Ward identities in phase space and their applications

Abstract

Starting from the phase space path integral, we have derived the Ward identities in canonical formalism for a system with regular and singular Lagrangian. This formulation differs from the traditional discussion based on path integral in configuration space. It is pointed out that the quantum canonical equations for systems with singular Lagrangians are different from the classical ones obtained from Dirac's conjecture. The preliminary applications of Ward identities in phase space to the Yang-Mills theory are given. Some relations among the proper vertices and propagators are deduced, the PCAC, AVV vertices and generalized PCAC expressions are also obtained. We have also pointed out that some authors in their early work had ignored the treatment of the constraints.