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arXiv:0710.3544 · International Journal of Geometric Methods in Modern Physics

WIGNER FUNCTIONS, FRESNEL OPTICS, AND SYMPLECTIC CONNECTIONS ON PHASE SPACE

J. M. Isidro

Published Aug 1, 2008• 1 citation
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Abstract

We prove that Wigner functions contain a symplectic connection. The latter covariantizes the symplectic exterior derivative on phase space. We analyze the role played by this connection and introduce the notion of local symplectic covariance of quantum-mechanical states. This latter symmetry is at work in the Schrödinger equation on phase space.

DOI
10.1142/s0219887808003089

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We discuss a family of quasidistributions (sss-ordered Wigner functions of Agarwal and Wolf [Phys. Rev. D 2, 2161 (1970); Phys. Rev. D 2, 2187 (1970); Phys. Rev. D 2, 2206 (1970)]) and its connection to the so-called phase space representation of the Schr\"odinger equation. It turns out that although Wigner functions satisfy the Schr\"odinger equation in phase space, they have a completely different interpretation.