Coherent states of theq-canonical commutation relations |
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Authors: | P. E. T. Jørgensen R. F. Werner |
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Affiliation: | (1) Department of Mathematics, University of Iowa, 52242 Iowa City, IA, USA;(2) FB Physik, Universität Osnabrück, D-49069 Osnabrück, Germany |
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Abstract: | For theq-deformed canonical commutation relationsa(f)a(g)=(1-q)f,g 1+qa(g)a(f) forf, g in some Hilbert space we consider representations generated from a vector satisfyinga(f)=>, where . We show that such a representation exists if and only if 1. Moreover, for <1 these representations are unitarily equivalent to the Fock representation (obtained for =0). On the other hand representations obtained for different unit vectors are disjoint. We show that the universal C*-algebra for the relations has a largest proper, closed, two-sided ideal. The quotient by this ideal is a naturalq-analogue of the Cuntz algebra (obtained forq=0). We discuss the conjecture that, ford<, this analogue should, in fact, be equal to the Cuntz algebra itself. In the limiting casesq=±1 we determine all irreducible representations of the relations, and characterize those which can be obtained via coherent states.Supported in part by the NSF(USA), and NATOAvailable by anonymous FTPfrom nostrom.physik.Uni-Osnabrueck.DE |
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