The moment map
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 on a symplectic manifold by (Hamiltonian) symplectomorphisms corresponds infinitesimally to a Lie algebra homomorphism from the Lie algebra to the Hamiltonian vector fields on . If this lifts to a coherent choice of Hamiltonians, hence to a Lie algebra homomorphism to the Poisson bracket, then, by dualization, this is equivalently a Poisson manifold homomorphism of the form
This 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.
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.
This section briefly reviews the notion of Hamiltonian vector fields on a symplectic manifold
The basic setup is the following: Let be a symplectic manifold with a Hamiltonian action of a Lie group . In particular that means that there is an action via symplectomorphisms (diffeomorphisms such that ). A vector field is symplectic if the corresponding flow preserves (again by pullbacks) . The symplectic vector fields form a Lie subalgebra of the Lie algebra of all smooth vector fields on with respect to the Lie bracket.
By the Cartan homotopy formula and closedness
where denotes the Lie derivative. Therefore a vector field is symplectic iff for some function , usually called Hamiltonian (function) for . Here is determined by up to a locally constant function. Such is called the Hamiltonian vector field corresponding to . The Poisson structure on is the bracket on functions may be given by
where there is a Lie bracket of vector fields on the right hand side.
For a connected symplectic manifold, there is an exact sequence of Lie algebras
See at Hamiltonian vector field – Relation to Poisson bracket.
Hamiltonian action and moment map
Let be a symplectic manifold and let be a Lie algebra. Write for the Poisson bracket Lie algebra underlying the corresponding Poisson algebra.
A Hamiltonian action of on is a Lie algebra homomorphism
The corresponding function
to the dual vector space of , defined by
is the corresponding moment map.
This follows by just unwinding the definitions.
In one direction, suppose that is a Lie homomorphism. Since the Lie-Poisson structure is fixed on linear functions on , it is sufficient to check that preserves the Poisson bracket on these. Consider hence two Lie algebra elements regarded as linear functions on . Noticing that on such linear functions the Lie-Poisson structure is given by the Lie bracket we have, using remark 1
and hence preserves the Poisson brackets.
Conversely, suppose that is a Poisson homomorphism. Then
and so is a Lie homomorphism.
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 is a symplectic manifold equipped with a “time evolution” Hamiltonian action given by a Hamiltonian and if is some Hamiltonian action with moment for which preserves the Hamiltonian in that the Poisson bracket vanishes
then of course also the time evolution of the moments vanishes
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.
Lecture notes and surveys include
Original articles include
Victor Guillemin, Shlomo Sternberg, Geometric asymptotics, AMS (1977) (online)
Michael Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), 1-15.
Michael Atiyah, Raoul Bott, The moment map and equivariant cohomology, Topology, Vol 23, No. 1 (1984) (pdf)
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.
Moment maps in higher geometry, Higher geometric prequantum theory, are discussed in
Relation to symplectic reduction
Reviews include for instance
Relation to equivariant cohomology
Relation to equivariant cohomology:
Generalization: group-valued moment maps
Anton Alekseev, Anton Malkin, Eckhard Meinrenken, Lie group valued moment maps, J. Differential Geom. Volume 48, Number 3 (1998), 445-495. euclid, MR1638045
Eckhard Meinrenken, Lectures on group-valued moment maps and Verlinde formulas, 35 pages, January 2012, pdf
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)