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Quantum Boundary Conditions for Torus Maps
Keating, J.P.; Mezzadri, F.; Robbins, J.M.
HPLBRIMS9821
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 twotorus 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 coordinates 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.
18 Pages
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