Quantal canonical symmetry for a constrained Hamiltonian system

Abstract

Starting from the phase-space generating functional of the Green's function for a constrained Hamiltonian system, the canonical Ward identities under the global symmetry transformation in phase space is deduced. The local transformation connected with this global symmetry transformation is studied, the conserved charges are obtained at quantum level if the effective canonical action is symmetric (the constraints are also invariant under the transformation) and, therefore, the canonical Noether theorem in the quantum case is obtained. The generalized canonical Ward identities under the local transformation has been deduced. We give a preliminary application to a system of an interacting polaron with a photon. The conserved charges and Ward identities for proper vertices are obtained, but we do not carry out the integration over the canonical momenta in the phase-space generating functional as usually performed.