Eur. Phys. J. C (2006) 47: 253-261
https://doi.org/10.1140/epjc/s2006-02557-y

Theoretical Physics

Classical and quantum q-deformed physical systems

¹ Dipartimento di Fisica, Politecnico di Torino, Torino, Italy
² Sezione di Torino, INFN-Istituto Nazionale di Fisica Nucleare, Torino, Italy
³ Sezione di Torino, INFM/CNR Istituto Nazionale di Fisica della Materia, Torino, Italy
⁴ Southern Illinois University, Edwardsville, IL, 62026, USA

^* e-mail: antonio.scarfone@polito.it

Received: 29 July 2005
Revised: 9 March 2006
Published online: 19 May 2006

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.