IJPAM: Volume 38, No. 1 (2007)

INFINITE DIMENSIONAL HEISENBERG GROUP
ALGEBRA AND FIELD-THEORETIC STRICT
DEFORMATION QUANTIZATION

Ernst Binz$^1$, Reinhard Honegger$^2$, Alfred Rieckers$^3$
$^1$Faculty of Mathematics and Informatics
University of Mannheim
A5/6, Mannheim, D-68131, GERMANY
e-mail: binz@rumms.uni-mannheim.de
$^{2,3}$Faculty of Theoretical Physics
University of Tübingen
Auf der Morgenstelle 14
Tübingen, D-72076, GERMANY
e-mail: alfred.rieckers@uni-tuebingen.de


Abstract.For arbitrary dimensional pre-symplectic test function spaces $(E, \gs)$ the discrete Heisenberg group C*-algebra is introduced and investigated. The (partially) regular and partially decomposable representations of the latter are analyzed. The Heisenberg group C*-algebra is shown to be *-isomorphic to a global C*-algebra generated by a continuous field of C*-Weyl algebras (incorporating simultaneously all values of the Planck parameter $\hbar\in\dR$). The Heisenberg group (algebra) approach is compared with the method of strict and continuous deformation quantization. A related quantization scheme, using the (possibly infinite dimensional) Heisenberg Lie algebra is outlined, where a correspondence between the quantum mechanical $\hbar$-sectors and certain leaves of the classical phase space shows up.

Received: April 30, 2007

AMS Subject Classification: 81T05, 58B25, 43A80

Key Words and Phrases: infinite dimensional Heisenberg group and group algebra, continuous field of C*-Weyl algebras, inequivalent representations, strict deformation quantization, infinite dimensional phase space

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2007
Volume: 38
Issue: 1