Representation Space

Asao Arai

For each integer n ≥ 2 and a parameter Λ = (θ, η) with θ and η being n × n real anti-symmetric matrices, a quantum phase space (QPS) (or a non-commutative phase space) with n degrees of freedom, denoted QPS n(Λ), is defined, where θ and η are parameters measuring non-commutativity of the QPS. Hilbert space representations of QPS n(Λ) are considered. A concept of quasi-Schrödinger representation of QPS n(Λ) is introduced. It is shown that there exists a general correspondence between representations of QPS n(Λ) and those of the canonical commutation relations with n degrees of freedom. Irreducibility of representations of QPS n(Λ) are investigated. A concept of Weyl representation of QPS n(Λ) is defined. It is proved that every Weyl representation of QPS n(Λ) on a separable Hilbert space is unitarily equivalent to a direct sum of a quasi-Schrödinger representation of the QPS n(Λ) (a uniqueness theorem). Finally representations of QPS n(Λ) which are not unitarily equivalent to any direct sum of a quasi-Schrödinger representation are described.

Edgard Elbaz

Hanitriarivo Rakotoson, Raoelina Andriambololona, Ravo Tokiniaina Ranaivoson +1 more

The aim of this paper is to present a study on the representations of coordinate, momentum and dispersion operators in the framework of a phase space representation of quantum mechanics that we have introduced and studied in previous works. We begin in the introduction section with a recall about the concept of representation of operators on wave function spaces. Then, we show that in the case of the phase space representation the coordinate and momentum operators can be represented either with differential operators or with matrices. The explicit expressions of both the differential operators and matrices representations are established. Multidimensional generalization of the obtained results are performed and phase space representation of dispersion operators are given.

Emilio d’Emilio, Luigi E. Picasso

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.

Harry A. Mavromatis