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Thermodynamic properties in higher-derivative electrodynamics

In this work, we study the thermodynamic properties of a photon gas in a heat bath within the context of higher-derivative electrodynamics, i.e., we study both Podolsky and its generalization with Lorentz violation. For doing so, we use the concept of the number of available states for the system in order to construct the so-called partition function. Next, we calculate the main thermodynamic functions: the Helmholtz free energy, the mean energy, the entropy, and the heat capacity. In particular, we verify that there exist significant changes in the heat capacity and the mean energy due to Lorentz violation. Additionally, the modification of the black body radiation and the correction to the Stefan-Boltzmann law in the context of the primordial inflationary universe are provided for both theories as well.


I. INTRODUCTION
The concept of the mass is a key issue in theoretical physics particularly within the context of particle physics. For instance, the so-called Higgs mechanism [1,2] is the most known approach to generate mass for the particles from a genuine gauge-invariant theory.
After all, we can highlight the fact that interactions between the constituents of matter are usually expressed in terms of a gauge theory which is supposed to be massless. With a different viewpoint, the presence of a massive vector field, commonly ascribed to the Proca's model, possesses many consequences well encountered in the literature [3,4]. Among them, since the electromagnetic interactions are described in terms of the symmetry group, namely U (1), all entire Quantum Electrodynamics should be reexamined if the massive modes were taken into account [5].
On the other hand, if one deals with the Podolsky electrodynamics [6][7][8], one obtains a remarkable feature, which is bringing about a massive mode without losing the so-called gauge symmetry. Moreover, it was also Podolsky who first attempted to describe the interpretation of this massive mode, i.e., it was depicted as a neutrino. In this theory, the propagator has two poles, one corresponding to the massless mode and the other one associated with the massive mode. In this sense, considering the classical approach, the latter has a feature of removing singularities associated with the pointlike self-energy. Nevertheless, if one regards quantum properties, one obtains the appearance of ghosts [9]. The appropriated gauge condition to address Podolsky electrodynamics is no longer the common Lorenz gauge but rather a modified one [10], being consistent with the existence of 5 degrees of freedom i.e., 2 of them related to the massless photon and the rest of 3 related to the massive mode [11]. Besides, it is worth mentioning that, in the presence of the Podolsky term, there exist other remarks involving quantum field theory in the context of renormalization [12], path integral quantization and fine-temperature approach [13][14][15], multipole expansions [16] and others [17][18][19][20][21].
About twenty years up to now, the Lorentz-violating contributions of mass dimensions 3 and 4 have been taken into account within both theoretical and phenomenological scenarios in the photon and lepton sectors [22]. Recently, theories with higher-dimension operators have received much attention after a generalized approach proposed by Mewes and Kostelecký in Ref. [23]. In the CPT-even photon sector, the leading-order contributions in an expansion in terms of additional derivatives are the dimension-6 ones. Hence, these are also the most prominent ones that could play a role in nature if higher-derivative Lorentz violation existed [11]. In such a way, there is a lack in the literature concerning the study of its respective thermodynamic properties. In this panorama, since it was given recent significance of working on a higher-order derivative theory, it is important to accomplish further investigations. Therefore, the physical implications of the thermodynamic properties of such theories should be taken into account in order to possibly address some fingerprints of a new physics which can be mapped into future electronic devices either in condensed matter physics or in statistical mechanics.
In this sense, this work provides a study in such a direction highlighting the behavior of the main thermodynamic functions such as the Helmholtz free energy, the mean energy, the entropy, and the heat capacity. It is worth mentioning that the corrections to the black body radiation and the so-called Stefan-Boltzmann law are analyzed as well.

