Phase Space Representations of Wavefunctions and Densities

Eric Heller

Juha-Pekka Pellonpää

Covariant phase observables are obtained by defining simple conditions for mappings from the set of phase wavefunctions (unit vectors of the Hardy space) to the set of phase probability densities. The existence of phase probability density for any phase wavefunction, the existence of interference effects and the natural phase shift covariance are those simple conditions. The non-localizability of covariant phase observables is proved.

Claudio Conti

Eduard Prugovečki

A stochastic phase space description of photon states is obtained by attaching to each circular polarization mode a probability amplitude at stochastic phase space points that are frontally localized. The ensuing formalism gives rise to a probability density at such points that transforms as a scalar under proper Lorentz transformations. There also exist conserved covariant currents associated with these probability densities. The wavefunctions corresponding to all extremal stochastic phase space representations can be embedded in a single Hilbert space, and give rise there to irreducible representations of the proper Poincaré group 𝒫↑+.

Anatoli Polkovnikov

Ravo Tokiniaina Ranaivoson, Raoelina Andriambololona, Hanitriarivo Rakotoson

This paper is a continuation of our previous works concerning coordinate, momentum, dispersion operators and phase space representation of quantum mechanics. It deals with the properties of momentum dispersion operator, its representations, eigenvalue equation and its relations with the wavefunctions in the phase space representation. After recalling some results from our previous works, we perform a study concerning the phase space wavefunctions and consider some example of them. Then we establish the eigenvalue equation for the differential operator corresponding to the momentum dispersion operator in the phase space representation. It is shown in particular that any wavefunction in phase space representation is solution of this equation.