Compressible Fluids with Maxwell-type equations, the minimal coupling with electromagnetic field and the Stefan-Boltzmann law
Albert C. R. Mendes, Flavio I. Takakura, Everton M. C. Abreu, Jorge Ananias Neto
aa r X i v : . [ c ond - m a t . o t h e r] J u l Compressible Fluids with Maxwell-type equations,the minimal coupling with electromagnetic field andthe Stefan-Boltzmann law
Albert C. R. Mendes, ∗ Flavio I. Takakura, † Everton M. C. Abreu,
2, 1, ‡ and Jorge Ananias Neto , § Departamento de F´ısica, Universidade Federal de Juiz de Fora,36036-330, Juiz de Fora - MG, Brazil Grupo de F´ısica Te´orica e Matem´atica F´ısica, Departamento de F´ısica,Universidade Federal Rural do Rio de Janeiro,23890-971, Serop´edica - RJ, Brazil (Dated: May 22, 2018)
Abstract
In this work we have obtained a higher-derivative Lagrangian for a charged fluid coupled withthe electromagnetic fluid and the Dirac’s constraints analysis was discussed. A set of first-classconstraints fixed by noncovariant gauge condition was obtained. The path integral formalismwas used to obtain the partition function for the corresponding higher-derivative Hamiltonian andthe Faddeev-Popov ansatz was used to construct an effective Lagrangian. Through the partitionfunction, a Stefan-Boltzmann type law was obtained.
PACS numbers: 03.50.Kk, 11.10.Ef, 47.10.-gKeywords: compressible fluid; electromagnetic background; Stefan-Boltzmann law ∗ Electronic address: albert@fisica.ufjf.br † Electronic address: takakura@fisica.ufjf.br ‡ Electronic address: [email protected] § Electronic address: jorge@fisica.ufjf.br . INTRODUCTION In recent papers the authors have discussed that, as an alternative way for the descriptionof fluid dynamics, concerning both the compressible fluids [1] and the equations of plasma [2],the better path would be through the recasting of the equations of motion to obtain a set ofMaxwell-type equations for the fluid. This transformation in the structure of the equationsof motion results in the generalization of the concept of charge and current connected to thedynamics of the fluid [3, 4]. The identification of what will be considered as a source termin the resulting theory depends on the choice of the objects which will form the main partof its new structure of fluid dynamics. In Lighthill’s work concerning the sound radiated bya fluid flow [5], the applied stress tensor was considered as the source of the radiation field.R. J. Thompson [2] recently introduced an extension of this new structure of the plasmaequations of motion, for each kind of fluid, from the equations of motion that describe suchsystem.The reason is to understand thermodynamical arguments in order to obtain how theenergy density ρ depends on the temperature T for a fluid’s equation of state given by p = ωρ . Besides, the Stefan-Boltzmann law has been widely discussed in the scenario ofblack holes thermodynamics [6], from where we know that the energy density is inverselyproportional to the temperature. More recently, the observed acceleration of the Universedemands the existence of a new component, the termed dark energy, which rules out allother forms of energy and has a negative pressure. The presence of such energy in theUniverse deserves detailed analysis, such as the consequences related to the application ofthe generalized second law [7] or the entropy bound [8]. Some elements, such as the phantomfield ( ω < − II. CANONICAL STRUCTURE
