https://doi.org/10.1140/epjp/i2019-12770-3
Regular Article
Coupled oscillators in non-commutative phase space: Path integral approach
1
LRPPS Laboratory, Department of Physics, University Kasdi Merbah of Ouargla, BP 511 Ouargla, 30000, Ouargla, Algeria
2
Theoretical Physics Laboratory, Department of Physics, University of Jijel, BP 98 Ouled Aissa, 18000, Jijel, Algeria
3
Institut de Recherche en Informatique, Mathématiques, Automatique et Signal, Université de Haute Alsace, Mulhouse, France
* e-mail: mewalid@yahoo.com
Received:
5
November
2018
Accepted:
20
May
2019
Published online:
22
August
2019
We present the path integral techniques in a non-commutative phase space and illustrate the calculation in the case of an exact problem of the coupled oscillator in two dimensions. The non-commutativity, with respect to Poisson (classical) and Heisenberg (quantum) brackets, in this phase space, is governed by two small constant parameters. They characterize the geometric deformation of this space. The path integral is formulated in a mixed representation due to the non-commutativity of the coordinates on one hand and those of the momentum on the other hand. Using a canonical linear transformation in this non-commutative phase space, we retrieve the commutative phase space properties by which the study becomes more suitable. The case of the non-commutative coupled harmonic oscillator in two dimensions is treated and the thermodynamics properties of the assembly of such oscillators are derived too.
© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature, 2019
