In a relativistic quantum mechanics, the exact solutions of the wave function are very important in the understanding of the physics that can be brought by such solutions.
In the conventional relativistic approach, the interaction of S=0 and S=1 hadrons with different nuclei has been described by the second-order Klein–Gordon (KG) equation for S=0 and Proca equation for S=1 particles. It has been well known that it is very difficult to tackle these second-order equations mathematically and to derive the physics behind them. Therefore, considerable interest in recent years has been devoted to examine the interactions of S=0 and S=1 hadrons with nuclei by using the first-order relativistic Duffin–Kemmer–Petiau (DKP) equation which is very similar to the first-order relativistic Dirac equation .
One important question related to DKP equation concerns the equivalence between its spin 0 and 1 sectors and the theories based on the second-order KG and Proca equations . The Dirac-like DKP equation is not new and dates back to the 1930s. Historically, the loss of interest in the DKP stems from the equivalence of the DKP approach to the Klein–Gordon (KG) and Proca descriptions in on-shell situations, in addition to the greater algebraic complexity of the DKP formulation. However, in 1970s, this supposed equivalence was question in several situations involving breaking of symmetries and hadronic possess, showing that in some cases, the DKP and KG theories can give different results. Moreover, the DKP equation appears to be richer than the KG equation if the interactions are introduced. In this context, alternative DKP-based models were proposed for the study of meson–nucleus interactions, yielding a better adjustment to the experimental data when compared to the KG-based theory . In the same direction, approximation techniques formerly developed in the context of nucleon-nucleus scattering were generalised, giving a good description for experimental data of meson-nucleus scattering . The deuteron-nucleus scattering was also studied using DKP equation, motivated by the fact that this theory suggest a spin1 structure from combining two . In the same context, we can cite the works of [6, 7] on the meson–nuclear interaction and the relativistic model of α-nucleus elastic scattering where they have been treated by the formalism of the DKP theory. Recently, there is a renewed interest in the DKP equation. It has been studied in the context of quantum chromodynamics (QCD) , covariant Hamiltonian formalism , in the causal approach [10, 11], in the context of five-dimensional Galilean invariance , in the scattering of K+ nucleus , in the presence of the Aharonov–Bohm potential [14, 15], in the Dirac oscillator interaction , in the study of thermodynamics properties , on the supersymmetric , and finally, in the presence of some shape of interactions [19–34]. Theses examples in some case break the equivalence between the theories based on the DKP equation and KG and Proca equations.
The Dirac oscillator was for the first time studied by Itô et al. . They considered a Dirac equation in which the momentum is replaced by with being the position vector, m the mass of particle, and ω the frequency of the oscillator. The interest in the problem was revived by Moshinsky and Szczepaniak , who gave it the name of Dirac oscillator (DO) because, in the nonrelativistic limit, it becomes a harmonic oscillator with a very strong spin-orbit coupling term. Physically, it can be shown that the (DO) interaction is a physical system, which can be interpreted as the interaction of the anomalous magnetic moment with a linear electric field [37, 38]. The electromagnetic potential associated with the DO has been found by Benitez et al. . The Dirac oscillator has attracted a lot of interest both because it provides one of the examples of the Dirac’s equation exact solvability and because of its numerous physical applications. As a relativistic quantum mechanics problem, the DO has been studied from many viewpoints, including the covariance properties, complete energy spectrum and corresponding wave functions, symmetry Lie algebra, shift operators, hidden supersymetry, conformal invariance properties, and the completeness of wave function. Relativistic many body problems with DO interactions have been extensively studied with the special emphasis on the mass spectra of mesons (quark-antiquark systems) and baryons (three quark systems). The dynamics of wave packets in a DO has been determined and a relation with the Jayne’s–Cumming model established. The (2+1) space time has also been shown to be an interesting framework for discussing the DO in connection with new phenomena (such as the quantum Hall effect and fractional statistics) in condensed matter physics. Fortunately, Franco-Villafane et al. , in order to vibrate this oscillator, exposed the proposal of the first experimental microwave realization of the one-dimensional DO.
