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


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.


Neural Network Statistical Physic Complex System Unit Vector Nonlinear Dynamics 
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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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