nLab
moment map

The moment map

Idea

A 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 GG on a symplectic manifold XX by (Hamiltonian) symplectomorphisms corresponds infinitesimally to a Lie algebra homomorphism from the Lie algebra 𝔤\mathfrak{g} to the Hamiltonian vector fields on XX. If this lifts to a coherent choice of Hamiltonians, hence to a Lie algebra homomorphism 𝔤(C (X),{,})\mathfrak{g} \to (C^\infty(X), \{-,-\}) to the Poisson bracket, then, by dualization, this is equivalently a Poisson manifold homomorphism of the form

μ:X𝔤 *. \mu : X \to \mathfrak{g}^* \,.

This μ\mu is called the moment map or momentum map of the Hamiltonian action.

The name derives from the special and historically first case of angular momentum in the dynamics of rigid bodies, see Examples - Angular momentum below.

Definition

The Preliminaries below review some basics of Hamiltonian vector fields. The definition of the moment map itself is below in Hamiltonian action and the moment map.

Preliminaries

This section briefly reviews the notion of Hamiltonian vector fields on a symplectic manifold

The basic setup is the following: Let (M,ω)(M,\omega) be a symplectic manifold with a Hamiltonian action of a Lie group GG. In particular that means that there is an action ν:G×MM\nu\colon G \times M \to M via symplectomorphisms (diffeomorphisms ν g\nu_g such that ν g *(ω)=ω\nu_g^*(\omega) = \omega). A vector field XX is symplectic if the corresponding flow preserves (again by pullbacks) ω\omega. The symplectic vector fields form a Lie subalgebra χ(M,ω)\chi(M,\omega) of the Lie algebra of all smooth vector fields χ(M)\chi(M) on MM with respect to the Lie bracket.

By the Cartan homotopy formula and closedness dω=0d \omega = 0

Xω=dι Xω \mathcal{L}_X \omega = d \iota_X \omega

where X\mathcal{L}_X denotes the Lie derivative. Therefore a vector field XX is symplectic iff ι(X)ω=dH\iota(X)\omega = d H for some function HC (M)H\in C^\infty(M), usually called Hamiltonian (function) for XX. Here XX is determined by HH up to a locally constant function. Such X=X HX = X_H is called the Hamiltonian vector field corresponding to HH. The Poisson structure on MM is the bracket {,}\{,\} on functions may be given by

{f,g}:=[X f,X g] \{ f, g\} := [X_f,X_g]

where there is a Lie bracket of vector fields on the right hand side.

For (M,ω)(M,\omega) a connected symplectic manifold, there is an exact sequence of Lie algebras

0R(C (M),{,})χ(M,ω)0. 0 \to \mathbf{R}\to (C^\infty(M), \{-,-\}) \to \chi(M,\omega) \to 0 \,.

See at Hamiltonian vector field – Relation to Poisson bracket.

Hamiltonian action and moment map

Let (X,ω)(X, \omega) be a symplectic manifold and let 𝔤\mathfrak{g} be a Lie algebra. Write (C (X),{,})(C^\infty(X), \{-,-\}) for the Poisson bracket Lie algebra underlying the corresponding Poisson algebra.

Definition

A Hamiltonian action of 𝔤\mathfrak{g} on (X,ω)(X, \omega) is a Lie algebra homomorphism

μ˜:𝔤(C (X),{,}). \tilde \mu \;\colon\; \mathfrak{g} \longrightarrow (C^\infty(X), \{-,-\}) \,.

The corresponding function

μ:X𝔤 * \mu \;\colon\; X \longrightarrow \mathfrak{g}^*

to the dual vector space of 𝔤\mathfrak{g}, defined by

μ:xμ˜()(x) \mu \;\colon\; x \mapsto \tilde \mu(-)(x)

is the corresponding moment map.

Remark

If one writes the evaluation pairing as

,:𝔤 *𝔤 \langle -,-\rangle : \mathfrak{g}^* \otimes \mathfrak{g} \to \mathbb{R}

then the equation characterizing μ\mu in def. 1 reads for all xXx \in X and v𝔤v \in \mathfrak{g}

μ(x),v=μ˜(v)(x). \langle \mu(x), v \rangle = \tilde \mu(v)(x) \,.

This is the way it is often written in the literature.

Notice that this in turn means that

μ˜(v)=μ *,v. \tilde \mu(v)= \mu^\ast \langle -,v\rangle \,.
Proposition

The following are equivalent

  1. the linear map underlying μ˜\tilde\mu in def. 1 is Lie algebra homomorphism;

  2. its dual μ\mu is a Poisson manifold homomorphism with respect to the Lie-Poisson structure on 𝔤 *\mathfrak{g}^\ast.

