Hindawi Advances in High Energy Physics Volume 2020, Article ID 5193692, 8 pages https://doi.org/10.1155/2020/5193692

Research Article Non-Abelian Gravitoelectromagnetism and Applications at Finite Temperature

A. F. Santos ,1 J. Ramos,2 and Faqir C. Khanna3

1Instituto de Física, Universidade Federal de Mato Grosso, 78060-900, Cuiabá, Mato Grosso, Brazil 2Faculty of Science, Burman University, Lacombe, Alberta, Canada T4L 2E5 3Department of Physics and Astronomy, University of Victoria, BC, Canada V8P 5C2

Correspondence should be addressed to A. F. Santos; alesandroferreira@fisica.ufmt.br

Received 22 January 2020; Revised 9 March 2020; Accepted 18 March 2020; Published 3 April 2020

Academic Editor: Michele Arzano

Copyright © 2020 A. F. Santos 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 SCOAP3.

Studies about a formal analogy between the gravitational and the electromagnetic fields lead to the notion of Gravitoelectromagnetism (GEM) to describe gravitation. In fact, the GEM equations correspond to the weak-field approximation of the gravitation field. Here, a non-abelian extension of the GEM theory is considered. Using the Thermo Field Dynamics (TFD) formalism to introduce temperature effects, some interesting physical phenomena are investigated. The non-abelian GEM Stefan- Boltzmann law and the Casimir effect at zero and finite temperatures for this non-abelian field are calculated.

