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The algebra of Weyl symmetrised polynomials and its quantum extension

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Abstract

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.

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Author notes

  1. I. M. Gelfand

    Present address: Moscow State University, Moscow, USA

  2. D. B. Fairlie

    Present address: University of Durham, UK

Authors and Affiliations

  1. Harvard University, 02138, Cambridge, MA, USA

    I. M. Gelfand & D. B. Fairlie

Authors
  1. I. M. Gelfand
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  2. D. B. Fairlie
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Additional information

Communicated by A. Jaffe

Research supported in part by the Department of Energy under Grant DE/FG02/88/ER25065, and by a grant from the Alfred P. Sloan Foundation and the Fulbright Commission

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Gelfand, I.M., Fairlie, D.B. The algebra of Weyl symmetrised polynomials and its quantum extension. Commun.Math. Phys. 136, 487–499 (1991). https://doi.org/10.1007/BF02099070

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  • DOI: https://doi.org/10.1007/BF02099070

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