Quantum symmetries and the Weyl-Wigner product of group representations

TL;DR

This paper explores the conditions under which phase-space representations of quantum symmetries can be factorized into Weyl-Wigner products of Hilbert space representations, revealing limitations and providing examples.

Contribution

It characterizes when phase-space representations can be expressed as Weyl-Wigner products, extending understanding of quantum symmetries in phase-space formulation.

Findings

Not all real, unitary phase-space representations correspond to automorphisms.

Factorization conditions for Weyl-Wigner products are established.

Examples illustrate cases where factorization is possible or not.

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.