1. Introduction between equations for the Newton and Coulomb laws and the interest has increased with the discovery of the The Standard Model (SM) is a non-abelian gauge theory with Lense-Thirring effect, where a rotating mass generates a symmetry group Uð1Þ × SUð2Þ × SUð3Þ. SM describes the- gravitomagnetic field [15–17]. Some experiments that oretically and experimentally three of the four fundamental study this effect have been developed, such as LAGEOS forces of nature, i.e., the electromagnetic, weak, and strong (Laser Geodynamics Satellites) and LAGEOS 2 [18], the forces. The electromagnetism is a Uð1Þ abelian gauge theory Gravity Probe B [19], and the mission LARES (Laser Relativity which has been tested to a high precision. The generalization Satellite) [20, 21]. ff of an abelian gauge theory to the non-abelian gauge theory The GEM theory may be analyzed by three di erent was proposed by Yang and Mills [1]. The last one approaches: (i) using the similarity between the linearized describes the electroweak unification and quantum chro- Einstein and Maxwell equations [22], (ii) a theory based on modynamics. The electroweak interaction is described by an an approach using tidal tensors [23], and (iii) the decompo- ℬ ∈ kl SUð2Þ × Uð1Þ group and while the SUð3Þ group satisfies sition of the Weyl tensor (Cijkl) into ij = 1/2 iklC0j and ℰ − the quantum chromodynamics [2–4]. ij = C0i0j, the gravitomagnetic and gravitoelectric compo- Gravity is not a part of SM. This implies that the SM is nents, respectively [24]. In this paper, the Weyl tensor not a fundamental theory that describes all fundamental approach is used. A Lagrangian formulation for GEM is interactions of nature. In this paper, an extension of non- developed [25], and a gauge transformation in GEM is stud- abelian gravity is discussed. Some applications of such a the- ied [26]. Here, an extension to non-abelian GEM fields is ory are developed. The gravitational theory studied here is introduced. Applications of the non-abelian GEM at finite the Gravitoelectromagnetism (GEM). GEM is an approach temperature are investigated. The temperature effects are based on describing gravity in a way analogous to the electro- introduced using Thermo Field Dynamics (TFD) formalism. magnetism [5–7]. Several studies about the GEM theory have There are two ways to introduce the temperature effect: been developed [8–14]. These ideas arise from the analogy- (i) using the imaginary time formalism [27] and (ii) using 2 Advances in High Energy Physics the real-time formalism [28–36]. In this paper, TFD formal- ℬ = curl A~ , ism is chosen. It is a real-time finite temperature formalism. 1 ∂A~ ð2Þ In this formalism, a thermal state is developed where the ℰ + = − grad φ, main objective is to interpret the statistical average of an c ∂t arbitrary operator as an expectation value in a thermal vac- uum. Two elements are necessary to construct this thermal where φ is the GEM counterpart of the electromagnetic state: (i) doubling of the original Hilbert space and (ii) the (EM) scalar potential ϕ. μνα use of Bogoliubov transformations. These are two Hilbert Defining ℱ as the gravitoelectromagnetic tensor, the fi spaces, the original space S and the tilde space ~S, which are GEM eld equations become related by a mapping, called the tilde conjugation rules, π while the Bogoliubov transformation consists in a rotation μνα 4 G να ∂μℱ = J , involving these two spaces that ultimately introduce the c ð3Þ temperature effects. μναhi ∂μG =0, The Stefan-Boltzmann law and the Casimir effect for the fi fi non-abelian GEM eld at nite temperature are calculated. να The Stefan-Boltzmann law describes the power radiated from where J depends on quantities ρi and Jij that are the mass a black body in terms of its temperature. The Casimir effect, and the current density, respectively. In addition, the gravi- proposed by H. Casimir [37], is a quantum phenomenon that toelectromagnetic tensor is defined as appears due to vacuum fluctuations of any quantum field. μνα μ να ν μα The results in this case may be at zero or finite temperatures. ℱ = ∂ A − ∂ A , ð4Þ This paper is organised as follows. In section II, a brief introduction to the abelian GEM Lagrangian formalism is and the dual GEM tensor is defined as presented. In section III, an extension to non-abelian GEM fi eld is developed. The energy-momentum tensor associated μνα 1 μνγσ αρ fi G = ∈ η − ℱ γσρ: ð5Þ to the non-abelian gauge eld is calculated. In section IV, 2 the TFD formalism is introduced. In section V, some applica- fi fi tions considering the non-abelian GEM eld at nite temper- Using these definitions, the GEM Lagrangian density is ature are analysed. (i) The Stefan-Boltzmann law is calculated. given as [25]. (ii) The Casimir effect at zero temperature is obtained, and (iii) the Casimir effect at finite temperature is calculated. In 1 μνα G να section VI, some concluding remarks are presented. ℒ = − ℱ μναℱ − J A να: ð6Þ G 16π c 2. Lagrangian Formulation of Abelian Gem This Lagrangian allows considering several gravitational In this section, an introduction to the Lagrangian formula- applications involving the graviton, such as interactions with tion of abelian GEM is presented. The GEM field equations, other fundamental particles. This makes it possible to study Maxwell-like equations, are several related topics. In this way, the GEM theory is described by two fields ℰij and ℬij, which are symmetric and traceless tensors of the ∂iℰij = −4πGρj, second order. These fields can be expressed in terms of the symmetric gravitoelectromagnetic potential Aμν [25, 26], ∂iℬij =0, analogous to that of electromagnetism Aμ. Thus, Aμν is the ∂ℰij π k 1 4 G ð1Þ fi ∈hikl∂ ℬlji + = − Jij, fundamental eld in GEM and naturally, it has two indices c ∂t c [25, 26]. 1 ∂ℬij It is important to note that GEM equations correspond to ∈hikl∂kℰlji + =0, the weak-field approximation of General Relativity. They do c ∂t not describe strong fields and, therefore, do not include the full Einstein equations. To be more specific, the abelian where G is the gravitational constant, εikl is the Levi-Civita GEM corresponds to the linear part of Einstein equations symbol, ρj is the vector mass density, Jij is the mass current and the non-abelian GEM corresponds up to the second order in the weak-field approach. density, and c is the speed of light. The quantities ℰij, ℬij, ij fi fi and J are the gravitoelectric eld, the gravitomagnetic eld, 3. Non-Abelian Gem and the mass current density, respectively. The symbol h⋯i fi denotes symmetrization of the rst and last indices, i.e., i Let us consider an extension of the GEM field to include the and j. non-abelian gauge transformations [38]. Then, in this sec- The fields ℰij and ℬij are expressed in terms of a tion, the Lagrangian for the non-abelian GEM field is pre- symmetric rank-2 tensor field, A~ , with components A ij, sented and the energy-momentum tensor associated to the such that non-abelian field is calculated. Advances in High Energy Physics 3

