A Parity-Conserving Canonical Quantization for the Baker's Map

Abstract

We present here a complete description of the quantization of the baker's map. The method we use is quite different from that used in Balazs and Voros [BV] and Saraceno [S]. We use as the quantum algebra of observables the operators generated by {exp(2 Pi ix),exp (2 Pi ip)} and construct a unitary propagator such that as Planck's constant tends to zero,the classical dynamics is returned. For Planck's constant satisfying the integrality condition 1/N with N even, and for periodic boundary conditions for the wave functions on the torus, we show that the dynamics can be reduced to the dynamics on an N-dimensional Hilbert space, and the unitary N by N matrix propagator is the same as given in [BV] except for a small correction of order Planck's constant. This correction is is shown to preserve the symmetry x->1-x and p->1-p of the classical map for periodic boundary conditions.