# Spin-1/2 Particles in Phase Space: Casimir Effect and Stefan-Boltzmann Law at Finite Temperature

Authors: R. G. G. Amorim; S. C. Ulhoa; J. S. da Cruz Filho; A. F. Santos; F. C. Khanna
Journal: Advances in High Energy Physics
 (2020)

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## Abstract

The Dirac field, spin 1/2 particles, is investigated in phase space. The Dirac propagator is defined. The Thermo Field Dynamics (TFD) formalism is used to introduce finite temperature. The energy-momentum tensor is calculated at finite temperature. The Stefan-Boltzmann law is established, and the Casimir effect is calculated for the Dirac field in phase space at zero and finite temperature. A comparative analysis with these results in standard quantum mechanics space is realized.


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## Body

## 1. Introduction

The Wigner function formalism [1, 2] and noncommutative geometry [3] play a fundamental role in the study of phase space quantum mechanics. The Wigner formalism enables a quantum operator, A, defined in the Hilbert space, S, to have an equivalent function of the type awq,p, in phase space Γ, using the Moyal-product or star-product (å). Such a formalism leads to the classical limit of a quantum theory. In fact, quantum mechanics is a noncommutative theory whose representation in phase space is an object of debate. The opposite question, i.e., for a given classical function, what is its quantum counterpart? It is solved by using the Weyl transformation which is formulated independent of the phase space. In fact, it can be established within the configuration space of the generalized coordinates. The phase space has a well-defined physical meaning. The Hamiltonian function is naturally identified with the energy of the system. Establishing a field theory in phase space sheds light on some obscure points in quantum mechanics. For instance, the quantum symmetries are better understood in the symplectic structure of phase space which is similar to the role of Lorentz transformation in the covariant formulation of special relativity. This theoretical framework has to include a finite temperature in order to be suitable for experiments.The star product has been employed for different objectives. In particular, it has been used for development of a nonrelativistic quantum mechanics formalism in terms of a phase space using the Galilean symmetry representation [4]. Thus, the Schrödinger equation is obtained. In this case, the wave function ψ=ψq,p is a quasiprobability amplitude defined in phase space and the Wigner function is obtained in an alternative way, i.e., by using fwq,p=ψ†åψ. The Dirac equation coupled with the electromagnetic field in phase space [5] and applications [6] has been obtained.Our goal is to explore the quasiprobability amplitude to study the effect of temperature using Thermo Field Dynamics (TFD) formalism [7–13] in a system for spin-1/2 particles. The principles of this theory are the duplication of the Fock space using the Bogoliubov transformations. The TFD formalism is used to study the Casimir effect, at zero and finite temperature. The scalar field in phase space has been studied [14], and some exclusive effects have been found at finite temperature. In addition, the Stefan-Boltzmann law for spin-1/2 particles in phase space is described in details.In Section2, the symplectic Dirac field is introduced. In Section 3, the Thermo Field Dynamics formalism is presented. In Section 4, the Stefan-Boltzmann law is established and the Casimir effect for the Dirac field is calculated in phase space at zero and finite temperature. In the last section, some concluding remarks are presented.


