# Non-Abelian Gravitoelectromagnetism and Applications at Finite Temperature

Authors: A. F. Santos; J. Ramos; Faqir C. Khanna
Journal: Advances in High Energy Physics
 (2020)

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## Abstract

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.


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## Body

## 1. Introduction

The Standard Model (SM) is a non-abelian gauge theory with symmetry groupU1×SU2×SU3. SM describes theoretically and experimentally three of the four fundamental forces of nature, i.e., the electromagnetic, weak, and strong forces. The electromagnetism is a U1 abelian gauge theory which has been tested to a high precision. The generalization of an abelian gauge theory to the non-abelian gauge theory was proposed by Yang and Mills [1]. The last one describes the electroweak unification and quantum chromodynamics. The electroweak interaction is described by an SU2×U1 group and while the SU3 group satisfies the quantum chromodynamics [2–4].Gravity is not a part of SM. This implies that the SM is not a fundamental theory that describes all fundamental interactions of nature. In this paper, an extension of non-abelian gravity is discussed. Some applications of such a theory are developed. The gravitational theory studied here is the Gravitoelectromagnetism (GEM). GEM is an approach based on describing gravity in a way analogous to the electromagnetism [5–7]. Several studies about the GEM theory have been developed [8–14]. These ideas arise from the analogybetween equations for the Newton and Coulomb laws and the interest has increased with the discovery of the Lense-Thirring effect, where a rotating mass generates a gravitomagnetic field [15–17]. Some experiments that study this effect have been developed, such as LAGEOS (Laser Geodynamics Satellites) and LAGEOS 2 [18], the Gravity Probe B [19], and the mission LARES (Laser Relativity Satellite) [20, 21].The GEM theory may be analyzed by three different approaches: (i) using the similarity between the linearized Einstein and Maxwell equations [22], (ii) a theory based on an approach using tidal tensors [23], and (iii) the decomposition of the Weyl tensor (Cijkl) into ℬij=1/2∈iklC0jkl and ℰij=−C0i0j, the gravitomagnetic and gravitoelectric components, respectively [24]. In this paper, the Weyl tensor approach is used. A Lagrangian formulation for GEM is developed [25], and a gauge transformation in GEM is studied [26]. Here, an extension to non-abelian GEM fields is introduced. Applications of the non-abelian GEM at finite temperature are investigated. The temperature effects are introduced using Thermo Field Dynamics (TFD) formalism.There are two ways to introduce the temperature effect: (i) using the imaginary time formalism [27] and (ii) using the real-time formalism [28–36]. In this paper, TFD formalism is chosen. It is a real-time finite temperature formalism. In this formalism, a thermal state is developed where the main objective is to interpret the statistical average of an arbitrary operator as an expectation value in a thermal vacuum. Two elements are necessary to construct this thermal state: (i) doubling of the original Hilbert space and (ii) the use of Bogoliubov transformations. These are two Hilbert spaces, the original space S and the tilde space S~, which are related by a mapping, called the tilde conjugation rules, while the Bogoliubov transformation consists in a rotation involving these two spaces that ultimately introduce the temperature effects.The Stefan-Boltzmann law and the Casimir effect for the non-abelian GEM field at finite temperature are calculated. The Stefan-Boltzmann law describes the power radiated from a black body in terms of its temperature. The Casimir effect, proposed by H. Casimir [37], is a quantum phenomenon that appears due to vacuum fluctuations of any quantum field. The results in this case may be at zero or finite temperatures.This paper is organised as follows. In section II, a brief introduction to the abelian GEM Lagrangian formalism is presented. In section III, an extension to non-abelian GEM field is developed. The energy-momentum tensor associated to the non-abelian gauge field is calculated. In section IV, the TFD formalism is introduced. In section V, some applications considering the non-abelian GEM field at finite temperature are analysed. (i) The Stefan-Boltzmann law is calculated. (ii) The Casimir effect at zero temperature is obtained, and (iii) the Casimir effect at finite temperature is calculated. In section VI, some concluding remarks are presented.


