Phase 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.

Peter Høyer

Jérôme Daligault

A Lagrangian, moving-with-the-fluid, picture in the quantum Wigner phase space is rigorously defined, introducing a state-dependent current in phase space. Contrary to its classical counterpart, the Lagrangian dynamics in quantum phase space is non-Hamiltonian and elementary phase-space volumes are not conserved. The consequences for the numerical implementation of a fluid trajectory method in quantum phase space are discussed.

Jens Peder Dahl

Claudio Conti

Leon W. Cohen

We formulate the standard quantum mechanical eigenvalue problem in quantum phase space. The equation obtained involves the c-function that corresponds to the quantum operator. We use the Wigner distribution for the phase space function. We argue that the phase space eigenvalue equation obtained has, in addition to the proper solutions, improper solutions. That is, solutions for which no wave function exists which could generate the distribution. We discuss the conditions for ascertaining whether a position momentum function is a proper phase space distribution. We call these conditions psi-representability conditions, and show that if these conditions are imposed, one extracts the correct phase space eigenfunctions. We also derive the phase space eigenvalue equation for arbitrary phase space distributions functions. © 2017 Wiley Periodicals, Inc.