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Projective representations of the inhomogeneous Hamilton group: Noninertial symmetry in quantum mechanics

Citation

Low, SG and Jarvis, PD and Campoamor-Stursberg, R, Projective representations of the inhomogeneous Hamilton group: Noninertial symmetry in quantum mechanics, Annals of Physics, 327, (1) pp. 74-101. ISSN 0003-4916 (2012) [Refereed Article]


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Copyright Statement

Copyright 2011 Elsevier Inc.

DOI: doi:10.1016/j.aop.2011.10.010

Abstract

Symmetries in quantum mechanics are realized by the projective representations of the Lie group as physical states are defined only up to a phase. A cornerstone theorem shows that these representations are equivalent to the unitary representations of the central extension of the group. The formulation of the inertial states of special relativistic quantum mechanics as the projective representations of the inhomogeneous Lorentz group, and its nonrelativistic limit in terms of the Galilei group, are fundamental examples. Interestingly, neither of these symmetries include the Weyl–Heisenberg group; the hermitian representations of its algebra are the Heisenberg commutation relations that are a foundation of quantum mechanics. The Weyl–Heisenberg group is a one dimensional central extension of the abelian group and its unitary representations are therefore a particular projective representation of the abelian group of translations on phase space. A theorem involving the automorphism group shows that the maximal symmetry that leaves the Heisenberg commutation relations invariant is essentially a projective representation of the inhomogeneous symplectic group. In the nonrelativistic domain, we must also have invariance of Newtonian time. This reduces the symmetry group to the inhomogeneous Hamilton group that is a local noninertial symmetry of the Hamilton equations. The projective representations of these groups are calculated using the Mackey theorems for the general case of a nonabelian normal subgroup.

Item Details

Item Type:Refereed Article
Keywords:noninertial symmetry, Hamilton group, Mackey representations, Born reciprocity, projective representations, semidirect product group
Research Division:Mathematical Sciences
Research Group:Mathematical Physics
Research Field:Mathematical Aspects of Classical Mechanics, Quantum Mechanics and Quantum Information Theory
Objective Division:Expanding Knowledge
Objective Group:Expanding Knowledge
Objective Field:Expanding Knowledge in the Physical Sciences
Author:Jarvis, PD (Dr Peter Jarvis)
ID Code:78000
Year Published:2012
Web of Science® Times Cited:1
Deposited By:Mathematics and Physics
Deposited On:2012-06-12
Last Modified:2013-04-03
Downloads:0

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