## 2. Lagrangian Formulation of Abelian Gem

In this section, an introduction to the Lagrangian formulation of abelian GEM is presented. The GEM field equations, Maxwell-like equations, are(1)∂iℰij=−4πGρj,∂iℬij=0,∈ikl∂kℬlj+1c∂ℰij∂t=−4πGcJij,∈ikl∂kℰlj+1c∂ℬij∂t=0,where G is the gravitational constant, εikl is the Levi-Civita symbol, ρj is the vector mass density, Jij is the mass current density, and c is the speed of light. The quantities ℰij, ℬij, and Jij are the gravitoelectric field, the gravitomagnetic field, and the mass current density, respectively. The symbol ⋯ denotes symmetrization of the first and last indices, i.e., i and j.The fieldsℰij and ℬij are expressed in terms of a symmetric rank-2 tensor field, A~, with components Aij, such that
(2)ℬ=curlA~,ℰ+1c∂A~∂t=−gradφ,where φ is the GEM counterpart of the electromagnetic (EM) scalar potential ϕ.Definingℱμνα as the gravitoelectromagnetic tensor, the GEM field equations become
(3)∂μℱμνα=4πGcJνα,∂μGμνα=0,where Jνα depends on quantities ρi and Jij that are the mass and the current density, respectively. In addition, the gravitoelectromagnetic tensor is defined as
(4)ℱμνα=∂μAνα−∂νAμα,and the dual GEM tensor is defined as
(5)Gμνα=12∈μνγσηαρ−ℱγσρ.Using these definitions, the GEM Lagrangian density is given as [25].
(6)ℒG=−116πℱμναℱμνα−GcJναAνα.This Lagrangian allows considering several gravitational applications involving the graviton, such as interactions with other fundamental particles. This makes it possible to study several related topics.In this way, the GEM theory is described by two fieldsℰij and ℬij, which are symmetric and traceless tensors of the second order. These fields can be expressed in terms of the symmetric gravitoelectromagnetic potential Aμν [25, 26], analogous to that of electromagnetism Aμ. Thus, Aμν is the fundamental field in GEM and naturally, it has two indices [25, 26].It is important to note that GEM equations correspond to the weak-field approximation of General Relativity. They do not describe strong fields and, therefore, do not include the full Einstein equations. To be more specific, the abelian GEM corresponds to the linear part of Einstein equations and the non-abelian GEM corresponds up to the second order in the weak-field approach.


