Quantum symmetries and the Weyl-Wigner product of group representations

A. J. Bracken, G. Cassinelli, J. G. Wood

Published:

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.