Recently [12], some of us have shown that a Lagrangian formulation for a compressiblefluid can be obtained, analogously to the one described by Marmanis concerning an incom-pressible fluid [3], resulting in a Maxwell-type action for the fluid considering the viscosity,given by L fluid = − T µν T µν , (1)2here T µν = ∂ µ U ν − ∂ ν U µ is the strength tensor of the fluid. The four-vector potential U µ ≡ ( U , ~U ), where U is the energy function and ~U is the average velocity field [12]. Thespacetime metric is η µν = ( − + ++).The Lagrangian density in Eq. (1) gives us the set of Maxwell-type equations for thehomogeneous case (no sources) [12], from which we can derive the main equations in fluiddynamics that are the equations for the vortex dynamics ~ω∂~ω∂t + ∇ × ( ~ω × ~U ) = ∇ × ~z , (2)where ~z = T ∇ s + ρ − ∇ σ , T is the temperature, s is the entropy per unit mass and ρ is thefluid density. And σ is the stress tensor [13].The right term in Eq. (2) that can be rewritten as ∇ × ~z = ∇ T × ∇ s + ∇ × ( ρ − ∇ σ ) = ~ Γ B + ~ Γ ν , (3)therefore, we have that ∂~ω∂t + ∇ × ( ~ω × ~U ) = ~ Γ B + ~ Γ ν , (4)which has, precisely, the standard form. In the above Eqs. (3)-(4), ~ Γ B and ~ Γ ν , explicitlydisplayed on the right hand side, are the possible sources of the vorticity ~ω , where ~ Γ B = ∇ T × ∇ s (5)that is equivalent to the traditional Biermann battery [14], and the second term ~ Γ ν = ∇ × ( ρ − ∇ σ ) , (6)which is connected to the viscosity.Considering initially that the system is simply composed of the electromagnetic field anda non-charged fluid, in this case the Lagrangian of the system is given by L = L F luid + L Maxwell = − T µν T µν − F µν F µν . (7)where we have not considered the presence of source terms.Let us now consider the case where the fluid is charged, where the interaction betweenthe fluid and the electromagnetic field will be introduced inside this last Lagrangian by usinga minimum coupling between the fluid’s velocity field ( U ( ǫ ) µ ) - here we have introduced theindex ( ǫ ) valid for each species (positively or negatively charged), where the generalizationto several species is straightforward - and the vector potential of the electromagnetic field( A µ ) is given by U ( ǫ ) µ −→ U ( ǫ ) µ + gA µ . (8)Thus, we have that T µν −→ T ( ǫ ) µν = ∂ µ U ( ǫ ) ν − ∂ ν U ( ǫ ) µ −→ T ( ǫ ) µν + gF µν (9)3hich, substituting in (7) gives us L = − T ( ǫ ) µν T µν ( ǫ ) −
14 (1 + g ) F µν F µν − gT µν F µν , (10)where the coupling constant is g = e ǫ /m ǫ , e ǫ is the charge and m ǫ is the mass of the charge.The Euler-Lagrange equations of motion are(1 + g ) ∂ µ F µν + g∂ µ T µν ( ǫ ) = 0 , (11)and it is easy to see that (10) is invariant under the gauge transformations, A µ → A µ + ∂ µ Λ, for the electromagnetic fields, and U ( ǫ ) µ → U ( ǫ ) µ + ∂ µ Λ, for the compressible fluid field.Concerning the potentials, U ( ǫ ) α and A α , the above equation reads(1 + g ) [ ✷ A µ − ∂ µ ∂ ν A ν ] + g (cid:2) ✷ U ( ǫ ) µ − ∂ µ ∂ ν U ν ( ǫ ) (cid:3) = 0 . (12)From now on, for simplicity, we will not use the species index, and much of what follows istrue for each species. The last term in (10) is exactly the interaction between the chargedfluid and the electromagnetic field applied. We can also observe that when the couplingconstant is zero (the fluid is non-charged) we can obtain the Lagrangian (7) again.We can rewrite the Lagrangian in Eq. (10) as L = − T µν T µν −