In order to obtain the non-abelian gauge transformation oneform pα is introduced [26]. The one form makes the for the GEM field, let us investigate the Dirac Lagrangian phase function to split into phase factors each associated with under global and local gauge transformations. The free Dirac one of the four directions in spacetime. μ Lagrangian is given as Using these results and replacing the derivative ∂ by the covariant derivative Dμ, the Dirac Lagrangian is gauge ℒ − ψ γ ∂μψ ψ ψ D = i ðÞx μ ðÞx + m ðÞx ðÞx , ð7Þ invariant, i.e., ℒ − ψ γ μψ ψ ψ : where ψðxÞ is a two-component column vector. This D = i ðÞx μD ðÞx + m ðÞx ðÞx ð15Þ Lagrangian is invariant under global SUð2Þ gauge transfor- mation given as In this formulation, three new gauge tensor fields are introduced. To write a full Lagrangian invariant under ψ′ðÞx = UψðÞx , ð8Þ local gauge transformation, a kinetic term of AμαðxÞ must be constructed. To do that, an analogue of the electromag- iH with U being a 2×2unitary matrix that is written as U = e , netic tensor Fμν is constructed. For obtaining the antisym- where H is a hermitian matrix. To study local gauge transfor- metric third-rank tensor of the gauge field, let us consider mation, more details are necessary. a covariant derivative (13). Then, Let us assume that the local gauge transformation is μ ν μνα ½ŠD , D = −igpασ · F , ð16Þ ψ′ðÞx = UxðÞψðÞx = eigHðÞ x ψðÞx , ð9Þ where where g is the coupling constant and HðxÞ is the hermitian 2×2matrix given by μνα ∂μ να − ∂ν μα εijk μα νβ: Fk = Ak Ak +2g pβAi Aj ð17Þ HxðÞ= σ · aðÞx , ð10Þ Then, the full Lagrangian that is invariant under local SU with aðxÞ being real functions of x and σ are Pauli matrices. ð2Þ gauge transformations is The Pauli matrices σiði = 1, 2, 3Þ are the generators of the  μ  1 μνα non-abelian group SUð2Þ satisfying the commutation rela- ℒ = −iψðÞx γμD ψðÞx + mψðÞx ψðÞx − Fμνα ⋅ F : D 16π tions ½σi, σjŠ =2iεijkσk. In a more compact form, aðxÞ is written as ð18Þ α fi aðÞx = pαb ðÞx , ð11Þ This Lagrangian describes two equal mass Dirac elds interacting with three massless tensor gauge fields. α where b ðxÞ are vectors associated to each of the four direc- In conclusion, GEM is an approach based on formulating tions in Minkowski spacetime and p are the components gravity in analogy to electromagnetism. In this way, GEM α fi of the one-form ~p. Then, the local gauge transformation becomes a gauge eld theory of gravity in contrast with the becomes geometric theory of General Relativity. Then, it is expected that SUð2Þ be the gauge symmetry group. It is the Weyl ten- σ bα ℰij ℬij ψ′ðÞx = eigpα · ðÞx ψðÞx : ð12Þ sors and that keep the connection of GEM to gravity. Now, let us determine the energy-momentum tensor fi The Dirac Lagrangian is not invariant under this local associated with the non-abelian GEM eld. μ ′ gauge transformation since the derivative ∂ ψ ðxÞ introduces 3.1. Energy-Momentum Tensor for Non-Abelian GEM. Here- a new term in the Lagrangian. In order to obtain an invariant after, the Lagrangian density for the free non-abelian GEM Lagrangian, a covariant derivative is defined as field is considered, i.e., μ ∂μ − σ Aμα D = igpα · ðÞx , ð13Þ 1 a μναa ℒ = − Fμνα F ð19Þ 16π where the tensor gauge field AμαðxÞ has three components Aμα μα μα μα ðxÞ = ðA1 ðxÞ, A2 ðxÞ, A3 ðxÞÞ and it transforms as The index a is summed over the generators of the gauge group and for an SUðNÞ group, one has a, b, μα A′ μα ∂μ α εijk μβ α: ⋯ 2 − : fi k ðÞx = Ak ðÞx + bk +2g pβAi bj ð14Þ c =1 N 1 Here, as a rst application of the non-abelian GEM, the self-interaction between the tensor μα fi An important note, there is one tensor gauge field A ðxÞ gauge elds is ignored. i Using the energy-momentum tensor definition, for each generator σi of the group SUð2Þ. Moreover, in the definition of the covariant derivative Dμ (Equation (13)), μν ∂ℒ ν μν the gauge field Aμν should appear to keep the local gauge T = ÀÁ∂ Aa − η ℒ , ð20Þ ∂∂ a λα invariance like in electromagnetism. In order to have it, the μAλα 4 Advances in High Energy Physics the energy-momentum tensor associated with the non- where fi abelian GEM eld is    λεωυ Δμν,λεωυ ′ Γμ νρα,λεωυ ′ − 1 ημνΓρσθ, ′ x, x = ρα, x, x ρσθ, x, x , μν 1 μa νλαa 1 μν a ρσθa 4 T = −ℱ ℱ + η ℱ ρσθℱ : ð21Þ 4 λα 4 ð28Þ

