Canonical Symmetries in Functional Integral Expression for Field Theories

Using functional integral quantization for the fields and starting from the generating functional of Green's function in canonical formalism, the generalized Ward identities in the phase space for the system with regular or singular Lagrangian are derived, and it is pointed out that the quantum canonical equations for a system with singular Lagrangian are different from canonical equations obtained by using Dirac's conjecture. The systems containing phonon, electron and photon can be described in terms of singular Lagrangian. The generating functional of Green's function is deduced and preliminary applications of generalized Ward identities are also given for such a system.