## 2. Spin-1/2 Field in Phase Space

A brief outline for spin-1/2 particles in phase space formalism is described. For this purpose, the following star operators in phase space are defined:(1)P^μ=pμå=pμ−i2∂∂qμ,(2)Q^μ=qμå=qμ+i2∂∂pμ,(3)M^μν=Q^μP^ν−Q^νP^μ,which satisfy the Heisenberg commutation relation Q^μ,P^ν=igμν, with gμν=diag−1,1,1,1. The Poincaré algebra has the form
(4)P^μ,P^ν=0,P^μ,M^νσ=igμνP^σ−gσμP^ν,M^μν,M^σρ=−igμρM^νσ−gνρM^μσ+gμσM^ρν−gνσM^ρμ.The operators in Equations ((1)–(3)) are defined on a Hilbert space, HΓ, associated with the phase space Γ. The operators P^μ and M^μν stand for translations, rotations, and boosts, respectively. Functions defined on the Hilbert space HΓ are defined as
(5)∫ϕ∗qμ,pμϕqμ,pμdqμdpμ<∞.The Casimir invariants areP^2=P^μP^μ and W^=W^μW^μ, where W^μ=1/2εμνσρM^νσP^ρ are Pauli-Lubansky matrices and εμνσρ is the Levi-Civita symbol.The Dirac equation in phase space is obtained using the invariant operatorγμP^μ. It is defined as
(6)γμP^μψq,p=mψq,p,γμpμ−i2∂∂qμψq,p=mψq,pwhere γμ are the Dirac matrices. The Lagrangian density for the Dirac equation is
(7)L=−i4∂ψ¯∂qμγμψ−ψ¯γμ∂ψ∂qμ−ψ¯m−γμpμψ,where ψ¯=ψ†γ0 and m is the mass of the particle.The Wigner function provides the physical interpretation [5] and is given as
(8)fWq,p=ψ¯q,påψq,p,where the star product is defined by
(9)aWq,påbWq,p=aWq,pexpiℏ2∂→∂q∂←∂p−∂→∂p∂←∂qbWq,p.Using the Noether theorem in phase space [5], the energy-momentum tensor for the Dirac field is
(10)θDμν=−i4−ψ¯γμ∂ψ∂qν+γμψ∂ψ¯∂qν−gμνL.Then, the Green function,GDq−q′,p−p′, is defined as
(11)i2γμ∂GDq−q′,p−p′∂qμ+m−γμpμGDq−q′,p−p′=δq−q′δp−p′,which may be written as
(12)12γμkμG~k,p−p′+m−γμpμG~k,p−p′=δp−p′,where G~k,p−p′=1/2π4∫d4qeikμqμ−q′μGDq−q′,p−p′. The propagator of the Dirac field is
(13)G~k,p−p′=δp−p′γμkμ/2−pμ+m.Then, Equation (12) is an algebraic expression which yields (13). Then, the Green function, GD, is
(14)GDq−q′,p−p′=∫d4k2π4e−ikμqμ−qμ′G~k,p−p′=∫d4k2π4e−ikμqμ−qμ′δp−p′γμkμ/2−pμ+m.TakingM=2m, the expression is
(15)GDq−q′,p−p′=2e−2iqμ−q′μpμ−p′μδp−p′∫d4k2π4e−ikμqμ−q′μγμkμ+M.This leads to the Green function for the Dirac equation in phase space(16)GDq−q′,p−p′=2e−2iqμ−q′μpμ−pμ′δp−p′i∂μγμ−MG0q−q′.G0q−q′ is defined as
(17)G0q−q′=−iM4π2K1−qμ−q′μqμ−q′μ−qμ−q′μqμ−q′μ,where κνq is the modified Bessel function. It should be noted that due to the dependence on the Dirac matrices, the Green’s function has matrix properties itself.


