Bicoherent States, Path Integrals and Systems with Constraints

In this work we discuss how dynamically constrained systems are handled in a path integral framework. We consider models with a finite number of degrees of freedom and with constraints that are either first class or second class. We first briefly review the techniques developed in the past to handle such systems. Then we describe in detail the method developed here which is summarized as follows: To account for the constraints we construct an appropriate projection operator. We use this projection operator, rather than the resolution of unity, at every time slice in building a path integral representation of the propagator. The derivation of the projection operator leads to the introduction of Bicoherent States and is an integral over properly weighted independent coherent-state bras and kets. The path integral representation of the propagator, built using bicoherent states, leads to a complex phase-space action. This complex action has twice as many 'labels' as the standard action, the imaginary part of which reduces to a surface term on the classical trajectories. Also, on the classical trajectories the real part of the action reduces to just the standard action. 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.