Noether Theorem and Poincaré-Cartan Integral Invariant in Quantum Case for a Constrained Hamilton System

Based on the phase-space generating functional of Green function for a constrained Hamiltonian system with finite degree of freedom, the Noether theorem in quantum case under the global symmetry in phase space is derived for such a system. According to the translation-invariance of generating functional in phase space, the Poincaré-Cartan integral invariant at the quantum level is deduced. The comparison of it with the classical results is discussed.