## 3. Thermo Field Dynamics Formalism

The Thermo Field Dynamics (TFD) is a thermal quantum field theory at finite temperature [7–13]. It has two basics elements: (i) doubling the degrees of freedom in a Hilbert space and (ii) the Bogoliubov transformation. The doubling of Hilbert space is given by the tilde (~) conjugate rules where the thermal space is ST=S⊗S, with S being the standard Hilbert space and S the tilde (dual) space. There is a mapping between the two spaces; i.e., the map between the tilde b~i and nontilde bi operators is defined by the following tilde conjugation rules:
(18)bibj~=b~ib~j,cbi+bj~=c∗b~i+b~j,bi†~=b~i†,b~i~=−ξbi,with ξ=−1 for bosons and ξ=+1 for fermions.The Bogoliubov transformation corresponds to a rotation of the tilde and nontilde variables. Using the doublet notation, for fermions leads to(19)ba=b1αb2α=bαb~†α=Bαbkb~†k,where b†,b~† are creation operators, b,b~ are destruction operators, and Bα is the Bogoliubov transformation given by
(20)Bα=uα−vαvαuα.Takingα=β (β≡1/kBT with kB being the Boltzmann constant and T the temperature), the thermal operators are written explicitly as
(21)bβ=uβbk−vβb~†k,b~β=uβb~k+vβb†k,b†β=uβb†k−vβb~k,b~†β=uβb~†k+vβbk.These thermal operators satisfy the algebraic rules(22)bpβ,bq†β=δ3p−q,b~pβ,b~q†β=δ3p−q,and other anticommutation relations are null. In addition, the quantities uβ and vβ are related to the Fermi distribution, i.e.,
(23)u2β=11+e−βωandv2β=11+eβω,such that v2β+u2β=1. The parameter α is associated with temperature, but, in general, it may be associated with other physical quantities. In general, a field theory on the topology ΓDd=S1d×ℝD−d with 1≤d≤D, is considered. D are the space-time dimensions, and d is the number of compactified dimensions. This establishes a formalism such that any set of dimensions of the manifold ℝD can be compactified, where the circumference of the nth S1 is specified by αn. The α parameter is assumed as the compactification parameter defined by α=α0,α1,⋯,αD−1. The effect of temperature is described by the choice α0≡β and α1,⋯,αD−1=0.Any field in the TFD formalism may be written in terms of theα parameter. As an example, the scalar field is considered. Then, the α-dependent scalar field becomes
(24)ϕq,p;α=Bαϕq,pB−1α,where the Bogoliubov transformation is used.Theα-dependent propagator for the scalar field is
(25)G0abq−q′,p−p′;α=i0,0~∣τϕaq,p;αϕbq′,p′;α∣0,0~,where τ is the time-ordering operator. Using 0α=Bα0,0~ leads to the Green function
(26)G0abq−q′,p−p′;α=i0ατϕaq,pϕbq′,p′0α=i∫d4k2π4e−ikq−q′p−p′G0abk;α,where
(27)G0abk;α=B−1k;αG0abkBk;α,with Bk;α being the Bogoliubov transformation and
(28)G0abk=G0k00G0∗k,where
(29)G0k=1k2−m2+iεis the scalar field propagator and m is the scalar field mass. Here, G0∗k is the complex conjugate of G0k.It is important to note that the physical quantities are given by the nontilde variables. Then, the physical Green functionG011k;α is written as
(30)G011k;α≡G0k;α=G0k+v2αG0∗k−G0k,where
(31)v2k;α=∑s=1d∑σs2s−1∑lσ1,⋯,lσs=1∞−ηs+∑r=1slσrexp−∑j=1sασjlσjkσjis the generalized Bogoliubov transformation [15], where d is the number of compactified dimensions, η=1−1 for fermions (bosons), σs denotes the set of all permutations with s elements, and k is the 4-momentum. In the next section, three different topologies are used [16]: (i) the topology Γ41=S1×ℝ3, where α=β,0,0,0. In this case, the time axis is compactified in S1, with circumference β; (ii) the topology Γ41 with α=0,0,0,i2d, where the compactification along the coordinate z is considered; and (iii) the topology Γ42=S1×S1×ℝ2 with α=β,0,0,i2d is used. In this case, the double compactification consists in time and the coordinate z. Then, thermal effects are considered for the Casimir effect and Stefan-Boltzmann law.


