Admissible states in quantum phase space
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 wave functions in terms of timedependent 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 byproduct Baker's converse construction is generalized both to an arbitrary stargenvalue equation, associated to a generic phase space symbol, as well as to the timedependent 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 quasidistributions. Finally, the formalism is illustrated through the simple examples of the harmonic oscillator and the free Gaussian wave packet. As a byproduct, we obtain in the former example an integral representation of the Hermite polynomials.
 Publication:

Annals of Physics
 Pub Date:
 September 2004
 DOI:
 10.1016/j.aop.2004.03.008
 arXiv:
 arXiv:hepth/0402008
 Bibcode:
 2004AnPhy.313..110D
 Keywords:

 03.65.Ca;
 03.65.Db;
 03.65.Ge;
 Formalism;
 Functional analytical methods;
 Solutions of wave equations: bound states;
 High Energy Physics  Theory
 EPrint:
 34 pages, Latex file