Symplectically Covariant Schrödinger Equation in Phase Space

Details Symplectically Covariant Schrödinger Equation in Phase Space Symplectically Covariant Schrödinger Equation in Phase Space

2005-05-27

arxiv.org/abs/math-ph/0505073v3

Physics

Mathematical Physics

To appear in J Phys A

Scientific paper

10.1088/0305-4470/38/42/007

A classical theorem of Stone and von Neumann says that the Schr\"{o}dinger representation is, up to unitary equivalences, the only irreducible representation of the Heisenberg group on the Hilbert space of square-integrable functions on configuration space. Using the Wigner-Moyal transform we construct an irreducible representation of the Heisenberg group on a certain Hilbert space of square-integrable functions defined on phase space. This allows us to extend the usual Weyl calculus into a phase-space calculus and leads us to a quantum mechanics in phase space, equivalent to standard quantum mechanics. We also briefly discuss the extension of metaplectic operators to phase space and the probabilistic interpretation of the solutions of the phase space Schr\"{o}dinger equation

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