## 4. Stefan-Boltzmann Law and Casimir Effect for the Dirac Field in Phase Space

The Stefan-Boltzmann law is calculated by analyzing the energy-momentum tensor given as(32)θDμνq,p=limq′,p′→q,pτ−i4−ψ¯′γμ∂ψ∂qν+γμψ∂ψ¯′∂qν′+gμνi4∂ψ¯∂qλγλψ′−ψ¯γλ∂ψ′∂qλ′+ψ¯m−γμpμψ′=limq′,p′→q,pΓμντψ¯′q′,p′ψq,p,where
(33)Γμν=−i4−γμ∂∂qν+γμ∂∂qν′−gμν∂∂qλγλ−γλ∂∂qλ′+gμνm−γμpμ.It should be noted that the fieldψ′ is the Dirac field in phase space as a function of the variables (q′,p′), i.e., ψ′≡ψq′,p′. The vacuum expectation value of the energy-momentum tensor is
(34)θDμνq,p=limq′,p′→q,pΓμν0∣τψ¯′q′,p′ψq,p∣0.The Dirac propagator in phase space is defined in Equation (16) as
(35)GDq−q′,p−p′=i0∣τψ¯′q′,p′ψq,p∣0.Then, the energy-momentum tensor has the form(36)θDμνq,p=−ilimq′,p′→q,pΓμν2e−2iqμ−q′μpμ−p′μδp−p′i∂μγμ−2mG0q−q′.The vacuum average of the energy-momentum tensor in terms ofα-dependent fields becomes
(37)θDμνabq,p;α=limq′,p′→q,p−iΓμν2e−2iqμ−q′μpμ−p′μδp−p′i∂μγμ−2mG0q−q′;α.In order to obtain measurable physical quantities at finite temperature, a renormalization procedure is carried out. The physical energy-momentum tensor is defined as(38)Tμνabq,p;α=θDμνabq,p;α−θDμνabq,p=−ilimq′,p′→q,pΓμνG¯Dabq−q′,p−p′;α,where
(39)G¯Dabq−q′,p−p′;α=GDabq−q′,p−p′;α−GDabq−q′,p−p′.Now, the Stefan-Boltzmann law and the Casimir effect in phase space are calculated at finite temperature.


### 4.1. Stefan-Boltzmann Law

The study of the Stefan-Boltzmann law in phase space corresponds to a choice of the parameterα. It is important to note that the parameter α is the compactification parameter that is defined as α=α0,α1,⋯,αD−1. The temperature effect is described by the choice α=β,0,0,0.The generalized Bogoliubov transformation, Equation (31), for these parameters is
(40)v2k,β=∑l=1∞−1l+1e−βk0l.The Green’s function for the Dirac field in phase space is(41)G¯Dabq−q′,p−p′;β=∑l=1∞−1l+1GD∗q−q′+iβln0,p−p′−GDq−q′−iβln0,p−p′,where n0=1,0,0,0 is a time-like vector. Then, the physical energy-momentum tensor is
(42)Tμν11β=−ilimq′,p′→q,p∑l=1∞−1l+1ΓμνGD∗q−q′+iβln0,p−p′−GDq−q′−iβln0,p−p′.In order to calculate the Stefan-Boltzmann law, takingμ=ν=0 leads to
(43)T0011β=−limp′→p∑l=1∞4mδp−p′−1l+1e−2p0−p′0βlπ2l3β3mlβ2p0−p′0κ02mlβ+K12mlβ3+2mlβ2+lβp0−p′01−mlβγ0.This is the Stefan-Boltzmann law for the Dirac field in phase space. It is worth pointing out that the resultT0011β~T4 is recovered by taking the limit of the momentum variable. This result in phase space is necessary to compare with experiments. In this sense, we can integrate over the momenta which explicitly yield
(44)T0011β=−∑l=1∞4m−1l+1βlπ2l3β3K12mlβ3+2mlβ2and take the limitm→0; then, the only remaining part is the factor ofK12mlβthat leads to the dependency β−4=T4, once the limit of Bessel function is taken. On the other hand, it is possible to project in the momentum space by integrating over coordinates. This process leads to a divergence which is of the same nature of the coordinate projection in the absence of temperature. Hence, a quantity in the momentum space analogous to the temperature is necessary, that is, the thermal energy. The introduction of TFD formalism introduces the role of temperature, but it can equally do the same for the thermal energy. Using phase space and TFD allows us to deal with systems where microscopic energy is dominant.