A. The model
The four-dimensional Lagrangian density of the Podolsky electrodynamics is written as [6][7][8] where F µν F µν = ∂ µ A ν − ∂ ν A µ is the usual Maxwell field strength, A µ is the vector field and θ is the Podolsky's parameter with mass dimension -1, and J µ is the conserved current. In addition, in Refs. [10,14] the authors accomplished a remarkable classical analysis of this theory i.e., they studied interparticle potential between sources, quantization, generalization of such a theory in the framework of Dirac's theory of constrained systems and others. In addition, the equation of motion can be written as Here, it must be pointed out that there exists a distinguished characteristic if compared with the Proca's theory which is the generation of a massive mode without losing its gauge symmetry. Besides, the propagator in the momentum space is given by where one verifies the presence of both Maxwell k 2 = 0 and Podolsky 1 − θ 2 k 2 = 0 poles.
The following dispersion relation is Furthermore, note that when one considers θ → 0, the standard dispersion relation is recovered i.e., k 2 = 0. Besides, Eq. (4) may be rewritten simply as where this equation shows that we have a different state equation which must change the thermodynamic properties of our system due to the fact that the relation between energy and momentum is no longer ascribed to be the usual one. It is worth to note that from Eq. (5), we choose the -1 configuration, since otherwise we will not have the contribution of parameter θ. Moreover, in what follows, we examine a photon gas in a volume Γ and instead of dealing with a quantizing momentum due to the boundary conditions, rather we assume a continuous momentum spectrum as it is commonly used in the literature [24][25][26].
We use the fact that the statistical mechanics tells us that the relation between energy and momentum has a remarkable aspect in evaluating the dependence of the pressure as a function of the energy density. In the next subsection, we procedure with the purpose of obtaining the microstates of the system in order to calculate the partition function which suffices to address all thermodynamic properties. In addition, it is noteworthy that in different contexts the thermodynamic functions were calculated as well [27][28][29][30][31].

B. Thermodynamic properties
We start with the construction of the partition function for the sake of obtaining the following thermodynamic properties i.e., the Helmholtz free energy, the mean energy, the entropy, and the heat capacity. In this sense, we use the traditional method for doing so; we use the concept of the number of microstates of the system [24]. Generically, it can be written as where ζ is the spin multiplicity which in our case will be considered as the photon sector i.e., ζ = 2. However, for the sake of simplicity, the above equation may be rewritten as follows where Γ is considered the volume of the thermal bath reservoir and dk being given by After substituting (5) and (8) in (7), we obtain and therefore we are properly able to write down the partition function analogously what the authors did in Ref. [26] as follows Using Eq.(10), we can obtain the thermodynamic functions per volume Γ, namely, the Helmholtz free energy F (β, θ), the mean energy U (β, θ), the entropy S(β, θ) and the heat capacity C V (β, θ) defined as follows: At the beginning, let us consider the mean energy which follows the spectral radiance defined by: considering E = hν where h is the Planck constant and ν is the frequency. Here, it is reasonable to investigate how the parameter θ affects our theory in the spectral radiation. Additionally, it has to be noted that, even though we explicit the constants h, k B , for performing the following calculations, we set them h = k B = 1. In this way, the plot of this configuration is shown in Fig. 1. For the sake of obtaining the well-established radiation constant of the Stefan-Boltzmann energy density i.e., u S = αT 4 , we consider (Eθ) 2 1 which leads to reproducing the well-established result in the literature [32]. On the other hand, in order to check how the coupling constant θ affects the new radiation constant, we procedure as The analysis will be accomplished via numerical calculations. The plots are shown in Fig.2 taking into account three different circumstances i.e., when θ is either a small or a huge number. Furthermore, the aspect of examining the limit when E 2 √ E 2 θ 2 − 1 θ is also regarded. The latter can be handily associated with the primordial inflationary universe, since we may deal with high temperature regime i.e., 10 13 GeV. Another interesting approach which it is worth being investigated is whether the thermodynamics functions bear with CMB (Cosmic Microwave Background) analysis. Nevertheless, this approach lies beyond the scope of the current work and will be addressed in a future upcoming manuscript though. Analogously, the remaining thermodynamic functions can be explicitly computed: Here, we study the Podolsky electrodynamics modified by the traceless LV dimension-6 term presented in Ref. [11]. In this work, the authors analyzed the classical aspects of such a theory taking into account unitariry and causality from the study of the respective propagator proceeding analogously as it is already established in the literature [33,34]. They consider the construction of a closed algebra using the prescription D µν = (B µ C ν −B ν C µ )/2, where B µ and C ν are constant background vectors which accounts for LV. In this sense, the Lagrangian density which represents this model is given by where θ is the same parameter defined previously, η is the constant coupling with dimension of mass [m] −1 and ξ is the gauge fixing parameter required to evaluate its respective propagator. Nevertheless, we focus only on the investigation of its dispersion relation presented in the poles of the propagator for the sake of calculating the following thermodynamic functions. Therefore, the poles are given by where Considering the complete isotropic sector in this theory, i.e., D µν = −D 00 × diag(3, 1, 1, 1) µν , Eq. (20) turns out to be written as As it was accomplished in the last section, in what follows, we calculate the microstates for this configuration which accounts for the Lorentz violation. Next, we procedure likewise.

