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A Canonical Quantization of the Baker's Map

✍ Scribed by Ron Rubin; Nathan Salwen

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
492 KB
Volume
269
Category
Article
ISSN
0003-4916

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✦ Synopsis

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 and Saraceno. We first construct a natural ``baker covering map'' on the plane R 2 . We then use as the quantum algebra of observables the subalgebra of operators on L 2 (R) generated by [exp(2?ix^), exp(2?ip^)]. We construct a unitary propagator such that as Γ„ 0 the classical dynamics is returned. For Planck's constant h=1Γ‚N, we show that the dynamics can be reduced to the dynamics on an N-dimensional Hilbert space, and the unitary N_N matrix propagator is the same as given by Balazs and Voros, except for a small correction of order h. This correction is shown to preserve the classical symmetry x Γ„ 1&x and p Γ„ 1& p in the quantum dynamics for periodic boundary conditions.

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