14 (1 + g ) F µν F µν − gU µ ∂ ν F µν , (13)where now we have a higher-derivative in the Maxwell sector represented by the last termin (11). Hence, there should be introduced another set of canonical pair (Σ µ = ∂ A µ , φ µ )to have a correct expanded phase space in order to proceed with the canonical analysis.Consequently, one can find the following Lagrangian L = 12 ( ˙ ~U − ∇ U ) + 14 a ( ∇ × ~U ) + 12 (1 + g )( ~ Σ − ∇ A ) + 14 (1 + g )( ∇ × ~A ) − g ~U · ( ∇ Σ − ˙ ~ Σ) − gU ( ∇ · ~ Σ − ∇ A ) − g ~U · ( ∇ × ∇ × ~A ) . (14)and the canonical Hamiltonian of the theory H c can be written as H c = Z d x ( p µ ˙ U µ + π µ ˙ A µ + φ µ ˙Σ µ − L ) , (15)where the momenta canonically conjugated to the fields U µ , A µ and Σ µ ≡ ˙ A µ , which can beconsidered as independent variables, defined respectively by p µ ≡ ∂ L ∂ ( ˙ U µ ) , (16) π µ ≡ ∂ L ∂ ( ˙ A µ ) − ∂ (cid:20) ∂ L ∂ ( ¨ A µ ) (cid:21) − ∂ k (cid:20) ∂ L ∂ ( ∂ ∂ k A µ ) + ∂ L ∂ ( ∂ k A µ ) (cid:21) , (17) φ µ ≡ ∂ L ∂ ( ¨ A µ ) , (18)4hich result in the following expressions p µ = T µ ,φ µ = gη µk U k , (19) π µ = (1 + g ) F µ − gη µk T k + gη µ ∂ k U k . So, using the equations (16)-(19) we can write the canonical Hamiltonian density H c as H c = π µ Σ µ + 12 ~p + ~p · ∇ U − a ( ∇ × ~U ) −
12 (1 + g )( ~ Σ − ∇ A ) −
12 (1 + g )( ∇ × ~A ) + g ~U · ∇ Σ + gU ( ∇ · ~ Σ − ∇ A ) + g ~U · ( ∇ × ∇ × ~A ) . (20)Therefore, by working out a pure Dirac analysis of the Hamiltonian (20) [15], we noticethat by the equations (19) we have obtained the set of constraints χ ≡ π − ∇ · φ ≈ , (21) χ ≡ φ ≈ , (22) χ ≡ ∇ · ~π ≈ , (23) χ ≡ p ≈ , (24) χ ≡ ∇ · ~p − g ( ∇ · ~ Σ − ∇ A ) ≈ , (25)where χ was obtained by the time evolution of χ , and χ , by the time evolution of χ .The set of constraints χ i , i = 1 , ..., ≈ ” means weak equality . Following Dirac’sprocedure, we have to choose five gauge conditions.These conditions can be suggested by many reasons and the most important one can bethe way it may simplify the theory. In Maxwell’s theory, the condition usually employed tobe the gauge fixing is the Coulomb gauge ∂ α A α = 0 . (26)However, concerning the theory described by the Lagrangian in Eq. (7), where a chargedcompressible fluid is immersed in an electromagnetic field, the condition (26) is not sufficientto promote the mentioned gauge fixing. In order to do that we need an extra condition. Inthis case, an appropriate choice could be the “Lorentz Gauge” for a compressible fluid [12],where ∂ α U α = 0 , (27)which is directly related to the condition relative to the compressibility of the fluid [12].Thus, considering Eqs. (26) and (27) we can obtain the following set of gauge conditionsΦ ≡ A ≈ , (28)Φ ≡ Σ ≈ A ≈ , (29)Φ ≡ ∇ · ~A ≈ , (30)Φ ≡ U − α ≈ , (31)Φ ≡ ∇ · ~U ≈ , (32)where α in Eq. (28) is a constant. This set of constraints constitutes an appropriatednoncovariant gauge condition which fixes the first-class constraints.5 II. PATH INTEGRAL FORMALISM
Now, we are able to write down the generating functional, or transition amplitude, Z = Z Dp ν DU ν Dφ ν D Σ ν Dπ ν DA ν det { χ a , Φ b } " Y n =1 δ [ χ n ] δ [Φ n ] exp (cid:18) i Z d x L c (cid:19) , (33)where the determinant between the first-class constraints (21)-(25) and the gauge-fixingcondition (28)-(32) has the form det { χ a , Φ b } = det (cid:2) ∇ (cid:3) , (34)which does not contain any field variables and it can be put within a normalization constant.Hence, L c = p µ ∂ t U µ + π µ ∂ t A µ + φ µ ∂ t Σ µ − H c . (35)Introducing Eqs. (20), (21)-(25), (28)-(32), (34) and (35) into (33), integrating over themomenta and field variables and, using the delta functional, we can obtain