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Coherent states of the q-canonical commutation relations
Journal article   Peer reviewed

Coherent states of the q-canonical commutation relations

P. E. T Jørgensen and R. F Werner
Communications in mathematical physics, Vol.164(3), pp.455-471
1994
DOI: 10.1007/BF02101486
url
https://arxiv.org/pdf/funct-an/9303002View
Open Access

Abstract

For the q-deformed canonical commutation relations a(f)a*(g)=(1-q)〈f,g〉 1+qa*(g)a(f) for f, g in some Hilbert space ℋ we consider representations generated from a vector Ω satisfying a(f)Ω=Ω, where φ{symbol}∈ℋ. We show that such a representation exists if and only if {norm of matrix}φ{symbol}{norm of matrix}≦1. Moreover, for {norm of matrix}φ{symbol}{norm of matrix}<1 these representations are unitarily equivalent to the Fock representation (obtained for φ{symbol}=0). On the other hand representations obtained for different unit vectors φ{symbol} 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 natural q-analogue of the Cuntz algebra (obtained for q=0). We discuss the conjecture that, for d<∞, this analogue should, in fact, be equal to the Cuntz algebra itself. In the limiting cases q=±1 we determine all irreducible representations of the relations, and characterize those which can be obtained via coherent states. © 1994 Springer-Verlag.
 Physics Quantum Mechanics Algebraic methods Classical and quantum physics: mechanics and fields Exact sciences and technology General Theory of quantized fields

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