Weyl and Wigner functions in an extended phase-space formalism

Abstract

The Wigner function $W(x,p)$ is related to the Weyl function $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{W}(X,P)$ through a two-dimensional Fourier transform. Several properties for these functions are proved which lead naturally to an extended phase space $x\ensuremath{-}p\ensuremath{-}X\ensuremath{-}P.$ Uncertainty relations for $\ensuremath{\delta}x\ensuremath{\delta}P$ and for $\ensuremath{\delta}p\ensuremath{\delta}X$ are studied. They reduce to the usual uncertainty relation in the special case of pure states, but are different for mixed states. Thermal density matrices obey these uncertainty relations as equalities.