the followingexpression for the transition amplitude Z = Z Du µ DA µ δ ( ∂ k U k ) δ ( ∂ s A s ) exp (cid:18) i Z d x L (cid:19) , (36)where L is given by Eq. (11).Using a straight-forward generalization of the Faddeev-Popov ansatz, we can go from anoncovariant gauge fixing form to a covariant one such as Z = Z DU µ DA µ δ (cid:20) ξ ∂ s U s − f (cid:21) δ (cid:20) ∂ s A s − f ′ (cid:21) exp (cid:18) i Z d x L (cid:19) , (37)where ξ = 0 and Λ = 0 are arbitrary real numbers and f = f ( x ) and f ′ = f ′ ( x ) are arbitraryreal functions.Now, since the generating functional is independent of f ( x ) and f ′ ( x ) we can integrate in f ( x ) with weight exp (cid:0) − i R d xf (cid:1) and in f ′ ( x ) with the weight exp (cid:0) − i R d xf ′ (cid:1) ˆ Z = Z Df Df ′ Zexp (cid:18) − i Z d xf (cid:19) exp (cid:18) − i Z d xf ′ (cid:19) = Z DcD ¯ c Z DbD ¯ b Z DU µ DA µ exp (cid:18) i Z d x L eff (cid:19) , (38)where c, ¯ c are the ghost fields by the Maxwell sector and b, ¯ b are the ghost fields for the fluidsector. The effective Lagrangian density L eff is defined by L eff = − T µν T µν −
14 (1+ g ) F µν F µν − gU µ ∂ ν F µν − ξ ( ∂ µ U µ ) − ( ∂ µ A µ ) − ξ c ∆¯ c − b ∆¯ b , (39)where ∆ = ∂ ν ∂ ν .In the next section we will analyze the theory in thermodynamic lequilibrium and we willwork with the partition function, which is the most important function in thermodynamics.From it, all the thermodynamical properties can be obtained, namely, pressure, particlenumbers, entropy and energy. 6 V. THE PARTITION FUNCTION OF THEORY
To obtain the partition function from the transition amplitude we have to carry out akind of “Euclideanization” of the time components of the vector fields, a compactificationof the Wick-rotated time coordinate, and to impose periodic boundary conditions ( P ) inthis coordinate for the fluid, the electromagnetic and ghost fields. Doing so, we can find thepartition function Z [ β ] = Z DcD ¯ c Z DbD ¯ bDU µ DA µ exp (cid:26) − Z β dx L E (cid:27) , (40)where β = 1 /T , T is the temperature, and Z β dx ≡ Z β dτ Z d x , (41)and L E = − T µν T µν −
14 (1+ g ) F µν F µν − gU µ ∂ ν F µν − ξ ( ∂ µ U µ ) − ( ∂ µ A µ ) − ξ c ∆¯ c − b ∆¯ b , (42)where L E is the so-called effective Euclidean Lagrangian density.So, the partition function takes the form Z [ β ] = Z P DcD ¯ c exp (cid:26)Z β dx (cid:20) ξ c ∆¯ c (cid:21)(cid:27) Z P DbD ¯ b exp (cid:26)Z β dx (cid:20) b ∆¯ b (cid:21)(cid:27) × Z P DU µ exp (cid:26) − Z β dx U µ O ( f ) µν U ν (cid:27) Z P DA µ exp (cid:26) − (1 + g )2 Z β dxA µ O ( M ) µν A ν (cid:27) × exp (cid:26)Z β dx gU µ ∂ ν F µν (cid:27) , (43)where the operators O ( f ) µν (for the fluid sector) and O ( M ) µν (for the Maxwell sector) are definedby O ( f ) µν = δ µν ∆ − ξ − ξ ∂ µ ∂ ν (44)and O ( M ) µν = δ µν ∆ − Λ − ∂ µ ∂ ν . (45)Note that the partition function in Eq. (43) have a cross term (the last one in (43)) for thefield of the fluid ( U µ ) and the Maxwell field ( A µ ), which does not allow a direct calculation.It is independent of the functional integral concerning these fields, such as the fields for theghosts that do not have the interaction for the fields A µ and U µ , where Z P DcD ¯ c exp (cid:26)Z β dx (cid:20) ξ c ∆¯ c (cid:21)(cid:27) ≡ det (cid:20) ξ ∆ (cid:21) , (46) Z P DbD ¯ b exp (cid:26)Z β dx (cid:20) b ∆¯ b (cid:21)(cid:27) ≡ det (cid:20)
