Abstract
For theq-deformed canonical commutation relationsa(f)a ✢ (g)=(1-q)〈f,g〉 1+qa ✢ (g)a(f) forf, g in some Hilbert spaceℋ we consider representations generated from a vector Ω satisfyinga(f)Ω=<f, ϕ>Ω, where ϕ∈ℋ. We show that such a representation exists if and only if ‖ϕ‖≦1. Moreover, for ‖ϕ‖<1 these representations are unitarily equivalent to the Fock representation (obtained for ϕ=0). On the other hand representations obtained for different unit vectors ϕ 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.
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Communicated by H. Araki
Supported in part by the NSF(USA), and NATO
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Jørgensen, P.E.T., Werner, R.F. Coherent states of theq-canonical commutation relations. Commun.Math. Phys. 164, 455–471 (1994). https://doi.org/10.1007/BF02101486
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DOI: https://doi.org/10.1007/BF02101486