### 4.2. Casimir Effect for the Dirac Field in Phase Space

Here, the choice isα=0,0,0,i2d, then
(45)v2d=∑l3=1∞−1l3+1e−i2dk3l3.The Green function is this case is(46)G¯Dabq−q′,p−p′;d=∑l3=1∞−1l3+1GD∗q−q′+2dl3n3,p−p′−GDq−q′−2dl3n3,p−p′,where n3=0,0,0,1 is a space-like vector. Then, the energy-momentum tensor is
(47)Tμν11d=−ilimq′,p′→q,p∑l3=1∞−1l3+1ΓμνGD∗q−q′+2dl3n3,p−p′−GDq−q′−2dl3n3,p−p′.By takingμ=ν=0, the Casimir energy for the Dirac field in phase space at zero temperature is
(48)T0011d=limp′→p∑l3=1∞m2δp−p′−1l3+1e4ipz−p′zdl3π2dl31+2p0−p′0γ0K12mdl3.And forμ=ν=3, the Casimir pressure in phase space is
(49)T3311d=limp′→p∑l3=1∞m2δp−p′−1l3+1e4ipz−p′zdl3π2dl3imγ3+pz−p′zκ22mdl3−2m−pz−p′zγ3K12mdl3.It reproduces the usual result whenm⟶0 and integrated over the momenta which means the projection on coordinate space. Then, only the factors of K1 is left; the limit of this part yields the dependency d−4. Here, the dependency on γ matrices should be viewed as part of the phase space formalism which is by its core matricial. This part does not survive once the projection on coordinates is performed, but it is part of the behavior in phase space. In order to be compared with experimental data, the projection on momentum space requires the introduction the thermal energy.


### 4.3. Casimir Effect for the Dirac Field in Phase Space at Finite Temperature

The effect of temperature is introduced by takingα=β,0,0,i2d. Then, the generalized Bogoliubov transformation becomes
(50)v2β,d=∑l=1∞−1l+1e−βk0l+∑l3=1∞−1l3+1e−i2dk3l3+2∑l,l3=1∞−1l+l3e−βk0l−i2dk3l3.The first two terms of these expressions correspond, respectively, to the Stefan-Boltzmann term and the Casimir effect atT=0. The third term is analyzed and it leads to the Green function
(51)G¯Dabq−q′,p−p′;β,d=∑l3=1∞−1l+l3GD∗q−q′+iβln0+2dl3n3,p−p′−GDq−q′−iβln0−2dl3n3,p−p′.Then, the Casimir energy at finite temperature is(52)T0011β,d=limp′→p∑l0,l3=1∞2mδp−p′−1l3+l0e−4ipz−p′zdl3−2p0−p′0βl04π24d2l32+β2l022κ0m4d2l32+β2l022m2dl321+2mβl0−βl023−2mβl0−24idβl0l3γ3+K1m4d2l32+β2l024d2l32+β2l021/2−4m2d2l322−βl0212−2mβl04−2mβl0+2dl328−2mβl0−8+6mβl0−48im2d3l33βl0γ3−3idβl0l3γ332+4m2β2l02,and the Casimir pressure at finite temperature is
(53)T3311β,d=limp′→p∑l0,l3=1∞2mδp−p′−1l3+l0e−4ipz−pz′dl3−2p0−p0′βl04π24d2l32+β2l022κ0m4d2l32+β2l024mβl02+4dl35dl3+6iβl0γ3+K1m4d2l32+β2l024d2l32+β2l021/2128m2d2l322+160d2l32+βl028+48m2d2l32+4m2βl04+6idβl0l332+4m24d2l32+β2l02γ3−4m4d2l32+β2l022+2mβl0+2i4mdl3+ipz−pz′βl0γ3κ2m4d2l32+β2l02.It should be noted that in the limitp⟶p′, both the Casimir energy and pressure are real quantities at zero and finite temperature. In the limit β⟶0, i.e., T⟶∞, the Casimir energy and pressure become
(54)T0011d=limp′→p∑l3=1∞mδp−p′−1l3+l0e−4ipz−pz′dl34π2dl33dl3κ02mdl3+1−2mdl32K12mdl3,T3311d=limp′→p∑l3=1∞mδp−p′−1l3+l0e−4ipz−p′zdl32π2dl33mdl33−8imdl3γ3κ02mdl3+3+4mdl3mdl3−2iγ3K12mdl3.It is important to note that in this limit, both the Casimir energy and pressure depend only on the distanced between the plates. The dependence on gamma matrices is not a problem since the formalism of the phase space is matrix. It leads to the conclusion that neither the energy nor the pressure are scalars but components of a tensor.


