The classical limit of the quantum baker's map

Abstract

This work concerns the finding of the semi-classical form of the coherent state representation for the class of quantum baker’s maps defined by Schack and Caves. It begins by introducing the finite-dimensional Hilbert space on which the quantum baker’s map is defined. Its pertinent features including the all important symmetry operators are introduced and given a full explanation. We also introduce the finite-dimensional phase space which will give the semi-classical limit a geometrical interpretation. For a D dimensional Hilbert space, the finite-dimensional phase space is found to be a grid with D2 points. Each point corresponds to a particular pair of position and momentum displacement operator eigenphases. We then detail the derivation of the finite-dimension version of the Wigner function, a quasi-distribution for the finite-dimensional phase space. We show that its most “irregular” feature, mainly its property of having more values than was thought necessary, can be explained by its correct behavior under the symmetry operations, a feature lacking in other Wigner candidates. However, even this special choice for the Wigner function proves unusable in the semi-classical limit as it is found to have a non-convergent limit. We then turn to another possible phase space function: the Q-function. It being necessary to find a suitable coherent state for the finite-dimensional Hilbert space, we begin by studying the properties of the periodically continued Gaussian states. These are the typical Weyl coherent states made periodic in both position and momentum such as to make them legitimate finite-dimensional states. Developing certain mathematical techniques allows us to show that they have compatible position and momentum representations, that a subset of them are complete and can be used to define a Q-function, and that this function obeys all of the symmetry properties. Finally, we use these coherent states to find a representation for the propagator of the quantum baker’s map. In the semi-classical limit, i.e. the large dimension limit, this representation is found, for most of the maps, to take a form of the exponentiation of the classical map’s generating function. This form was predicted long ago by Van Vleck as indicator of an operator’s classical limit. Therefore, we assert that these maps limit to the classically chaotic baker’s map. In certain limiting schemes, however, the Schack-Caves maps do not reach this form and must be given a different interpretation.