## 3. Non-Abelian Gem

Let us consider an extension of the GEM field to include the non-abelian gauge transformations [38]. Then, in this section, the Lagrangian for the non-abelian GEM field is presented and the energy-momentum tensor associated to the non-abelian field is calculated.In order to obtain the non-abelian gauge transformation for the GEM field, let us investigate the Dirac Lagrangian under global and local gauge transformations. The free Dirac Lagrangian is given as(7)ℒD=−iψ¯xγμ∂μψx+mψ¯xψx,where ψx is a two-component column vector. This Lagrangian is invariant under global SU2 gauge transformation given as
(8)ψ′x=Uψx,with U being a 2×2 unitary matrix that is written as U=eiH, where H is a hermitian matrix. To study local gauge transformation, more details are necessary.Let us assume that the local gauge transformation is(9)ψ′x=Uxψx=eigHxψx,where g is the coupling constant and Hx is the hermitian 2×2 matrix given by
(10)Hx=σ·ax,with ax being real functions of x and σ are Pauli matrices. The Pauli matrices σii=1,2,3 are the generators of the non-abelian group SU2 satisfying the commutation relations σi,σj=2iεijkσk. In a more compact form, ax is written as
(11)ax=pαbαx,where bαx are vectors associated to each of the four directions in Minkowski spacetime and pα are the components of the one-form p~. Then, the local gauge transformation becomes
(12)ψ′x=eigpασ·bαxψx.The Dirac Lagrangian is not invariant under this local gauge transformation since the derivative∂μψ′x introduces a new term in the Lagrangian. In order to obtain an invariant Lagrangian, a covariant derivative is defined as
(13)Dμ=∂μ−igpασ·Aμαx,where the tensor gauge field Aμαx has three components Aμαx=A1μαx,A2μαx,A3μαx and it transforms as
(14)A′kμαx=Akμαx+∂μbkα+2gεijkpβAiμβbjα.An important note, there is one tensor gauge fieldAiμαx for each generator σi of the group SU2. Moreover, in the definition of the covariant derivative Dμ (Equation (13)), the gauge field Aμν should appear to keep the local gauge invariance like in electromagnetism. In order to have it, the oneform pα is introduced [26]. The one form makes the phase function to split into phase factors each associated with one of the four directions in spacetime.Using these results and replacing the derivative∂μ by the covariant derivative Dμ, the Dirac Lagrangian is gauge invariant, i.e.,
(15)ℒD=−iψ¯xγμDμψx+mψ¯xψx.In this formulation, three new gauge tensor fields are introduced. To write a full Lagrangian invariant under local gauge transformation, a kinetic term ofAμαx must be constructed. To do that, an analogue of the electromagnetic tensor Fμν is constructed. For obtaining the antisymmetric third-rank tensor of the gauge field, let us consider a covariant derivative (13). Then,
(16)Dμ,Dν=−igpασ·Fμνα,where
(17)Fkμνα=∂μAkνα−∂νAkμα+2gεijkpβAiμαAjνβ.Then, the full Lagrangian that is invariant under localSU2 gauge transformations is
(18)ℒD=−iψ¯xγμDμψx+mψ¯xψx−116πFμνα⋅Fμνα.This Lagrangian describes two equal mass Dirac fields interacting with three massless tensor gauge fields.In conclusion, GEM is an approach based on formulating gravity in analogy to electromagnetism. In this way, GEM becomes a gauge field theory of gravity in contrast with the geometric theory of General Relativity. Then, it is expected thatSU2 be the gauge symmetry group. It is the Weyl tensors ℰij and ℬij that keep the connection of GEM to gravity.Now, let us determine the energy-momentum tensor associated with the non-abelian GEM field.


### 3.1. Energy-Momentum Tensor for Non-Abelian GEM

Hereafter, the Lagrangian density for the free non-abelian GEM field is considered, i.e.,(19)ℒ=−116πFμναaFμναaThe indexa is summed over the generators of the gauge group and for an SUN group, one has a,b,c=1⋯N2−1. Here, as a first application of the non-abelian GEM, the self-interaction between the tensor gauge fields is ignored.Using the energy-momentum tensor definition,(20)Tμν=∂ℒ∂∂μAλαa∂νAλαa−ημνℒ,the energy-momentum tensor associated with the non-abelian GEM field is
(21)Tμν=14−ℱλαμaℱνλαa+14ημνℱρσθaℱρσθa.To avoid a product of field operators at the same spacetime point, the energy-momentum tensor is written as(22)Tμνx=14πlimx′→xτ−ℱλαμaxℱνλαax′+14ημνℱρσθaxℱρσθax′where τ is the time order operator.The quantization of the non-abelian GEM field requires that(23)πκλa=∂ℒ∂∂0Aκλa=−14πF0κλa.the commutation relation0 and divA~=∂iAij=0, the covariant quantization is carried out and the commutation relation is
(24)Aijax,t,πklbx′,t=i2δikδjl−δilδjk−1∇2δjl∂i∂k−−δjk∂i∂l−δil∂j∂k+δik∂j∂lδ3x−x′δab.Other commutation relations are zero.In order to write the energy-momentum tensor, let us consider(25)τℱακγaxℱμνρax′=ℱακγaxℱμνρax′θx0−x0′+ℱμνρax′ℱακγaxθx0−x0′,with θx0−x0′ being the step function. In the calculations that follow, we use the commutation relation, Equation (24), and
(26)∂μθx0−x0′=n0μδx0−x0′,where n0μ=1,0,0,0 is a time-like vector.Using these definitions, the energy-momentum tensor for the non-abelian GEM field becomes(27)Tμνx=−14πlimx′→xΔμν,λεωυx,x′τAλεaxAωυax′,where
(28)Δμν,λεωυx,x′=Γμρα,νρα,λεωυx,x′−14ημνΓρσθ,ρσθ,λεωυx,x′,with
(29)Γακγ,μνρ,λεωυx,x′=gκλgεγ∂α−gαλgεγ∂κgνωgρυ∂′μ−gμωgρυ∂′ν.The vacuum expectation value of the energy-momentum tensor leads to the expression(30)Tμνx=0∣Tμνx∣0=−14πlimx′→xΔμν,λεωυx,x′0τAλεaxAωυax′0,where the graviton propagator is
(31)0τAλεaxAωυax′0=δab0τAλεaxAωυbx′0=iδabDλεωυabx−x′,with
(32)Dλεωυab=12δabgλωgευ+gλυgεω−gλεgωυG0x−x′,and G0x−x′ is the massless scalar field propagator. Then, the vacuum expectation value of Tμνx becomes
(33)Tμνx=−3i8πlimx′→xΓμνx,x′G0x−x′,with
(34)Γμνx,x′=8∂μ∂′ν−14ημν∂ρ∂′ρ.Now, the main objective is to study the effects due to temperature and spatial compactification in Equation (33). To achieve such an objective, the Thermo Field Dynamics formalism is used.