Proof

This follows by just unwinding the definitions.

In one direction, suppose that μ˜\tilde \mu is a Lie homomorphism. Since the Lie-Poisson structure is fixed on linear functions on 𝔤 *\mathfrak{g}^\ast, it is sufficient to check that μ *\mu^\ast preserves the Poisson bracket on these. Consider hence two Lie algebra elements v 1,v 2𝔤v_1, v_2 \in \mathfrak{g} regarded as linear functions ,v i\langle -,v_i\rangle on 𝔤 *\mathfrak{g}^\ast. Noticing that on such linear functions the Lie-Poisson structure is given by the Lie bracket we have, using remark 1

μ *{,v 1,,v 2} =μ *,[v 1,v 2] =μ˜([v 1,v 2]) ={μ˜(v 1),μ˜(v 2)} ={μ *,v 1,μ *,v 2} \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}

and hence μ *\mu^\ast preserves the Poisson brackets.

Conversely, suppose that μ\mu is a Poisson homomorphism. Then

μ˜[v 1,v 2] =μ *,[v 1,v 2] =μ *{,v 1,,v 2} ={μ *,v 1,μ *,v 2} ={μ˜(v 1),μ˜(v 2)} \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}

and so μ˜\tilde \mu is a Lie homomorphism.

Examples

Angular moment

(…)

Properties

Relation to conserved quantities

The values of the moment map for each given Lie algebra generator may be regarded as the conserved currents given by a Hamiltonian Noether theorem.

Specifically if (X,ω)(X,\omega) is a symplectic manifold equipped with a “time evolution” Hamiltonian action 𝔓𝔬𝔦𝔰𝔰𝔬𝔫(X,ω)\mathbb{R} \to \mathfrak{Poisson}(X,\omega) given by a Hamiltonian GG and if 𝔤𝔓𝔬𝔦𝔰𝔰𝔬𝔫(X,ω)\mathfrak{g} \to \mathfrak{Poisson}(X,\omega) is some Hamiltonian action with moment Φ(ξ)\Phi(\xi) for ξ𝔤\xi \in \mathfrak{g} which preserves the Hamiltonian in that the Poisson bracket vanishes

{Φ ξ,H}=0 \{\Phi^\xi, H\} = 0

then of course also the time evolution of the moments vanishes

ddtΦ ξ={H,Φ ξ}=0. \frac{d}{d t} \Phi^\xi = \{H, \Phi^\xi\} = 0 \,.

See at Noether theorem – In terms of moment maps/Hamiltonian Noether theorem.

Relation to constrained mechanics

In the context of constrained mechanics? the components of the moment map (as the Lie algebra argument varies) are called first class constraints. See symplectic reduction for more.

The moment map is a crucial ingredient in the construction of Marsden–Weinstein symplectic quotients and in other variants of symplectic reduction.

References

General

Lecture notes and surveys include

Original articles include

Further developments are in

  • M. Spera, On a generalized uncertainty principle, coherent states and the moment map, J. of Geometry and Physics 12 (1993) 165-182, MR94m:58097, doi

  • Ctirad Klimčík, Pavol Ševera, T-duality and the moment map, IHES/P/96/70, hep-th/9610198; Poisson-Lie T-duality: open strings and D-branes, CERN-TH/95-339. Phys.Lett. B376 (1996) 82-89, hep-th/9512124

  • A. Cannas da Silva, Alan Weinstein, Geometric models for noncommutative algebras, Berkeley Math. Lec. Notes Series, AMS 1999, (pdf)

  • Friedrich Knop, Automorphisms of multiplicity free Hamiltonian manifolds, arxiv/1002.4256

  • W. Crawley-Boevey, Geometry of the moment map for representations of quivers, Compositio Math. 126 (2001), no. 3, 257-293.

See also

Moment maps in higher geometry, Higher geometric prequantum theory, are discussed in

Relation to symplectic reduction

Reviews include for instance

p. 26 of

Relation to equivariant cohomology

Relation to equivariant cohomology:

Generalization: group-valued moment maps

The relation between moment maps and conserved currents/Noether's theorem is amplied for instance in

  • Huijun Fan, Lecture 8, Moment map and symplectic reduction (pdf)

Revised on September 3, 2014 03:49:33 by Urs Schreiber (82.136.246.44)