Necessary and sufficient conditions for a phase-space function to be a Wigner distribution

Francis J. Narcowich and R. F. O’Connell
Phys. Rev. A 34, 1 – Published 1 July 1986
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Abstract

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 function’s ‘‘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.

  • Received 13 February 1986

DOI:https://doi.org/10.1103/PhysRevA.34.1

©1986 American Physical Society

Authors & Affiliations

Francis J. Narcowich

  • Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012

R. F. O’Connell

  • Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803-4001

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Vol. 34, Iss. 1 — July 1986

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