1Λ ∆ (cid:21) . (47)7o, we will calculate the functional integral for the Maxwell field considering the cross termas an external source term for the electromagnetic field by doing the following transformationin the field A µ A µ −→ A µ (1 + g ) − / . (48)Thus we have Z P DA µ exp (cid:26) − (1 + g )2 Z β dxA µ O ( M ) µν A ν + Z β dx A µ g∂ ν T µν (cid:27) = Z P DA µ exp (cid:26) − Z β dxA µ O ( M ) µν A ν + Z β dx A µ g (1 + g ) / ∂ ν T µν (cid:27) == Z P DA µ exp (cid:26) − Z β dxA µ O ( M ) µν A ν + Z β dx A µ J ( f ) µ (cid:27) , (49)where J ( f ) µ (the above index ( f ) means source term due to the coupling with the fluid) isdefined as being J ( f ) µ = g (1 + g ) / ∂ ν T µν . (50)Hence, after integration in the gauge field A µ , we have that Z P DA µ exp (cid:26)(cid:26)(cid:26) − Z β dxA µ O ( M ) µν A ν + Z β dx A µ J ( f ) µ (cid:27)(cid:27)(cid:27) (51)= (cid:2) Det (cid:0) O Mµν (cid:1)(cid:3) − / Z β dx J µ ( f ) (cid:0) O ( M ) µν (cid:1) − J ν ( f ) , where “Det” in (51) means the determinant in both Euclidean space-time and the Hilbertspace.So, introducing (46), (47) , (50) and (51) into (43) the partition function is given by Z [ β ] = det (cid:20) ξ ∆ (cid:21) det (cid:20)
1Λ ∆ (cid:21) (cid:2)
Det (cid:0) O Mµν (cid:1)(cid:3) − / × Z P DU µ exp (cid:26)Z β dx (cid:20) − U µ O ( f ) µν U ν + 12 g g U µ ( δ µν ∆ − ∂ µ ∂ ν ) U ν (cid:21)(cid:27) (52)or, using the operator defined in (44), we have after some steps that Z [ β ] = det (cid:20) ξ ∆ (cid:21) det (cid:20)
1Λ ∆ (cid:21) (cid:2)
Det (cid:0) O Mµν (cid:1)(cid:3) − / × Z P DU µ exp (cid:26)Z β dx (cid:20) − U µ (cid:20)(cid:18) − g g (cid:19) δ µν ∆ − (cid:18) ξ − ξ − g g (cid:19) ∂ µ ∂ ν (cid:21) U ν (cid:21)(cid:27) = det (cid:20) ξ ∆ (cid:21) det (cid:20)
1Λ ∆ (cid:21) (cid:2)
Det (cid:0) O Mµν (cid:1)(cid:3) − / Z P DU µ exp (cid:26) − Z β dx U µ O µν U ν (cid:27) , (53)where the operator O is defined by O µν ≡ (cid:18) − g g (cid:19) δ µν ∆ − (cid:18) ξ − ξ − g g (cid:19) ∂ µ ∂ ν . (54)8hus, we obtain that Z [ β ] = det (cid:20)
1Λ ∆ (cid:21) (cid:2)
Det (cid:0) O Mµν (cid:1)(cid:3) − / det (cid:20) ξ ∆ (cid:21) [Det ( O µν )] − / . (55)Since the temperature does not depend on ξ or Λ, it can be included into the normalizationconstant. Hence, we can write the partition function after evaluating the determinant in theEuclidean space-time as Z [ β ] = [det (∆)] − (1 + g ) − [det (∆)] − . (56)We note that the partition function is a product of determinants of the form [det(∆ + m j )] ( − n j ) / , with j = 1 and 2. Each one of these terms describes a gas of free particleswith mass m j and n j as being the degrees of freedom (DOF). We identify the first of thesedeterminants as a partition function for massless particles with two DOF’s, i.e., the Maxwellphotons. On the other hand, the second determinant is the partition function for the fluidwith two DOF’s. In [12], in appendix A, we have made a brief discussion about the fluidDOF’s.In order to evaluate the determinants, we note that the equationdet(∆) = Y n,~p β ( ω n + ~p ) (57)and, using this identity, the logarithm of the partition function can be written asln[ Z ( β )] = − X n,~p ln[ β ( ω n + ~p )] − X n,~p ln[(1 + g ) − β ( ω n + ~p )] . (58)Now, evaluating the sum in n , and passing to the continuous in momentum space, wehave thatln[ Z ( β, V ] = − V Z d p (2 π ) ln(1 − e − βp ) (cid:12)(cid:12)(cid:12) Maxwell − V Z d p (2 π ) ln(1 − e − β ′ p ) (cid:12)(cid:12)(cid:12) Fluid , (59)where β ′ = β (1 + g ) − , and we findln[ Z ( β, V ] = π Vβ + π Vβ ′ = π Vβ + π Vβ (1 + g ) . (60)The first term in Eq. (60) is the usual Planck result associated with the free