## 3.1. Energy-Momentum Tensor for Non-Abelian GEM

Hereafter, the Lagrangian density for the free non-abelian GEM field is considered, i.e.,(19)ℒ=−116πFμναaFμναaThe indexa is summed over the generators of the gauge group and for an SUN group, one has a,b,c=1⋯N2−1. Here, as a first application of the non-abelian GEM, the self-interaction between the tensor gauge fields is ignored.Using the energy-momentum tensor definition,(20)Tμν=∂ℒ∂∂μAλαa∂νAλαa−ημνℒ,the energy-momentum tensor associated with the non-abelian GEM field is
(21)Tμν=14−ℱλαμaℱνλαa+14ημνℱρσθaℱρσθa.To avoid a product of field operators at the same spacetime point, the energy-momentum tensor is written as(22)Tμνx=14πlimx′→xτ−ℱλαμaxℱνλαax′+14ημνℱρσθaxℱρσθax′where τ is the time order operator.The quantization of the non-abelian GEM field requires that(23)πκλa=∂ℒ∂∂0Aκλa=−14πF0κλa.the commutation relation0 and divA~=∂iAij=0, the covariant quantization is carried out and the commutation relation is
(24)Aijax,t,πklbx′,t=i2δikδjl−δilδjk−1∇2δjl∂i∂k−−δjk∂i∂l−δil∂j∂k+δik∂j∂lδ3x−x′δab.Other commutation relations are zero.In order to write the energy-momentum tensor, let us consider(25)τℱακγaxℱμνρax′=ℱακγaxℱμνρax′θx0−x0′+ℱμνρax′ℱακγaxθx0−x0′,with θx0−x0′ being the step function. In the calculations that follow, we use the commutation relation, Equation (24), and
(26)∂μθx0−x0′=n0μδx0−x0′,where n0μ=1,0,0,0 is a time-like vector.Using these definitions, the energy-momentum tensor for the non-abelian GEM field becomes(27)Tμνx=−14πlimx′→xΔμν,λεωυx,x′τAλεaxAωυax′,where
(28)Δμν,λεωυx,x′=Γμρα,νρα,λεωυx,x′−14ημνΓρσθ,ρσθ,λεωυx,x′,with
(29)Γακγ,μνρ,λεωυx,x′=gκλgεγ∂α−gαλgεγ∂κgνωgρυ∂′μ−gμωgρυ∂′ν.The vacuum expectation value of the energy-momentum tensor leads to the expression(30)Tμνx=0∣Tμνx∣0=−14πlimx′→xΔμν,λεωυx,x′0τAλεaxAωυax′0,where the graviton propagator is
(31)0τAλεaxAωυax′0=δab0τAλεaxAωυbx′0=iδabDλεωυabx−x′,with
(32)Dλεωυab=12δabgλωgευ+gλυgεω−gλεgωυG0x−x′,and G0x−x′ is the massless scalar field propagator. Then, the vacuum expectation value of Tμνx becomes
(33)Tμνx=−3i8πlimx′→xΓμνx,x′G0x−x′,with
(34)Γμνx,x′=8∂μ∂′ν−14ημν∂ρ∂′ρ.Now, the main objective is to study the effects due to temperature and spatial compactification in Equation (33). To achieve such an objective, the Thermo Field Dynamics formalism is used.


