Advances in High Energy Physics

Volume 2015 (2015), Article ID 901675, 9 pages

http://dx.doi.org/10.1155/2015/901675

## On the Thermodynamic Properties of the Spinless Duffin-Kemmer-Petiau Oscillator in Noncommutative Plane

Department of Physics, Guizhou University, Guiyang 550025, China

Received 15 July 2015; Revised 30 September 2015; Accepted 4 October 2015

Academic Editor: Elias C. Vagenas

Copyright © 2015 Zhi Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

The Duffin-Kemmer-Petiau oscillator for spin 0 particle in noncommutative plane is analyzed and the energy eigenvalue of the system is obtained by employing the functional analysis method. Furthermore, the thermodynamic properties of the noncommutative DKP oscillator are investigated via numerical method and the influence of noncommutative space on thermodynamic functions is also discussed. We show that the energy spectrum and corresponding thermodynamic functions of the considered physical systems depend explicitly on the noncommutative parameter which characterizes the noncommutativity of the space.

#### 1. Introduction

The Duffin-Kemmer-Petiau (DKP) equation introduced by Kemmer et al. [1–3] in the 1930s is a relativistic wave equation which describes spin 0 and spin 1 bosons in the description of the standard model. The DKP equation for spin 0 hadrons with nuclei is closely linked to the conventional second-order Klein-Gordon (KG) equation, which has more complex algebraic structure than the latter [4], and the DKP equation enjoys a richness of couplings not capable of being expressed in the Klein-Gordon and Proca equations [5, 6]. In addition, several situations researched in past years involving hadronic process and breaking of symmetries show that in some situations the DKP equation and KG equation can give some different results. For these reasons, there has been a growing interest in studying the DKP equation [7–13].

On the other hand, the noncommutative (NC) geometry has been put forward in both quantum mechanics and field theory in past years, which is growing rapidly. The coordinate noncommutativity was first introduced by Snyder [14] as a regularized mechanism to improve the problem of infinite self-energy in quantum field theory. But with the development of renormalization theory, this concept became not popular until Connes [15] investigated Yang-Mills theories in noncommutative (NC) space. In recent years, there has been a renewed interest in NC geometry because of the discovery in string theory and matrix model of M-theory [16]. When studying NC space, one considers the coordinate to be noncommutative, where the coordinate operators satisfy the commutation relation , with being a constant antisymmetric matrix, which represent the noncommutative property of NC space, and play an analogous role with in the usual quantum mechanics. Recently, various aspects of both NC classical [17] and quantum [18] mechanics have been investigated by large number of papers, devoted to exploring the role of NC parameter in the physical observables. For example, classical Newton mechanics in the noncommutative space was studied in [19, 20], a particle confined by a quadratic potential in the generalized noncommutative plane was investigated from both the classical and the quantum aspects in [21], the noncommutative harmonic oscillators were discussed in detail in [22–24], and [25] studied the Aharonov-Bohm effect in a class of noncommutative theories.

The DKP equation in the presence of the Dirac oscillator interaction called DKP oscillator for spin zero and spin one particles has also been investigated in the NC formalism in publications [26–28], with the purpose of obtaining the energy spectrum and studying the noncommutative effect on relativistic quantum mechanics. In addition, the studies of thermal properties of the quantum oscillator have been carried out in both commutative and NC space by several authors these years [29–34]. By employing the numerical method based on the Euler-MacLaurin formula, one can calculate the associated partition function and obtain the thermodynamic functions, such as the free energy, mean energy, heat capacity, and entropy. In spite of several papers that have been published concerning the thermodynamics properties of quantum oscillator, as far as we know they have not reported the case of DKP oscillator for spin 0 particle in NC space. It will be interesting to study thermodynamic properties of the spinless DKP oscillator in NC formalism.

This work is organized as follows: In Section 2 we briefly review the NC quantum mechanics, where the NC space is introduced. Subsequently, we analyze the solutions of the DKP oscillator for spin 0 particle in NC plane and obtain the corresponding energy spectrum. The thermodynamic properties of the NC DKP oscillator are investigated by employing the Euler-MacLaurin method in Section 3. To have an intuitive understanding for the statistical properties of the physical system, we depict the numerical results of the thermodynamic functions with several figures and discuss the effect of the NC parameter on thermal properties in Section 4. In addition, we also discuss the case of low temperature limit based on the Hurwitz zeta function formula. Finally, the work is summarized in last section.

#### 2. The DKP Oscillator for Spin 0 Particle in NC Plane

##### 2.1. Review of the NC Formalism

In NC space, the canonical variables satisfy the following commutation relations:with being the antisymmetric NC parameter, representing the noncommutativity of the space, and , are the coordinate and momentum operators in the NC space.

One way to deal with the problem of noncommutative space is via the star product or Moyal-Weyl product on the commutative space functions:where and are arbitrary infinitely differentiable functions.

Then the time-independent Schrödinger equation in the NC space can be written in the formThe Moyal-Weyl product may be replaced by a Bopp shift of the formwhere and are the position and momentum operators in the usual quantum mechanics which meet the canonical Heisenberg commutation relations.

Then, via above Bopp shift, the Schrödinger equation can be written asTherefore, (5) is actually defined on the commutative space, and the NC effect may be calculated from the term that contains .

##### 2.2. The Solution of the DKP Oscillator for Spin 0 Particle in NC Plane

The relativistic DKP equation for a free boson of mass is given by [35, 36]with and being the DKP matrices which satisfy the following algebra relation:where is the metric tensor in Minkowski space. For spin 0 particle, are matrices written aswith , , and being , , and zero matrices, respectively. Other matrices in (8) are given as follows:Introducing the nonminimal substitution in (6), we get the DKP oscillator aswith being the oscillator frequency and .

It is easy to get the 2D DKP oscillator from the above equation:Considering the NC formalism and via the Bopp shift (4), the DKP oscillator of (11) can be written asFor a boson of spin 0, the spinor is a vector with five components [9, 10], which has the form .

Substituting into (12), one can obtain the following five coupled equations:Combination of (13) givesUsing polar coordinates, the component can be written asand substituting it into (14), we can obtain the radial equationwithIntroducing a new variable , one can get the following second-order differential equation:The equation can be analyzed asymptotically at the critical points and , allowing us to construct an equation whose form is the same as the confluent hypergeometric equation. This analysis shows that it is reasonable to write asSubstituting the above function into (18), one obtainsObviously, this is confluent hypergeometric equation whose solution is the confluent hypergeometric functionTo obtain the solution of (20), the polynomial series should be convergent. It requires the independent term to be negative integer [37]:Through a simple calculation we obtain the energy of the system as follows:The eigenvalues of considered physical system depend distinctly on the NC parameter which characterizes the noncommutativity of the space. We find that the energy spectrum is exact and not degenerate due to the NC effect.

Then can be written in the form of the confluent hypergeometric function aswhere has already been given above, and through straightforward calculation, the expression of other components of the spinor and the final form can be obtained easily.

#### 3. Thermodynamic Properties of the DKP Oscillator for Spin 0 Particle in NC Plane

In this section, we study the thermodynamic properties of above DKP oscillator. The energy eigenvalues of the DKP oscillator in NC space arewith

In Figure 1, we plot the energy spectra versus quantum number for different values of the NC parameter . Positive- and negative-energy levels correspond to the case of particle and antiparticle, respectively. We see that the energy increases monotonically with quantum number and the tendency of the spectrum can be observed for large quantum numbers. The effect of the NC parameter on the energy levels is observable, where corresponding to the case of commutative space.