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Classical and quantum q-deformed physical systems

Theoretical Physics

Abstract

On the basis of non-commutative q-calculus, we investigate a q-deformation of the classical Poisson bracket in order to formulate a generalized q-deformed dynamics in the classical regime. The obtained q-deformed Poisson bracket appears invariant under the action of the q-symplectic group of transformations. Within this framework we introduce the q-deformed Hamilton equations and we derive the evolution equation for some simple q-deformed mechanical systems governed by a scalar potential dependent only on the coordinate variable. It appears that the q-deformed Hamiltonian, which is the generator of the equation of motion, is generally not conserved in time but, in correspondence, a new constant of motion is generated. Finally, by following the standard canonical quantization rule, we compare the well-known q-deformed Heisenberg algebra with the algebra generated by the q-deformed Poisson bracket.

Keywords

Quantum Group Poisson Bracket Leibniz Rule Canonical Quantization Heisenberg Algebra 
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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • A. Lavagno
    • 1
    • 2
  • A.M. Scarfone
    • 1
    • 3
  • P. Narayana Swamy
    • 4
  1. 1.Dipartimento di FisicaPolitecnico di TorinoTorinoItaly
  2. 2.Sezione di TorinoINFN-Istituto Nazionale di Fisica NucleareTorinoItaly
  3. 3.Sezione di TorinoINFM/CNR Istituto Nazionale di Fisica della MateriaTorinoItaly
  4. 4.Southern Illinois UniversityEdwardsvilleUSA

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