To avoid a product of field operators at the same space- with time point, the energy-momentum tensor is written as  ακγ μνρ λεωυ κλ εγ α αλ εγ κ Γ , , x, x′ = g g ∂ − g g ∂ hn   μν 1 τ −ℱ μa ℱ νλαa ′ μ ν ð29Þ T ðÞx = lim λαðÞx x νω ρυ∂′ − μω ρυ∂′ : 4π ′→ Á g g g g x x  io ð22Þ 1 μν a ρσθa ′ + η ℱ ρσθðÞx ℱ x 4 The vacuum expectation value of the energy-momentum tensor leads to the expression where τ is the time order operator. μν μν The quantization of the non-abelian GEM field requires hiT ðÞx =0hi∣ T ðÞx ∣ 0 that noDE hi 1 μν,λεωυ ′ a a ′ = − lim Δ x, x 0 τ AλεðÞx Aωυ x 0 , 4π x′→x ∂ℒ 1 πκλa − 0κλa: ð30Þ = ∂∂ a = π F ð23Þ ðÞ0Aκλ 4 where the graviton propagator is ~ ∂ ij the commutation relation0 and div A = iA =0, the DE hi DE hi covariant quantization is carried out and the commutation a a ′ ab a b ′ 0 τ AλεðÞx Aωυ x 0 = δ 0 τ AλεðÞx Aωυ x 0 relation is  ab ab ′ hi h = iδ Dλεωυ x − x , ija πklb ′ i δikδjl − δilδjk A ðÞx, t , x , t = ð31Þ 2  − 1 δjl∂i∂k−−δjk∂i∂l − δil∂j∂k 2 with ∇ i  + δik∂j∂lÞ δ3 x − x′ δab:  ab 1 δab − − ′ Dλεωυ = ðÞgλωgευ + gλυgεω gλεgωυ G0 x x , ð32Þ ð24Þ 2 and G ðx − x′Þ is the massless scalar field propagator. Then, Other commutation relations are zero. 0 μν the vacuum expectation value of T ðxÞ becomes In order to write the energy-momentum tensor, let us consider no μν − 3i Γμν ′ − ′ hi  hiT ðÞx = lim x, x G0 x x , ð33Þ 8π ′→ τ ℱ ακγa ℱ μνρa ′ ℱ ακγa ℱ μνρa ′ θ − ′ x x ðÞx x = ðÞx x x0 x0   ℱ μνρa ′ ℱ ακγa θ − ′ with + x ðÞx x0 x0 ,  μν ′ μ ′ν 1 μν ρ ′ ð25Þ Γ x, x =8 ∂ ∂ − η ∂ ∂ ρ : ð34Þ 4 with θðx − x ′ Þ being the step function. In the calculations 0 0 ff that follow, we use the commutation relation, Equation Now, the main objective is to study the e ects due to tem- fi (24), and perature and spatial compacti cation in Equation (33). To  achieve such an objective, the Thermo Field Dynamics for- ∂μθ − ′ μδ − ′ malism is used. x0 x0 = n0 x0 x0 , ð26Þ 4. Thermo Field Dynamics (TFD) Formalism μ where n0 = ð1, 0, 0, 0Þ is a time-like vector. fi Here, the Thermo Field Dynamics (TFD) formalism is intro- Using these de nitions, the energy-momentum tensor fi fi for the non-abelian GEM field becomes duced. TFD is a quantum eld theory at nite temperature [31–36]. In this formalism, the statistical average of any nohi operator is equal to its expected value in a thermal vac- μν 1 μν,λεωυ ′ a a ′ T ðÞx = − lim Δ x, x τ AλεðÞx Aωυ x , uum. For this equality to be true, two main elements are 4π ′→ x x required, i.e., (i) doubling of the original Hilbert space ð27Þ and (ii) the Bogoliubov transformation. Advances in High Energy Physics 5

