Quantum symmetries and the Weyl-Wigner product of group representations

Abstract

In the usual formulation of quantum mechanics, groups of automorphisms of quantum states have ray representations by unitary and antiunitary operators on complex Hilbert space, in accordance with Wigner's Theorem. In the phase-space formulation, they have real, true unitary representations in the space of square-integrable functions on phase-space. Each such phase-space representation is a Weyl-Wigner product of the corresponding Hilbert space representation with its contragredient, and these can be recovered by `factorising' the Weyl-Wigner product. However, not every real, unitary representation on phase-space corresponds to a group of automorphisms, so not every such representation is in the form of a Weyl-Wigner product and can be factorised. The conditions under which this is possible are examined. Examples are presented.

Latex2e file, 37 pages

References in corpus (2)

• Deformation Quantization: Quantum Mechanics Lives and Works in Phase-Space
• The Moyal-Lie Theory of Phase Space Quantum Mechanics

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• $\hslash$-independent Universality of the Quantum-Classical Canonical Transformations