The Feynman path integral for constrained systems
The paper under review discusses how to construct a path integral representation of the propagator for a canonically quantized constrained system. Given a classical Lagrangian expressed in terms of an overdetermined set of coordinates one determines the classical Hamiltonian H and constraints \(\phi_ a\). In the canonical quantization they are promoted to the quantum Hamiltonian \({\mathfrak H}\) and constraint operators \(\phi_ a\) such that \({\mathfrak H}\) acts on a restricted Hilbert space consisting of the states \(| \psi >\) satisfying \(\phi_ a| \psi >=0\). Then reexpressing the Hamiltonian as a Weyl-ordered function of an ``effective'' momentum and position operators appropriate to this Hilbert space, one obtains a phase space path integral representation for the propagator \(\exp [-i(t'-t){\mathfrak H}]\) on these states. The ``effective'' Hamiltonian appearing in this path integral differs from the original classical Hamiltonian H. For related problems see also \textit{S. V. Shabanov}, J. Phys. A, Math. Gen. 24, No.6, 1199-1214 (1991; see the review above).
scientific article; zbMATH DE number 4209123
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