A Canonical Quantization of the Baker's Map

Details A Canonical Quantization of the Baker's Map A Canonical Quantization of the Baker's Map

1998-07-17

arxiv.org/abs/quant-ph/9807045v1

Annals Phys. 269 (1998) 159-181

Physics

Quantum Physics

27 pages, 3 figures. Annals of Physics, to appear

Scientific paper

10.1006/aphy.1998.5845

We present here a canonical quantization for the baker's map. The method we use is quite different from that used in Balazs and Voros (ref. \QCITE{cite}{}{BV}) and Saraceno (ref. \QCITE{cite}{}{S}). We first construct a natural ``baker covering map'' on the plane $\QTO{mathbb}{\mathbb{R}}^{2}$. We then use as the quantum algebra of observables the subalgebra of operators on $L^{2}(\QTO{mathbb}{\mathbb{R}}) $ generated by $\left\{\exp (2\pi i\hat{x}) ,\exp (2\pi i\hat{p}) \right\} $ . We construct a unitary propagator such that as $\hbar \to 0$ the classical dynamics is returned. For Planck's constant $h=1/N$, we show that the dynamics can be reduced to the dynamics on an $N$-dimensional Hilbert space, and the unitary $N\times N$ matrix propagator is the same as given in ref. \QCITE{cite}{}{BV} except for a small correction of order $h$. This correction is shown to preserve the classical symmetry $x\to 1-x$ and $p\to 1-p$ in the quantum dynamics for periodic boundary conditions.

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