## 4.1. Stefan-Boltzmann Law

The study of the Stefan-Boltzmann law in phase space corresponds to a choice of the parameterα. It is important to note that the parameter α is the compactification parameter that is defined as α=α0,α1,⋯,αD−1. The temperature effect is described by the choice α=β,0,0,0.The generalized Bogoliubov transformation, Equation (31), for these parameters is
(40)v2k,β=∑l=1∞−1l+1e−βk0l.The Green’s function for the Dirac field in phase space is(41)G¯Dabq−q′,p−p′;β=∑l=1∞−1l+1GD∗q−q′+iβln0,p−p′−GDq−q′−iβln0,p−p′,where n0=1,0,0,0 is a time-like vector. Then, the physical energy-momentum tensor is
(42)Tμν11β=−ilimq′,p′→q,p∑l=1∞−1l+1ΓμνGD∗q−q′+iβln0,p−p′−GDq−q′−iβln0,p−p′.In order to calculate the Stefan-Boltzmann law, takingμ=ν=0 leads to
(43)T0011β=−limp′→p∑l=1∞4mδp−p′−1l+1e−2p0−p′0βlπ2l3β3mlβ2p0−p′0κ02mlβ+K12mlβ3+2mlβ2+lβp0−p′01−mlβγ0.This is the Stefan-Boltzmann law for the Dirac field in phase space. It is worth pointing out that the resultT0011β~T4 is recovered by taking the limit of the momentum variable. This result in phase space is necessary to compare with experiments. In this sense, we can integrate over the momenta which explicitly yield
(44)T0011β=−∑l=1∞4m−1l+1βlπ2l3β3K12mlβ3+2mlβ2and take the limitm→0; then, the only remaining part is the factor ofK12mlβthat leads to the dependency β−4=T4, once the limit of Bessel function is taken. On the other hand, it is possible to project in the momentum space by integrating over coordinates. This process leads to a divergence which is of the same nature of the coordinate projection in the absence of temperature. Hence, a quantity in the momentum space analogous to the temperature is necessary, that is, the thermal energy. The introduction of TFD formalism introduces the role of temperature, but it can equally do the same for the thermal energy. Using phase space and TFD allows us to deal with systems where microscopic energy is dominant.


## 4.2. Casimir Effect for the Dirac Field in Phase Space

Here, the choice isα=0,0,0,i2d, then
(45)v2d=∑l3=1∞−1l3+1e−i2dk3l3.The Green function is this case is(46)G¯Dabq−q′,p−p′;d=∑l3=1∞−1l3+1GD∗q−q′+2dl3n3,p−p′−GDq−q′−2dl3n3,p−p′,where n3=0,0,0,1 is a space-like vector. Then, the energy-momentum tensor is
(47)Tμν11d=−ilimq′,p′→q,p∑l3=1∞−1l3+1ΓμνGD∗q−q′+2dl3n3,p−p′−GDq−q′−2dl3n3,p−p′.By takingμ=ν=0, the Casimir energy for the Dirac field in phase space at zero temperature is
(48)T0011d=limp′→p∑l3=1∞m2δp−p′−1l3+1e4ipz−p′zdl3π2dl31+2p0−p′0γ0K12mdl3.And forμ=ν=3, the Casimir pressure in phase space is
(49)T3311d=limp′→p∑l3=1∞m2δp−p′−1l3+1e4ipz−p′zdl3π2dl3imγ3+pz−p′zκ22mdl3−2m−pz−p′zγ3K12mdl3.It reproduces the usual result whenm⟶0 and integrated over the momenta which means the projection on coordinate space. Then, only the factors of K1 is left; the limit of this part yields the dependency d−4. Here, the dependency on γ matrices should be viewed as part of the phase space formalism which is by its core matricial. This part does not survive once the projection on coordinates is performed, but it is part of the behavior in phase space. In order to be compared with experimental data, the projection on momentum space requires the introduction the thermal energy.


