Path Integral Quantization for a Toroidal Phase Space

A Wiener-regularized path integral is presented as an alternative way to formulate Berezin-Toeplitz quantization on a toroidal phase space. Essential to the result is that this quantization prescription for the torus can be constructed as an induced representation from anti-Wick quantization on its covering space, the plane. When this construction is expressed in the form of a Wiener-regularized path integral, symmetrization prescriptions for the propagator emerge similar to earlier path-integral formulas on multiply-connected configuration spaces. 1 . INTRODUCTION With the notion of “quantization” we associate the construction of quantum systems that are in correspondence with a given classical system. The various quantization prescriptions usually specify some Hilbert space and a mapping of suitable classical observables to operators on this Hilbert space. In Schrödinger’s prescription for canonical quantization, the vectors in the Hilbert space are square-integrable functions on classical configuration space, and their dynamics is derived from a partial differential equation commonly known as Schrödinger’s equation. There are other quantization schemes in which the Hilbert space consists of functions on the classical phase space, and among these, coherent states often play an important role [8, 2, 5]. An alternative way to express quantization is by path integration. Thanks to the Feynman-Kac formula [17], Schrödinger’s approach can be associated with a Wiener integral over paths in configuration-space. Similarly, a formula of Daubechies and Klauder [6] relates anti-Wick quantization to a so-called Wiener-regularized path integral, that Also at the Department of Physics.