The thermal properties of the one-dimensional DKP equation in a DO interaction was at first considered by . The author has treated the case of spin-1 particles by using a different formalism that used here. This difference has been well explained in the study of Boumali and Chetouani . Encouraged by the experimental realization of a one-dimensional DO, we are interested in the calculations of the thermodynamics properties for the one-dimensional DKP oscillator in all range of temperatures for both spin-0 and spin-1 particles using a method based on the zeta function [41, 42]. This method is used by Dariescu and Dariescu  with the aim of calculating the partition function in the case of the graphene. We note here that the zeta function is applied successfully in different areas of physics, and the examples vary from ordinary quantum and statistical mechanics to quantum field theory . This study is organised as follows: in Section 2, we review the solutions of the one-dimensional DKP oscillator. In Section 3, our numerical results and discussions are described. Finally, Section 4 will be a conclusion.
2 Review of the Solutions of the One-Dimensional DKP Oscillator
2.1 Free DKP Equation
The free DKP equation is given by [45–47]
where ψ is the boson wave function, and M is the mass and the β are matrices which satisfy the following relations:
where gμν=diag(1, –1, –1, –1) is the metric of the Minkowski space. There are 126 linearly independent elements for this algebra. The rank of the matrices turns out to be 16 and the corresponding representation can be decomposed into three irreducible representations of the dimensions 1, 5, and 10. The one-dimensional representation is trivial, while the five- and 10-dimensional representations describe scalar (spin-0) and vector (spin-1) particles, respectively. The advantage of the DKP equation is that it gives a unified description of spin-0 and spin-1 systems. The β are 5×5 matrices in the spin-0 representation and 10×10 matrices in the spin-1 representation. In our case, the βμ matrices are chosen as follows:
For the five-dimensional representation
For the 10-dimensional representation
2.2 Review of the Eigensolutions of One-Dimensional DKP Oscillator
2.2.1 Case of Spin-1
The one-dimensional DKP equation in the presence of a DO potential is [48, 49]
where with (η0)2=I10×10, and I10×10 is the 10×10 unit matrix, and m is the mass of particles. The stationary state ψ is a 10-component wave function of the DKP equation, which can be written as follows:
Following [48, 49], the eigensolutions are decomposed in a two possible cases.
In the first case, where we call it as the even solutions (corresponds to the case when a=−n and B=0 in ), we have
In the the second case, where we denote it as the odd solutions (corresponds to the case when and A=0 in ), we get
The other components are derived from ψ ((17) and (18) in ).
2.2.2 Case of Spin-0
The one-dimensional DKP oscillator for the particles of spin-0 is described by the following equation:
where with (η0)2=I5×5, I5×5 is the 5×5 unit matrix, and m0 is the mass of particles. The stationary state ϕ is a five-component wave function of the DKP equation, which can be written by the following:
As in the case of spin-1, the eigensolutions can be summarised as follows [48, 49]:
For the even solutions case, the eigensolutions are given as follows:
On the other hand, for the odd solutions, we get
Now, and following the (11) and (13), for the particles of spin-1, and (17) and (19), for particles of spin-0, we start to the determinations of all thermodynamics properties of our oscillator in question.
3 Thermal Properties of the Relativistic Harmonic Oscillator
Before we study the thermodynamic properties of the DKP oscillator, we note that the form of the spectrum of energy for both spin-1 and spin-0 particles are the same. Consequently, we are looking, first, for the numerical thermal quantities for the case of particles of spin-1. Then, we extend these results to the case of spin-0 particles.
Also, in order to extract these properties of our oscillator, we will only restrict ourselves to stationary states of positive energy. The reason is threefold: (i) according to the following works of [50–52], the DKP equation possesses an exact Foldy–Wouthuysen transformation (FWT): so, the positive- and negative-energy solutions never mix.
(ii) The (11) and (17) (even solutions) illustrate the form of the energy as a function of n. From Figure 1, one observes that all the energy levels emerge from the positive (negative)-energy continuum so that it is plausible to identify them with particle (antiparticle) levels. Furthermore, it is noticeable from this figure that for positive-energy spectrum one finds that the lowest quantum numbers correspond to the lowest eigenenergies, as it should for particle energy levels. On the other hand, for negative-energy spectrum, this presents a similar behavior but the highest energy levels are labelled by the lowest quantum numbers and are to be identified with antiparticle levels .
Finally, (iii) it is correct that the case of bosons is very different to the one of fermions: all bosons can occupied the same level, contrary to the case of fermions where they are controlled by the Pauli principle. In spite of that, we can extending the method of filling the Dirac negative energy sea for the case of fermions to the case of bosons (Nielsen and Ninomiya [54, 55]): as mentioned by these authors, the bosons states are divided into two sectors: the usual positive sector consisting of states with a positive number of bosons, and the negative sector, consisting of states with a negative number of bosons. Once a state enters the negative sector, it cannot return to the usual positive sector through an ordinary interaction, due to the presence of a barrier.
