Symplectically Covariant Schrödinger Equation in Phase Space

Abstract

A classical theorem of Stone and von Neumann says that the Schrödinger 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 Schrödinger equation

To appear in J Phys A

References in corpus (2)

• Path integration over closed loops and Gutzwiller's trace formula
• Extended Weyl Calculus and Application to the Phase-Space Schrödinger Equation

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• Asymptotic Approximations for the Phase Space Schrodinger Equation
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• Star products on generalised complex manifolds
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