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Quantum Boundary Conditions for Torus Maps
Keating, J.P.; Mezzadri, F.; Robbins, J.M.
Keyword(s): quantum mechanics; torus maps
Abstract: Please Note. This abstract contains mathematical formulae which cannot be represented here. The quantum states of a dynamical system whose phase space is the two-torus are periodic up to phase factors under translations by the fundamental periods of the torus in the position and momentum representations. These phases, 1 and 2, are conserved quantities of the quantum evolution. We show that for a large and important class of quantum maps, 1 and 2, are restricted to being the co-ordinates of the fixed points of the automorphism induced on the fundamental group of the torus by the underlying classical dynamics. As a consequence, if the classical map commutes with lattice translations in R2 it can be quantized for any choice of the phases, but otherwise it can be quantized for only a finite set. This result is a special case of a more general condition on the phases, which is also derived. The cat maps, perturbed cat maps, and the kicked Harper map are discussed as specific examples.
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