fi S S ⊗ S~ S~ S s This doubling is de ned as T = , where and "# d ∞ s+ 〠 lσ s r are the tilde and original Hilbert space, respectively. The 2 s−1 σ v ðÞk ; α = 〠 〠 2 〠 ðÞ−η r=1 exp − 〠 ασ lσ k j , Bogoliubov transformation corresponds to a rotation of the j j s=1 fgσ lσ ,⋯,lσ =1 j=1 tilde and non-tilde variables which introduces the thermal s 1 s effects. To understand this doubling of Hilbert space, let ð42Þ us consider with d being the number of compactified dimensions, η =1 ! ! ð−1Þ for fermions (bosons), fσ g denotes the set of all combi- dðÞα dkðÞ s = ℬðÞα , ð35Þ nations with s elements and k is the 4-momentum. ~† ~† d ðÞα d ðÞk For the doubled notation, the vacuum expectation value of the energy-momentum tensor of the non-abelian GEM is where ℬðαÞ is the Bogoliubov transformation given as DEno  ! μνðÞab α − 3i Γμν ′ ðÞab − ′ α : T ðÞx ; = lim x, x G0 x x ; uðÞα −vðÞα 8π x′→x ℬðÞα = ð36Þ −vðÞα uðÞα ð43Þ In order to obtain a physical (renormalized) energy- with momentum tensor, the standard Casimir prescription is used. Then, αω − v2ðÞα = ðÞe − 1 1, u2ðÞα =1+v2ðÞα : ð37Þ DEDE μν μν μν T ðÞab ðÞx ; α = T ðÞab ðÞx ; α − T ðÞab ðÞx : ð44Þ The parameter α is the compactification parameter fi α α α ⋯α ω de ned by = ð 0, 1, D−1Þ and is energy. The tem- In this form, a measurable physical quantity is given as ff α ≡ β perature e ect is described by the choice 0 and no  α ⋯α α β μν 3i μν 1, D−1 =0. In this case, with = , the quantities T ðÞab α − Γ ′ ðÞab − ′ α 2 2 ðÞx ; = lim x, x G0 x x ; , v ðβÞ and u ðβÞ are related to the Bose distribution. 8π x′→x In order to introduce an application of TFD formalism, ð45Þ let us consider the free scalar field propagator. Then, in a doublet notation, it is given as where DE hi 

