; x \mapsto \tilde \mu(-)(x) \end{displaymath} is the corresponding \textbf{moment map}. \end{defn} \begin{remark} \label{Notation}\hypertarget{Notation}{} If one writes the evaluation pairing as \begin{displaymath} \langle -,-\rangle : \mathfrak{g}^* \otimes \mathfrak{g} \to \mathbb{R} \end{displaymath} then the equation characterizing $\mu$ in def. \ref{MomentMap} reads for all $x \in X$ and $v \in \mathfrak{g}$ \begin{displaymath} \langle \mu(x), v \rangle = \tilde \mu(v)(x) \,. \end{displaymath} This is the way it is often written in the literature. Notice that this in turn means that \begin{displaymath} \tilde \mu(v)= \mu^\ast \langle -,v\rangle \,. \end{displaymath} \end{remark} \begin{prop} \label{EquivalenceOfLieAndPoissonFormulation}\hypertarget{EquivalenceOfLieAndPoissonFormulation}{} The following are equivalent \begin{enumerate}% \item the linear map underlying $\tilde\mu$ in def. \ref{MomentMap} is [[Lie algebra]] [[homomorphism]]; \item its dual $\mu$ is a [[Poisson manifold]] [[homomorphism]] with respect to the [[Lie-Poisson structure]] on $\mathfrak{g}^\ast$. \end{enumerate} \end{prop} \begin{proof} This follows by just unwinding the definitions. In one direction, suppose that $\tilde \mu$ is a Lie homomorphism. Since the [[Lie-Poisson structure]] is fixed on linear functions on $\mathfrak{g}^\ast$, it is sufficient to check that $\mu^\ast$ preserves the Poisson bracket on these. Consider hence two Lie algebra elements $v_1, v_2 \in \mathfrak{g}$ regarded as linear functions $\langle -,v_i\rangle$ on $\mathfrak{g}^\ast$. Noticing that on such linear functions the Lie-Poisson structure is given by the Lie bracket we have, using remark \ref{Notation} \begin{displaymath} \begin{aligned} \mu^\ast \{\langle -,v_1\rangle, \langle -,v_2\rangle\} &= \mu^\ast \langle-,[v_1,v_2]\rangle \\ & = \tilde \mu([v_1,v_2]) \\ & = \{\tilde\mu(v_1), \tilde\mu(v_2)\} \\ & = \left\{ \mu^\ast \langle -,v_1\rangle, \mu^\ast \langle -,v_2\rangle \right\} \end{aligned} \end{displaymath} and hence $\mu^\ast$ preserves the Poisson brackets. Conversely, suppose that $\mu$ is a Poisson homomorphism. Then \begin{displaymath} \begin{aligned} \tilde\mu [v_1,v_2] &= \mu^\ast \langle -, [v_1,v_2]\rangle \\ & = \mu^\ast \{\langle -,v_1\rangle, \langle -,v_2\rangle\} \\ & = \left\{ \mu^\ast \langle -, v_1\rangle, \mu^\ast \langle -, v_2\rangle \right\} \\ & = \left\{ \tilde\mu(v_1), \tilde\mu(v_2) \right\} \end{aligned} \end{displaymath} and so $\tilde \mu$ is a Lie homomorphism. \end{proof} \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{AngularMomentum}{}\subsubsection*{{Angular moment}}\label{AngularMomentum} Consider the action of SO(3) on $\mathbb{R}^3$, which induces a Hamiltonian action on $T^*\mathbb{R}^3\cong\mathbb{R}^3\times\mathbb{R}^3$ via \begin{displaymath} (q,p)\xrightarrow{A\in\text{SO(3)}}(Aq,pA^{-1}) \end{displaymath} where $q$ is a column vector and $p$ is a row vector. Then the moment map for this Hamiltonian action is \begin{displaymath} \mu\colon T^*(\mathbb{R}^3)\to \mathfrak{so}(3)^*,\quad \left\langle\mu(q,p),\;\vec\theta\cdot\begin{pmatrix}\Omega_1\\\Omega_2\\\Omega_3\end{pmatrix} \right\rangle\to (\vec{q}\times \vec p)\cdot\vec{\theta} \end{displaymath} where \begin{displaymath} \Omega_1=\begin{pmatrix}0&0&0\\0&0&-1\\0&1&0\end{pmatrix},\quad\Omega_2=\begin{pmatrix}0&0&1\\0&0&0\\-1&0&0\end{pmatrix},\quad \Omega_3=\begin{pmatrix}0&-1&0\\1&0&0\\0&0&0\end{pmatrix} \end{displaymath} If we choose $\Omega_1,\Omega_2,\Omega_3$ as an orthonormal basis of $\mathfrak{so}(3)$ and then identify $\mathfrak{so}(3)\cong\mathfrak{so}(3)^*\cong\mathbb{R}^3$, then $\mu(q,p)=\vec q\times\vec p$, which is the angular momentum. \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{relation_to_conserved_quantities}{}\subsubsection*{{Relation to conserved quantities}}\label{relation_to_conserved_quantities} The values of the moment map for each given Lie algebra generator may be regarded as the [[conserved current|conserved currents]] given by a \emph{Hamiltonian [[Noether theorem]]}. Specifically if $(X,\omega)$ is a [[symplectic manifold]] equipped with a ``time evolution'' [[Hamiltonian action]] $\mathbb{R} \to \mathfrak{Poisson}(X,\omega)$ given by a [[Hamiltonian]] $H$ and if $\mathfrak{g} \to \mathfrak{Poisson}(X,\omega)$ is some [[Hamiltonian action]] with moment $\Phi(\xi)$ for $\xi \in \mathfrak{g}$ which preserves the Hamiltonian in that the [[Poisson bracket]] vanishes \begin{displaymath} \{\Phi^\xi, H\} = 0 \end{displaymath} then of course also the time evolution of the moments vanishes \begin{displaymath} \frac{d}{d t} \Phi^\xi = \{H, \Phi^\xi\} = 0 \,. \end{displaymath} See at \emph{\href{Noether%27s+theorem#HamiltonianNoetherTheorem}{Noether theorem -- In terms of moment maps/Hamiltonian Noether theorem}}. \hypertarget{RelationToConstrainedMechanics}{}\subsubsection*{{Relation to constrained mechanics}}\label{RelationToConstrainedMechanics} In the context of [[constrained mechanics]] the components of the moment map (as the Lie algebra argument varies) are called [[first class constraints]]. See \emph{[[symplectic reduction]]} for more. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} The moment map is a crucial ingredient in the construction of Marsden--Weinstein [[symplectic quotients]] and in other variants of symplectic reduction. \hypertarget{references}{}\subsection*{{References}}\label{references} The concept is originally due to [[Jean-Marie Souriau]]. \hypertarget{general}{}\subsubsection*{{General}}\label{general} \begin{itemize}% \item [[Victor Guillemin]], [[Yael Karshon]], [[Viktor Ginzburg]], \emph{Moment Maps, Cobordisms, and Hamiltonian Group Actions}, Mathematical Surveys and Monographs, volume 98 \end{itemize} Lecture notes and surveys include \begin{itemize}% \item [[Joel W. Robbin]], \emph{The moment map}, an exposition, \href{http://www.math.wisc.edu/~robbin/moment.pdf}{pdf} \item [[Nicole Berline]], [[Michèle Vergne]], \emph{Hamiltonian manifolds and moment maps} (\href{http://www.math.polytechnique.fr/~berline/cours-Fudan.pdf}{pdf}) \end{itemize} Original articles include \begin{itemize}% \item [[Victor Guillemin]], [[Shlomo Sternberg]], \emph{Geometric asymptotics}, AMS (1977) (\href{http://www.ams.org/online_bks/surv14}{online}) \item [[Michael Atiyah]], \emph{Convexity and commuting Hamiltonians}, Bull. London Math. Soc. \textbf{14} (1982), 1-15. \item [[Michael Atiyah]], [[Raoul Bott]], \emph{The moment map and equivariant cohomology}, Topology, Vol 23, No. 1 (1984) (\href{https://www.math.sunysb.edu/~mmovshev/MAT570Spring2008/BOOKS/atiyahbott_moment.pdf}{pdf}) \end{itemize} Further developments are in \begin{itemize}% \item M. Spera, \emph{On a generalized uncertainty principle, [[coherent state]]s and the moment map}, J. of Geometry and Physics \textbf{12} (1993) 165-182, \href{http://www.ams.org/mathscinet-getitem?mr=1237511}{MR94m:58097}, \item [[Ctirad Klimcik]], [[Pavol Severa]], \emph{[[T-duality]] and the moment map}, IHES/P/96/70, \href{http://arxiv.org/abs/hep-th/9610198}{hep-th/9610198}; \emph{Poisson-Lie T-duality: open strings and D-branes}, CERN-TH/95-339. Phys.Lett. B376 (1996) 82-89, \href{http://arxiv.org/abs/hep-th/9512124}{hep-th/9512124} \item A. Cannas da Silva, [[Alan Weinstein]], \emph{Geometric models for noncommutative algebras}, Berkeley Math. Lec. Notes Series, AMS 1999, (\href{http://math.berkeley.edu/%7Ealanw/Models.pdf}{pdf}) \item Friedrich Knop, \emph{Automorphisms of multiplicity free Hamiltonian manifolds}, \href{http://arxiv.org/abs/1002.4256}{arxiv/1002.4256} \item W. Crawley-Boevey, \emph{Geometry of the moment map for representations of quivers}, Compositio Math. \textbf{126} (2001), no. 3, 257-293. \end{itemize} See also \begin{itemize}% \item wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Moment_map}{moment map}} \end{itemize} Moment maps in [[higher geometry]], [[schreiber:Higher