nLab
gauge reduction

Contents

Idea

Given a presymplectic manifold (X,ω)(X,\omega), the quotient (if it exists suitably) by the flow of the kernel of the presymplectic form ω:TXT *X\omega : T X \to T^* X is the symplectic manifold obtained by gauge reduction. Also called presymplectic reduction

In the interpretation in physics this takes one from a covariant phase space to a reduced phase space.

In constrast, in symplectic reduction one takes the quotient of a pre-specified Hamiltonian action and restricts to the 0-locus of the corresponding Hamiltonians (the momentum map).

Reductions of (pre-)symplectic manifolds:

symplectic geometryphysics
presymplectic manifoldcovariant phase space
\downarrow gauge reduction\downarrow quotient by gauge symmetry
symplectic manifoldreduced phase space
\downarrow symplectic reduction\downarrow quotient by global symmetry
symplectic manifoldreduced phase space

References

Around page 11 of

  • A. Echeverría-Enríquez, M.C. Muñoz-Lecanda, N. Román-Roy, Reduction of Presymplectic Manifolds with Symmetry (arXiv:math-ph/9911008)
Revised on September 16, 2013 01:56:04 by Urs Schreiber (89.204.139.122)