ðÞab − ′ α ~ τϕa α ϕb ′ α ~ ðÞab − ′ α ðÞab − ′ α − ðÞab − ′ : G0 x x ; = i 0, 0 ðÞx ; x ; 0, 0 , G0 x x ; = G0 x x ; G0 x x ð46Þ ð38Þ In the next section, some applications for different α − choices of parameter are developed. where ϕðx ; αÞ = ℬðαÞϕðxÞℬ 1ðαÞ and a, b =1,2. Then, 5. Some Applications ð 4 d k − − ′ ðÞab − ′ α ikðÞ x x ðÞab α In this section, applications, which consider the temperature G0 x x ; = i 4 e G0 ðÞk ; , ð39Þ ðÞ2π effects and spatial compactifications, are calculated. 5.1. Stefan-Boltzmann Law. As a first application, consider where the thermal effect that appears for α = ðβ,0,0,0Þ. In this case, the generalized Bogoliubov transformation becomes ðÞ11 α ≡ α 2 α − ∗ G0 ðÞk ; G0ðÞk ; = G0ðÞk + v ðÞk ; ½ŠG0ðÞk G0 ðÞk , ∞ 2 −β 0 ð40Þ v ðÞβ = 〠 e k j0 : ð47Þ

j0=1 with Then, the Green function is given as ð 1 4 ∞ G ðÞk = , ðÞ11 d k −ik x−x′ −βk0 j ∗ 0 2 − 2 ε G x − x′ ; α = e ðÞ〠 e 0 ½ŠG ðÞk − G ðÞk , k m + i ð41Þ 0 π 4 0 0 ÀÁ ðÞ2 j =1 − ∗ π δ 2 − 2 : 0 G0ðÞk ½ŠG0 ðÞk =2 i k m ∞  〠 − ′ − β =2 G0 x x i j0n0 , j =1 The parameter v2ðk ; αÞ is the generalized Bogoliubov 0 transformation [39]. It is defined as ð48Þ 6 Advances in High Energy Physics

μ tion. With these conditions, the energy-momentum tensor where n0 = ð1, 0, 0, 0Þ. Then, the energy-momentum tensor at finite temperature is becomes ()() ∞  ∞  ν μν 6i μ μ 1 μν ρ μνðÞ11 6i μ ′ 1 μν ρ ′ ′ T ðÞ11 − 〠 ∂ ∂′ − ∂ ∂′ − ′ − : T ðÞx ; β = − lim 〠 ∂ ∂ − g ∂ ∂ ρ G x − x − iβj n : ðÞx ; d = lim g ρ G0 x x 2dl3z π ′ 0 0 0 π → ′ x →x 4 x x l =1 4 j0=1 3 ð49Þ ð54Þ

For μ = ν =0, the Casimir energy to the non-abelian Using the Riemann Zeta function, i.e., field case is ∞ 1 π4 π ζðÞ4 = 〠 = , ð50Þ EdðÞ≡ T 00ðÞ 11 ðÞx ; d = − , ð55Þ j4 90 4 j0=1 0 480d fi and for μ = ν =3, the Casimir pressure for the non- the Stefan-Boltzmann law for the non-abelian GEM eld is fi obtained as abelian GEM eld is π π 33ðÞ 11 00ðÞ 11 4 Pd ≡ T x ; d = − : ð56Þ ETðÞ≡ T ðÞx ; β = T : ð51Þ ðÞ ðÞ 4 10 160d

Note that the energy density of the non-abelian gauge The negative sign shows that the Casimir force between fields is similar to the abelian field case. the plates is attractive, similar to the case of the electromag- Here, the numeric value is multiplied by the group gener- netic field and of the abelian GEM field. ator number. 5.3. Casimir Effect at Finite Temperature. For α = ðβ,0,0, ff fi 5.2. Casimir Effect at Zero Temperature. Here, α = ð0, 0, 0, iLÞ i2dÞ, the temperature e ects and spatial compacti cations is chosen and the Bogoliubov transformation is are considered. In this case, the Bogoliubov transformation becomes ∞ − 3 ÀÁÀÁÀÁ ÀÁÀÁ 2 〠 iLk l3 : v ðÞL = e ð52Þ v2 k0, k3 ; β, d = v2 k0 ; β + v2 k3 ; d +2v2 k0 ; β v2 k3 ; d l3=1 ∞ ∞ ∞ −β 0 − 3 −β 0 − 3 = 〠 e k j0 + 〠 e iLk l3 +2 〠 e k j0 iLk l3 : The Green function is j0=1 l3=1 j0,l3=1 ∞  ð57Þ ðÞ11 − ′ 〠 − ′ − : G x x ; L =2 G0 x x Ll3z ð53Þ 0 fi l3=1 The Green function, corresponding to the rst two terms, is given in Equation (48) and in Equation (53), fi A sum over l3, for L =2d,denes the nontrivial part respectively. The Green function associated with the third of the Green function with the Dirichlet boundary condi- term is

