Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 4 (2008), 047, 37 pages      arXiv:0805.4536      https://doi.org/10.3842/SIGMA.2008.047
Contribution to the Special Issue on Deformation Quantization

Field-Theoretic Weyl Deformation Quantization of Enlarged Poisson Algebras

Reinhard Honegger, Alfred Rieckers and Lothar Schlafer
Institut für Theoretische Physik, Universität Tübingen, Auf der Morgenstelle 14, D-72076 Tübingen, Germany

Received December 20, 2007, in final form May 06, 2008; Published online May 29, 2008

Abstract
C*-algebraic Weyl quantization is extended by allowing also degenerate pre-symplectic forms for the Weyl relations with infinitely many degrees of freedom, and by starting out from enlarged classical Poisson algebras. A powerful tool is found in the construction of Poisson algebras and non-commutative twisted Banach-*-algebras on the stage of measures on the not locally compact test function space. Already within this frame strict deformation quantization is obtained, but in terms of Banach-*-algebras instead of C*-algebras. Fourier transformation and representation theory of the measure Banach-*-algebras are combined with the theory of continuous projective group representations to arrive at the genuine C*-algebraic strict deformation quantization in the sense of Rieffel and Landsman. Weyl quantization is recognized to depend in the first step functorially on the (in general) infinite dimensional, pre-symplectic test function space; but in the second step one has to select a family of representations, indexed by the deformation parameter h. The latter ambiguity is in the present investigation connected with the choice of a folium of states, a structure, which does not necessarily require a Hilbert space representation.

Key words: Weyl quantization for infinitely many degrees of freedom; strict deformation quantization; twisted convolution products on measure spaces; Banach-*- and C*-algebraic methods; partially universal representations.

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