Moment map

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In mathematics, specifically in symplectic geometry, the momentum map (or moment map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The moment map generalizes the classical notions of linear and angular momentum. It is an essential ingredient in various constructions of symplectic manifolds, including symplectic (Marsden–Weinstein) quotients, discussed below, and symplectic cuts and sums.

Formal definition[edit]

Let M be a manifold with symplectic form ω. Suppose that a Lie group G acts on M via symplectomorphisms (that is, the action of each g in G preserves ω). Let be the Lie algebra of G, its dual, and

the pairing between the two. Any ξ in induces a vector field ρ(ξ) on M describing the infinitesimal action of ξ. To be precise, at a point x in M the vector is

where is the exponential map and denotes the G-action on M.[1] Let denote the contraction of this vector field with ω. Because G acts by symplectomorphisms, it follows that is closed for all ξ in .

A moment map for the G-action on (M, ω) is a map such that

for all ξ in . Here is the function from M to R defined by . The moment map is uniquely defined up to an additive constant of integration.

A moment map is often also required to be G-equivariant, where G acts on via the coadjoint action. If the group is compact or semisimple, then the constant of integration can always be chosen to make the moment map coadjoint equivariant; however in general the coadjoint action must be modified to make the map equivariant (this is the case for example for the Euclidean group). The modification is by a 1-cocycle on the group with values in , as first described by Souriau (1970).

Hamiltonian group actions[edit]

The definition of the moment map requires to be closed. In practice it is useful to make an even stronger assumption. The G-action is said to be Hamiltonian if and only if the following conditions hold. First, for every ξ in the one-form is exact, meaning that it equals for some smooth function

If this holds, then one may choose the to make the map linear. The second requirement for the G-action to be Hamiltonian is that the map be a Lie algebra homomorphism from to the algebra of smooth functions on M under the Poisson bracket.

If the action of G on (M, ω) is Hamiltonian in this sense, then a moment map is a map such that writing defines a Lie algebra homomorphism satisfying . Here is the vector field of the Hamiltonian , defined by

Examples of moment maps[edit]

In the case of a Hamiltonian action of the circle , the Lie algebra dual is naturally identified with , and the moment map is simply the Hamiltonian function that generates the circle action.

Another classical case occurs when is the cotangent bundle of and is the Euclidean group generated by rotations and translations. That is, is a six-dimensional group, the semidirect product of and . The six components of the moment map are then the three angular momenta and the three linear momenta.

Let be a smooth manifold and let be its cotangent bundle, with projection map . Let denote the tautological 1-form on . Suppose acts on . The induced action of on the symplectic manifold , given by for is Hamiltonian with moment map for all . Here denotes the contraction of the vector field , the infinitesimal action of , with the 1-form .

The facts mentioned below may be used to generate more examples of moment maps.

Some facts about moment maps[edit]

Let be Lie groups with Lie algebras , respectively.

1. Let be a coadjoint orbit. Then there exists a unique symplectic structure on such that inclusion map is a moment map.

2. Let act on a symplectic manifold with a moment map for the action, and be a Lie group homomorphism, inducing an action of on . Then the action of on is also Hamiltonian, with moment map given by , where is the dual map to ( denotes the identity element of ). A case of special interest is when is a Lie subgroup of and is the inclusion map.

3. Let be a Hamiltonian -manifold and a Hamiltonian -manifold. Then the natural action of on is Hamiltonian, with moment map the direct sum of the two moment maps and . Here , where denotes the projection map.

4. Let be a Hamiltonian -manifold, and a submanifold of invariant under such that the restriction of the symplectic form on to is non-degenerate. This imparts a symplectic structure to in a natural way. Then the action of on is also Hamiltonian, with moment map the composition of the inclusion map with 's moment map.

Symplectic quotients[edit]

Suppose that the action of a compact Lie group G on the symplectic manifold (M, ω) is Hamiltonian, as defined above, with moment map . From the Hamiltonian condition it follows that is invariant under G.

Assume now that 0 is a regular value of μ and that G acts freely and properly on . Thus and its quotient are both manifolds. The quotient inherits a symplectic form from M; that is, there is a unique symplectic form on the quotient whose pullback to equals the restriction of ω to . Thus the quotient is a symplectic manifold, called the Marsden–Weinstein quotient, symplectic quotient or symplectic reduction of M by G and is denoted . Its dimension equals the dimension of M minus twice the dimension of G.

See also: GIT quotient, Quantization commutes with reduction.

See also[edit]

Notes[edit]

  1. ^ The vector field ρ(ξ) is called sometimes the Killing vector field relative to the action of the one-parameter subgroup generated by ξ. See, for instance, (Choquet-Bruhat & DeWitt-Morette 1977)

References[edit]

  • J.-M. Souriau, Structure des systèmes dynamiques, Maîtrises de mathématiques, Dunod, Paris, 1970. ISSN 0750-2435.
  • S. K. Donaldson and P. B. Kronheimer, The Geometry of Four-Manifolds, Oxford Science Publications, 1990. ISBN 0-19-850269-9.
  • Dusa McDuff and Dietmar Salamon, Introduction to Symplectic Topology, Oxford Science Publications, 1998. ISBN 0-19-850451-9.
  • Choquet-Bruhat, Yvonne; DeWitt-Morette, Cécile (1977), Analysis, Manifolds and Physics, Amsterdam: Elsevier, ISBN 978-0-7204-0494-4 
  • Ortega, Juan-Pablo; Ratiu, Tudor S. (2004). Momentum maps and Hamiltonian reduction. Progress in Mathematics 222. Birkhauser Boston. ISBN 0-8176-4307-9.