geometric prequantum theory]], are discussed in \begin{itemize}% \item [[Yael Fregier]], [[Chris Rogers]], [[Marco Zambon]], \emph{Homotopy moment maps} (\href{http://arxiv.org/abs/1304.2051}{arXiv:1304.2051}) \end{itemize} \hypertarget{relation_to_symplectic_reduction}{}\subsubsection*{{Relation to symplectic reduction}}\label{relation_to_symplectic_reduction} Reviews include for instance \begin{itemize}% \item [[José Figueroa-O'Farrill]], p. 26 of chapter 2+3 of PhD thesis (\href{http://empg.maths.ed.ac.uk/Activities/BRST/Ch2+3PhD.pdf}{pdf}) \end{itemize} \hypertarget{relation_to_equivariant_cohomology}{}\subsubsection*{{Relation to equivariant cohomology}}\label{relation_to_equivariant_cohomology} Relation to [[equivariant cohomology]]: \begin{itemize}% \item [[Michael Atiyah]], [[Raoul Bott]], \emph{The moment map and equivariant cohomology}, Topology vol. 23 No. 1 1984 (\href{https://www.math.sunysb.edu/~mmovshev/MAT570Spring2008/BOOKS/atiyahbott_moment.pdf}{pdf}) \end{itemize} \hypertarget{generalization_groupvalued_moment_maps}{}\subsubsection*{{Generalization: group-valued moment maps}}\label{generalization_groupvalued_moment_maps} \begin{itemize}% \item [[Anton Alekseev]], Anton Malkin, [[Eckhard Meinrenken]], \emph{Lie group valued moment maps}, J. Differential Geom. Volume 48, Number 3 (1998), 445-495. \href{http://projecteuclid.org/euclid.jdg/1214460860}{euclid}, \href{http://www.ams.org/mathscinet-getitem?mr=1638045}{MR1638045} \item [[Eckhard Meinrenken]], \emph{Lectures on group-valued moment maps and Verlinde formulas}, 35 pages, January 2012, \href{http://www.math.toronto.edu/mein/research/NotreLectures.pdf}{pdf} \end{itemize} The relation between moment maps and [[conserved currents]]/[[Noether's theorem]] is amplied for instance in \begin{itemize}% \item Huijun Fan, \emph{Lecture 8, Moment map and symplectic reduction} (\href{http://www.math.pku.edu.cn/teachers/fanhj/courses/symp-8.pdf}{pdf}) \end{itemize} \hypertarget{ReferencesInThermodynamics}{}\subsubsection*{{In thermodynamics}}\label{ReferencesInThermodynamics} Since moment maps generalize [[energy]]-functionals, they provide a covariant formulation of [[thermodynamics]]: \begin{itemize}% \item [[Jean-Marie Souriau]], \emph{Thermodynamique et g\'e{}om\'e{}trie}, Lecture Notes in Math. 676 (1978), 369--397 (\href{http://www-lib.kek.jp/cgi-bin/kiss_prepri.v8?KN=197810025}{scan}) \item [[Patrick Iglesias-Zemmour]], [[Jean-Marie Souriau]] \emph{Heat, cold and Geometry}, in: M. Cahen et al (eds.) Differential geometry and mathematical physics, 37-68, D. Reidel 1983 (\href{http://www.jmsouriau.com/Heat_Cold_And_Geometry_1983.htm}{web}, \href{http://www.jmsouriau.com/Publications/JMSouriau-PIglesias-HeatColdAndGeometry1983.pdf}{pdf}, \href{https://doi.org/10.1007/978-94-009-7022-9_5}{doi:978-94-009-7022-9\_5}) \item [[Jean-Marie Souriau]], chapter IV ``Statistical mechanics'' of \emph{Structure of dynamical systems. A symplectic view of physics} . Translated from the French by C. H. Cushman-de Vries. Translation edited and with a preface by R. H. Cushman and G. M. Tuynman. Progress in Mathematics, 149. Birkh\"a{}user Boston, Inc., Boston, MA, 1997 \item [[Patrick Iglesias-Zemmour]], \emph{Essai de «thermodynamique rationnelle» des milieux continus}, Annales de l'I.H.P. Physique théorique, Volume 34 (1981) no. 1, p. 1-24 (\href{http://www.numdam.org/item/AIHPA_1981__34_1_1_0}{numdam:AIHPA\_1981\_\_34\_1\_1\_0}) \end{itemize} Review includes \begin{itemize}% \item Charles-Michel Marle, \emph{From tools in symplectic and Poisson geometry to Souriau's theories of statistical mechanics and thermodynamics}, Entropy 2016, 18(10), 370 (\href{https://arxiv.org/abs/1608.00103}{arXiv:1608.00103}) \end{itemize} [[!redirects moment maps]] [[!redirects momentum map]] [[!redirects momentum maps]] [[!redirects ∞-moment map]] [[!redirects ∞-moment maps]] [[!redirects infinity-moment map]] [[!redirects infinity-moment maps]] \end{document}