On the Quantum Baker's Map and its Unusual Traces

The quantum baker’s map is the quantization of a simple classically chaotic system, and has many generic features that have been studied over the last few years. While there exists a semiclassical theory of this map, a more rigorous study of the same revealed some unexpected features which indicated that correction terms of the order of log(h) had to be included in the periodic orbit sum. Such singular semiclassical behaviour was also found in the simplest traces of the quantum map. In this note we study the quantum mechanics of a baker’s map which is obtained by reflecting the classical map about its edges, in an effort to understand and circumvent these anomalies. This leads to a real quantum map with traces that follow the usual Gutzwiller-Tabor like semiclassical formulae. We develop the relevant semiclassical periodic orbit sum for this map which is closely related to that of the usual baker’s map, with the important difference that the propagators leading to this sum have no anomalous traces.