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arXiv:1704.02946 · Annals of Physics

A representation of Weyl–Heisenberg Lie algebra in the quaternionic setting

B. Muraleetharan, Kengatharam Thirulogasanthar, Irene Sabadini

Published Aug 3, 2017• 26 citations
arXivHTMLPDF
DOI
10.1016/j.aop.2017.07.014

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The classical limit of a quantum system with one degree of freedom is examined in terms of the contraction of the underlying non-compact kinematical algebra w 1 (the Weyl-Heisenberg algebra) to the three-dimensional Abelian algebra t 3 . An appropriate definition of the contraction of a Lie algebra and of a sequence of its representations is given. For some quantum systems with simple explicitly integrable dynamics, it is shown how the classical Poisson bracket and classical trajectories are obtained in the limit. Each trajectory is associated with a one-dimensional representation of t 3 within a direct integral of such representations in a Hilbert space. Each of the corresponding generalized one dimensional subspaces is stable under the action of the limiting dynamics, and a superselection rule arises naturally between any two such subspaces.