Quantum Poincaré-Cartan Integral Invariant for Singular System

In order to study the symmetry of a singular system, based on phase-space generation of Green function of a singular system with finite degrees of freedom, an integral invariant of quantum Poincare-Cartan (excluding ground state sign |0>) is established. The relationship between the Poincare-Cartan integral invariant and the Hamilton-Jacobi equation is discussed at the quantum level. It is pointed that the Hamilton-Jacobi equation can be deduced from integral invariant of the quantum Poincare-Cartan.