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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Références
Kastler, D.: Commun. math. Phys.1, 14 (1965).
Mackey, G. W.: Acta Math.99, 265 (1958).
Wigner, E.: Phys. Rev.40, 749 (1932).
Moyal, J. E.: Proc. Cambridge Phil. Soc.45, 99 (1949).
Baker, G. A.: Phys. Rev.109, 2198 (1958).
Bargmann, V.: Commun. Pure and Appl. Math.14, 187 (1961).
Segal, I.: Illinois J. Math.6, 500 (1962).
Michel, L.: Invariance in Quantum Mechanics and Group extensions. Manuscript, I.H.E.S. Bures-sur-Yvette. Seine et Oise, France.
Chevalley, C.: Théorie des groupes de Lie. Tome III. Actualités Scientifiques et Industrielles. Paris: Hermann 1955.
Dixmier, J.: LesC*-alèbres et leurs représentation. Paris: Gauthier-Villars 1964.
Fell, J. M. G.: Proc. Am. Math. Soc.13, 93 (1962).
Hewitt, E., andK. A. Ross: Abstract Harmonic Analysis. Berlin-Göttingen-Heidelberg: Springer 1963.
Rudin, W.: Fourier Analysis on Groups. New York: John Wiley 1962.
Dixmier, J.: Les Algèbres d'opérateurs dans l'Espace Hilbertien. Paris: Gauthier-Villars 1957.
Weyl, H.: The Theory of Groups and Quantum Mechanics. London: Methnen 1931.
Bourbaki, N.: Espaces vectoriels topologiques. Fascicule de Résultats. Actualités Scientifiques et Industrielles. Paris: Hermann 1955.
Rickart, C. E.: General Theory of Banach Algebras. Princeton-London-Toronto: Van Nostrand (1960).
Schwartz, L.: Théorie des Distributions. Tome II. Actualités Scientifiques et Industrielles. Paris: Hermann 1959.
Souriau, J. M.: Commun. math. Phys1, 374–398 (1966).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Loupias, G., Miracle-Sole, S. C*-Algèbres des systèmes canoniques. I. Commun.Math. Phys. 2, 31–48 (1966). https://doi.org/10.1007/BF01773339
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01773339