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arXiv:0807.2510 · Journal of Physics A Mathematical and Theoretical

An interacting particles system revisited in the framework of theq-deformed algebra

Antonio Maria Scarfone, P Narayana Swamy

Published Jun 16, 2008• 30 citations
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Abstract

We discuss the possibility of interpreting a q-deformed non-interacting system as incorporating the effects of interactions among its particles. This can be accomplished, for instance, in an ensemble of $q$-Bosons by means of the virial expansion of a real gas in powers of the deformed parameter. The lowest order virial coefficient reduces to the case of the standard, non-interacting Bose gas, while the higher order virial coefficients contain effects arising from the interaction. The same picture can be drawn in a quantum mechanical system where it is shown that the q-deformed momentum can be expanded in a series contains high-order powers of the standard quantum phase-space variables. Motivated by this result, we introduce, in the classical framework, a transformation relating the momentum of a free system with the momentum of an interacting system. It is shown that the canonical quantization applied to the interacting system imply a q-deformed quantization for the free system.

DOI
10.1088/1751-8113/41/27/275211

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We discuss the possibility of interpreting a q-deformed non-interacting system as incorporating the effects of interactions among its particles. This can be accomplished, for instance, in an ensemble of qqq-Bosons by means of the virial expansion of a real gas in powers of the deformed parameter. The lowest order virial coefficient reduces to the case of the standard, non-interacting Bose gas, while the higher order virial coefficients contain effects arising from the interaction. The same picture can be drawn in a quantum mechanical system where it is shown that the q-deformed momentum can be expanded in a series contains high-order powers of the standard quantum phase-space variables. Motivated by this result, we introduce, in the classical framework, a transformation relating the momentum of a free system with the momentum of an interacting system. It is shown that the canonical quantization applied to the interacting system imply a q-deformed quantization for the free system.
In the framework of the q -deformed oscillator algebra we investigate the nature of the interaction among the particles of a free q -boson gas with a view to interpreting the interaction in terms of the general properties of the q -deformation. It is shown that many thermodynamic quantities characterizing the q -boson system can be expressed as power series in the number of particles operator or as a series in the deformation parameter q , implying that the q -deformation can originate from the interaction among the particles of the ensemble. As an example, we derive the virial expansion of the q -boson gas, showing that the lowest order coefficient reduces to the standard virial coefficient of an undeformed boson gas, while the higher order coefficients contain effects arising from interaction.
In the study of many-particle systems both the interaction of particles can be essential and such feature as their internal (composite) structure. To describe these aspects, the theory of deformed oscillators is very efficient. Viewing the particles as p,q-deformed bosons, in the corresponding p,q-Bose gas model we obtain in explicit form virial expansion along with the 2nd to 5th virial coefficients. The obtained virial coefficients depend on the deformation parameters p,q in the form symmetric under p<->q, and at p->1, q->1 turn into those known for usual bosons. Besides real parameters, we analyze the case of complex mutually conjugate p and q and find interesting implications. Also, the critical temperature is derived (for the p,q-Bose gas) and compared with the T_c of standard case of Bose condensation. Similar results are presented for the deformed Bose gas model of the Tamm-Dancoff type.
In the study of many-particle systems both the interaction of particles can be essential and such feature as their internal (composite) structure. To describe these aspects, the theory of deformed oscillators is very efficient. Viewing the particles as p, q-deformed bosons, in the corresponding p, q-Bose gas model we obtain in explicit form virial expansion along with the 2nd to 5th virial coefficients. The obtained virial coefficients depend on the deformation parameters p, q in the form symmetric under p ↔ q, and at p → 1, q → 1 turn into those known for usual bosons. Besides real parameters, we analyze the case of complex mutually conjugate p and q and find interesting implications. Also, the critical temperature is derived (for the p, q-Bose gas) and compared with the Tc of standard case of bosons condensation. Similar results are presented for the deformed Bose gas model of the Tamm–Dancoff (TD) type.
We consider the deformed Bose gas model with the deformation structure function that is the combination of a q-deformation and a quadratically polynomial deformation. Such a choice of the unifying deformation structure function enables us to describe the interacting gas of composite (two-fermionic or two-bosonic) bosons. Using the relevant generalization of the Jackson derivative, we derive a two-parametric expression for the total number of particles, from which the deformed virial expansion of the equation of state is obtained. The latter is interpreted as the virial expansion for the effective description of a gas of interacting composite bosons with some interaction potential.
Two approaches to the construction of thermodynamics in the framework of the q- and м-formalisms, which correspond to certain deformations of the algebra of the creation–annihilation operators, have been considered. By comparing the obtained results, an approximate, independent of the space dimension, correspondence was revealed between the second virial coefficients for the ideal q- and м-deformed Bose gases. The corresponding discrepancy arises only at the level of the third virial coefficient. A method for emulating the м-deformed Bose gas up to the third virial coefficient inclusive by means of the two-parametric nonadditive Polychronakos statistics is demonstrated.
Dynamical systems associated with a q-deformed two dimensional phase space are studied as effective dynamical systems described by ordinary variables.In quantum theory, the momentum operator in such a deformed phase space becomes a difference operator instead of the differential operator.Then, using the path integral representation for such a dynamical system, we derive an effective short-time action, which contains interaction terms even for a free particle with q-deformed phase space.We also analyze the eigenvalue problem for a particle with q-deformed phase space confined in a compact space.Under some boundary conditions of the compact space, there arises fairly different structures from that of the q = 1 case in the energy spectrum of the particle and in the corresponding eigenspace.