Bicoherent states and path integrals for systems with first-class constraints

Abstract

We study the path integral for a model with a finite number of degrees of freedom and two first-class constraints. To account for the constraints, we construct the appropriate projection operator, and, rather than the resolution of unity, use it at every time slice in the building of the coherent-state path-integral representation of the propagator. The derivation of the projection operator leads to the introduction of bicoherent states and is built by integration over properly-weighted, independent coherent-state bras and kets. The construction of the propagator using bicoherent states leads to a phase space action, which, in general, is complex and has twice as many labels as there are in the standard classical phase space action. The imaginary part of the complex action reduces to a surface term on the classical trajectories. We argue that the projection operator leads to the correct measure in the path-integral representation of the propagator. The measure, which is path dependent, is `modulated' by the imaginary part of the action.