B. Thermodynamic properties
The number of microstates per volume is  (23) and using the definitions established in (11), we calculate the Helmholtz free energy, the mean energy, the entropy, and the heat capacity. At the beginning, we devote our attention to the spectral radiance as we did before. The respective plot ofχ as a function of frequency ν for different values of η is shown in Fig. 6. In agreement with the previous section, in which we accomplished the correction to the Stefan-Boltzmann law exhibited in Eq. (15), we step forward likewiseᾱ ≡Ū (β, η, θ, D 00 )β 4 .
Again, this analysis will be performed via numerical calculations in the context of primordial temperature of the universe. To perform such calculation, we need to obtain the behavior of the mean energy. In this way, with the spectral radiance (plotted in Fig. 6) being given bȳ Here, the remaining thermodynamics functions are provided bellow: the Helmholtz free energy the entropȳ and finally, the heat capacitȳ

IV. RESULTS AND DISCUSSIONS
Initially, let us focus our attention on the pure Podolsky electrodynamics. Indeed, we determine the expression of the spectral radiance χ(ν) exhibited in (13) with its plot displayed in Fig. 1. From it, we can see that χ(ν) is sensitive to changes of θ i.e., θ= 10, θ= 30, θ= 50, θ= 70, θ= 90. We make a comparison of these different values to the black body radiation with the Maxwell theory. We note that the latter has its respective spectral radiance smaller than the Podolsky one. It is important to note that to accomplish such an analysis, we need to consider the limit where E 2 √ E 2 θ 2 − 1 θ. Next, we do the plot of Fig. 2 which shows the correction to the so-called Stefan-Boltzmann law represented by parameter α(θ) considering the temperature in the early inflationary universe i.e., β = 10 − electrodynamics. Next, we analyze Fig. 7, which is the compilation of all thermodynamic functions in order to provide a concise discussion on this recent electrodynamics. Considering the plot toᾱ(η) for huge values of η, we obtain an expressive positive inclination of such a curve which differs from the pure Podolsky theory due to its increase characteristic. A similar analysis is shown inS(η), but for a different range of η though i.e., 0 ≤ η ≤ 100.
Additionally, it has the same behaviour encountered in the plot of S(θ). In the same range of η, we verify that there exists a negative curve for Helmholtz free energyF (η) as well being in agreement with the pure Podolsky case. Next, for the case of the heat capacityC V (η), it is displayed a linear increase with a positive angular coefficient when η starts to increase.
In addition, the behavior of this curve is completely different from the heat capacity C V (θ) exhibited in the Podolsky electrodynamics.

V. CONCLUSION
As a matter of fact, this paper aimed at studying the thermodynamic properties of a photon gas in a heat bath when higher-derivative theories of electrodynamics are taken into account. In this way, we computed the microstates of the system for obtaining the partition function. With its possession, it allowed us to investigate the behavior of the main thermodynamic functions i.e., the Helmholtz free energy, the mean energy, the entropy, and the heat capacity. It is worth to be mentioned that all analysis were performed in the context of the primordial temperature of the universe i.e., 10 13 GeV. Moreover, we proposed a correction to the black body radiation in terms of the parameters θ and η which account for Podolsky and its generalization with Lorentz violation respectively.
Besides, we suggested corrections to the Stefan-Boltzmann law for both theories. In the case ofα(θ), we saw that, depending on the considerations ascribed to the mean energy, we may found different behavior of such curves. Likewise, the Helmholtz F (θ), the entropy S(θ), and the heat capacity C V (θ) can also have three different curves depending on the initial circumstances ascribed to them.
On the other hand, the parameterᾱ(η) showed that there existed an accentuated increase as far as positive values of η were taken into account. The Helmholtz free energyF (η) started to decrease when η increased. BothS(η) andC V increased for positive changes of η, but the latter exhibited an increase with a constant angular coefficient though.
Finally, the physical consequences of the thermodynamic functions of such theories should be taken into account in order to possibly address some fingerprints of a new physics which might be useful in future electronic devices in condensed matter physics for instance. Within the context of Lorentz violation, as future perspective, we could analyze the behaviour of the thermodynamic properties for the generalized Lee-Wick electrodynamics. Additionally, knowing if such a theory is in agreement with either Cosmic Microwave Background regime or spectral radiance for the electroweak epoch of the Universe seems to be some further interesting investigations as well. These and others ideas are now under development.