Thus, according to these arguments, we can assume that only particles with positive energy are available in order to determine the thermodynamic properties of our oscillator in question.
In order to obtain all thermodynamic quantities of our oscillator in question, we concentrate, at first, on the calculation of the partition function Z.
Given the energy spectrum, we can define the partition via
where kB is the Boltzmann constant and
Following the above discussion about the eigensolutions of the DKP oscillator, we consider two possible partition function.
3.1.1 Even Partition Function Zeven
With the following substitutions,
and by using the (11), the partition function becomes
where and r is a parameter that controls the nonrelativistic regime.
Following the same procedure as in [41–43], and using the inverse Mellin transformation of e–x, with
the sum in (24) is transformed into
with and Γ(s) and are, respectively, the Euler and Hurwitz zeta function.
Applying the residues theorem, for the two poles s=0 and s=2, the desired partition function is written down in terms of the Hurwitz zeta function as follows:
Now, using that 
the final partition function is transformed into
3.1.2 Odd Partition Function Zodd
According to the (13), the odd partition function is defined by
With the following substitutions,
and by the same strategy used above, we obtain
Thus, and from both definitions of partition function, the most important thermal quantities, which are are the free energy F, the mean energy U, the entropy S and the specific heat C, can be obtained by using the following expressions:
3.2 Numerical Results and Discussions
We are now ready to present and discuss our results concerning the thermal quantities of the one-dimensional DKP oscillator. All quantities are reported as a function of the reduce temperature τ for the relativistic case with r=1.
The behavior of all thermodynamics quantities for the spin-1 particles, for both solutions, are shown in Figure 2. The difference can be revealed in the curve of the specific heat: when the odd solutions tends very rapidly to the 2KB limit, the even solutions depicted a curvature around a certain temperature τ0. From the curve of the numerical entropy function, no abrupt change, around this temperature, has been identified. This means that the curvature, which is observed in the specific heat curve, does not exhibit or indicate an existence of a phase transition around a τ0 temperature.
Figure 3 depicted the thermal quantities of spin-0 particles: the same argument used in the spin-1 particles can be extended here.
Finally, the Hurwitz zeta function method allows us to compute the vacuum expectation value of the energy . The last is defined by 
Following the above description about the eigensolutions in Section 2, we define two vacuum expectation values of the energy. Therefore, we start with the case of spin-1 particles, and then, the results can be extended automatically for the case of spin-0 particles.
Thus, two possible case can be distinguished as follows:
For the even solutions, the vacuum value of energy, in terms of Hurwitz zeta function, is written as
For the odd solutions, we have
Now, using the asymptotic series corresponding to the Hurwitz zeta function (Appendix), (37) and (38) become
for even solutions, and
for odd solutions.
In Figure 4, we show the as a function of the parameter r for both cases. We can see that the vacuum expectation value of the energy depend on the parameter r.
In this work, we have studied the thermodynamic properties of the one-dimensional DKP oscillator using the Hurwitz zeta function method. According to the work of [48, 49], we distinguish two possible cases concerning the partition function: the even and the odd partition function. Consequently, we have two cases of the thermal properties for both particles of our oscillator, which are depicted in Figures 2 and 3. Also, from the Hurwitz zeta function, we have calculated the vacuum expectation value of the energy of our oscillator. As we can see from Figure 4, the last depends on the parameter r.
Appendix: Some Properties of Zeta Function
The Riemann zeta function is defined by
Nowadays, the Riemann zeta function is just one member of a whole family of zeta function’s (Hurwitz, Epstein, and Selberg). The most important of them is the Hurwitz zeta function ζH given by
where 0<a ≤ 1, is a well-defined series when ℜe(s)>1, and can be analytically continued to the whole-complex plane with one singularity, a simple pole with residue 1 at s=1.
An integral representation is
It can be shown that ζH (s, α) has only one singularity, namely a simple pole at s=1 with residue 1 and that it can be analytically continued to the rest of the complex s-plane.
Also, we can shown that ζH (s, α) have the following properties:
Br(a) being the Bernoulli polynomials. The asymptotic series corresponding to the Hurwitz zeta function is given by the following:
with Bk are Bernoulli’s numbers.
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Published Online: 2015-08-22
Published in Print: 2015-10-01