## 4.3. Casimir Effect for the Dirac Field in Phase Space at Finite Temperature

The effect of temperature is introduced by takingα=β,0,0,i2d. Then, the generalized Bogoliubov transformation becomes
(50)v2β,d=∑l=1∞−1l+1e−βk0l+∑l3=1∞−1l3+1e−i2dk3l3+2∑l,l3=1∞−1l+l3e−βk0l−i2dk3l3.The first two terms of these expressions correspond, respectively, to the Stefan-Boltzmann term and the Casimir effect atT=0. The third term is analyzed and it leads to the Green function
(51)G¯Dabq−q′,p−p′;β,d=∑l3=1∞−1l+l3GD∗q−q′+iβln0+2dl3n3,p−p′−GDq−q′−iβln0−2dl3n3,p−p′.Then, the Casimir energy at finite temperature is(52)T0011β,d=limp′→p∑l0,l3=1∞2mδp−p′−1l3+l0e−4ipz−p′zdl3−2p0−p′0βl04π24d2l32+β2l022κ0m4d2l32+β2l022m2dl321+2mβl0−βl023−2mβl0−24idβl0l3γ3+K1m4d2l32+β2l024d2l32+β2l021/2−4m2d2l322−βl0212−2mβl04−2mβl0+2dl328−2mβl0−8+6mβl0−48im2d3l33βl0γ3−3idβl0l3γ332+4m2β2l02,and the Casimir pressure at finite temperature is
(53)T3311β,d=limp′→p∑l0,l3=1∞2mδp−p′−1l3+l0e−4ipz−pz′dl3−2p0−p0′βl04π24d2l32+β2l022κ0m4d2l32+β2l024mβl02+4dl35dl3+6iβl0γ3+K1m4d2l32+β2l024d2l32+β2l021/2128m2d2l322+160d2l32+βl028+48m2d2l32+4m2βl04+6idβl0l332+4m24d2l32+β2l02γ3−4m4d2l32+β2l022+2mβl0+2i4mdl3+ipz−pz′βl0γ3κ2m4d2l32+β2l02.It should be noted that in the limitp⟶p′, both the Casimir energy and pressure are real quantities at zero and finite temperature. In the limit β⟶0, i.e., T⟶∞, the Casimir energy and pressure become
(54)T0011d=limp′→p∑l3=1∞mδp−p′−1l3+l0e−4ipz−pz′dl34π2dl33dl3κ02mdl3+1−2mdl32K12mdl3,T3311d=limp′→p∑l3=1∞mδp−p′−1l3+l0e−4ipz−p′zdl32π2dl33mdl33−8imdl3γ3κ02mdl3+3+4mdl3mdl3−2iγ3K12mdl3.It is important to note that in this limit, both the Casimir energy and pressure depend only on the distanced between the plates. The dependence on gamma matrices is not a problem since the formalism of the phase space is matrix. It leads to the conclusion that neither the energy nor the pressure are scalars but components of a tensor.


## 5. Conclusion

The Dirac field in phase space is considered. Using the Dirac equation, the propagator for spin-1/2 particles is calculated. This form of the propagator is similar to that in the usual quantum mechanics. The TFD results are obtained by using the temperature effects in the Dirac propagator. TFD, a real-time finite temperature formalism, is a thermal quantum field theory. Using this formalism, a physical (renormalized) energy-momentum tensor is defined. Then, the Stefan-Boltzmann law in phase space and the Casimir effect are calculated at finite temperature. The results lead to the usual results for the Dirac field when they are projected in the quantum field theory space. The TFD formalism allows studying the finite temperature effects in phase space. On the other hand, such a formalism also may be used to explore the role of a thermal energy which is possibly related to the fermionic feature of the field.



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*Source: 7032834-2020-06-14.xml*