## 4. Thermo Field Dynamics (TFD) Formalism

Here, the Thermo Field Dynamics (TFD) formalism is introduced. TFD is a quantum field theory at finite temperature [31–36]. In this formalism, the statistical average of any operator is equal to its expected value in a thermal vacuum. For this equality to be true, two main elements are required, i.e., (i) doubling of the original Hilbert space and (ii) the Bogoliubov transformation.This doubling is defined asST=S⊗S~, where S~ and S are the tilde and original Hilbert space, respectively. The Bogoliubov transformation corresponds to a rotation of the tilde and non-tilde variables which introduces the thermal effects. To understand this doubling of Hilbert space, let us consider
(35)dαd~†α=ℬαdkd~†k,where ℬα is the Bogoliubov transformation given as
(36)ℬα=uα−vα−vαuαwith
(37)v2α=eαω−1−1,u2α=1+v2α.The parameterα is the compactification parameter defined by α=α0,α1,⋯αD−1 and ω is energy. The temperature effect is described by the choice α0≡β and α1,⋯αD−1=0. In this case, with α=β, the quantities v2β and u2β are related to the Bose distribution.In order to introduce an application of TFD formalism, let us consider the free scalar field propagator. Then, in a doublet notation, it is given as(38)G0abx−x′;α=i0,0~τϕax;αϕbx′;α0,0~,where ϕx;α=ℬαϕxℬ−1α and a,b=1,2. Then,
(39)G0abx−x′;α=i∫d4k2π4e−ikx−x′G0abk;α,where
(40)G011k;α≡G0k;α=G0k+v2k;αG0k−G0∗k,with
(41)G0k=1k2−m2+iε,G0k−G0∗k=2πiδk2−m2.The parameterv2k;α is the generalized Bogoliubov transformation [39]. It is defined as
(42)v2k;α=∑s=1d∑σs2s−1∑lσ1,⋯,lσs=1∞−ηs+∑r=1slσrexp−∑j=1sασjlσjkσj,with d being the number of compactified dimensions, η=1−1 for fermions (bosons), σs denotes the set of all combinations with s elements and k is the 4-momentum.For the doubled notation, the vacuum expectation value of the energy-momentum tensor of the non-abelian GEM is(43)Tμνabx;α=−3i8πlimx′→xΓμνx,x′G0abx−x′;α.In order to obtain a physical (renormalized) energy-momentum tensor, the standard Casimir prescription is used. Then,(44)Tμνabx;α=Tμνabx;α−Tμνabx.In this form, a measurable physical quantity is given as(45)Tμνabx;α=−3i8πlimx′→xΓμνx,x′G¯0abx−x′;α,where
(46)G¯0abx−x′;α=G0abx−x′;α−G0abx−x′.In the next section, some applications for different choices of parameterα are developed.


## 5. Some Applications

In this section, applications, which consider the temperature effects and spatial compactifications, are calculated.