Maxwelland it gives the usual Stefan-Boltzmann law. The second term corresponds to the fluidStefan-Boltzmann type law [16]. This is a very interesting result, which allows us to derivesome properties of our system. Now, starting from (60), which can be rewritten asln[ Z ( β, V ] = π Vβ [1 + (1 + g ) ] , (61)we can obtain the energy density defined by ρ = k B T V ∂∂T ln[ Z ( β, V ]= k B π T [1 + (1 + g ) ] , (62)9here k B is the Boltzmann constant, and the pressure defined by p = k B T ∂∂V ln[ Z ( β, V ]= 13 k B π T [1 + (1 + g ) ] . (63)Notice that we can obtain a relationship between these quantities given by p = 13 ρ ( g ) , (64)which is a equation of state, that depends on the coupling constant g .It is also interesting to note that is a particular case of the “gamma-law” equation of state p = ( γ − ρ , (65)where the index γ = 4 / ρ ( g ) = η ( g ) T , (66)where η depends on the coupling constant g . We will comment this result in the next section. V. CONCLUSIONS
The current fluid dynamics literature has several motivations that keep the interest inthis subject at high levels during this last decades. The interaction with the electromagneticfields is one of these motivations.In this work we have provided a constraint analysis of the Lagrangian system formedby this kind of interaction. The final Lagrangian has higher derivatives five first-class con-straints. The Lorentz gauge fixing for compressible fluids was used and the set of gaugetransformations was described.We have used statistical elements such as the partition function, and the Faddeev-Popovansatz, to obtain an equation of state similar to the dark energy model for the acceleratedUniverse one. The final result is the fourth power temperature dependence, which shows adirect analogy to the Stefan-Boltzmann law, where the coefficient term is a function of thecoupling constant.
VI. ACKNOWLEDGMENTS
The authors thank CNPq (Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico),Brazilian scientific support federal agency, for partial financial support, Grants numbers302155/2015-5, 302156/2015-1 and 442369/2014-0 and E.M.C.A. thanks the hospitality ofTheoretical Physics Department at Federal University of Rio de Janeiro (UFRJ), where partof this work was carried out. [1] T. Kambe, Fluid Dyn. Res. , 055502 (2010).
2] R.J. Thompson and T.M. Moeller, Phys. of Plasmas , 010702 (2012); , 082116 (2012).[3] H . Marmanis, Phys. Fluids. , 1428 (1998).[4] R. Jackiw, V. P. Nair, S. Y. Pi, and A. P. Plolychronakus, J. Phys. A , R327 (2004).[5] M. J. Lighthill, Proc. R. Soc. A , 564 (1952); , 1 (1954).[6] S. W. Hawking, Phys. Rev. D , 191 (1976).[7] J. D. Bekenstein, Phys. Rev. D , 3292 (1974).[8] J. D. Bekenstein, Phys. Rev. D , 287 (1981).[9] J. A. de Freitas Pacheco and J. E. Horvath, Class. Quant. Grav. , 5427 (2007).[10] P. F. Gonz´alez-Dias and C. L. Sig¨uenza, Nucl. Phys. B , 363 (2004).[11] A.C.R. Mendes, C. Neves. W. Oliveira and F.I. Takakura, Braz. J. Phys. , 346 (2003).[12] E.M.C. Abreu, J.A. Neto, A.C.R. Mendes and N. Sasaki, Phys. Rev. D , 125011 (2015).[13] L.D. Landau and E.M. Lifshits, Fluid Mechanics , Pergamon Press, Oxford, 1980.[14] L. Biermann, Z. Naturforsch. Teil A, , 65 (1950).[15] P.A.M. Dirac, Lectures on Quantum Mechanics, Yeshiva University, New York, 1964.[16] J.A.S Lima and A. Maia, Jr, Phys. Rev. D , 5628 (1995); J.A.S Lima and J. Santos, Int. J.Theor. Phys. , 127 (1995); J.A.E. Carrillo, J.A.S Lima and A. Maia, Int. J. Theor. Phys. , 2013 (1996)., 2013 (1996).