nLab symplectic reduction

Idea

This entry is being splitt-off from BRST-BV formalism. For the moment see the discussion there at Hamiltonian BFV.

Properties

• Guillemin-Sternberg geometric quantization conjecture

Related concepts

• Hamiltonian action

The analog for presymplectic manifolds is presymplectic reduction.

The analog for Poisson manifolds is Poisson reduction.

Reductions of (pre-)symplectic manifolds:

symplectic geometry	physics
presymplectic manifold	covariant phase space
$\downarrow$ gauge reduction	$\downarrow$ quotient by gauge symmetry
symplectic manifold	reduced phase space
$\downarrow$ symplectic reduction	$\downarrow$ quotient by global symmetry
symplectic manifold	reduced phase space

• quantization commutes with reduction

References

The original influential article is

• Jerrold Marsden, Alan Weinstein, Reduction of symplectic manifolds with symmetry, Reports on Math. Phys., 5, (1974), 121-130 (web)

Reviews include

• José Figueroa-O'Farrill, chapter 2+3 of PhD thesis (pdf)

Textbook accounts include

• Bertram Kostant, Shlomo Sternberg, Symplectic reduction, BRS cohomology and infinite-dimensional Clifford algebras, Annals of Physics 176:1 (15 May 1987) pp. 49–113 MR893479 doi