Wu, Pierre S. Striehl, Mara G. Prentiss'
title: 'A Weyl function approach to matter-wave coherence and Talbot-Lau effects'
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Introduction
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In quantum phase space methods, operators in a Hilbert space are mapped to functions in a “mock” phase space [@mockphasespace84]. One example is the Wigner representation which is particularly useful for establishing quantum-classical correspondence. In the Wigner representation, the Wigner function represents a density matrix operator $\hat \rho$ with a quasi-probability distribution in classical phase space. The Wigner functions are conveniently applied in semiclassical methods [@wignerheller76; @berrywigner77], to visualize or characterize quantum states, and experimentally allows one to recover information of quantum systems through tomographic measurements of their phase-space distributions  [@wignertomo93].

In the related Weyl function representation [@mockphasespace84; @dispoperators], the Hilbert space operators are mapped to functions in a “reciprocal phase space” where each function is simply a Fourier transform of those in the Wigner representation. In particular, Weyl function $W({\bf q},{\bf x})$ is the Fourier transform of the Wigner function $w({\bf r},{\bf p})$ over the phase space coordinates $({\bf r},{\bf p})$ [@analyticrepresentations; @weylChountasis]. The Weyl function can also be expressed as: $$\begin{array}{c}
W({\bf q},{\bf x})={\bf Tr}(\hat \rho \hat D({\bf q},{\bf x})),
\\ \hat D({\bf q},{\bf x})  = e^{i 