Wigner’s Symmetry Representation Theorem

Gerardo Ortíz, Chinmay Giridhar, Philipp Vojta +2 more

David John Baker, Hans Halvorson

Anthony J. Bracken, G Cassinelli, J G Wood

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 'factorizing' 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 factorized. The conditions under which this is possible are examined. Examples are presented.

Georges Chevalier

Sibel Başkal, Young S Kim, Marilyn E. Noz

Ninnat Dangniam, Christopher Ferrie

Bochner's theorem gives the necessary and sufficient conditions on a function such that its Fourier transform corresponds to a true probability density function.In the Wigner phase space picture, quantum Bochner's theorem gives the necessary and sufficient conditions on a function such that it is a quantum characteristic function of a valid (and possibly mixed) quantum state and such that its Fourier transform is a true probability density.We extend this theorem to discrete phase space representations which possess enough symmetry.More precisely, we show that discrete phase space representations that are built on projective unitary representations of abelian groups, with a slight restriction on admissible two-cocycles, enable a quantum Bochner's theorem.