Optical realization of the baker's transformation

Abstract

The quantization of a completely chaotic system-the baker's transformation-is investigated using the fact that the relationship between classical and quantum mechanics is directly analogous to the one between ray and wave optics. A class of optical systems is constructed for which the ray dynamics is governed by the baker map. Applying wave-optical methods to these physical systems then gives, in the standard way, their corresponding quantum dynamics. The result is that in certain cases the quantum propagator is identical to that found previously using an ad hoc mathematical quantization procedure. However, this is not always so-for other systems with the same classical (ray) limit there are important differences. In particular, the propagator need not be block-diagonal in its q-p' representation, as had previously been assumed, although it always tends to this basic form in the semiclassical limit.