ð 4 ∞ ∞  ðÞ11 d k − − ′ −β 0 − 3 ∗  − ′ β ikðx x Þ 〠 k j0 iLk l3 − 〠 − ′ − β − : G x x ; , d =2 e e ½ŠG0ðÞk G ðÞk =4 G0 x x i j n 2dl3z ð58Þ 0 ðÞ2π 4 0 0 j0,l3=1 j0,l3=1

Then, the Casimir energy and pressure at finite tempera- π π ture are given, respectively, by PðÞβ, d = T 33ðÞ 11 ðÞβ, d = − 30β4 160d4 ∞ 2 2 ð59Þ π π 6 ðÞβj − 32ðÞdl 00ðÞ 11 〠 ÂÃ0 3 : EðÞβ, d = T ðÞβ, d = − + 3 3 β4 4 π β 2 2 10 480d j0,l3=1 ðÞj0 +2ðÞdl3 ∞ 6 3ðÞβj 2 − ðÞ2dl 2 + 〠 ÂÃ0 3 , π3 β 2 2 3 fi j0,l3=1 ðÞj0 +2ðÞdl3 Note that the rst and second terms are the Stefan- Boltzmann law and Casimir effect at zero temperature, Advances in High Energy Physics 7 respectively, while the third term corresponds to the Casimir [4] N. Brambilla, S. Eidelman, P. Foka et al., “QCD and strongly effect at finite temperature. coupled gauge theories: challenges and perspectives,” Euro- In the last case, both effects, temperature and spatial pean Physical Journal C: Particles and Fields, vol. 74, no. 10, compactification, are present. p. 2981, 2014. [5] J. C. Maxwell, “A dynamical theory of the electromagnetic field,” Philosophical Transactions of the Royal Society of 6. Conclusion London, vol. 155, pp. 459–512, 1865. “ The non-abelian GEM field is investigated. First, the [6] O. Heaviside, A gravitational and electromagnetic analogy. ” Lagrangian formulation for the abelian GEM field is pre- Part II, The Electrician, vol. 31, p. 259, 1893. “ sented. Then, using the principle of local gauge invariance, [7] O. Heaviside, A gravitational and electromagnetic analogy. ” an extension of the non-abelian GEM field is constructed. Part I, The Electrician, vol. 31, pp. 281-282, 1893. “ The symmetry group for the non-abelian GEM is group [8] A. Matte, Sur De Nouvelles Solutions Oscillatoires Des Equa- ” SUð2Þ. The abelian and non-abelian GEMs have a corre- tions De La Gravitation, Canadian Journal of Mathematics, fi vol. 5, pp. 1–16, 1953. spondence with the weak- eld approach of General Relativ- “ ity. The abelian GEM has a structure equivalent to the [9] W. B. Campbell and T. A. Morgan, Debye potentials for the fi fi gravitational field,” Physica, vol. 53, no. 2, pp. 264–288, 1971. weak- eld approximation of rst-order and non-abelian “ Weyl GEM is equivalent to the weak-field approximation [10] W. B. Campbell, The linear theory of gravitation in the radi- ation gauge,” General Relativity and Gravitation, vol. 4, no. 2, up to the second order. For simplicity, the self-interaction pp. 137–147, 1973. terms of the non-abelian gauge field are ignored. Then, the [11] W. B. Campbell and T. A. Morgan, “Maxwell form of the linear energy-momentum tensor is calculated. The TFD formalism ” ff theory of gravitation, American Journal of Physics, vol. 44, is used to introduce thermal e ects. This formalism requires no. 4, pp. 356–365, 1976. two basic ingredients: the doubling of the Hilbert space and [12] W. B. Campbell, J. Macek, and T. A. Morgan, “Relativistic the Bogoliubov transformation. 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