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Communications in Mathematical Physics

, Volume 164, Issue 3, pp 455–471 | Cite as

Coherent states of theq-canonical commutation relations

  • P. E. T. Jørgensen
  • R. F. Werner
Article

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 Ω satisfyinga(f)Ω=<f, ϕ>Ω, where ϕ∈. 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.

Keywords

Neural Network Statistical Physic Complex System Unit Vector Nonlinear Dynamics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • P. E. T. Jørgensen
    • 1
  • R. F. Werner
    • 2
  1. 1.Department of MathematicsUniversity of IowaIowa CityUSA
  2. 2.FB PhysikUniversität OsnabrückOsnabrückGermany

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