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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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