### 5.1. Stefan-Boltzmann Law

As a first application, consider the thermal effect that appears forα=β,0,0,0. In this case, the generalized Bogoliubov transformation becomes
(47)v2β=∑j0=1∞e−βk0j0.Then, the Green function is given as(48)G¯011x−x′;α=∫d4k2π4e−ikx−x′∑j0=1∞e−βk0j0G0k−G0∗k,=2∑j0=1∞G0x−x′−iβj0n0,where n0μ=1,0,0,0. Then, the energy-momentum tensor at finite temperature is
(49)Tμν11x;β=−6iπlimx′→x∑j0=1∞∂μ∂′ν−14gμν∂ρ∂′ρG0x−x′−iβj0n0.Using the Riemann Zeta function, i.e.,(50)ζ4=∑j0=1∞1j04=π490,the Stefan-Boltzmann law for the non-abelian GEM field is obtained as
(51)ET≡T0011x;β=π10T4.Note that the energy density of the non-abelian gauge fields is similar to the abelian field case.Here, the numeric value is multiplied by the group generator number.


### 5.2. Casimir Effect at Zero Temperature

Here,α=0,0,0,iL is chosen and the Bogoliubov transformation is
(52)v2L=∑l3=1∞e−iLk3l3.The Green function is(53)G¯011x−x′;L=2∑l3=1∞G0x−x′−Ll3z.A sum overl3, for L=2d, defines the nontrivial part of the Green function with the Dirichlet boundary condition. With these conditions, the energy-momentum tensor becomes
(54)Tμν11x;d=−6iπlimx→x′∑l3=1∞∂μ∂′μ−14gμν∂ρ∂ρ′G0x−x′−2dl3z.Forμ=ν=0, the Casimir energy to the non-abelian field case is
(55)Ed≡T0011x;d=−π480d4,and for μ=ν=3, the Casimir pressure for the non-abelian GEM field is
(56)Pd≡T3311x;d=−π160d4.The negative sign shows that the Casimir force between the plates is attractive, similar to the case of the electromagnetic field and of the abelian GEM field.


### 5.3. Casimir Effect at Finite Temperature

Forα=β,0,0,i2d, the temperature effects and spatial compactifications are considered. In this case, the Bogoliubov transformation becomes
(57)v2k0,k3;β,d=v2k0;β+v2k3;d+2v2k0;βv2k3;d=∑j0=1∞e−βk0j0+∑l3=1∞e−iLk3l3+2∑j0,l3=1∞e−βk0j0−iLk3l3.The Green function, corresponding to the first two terms, is given in Equation (48) and in Equation (53), respectively. The Green function associated with the third term is
(58)G¯011x−x′;β,d=2∫d4k2π4e−ikx−x′∑j0,l3=1∞e−βk0j0−iLk3l3G0k−G0∗k=4∑j0,l3=1∞G0x−x′−iβj0n−2dl3z.Then, the Casimir energy and pressure at finite temperature are given, respectively, by(59)Eβ,d=T0011β,d=π10β4−π480d4+6π3∑j0,l3=1∞3βj02−2dl32βj02+2dl323,Pβ,d=T3311β,d=π30β4−π160d4+6π3∑j0,l3=1∞βj02−32dl32βj02+2dl323.Note that the first and second terms are the Stefan-Boltzmann law and Casimir effect at zero temperature, respectively, while the third term corresponds to the Casimir effect at finite temperature.In the last case, both effects, temperature and spatial compactification, are present.


## 5.1. Stefan-Boltzmann Law

As a first application, consider the thermal effect that appears forα=β,0,0,0. In this case, the generalized Bogoliubov transformation becomes
(47)v2β=∑j0=1∞e−βk0j0.Then, the Green function is given as(48)G¯011x−x′;α=∫d4k2π4e−ikx−x′∑j0=1∞e−βk0j0G0k−G0∗k,=2∑j0=1∞G0x−x′−iβj0n0,where n0μ=1,0,0,0. Then, the energy-momentum tensor at finite temperature is
(49)Tμν11x;β=−6iπlimx′→x∑j0=1∞∂μ∂′ν−14gμν∂ρ∂′ρG0x−x′−iβj0n0.Using the Riemann Zeta function, i.e.,(50)ζ4=∑j0=1∞1j04=π490,the Stefan-Boltzmann law for the non-abelian GEM field is obtained as
(51)ET≡T0011x;β=π10T4.Note that the energy density of the non-abelian gauge fields is similar to the abelian field case.Here, the numeric value is multiplied by the group generator number.


