Communications in Mathematical Physics

, Volume 136, Issue 3, pp 487–499 | Cite as

The algebra of Weyl symmetrised polynomials and its quantum extension

  • I. M. Gelfand
  • D. B. Fairlie


The Algebra of Weyl symmetrised polynomials in powers of Hamiltonian operatorsP andQ which satisfy canonical commutation relations is constructed. This algebra is shown to encompass all recent infinite dimensional algebras acting on two-dimensional phase space. In particular the Moyal bracket algebra and the Poisson bracket algebra, of which the Moyal is the unique one parameter deformation are shown to be different aspects of this infinite algebra. We propose the introduction of a second deformation, by the replacement of the Heisenberg algebra forP, Q with aq-deformed commutator, and construct algebras ofq-symmetrised Polynomials.


Neural Network Phase Space Complex System Nonlinear Dynamics Quantum Computing 
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Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • I. M. Gelfand
    • 1
  • D. B. Fairlie
    • 1
  1. 1.Harvard UniversityCambridgeUSA

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