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Communications in Mathematical Physics

, Volume 2, Issue 1, pp 31–48 | Cite as

C*-Algèbres des systèmes canoniques. I

  • G. Loupias
  • S. Miracle-Sole
Article

Abstract

The “twisted convolution” associated with the Weyl form of the canonical commutation relations forn degrees of freedom is decribed using ordinary convolution on a nilpotent central extension of additive phase space by the one-dimensional torus. Twisted convolution determines severalC*-algebras of quantum mechanical observables amongst which we study especially the algebra ℒ2(\(\mathfrak{E}\), σ) consisting of the ℒ2-functions on phase space and mapped isometrically onto the Hilbert-Schmidt-operators by the Schrödinger representation. The two last sections of the paper deal with “phase space quantum mechanics” from the point of view of twisted convolution: theWigner-Moyal formalism and the entire function formalism ofBargmann andSegal.

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Copyright information

© Springer-Verlag 1966

Authors and Affiliations

  • G. Loupias
    • 1
  • S. Miracle-Sole
    • 1
  1. 1.Physique ThéoriqueUniversité D'Aix-MarseilleFrance

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