The Feynman path integral for constrained systems

Abstract

I describe how to construct the Feynman path integral for a constrained system. Given a classical Hamiltonian one constructs a quantum Hamiltonian by the usual procedure of promoting the position and momentum functions to quantum operators. In a constrained system the quantum Hamiltonian acts on a restricted Hilbert space, which I describe. One must reexpress the Hamiltonian in terms of canonical position and momentum operators appropriate to the restricted Hilbert space. Finally, one must make a correspondence between these canonical operators and the classical functions which appear in the path integral. The naive procedure of simply demoting the quantum operators to functions is incorrect.