## 5.2. Casimir Effect at Zero Temperature

Here,α=0,0,0,iL is chosen and the Bogoliubov transformation is
(52)v2L=∑l3=1∞e−iLk3l3.The Green function is(53)G¯011x−x′;L=2∑l3=1∞G0x−x′−Ll3z.A sum overl3, for L=2d, defines the nontrivial part of the Green function with the Dirichlet boundary condition. With these conditions, the energy-momentum tensor becomes
(54)Tμν11x;d=−6iπlimx→x′∑l3=1∞∂μ∂′μ−14gμν∂ρ∂ρ′G0x−x′−2dl3z.Forμ=ν=0, the Casimir energy to the non-abelian field case is
(55)Ed≡T0011x;d=−π480d4,and for μ=ν=3, the Casimir pressure for the non-abelian GEM field is
(56)Pd≡T3311x;d=−π160d4.The negative sign shows that the Casimir force between the plates is attractive, similar to the case of the electromagnetic field and of the abelian GEM field.


## 5.3. Casimir Effect at Finite Temperature

Forα=β,0,0,i2d, the temperature effects and spatial compactifications are considered. In this case, the Bogoliubov transformation becomes
(57)v2k0,k3;β,d=v2k0;β+v2k3;d+2v2k0;βv2k3;d=∑j0=1∞e−βk0j0+∑l3=1∞e−iLk3l3+2∑j0,l3=1∞e−βk0j0−iLk3l3.The Green function, corresponding to the first two terms, is given in Equation (48) and in Equation (53), respectively. The Green function associated with the third term is
(58)G¯011x−x′;β,d=2∫d4k2π4e−ikx−x′∑j0,l3=1∞e−βk0j0−iLk3l3G0k−G0∗k=4∑j0,l3=1∞G0x−x′−iβj0n−2dl3z.Then, the Casimir energy and pressure at finite temperature are given, respectively, by(59)Eβ,d=T0011β,d=π10β4−π480d4+6π3∑j0,l3=1∞3βj02−2dl32βj02+2dl323,Pβ,d=T3311β,d=π30β4−π160d4+6π3∑j0,l3=1∞βj02−32dl32βj02+2dl323.Note that the first and second terms are the Stefan-Boltzmann law and Casimir effect at zero temperature, respectively, while the third term corresponds to the Casimir effect at finite temperature.In the last case, both effects, temperature and spatial compactification, are present.


## 6. Conclusion

The non-abelian GEM field is investigated. First, the Lagrangian formulation for the abelian GEM field is presented. Then, using the principle of local gauge invariance, an extension of the non-abelian GEM field is constructed. The symmetry group for the non-abelian GEM is groupSU2. The abelian and non-abelian GEMs have a correspondence with the weak-field approach of General Relativity. The abelian GEM has a structure equivalent to the weak-field approximation of first-order and non-abelian Weyl GEM is equivalent to the weak-field approximation up to the second order. For simplicity, the self-interaction terms of the non-abelian gauge field are ignored. Then, the energy-momentum tensor is calculated. The TFD formalism is used to introduce thermal effects. This formalism requires two basic ingredients: the doubling of the Hilbert space and the Bogoliubov transformation. With this formalism, the vacuum expectation value of the energy-momentum tensor is obtained and thus, some applications for the non-abelian GEM field are investigated. The Stefan-Boltzmann law and the Casimir effect at finite temperature are calculated. Our results show that the non-abelian quantities are similar to the abelian quantities. The main difference consists in the fact that the non-abelian results are equal to the abelian result multiplied by the number of gauge fields. These results are similar to the case of the electromagnetic field. For example, the non-abelian GEM Casimir effect is attractive as the electromagnetic case. In addition, calculations involving the SU2 group and GEM have not been done in the literature. This work is the first to introduce this type of approach.



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*Source: 5193692-2020-04-03.xml*