Admissible states in quantum phase space

TL;DR

This paper characterizes admissible phase space functionals as quantum states, deriving conditions for pure and mixed states, and explores their evolution, correlations, and representations within Wigner quantum mechanics.

Contribution

It provides necessary and sufficient conditions for phase space quantum states, extending Baker's construction, and characterizes the structure of Wigner quantum mechanics.

Findings

Pure state condition preserved by Moyal evolution

Explicit formulas for wavefunctions in phase space

Heisenberg uncertainty relations derived from mixed state conditions

Abstract

We address the question of which phase space functionals might represent a quantum state. We derive necessary and sufficient conditions for both pure and mixed phase space quantum states. From the pure state quantum condition we obtain a formula for the momentum correlations of arbitrary order and derive explicit expressions for the wavefunctions in terms of time dependent and independent Wigner functions. We show that the pure state quantum condition is preserved by the Moyal (but not by the classical Liouville) time evolution and is consistent with a generic stargenvalue equation. As a by-product Baker's converse construction is generalized both to an arbitrary stargenvalue equation, associated to a generic phase space symbol, as well as to the time dependent case. These results are properly extended to the mixed state quantum condition, which is proved to imply the Heisenberg uncertainty relations. Globally, this formalism yields the complete characterization of the kinematical structure of Wigner quantum mechanics. The previous results are then succinctly generalized for various quasi-distributions. Finally, the formalism is illustrated through the simple examples of the harmonic oscillator and the free Gaussian wave packet. As a by-product, we obtain in the former example an integral representation of the Hermite polynomials.