Necessary and sufficient conditions for a phase-space function to be a Wigner distribution
We discuss two sets of conditions that are necessary and sufficient for a function defined on phase space to be a Wigner distribution function (WDF). The first set is well known and involves the function itself; the second set is less familiar and involves the functions symplectic Fourier transform. After explaining why these two sets are equivalent, we explore some properties and applications of the second one. Among other things, we show that that set includes the position-momentum uncertainty relations as a special case, and in doing so we give a new derivation of them. This derivation itself serves as the starting point for the discussion of a quantum-mechanical moment problem. It also enables us to exhibit a real-valued phase-space function that obeys the uncertainty relations but that is not a WDF. © 1986 The American Physical Society.
Publication Source (Journal or Book title)
Physical Review A
Narcowich, F., & Oconnell, R. (1986). Necessary and sufficient conditions for a phase-space function to be a Wigner distribution. Physical Review A, 34 (1), 1-6. https://doi.org/10.1103/PhysRevA.34.1