Standard Model Extension and Casimir effect for fermions at finite temperature

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Standard

Model

Extension

and

Casimir

effect

for

fermions

at

finite

temperature

A.F. Santos

a

,

b

,

∗

,

Faqir

C. Khanna

b

,

c

a_{InstitutodeFísica,UniversidadeFederaldeMatoGrosso,78060-900,Cuiabá,MatoGrosso,Brazil} b_{DepartmentofPhysicsandAstronomy,UniversityofVictoria,3800FinnertyRoad,Victoria,BC,Canada} c_{DepartmentofPhysics,UniversityofAlberta,T6J2J1,Edmonton,Alberta,Canada}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received5July2016

Receivedinrevisedform20September 2016

Accepted22September2016 Availableonline28September2016 Editor:M.Cvetiˇc

Keywords: Casimireffect

StandardModelExtension Finitetemperature

LorentzandCPTsymmetriesarefoundationsforimportantprocessesinparticlephysics.Recentstudies in Standard Model Extension (SME) athigh energy indicate that these symmetries may be violated. Modifications inthe lagrangianare necessarytoachieve ahermitian hamiltonian.The fermion sector ofthe standard model extension isused to calculatethe effects ofthe Lorentz and CPTviolationon theCasimireffectatzeroandfinitetemperature.TheCasimireffectandStefan–Boltzmannlawatfinite temperaturearecalculatedusingthethermofielddynamicsformalism.

©2016TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

StandardModel (SM) hasbeen highly successful in predicting interactionamongquarksatenergyuptoafewTeV.Inweak inter-actionsbreakdownofParity

[1,2]

andCPsymmetry

[3]

hasbeen observed at low energies. String theory in higher dimensions is possiblefor particlephysics athigh energies. Such atheory may have violation of Lorentz and CPT symmetry. At some range of higherenergies,can therebe abreak downinvarianceproperties like Lorentz invariance and CPT symmetry of the SM [4]? Such an extension of the Standard Model (SME) has been applied to severalprocesses in orderto get an estimate ofthe break down ofsymmetries. Such violations havealso been found to occur in loopquantumgravity

[5]

,noncommutativetheories

[6]

,spacetimes withanontrivialtopology

[7]

,amongothers.

ThegeneraltheoryoftheSME

[8,9]

includestheknownphysics oftheSMplusallpossibletermsthatviolateLorentzandCPT sym-metry.Inaddition,theSMEisdividedintotwoparts:(i)the mini-malversionrestrictedtopowercountingrenormalizableoperators and(ii)thenonminimalversion which alsoincludesoperators of higherdimensions.Inthispaperourinterestisinthefermion sec-torthatisbasedonaminimalextendedQuantumElectrodynamics

*

Correspondingauthor.

E-mailaddresses:alesandroferreira@fisica.ufmt.br(A.F. Santos),khannaf@uvic.ca (F.C. Khanna).

(EQED) that is part of SME. This EQEDinvolves modifications of theusualQEDinbothfermionandphotonsectors.Therelativistic lagrangian thatdescribes fermionsin SME doesnot implya her-mitianhamiltonian.Inordertoresolvethisproblemaredefinition of thefield isneeded andthishas beenachieved [10]. Thiswill be utilized in the present development. Our aimhere is to pro-videtheoreticalpredictionsregardingthequantumvacuuminthis EQED.Weconcentrateoncalculatingtheeffectsofthese modifica-tionsontheCasimirforceinthefermionsector.

The Casimir effect consists in the calculation of the vacuum energy density of a quantum field in the presence of boundary conditions. H. Casimir [11] was the first to analyze the vacuum fluctuationoftheelectromagneticfieldconfinedbetweentwo con-ductingparallelplates.Theeffectwas anattractiveforce between theplates.Sparnaay

[12]

madethefirstexperimental observation withcorrectsignandmagnitude. Subsequentexperiments

[13,14]

haveestablishedthiseffecttoahighdegreeofaccuracy.This phe-nomenon has been applied to micro- and nanotechnologies [15, 16]andsuperconductorsathightemperatures

[17,18]

.TheCasimir effect for fermions at zeroand finite temperature also has been investigated [19–21].This effectfor fermions isinteresting when thestructureofprotoninparticlephysicsisconsidered,in partic-ularforthe phenomenologicalbagmodel.Quarksandgluonsare confinedinthebag. Inthispaperwederive theCasimireffectat finitetemperature consideringthe fermionsector ofthe EQEDof theSME.

http://dx.doi.org/10.1016/j.physletb.2016.09.049

0370-2693/©2016TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

There are three different, but equivalent, formalisms to intro-duce temperatureeffects ina quantumfield theory.(i) The Mat-subara formalism, the imaginary time formalism, [22] which is based on a substitution of time, t, by a complex time, i

τ

. Since thetime variableis exchanged fortemperature, thismethod isa goodtoolforstudyingsystemsatequilibrium.(ii)Theclosedtime path formalism [23] is a real time formalism at finite tempera-ture.Thisprocedurecanbeusedtodescribebothequilibriumand non-equilibriumphenomena.Inaddition,leadstoadoublingofthe degreesoffreedom,suchthattheGreenfunctionsarerepresented byatwodimensionalmatrixstructure.(iii)TheThermoField Dy-namics(TFD)isa realtime finitetemperatureformalism

[24–29]

. Thethermalvacuum,

|

0(β)



,belongstotheFockspace

S

T that is adirectproductoftheoriginalFockspace

S

andanindependent identicalcopyofit

S

˜

(tildesystem).Inthisformalismthe statisti-calaverageofan observable

A

isexpressedasathermalvacuum expectation value i.e.,



A

= 

0(β)

|

A|

0(β)



, where

β

=

_k1

BT, and T is the temperature andkB isthe Boltzmann constant (weuse kB

= ¯

h

=

c

=

1). The map between thetilde A

˜

i andnon-tilde Ai operators is definedby the following tilde (or dual) conjugation rules:

(

AiAj

)

∼

= ˜

AiA

˜

j

,

(

c Ai

+

Aj

)

∼

=

c∗A

˜

i

+ ˜

Aj

,

(

A†_i

)

∼

= ˜

Ai †

,

( ˜

Ai

)

∼

= −ξ

Ai

,

(1)

with

ξ

= −

1 forbosonsand

ξ

= +

1 forfermions.Thetemperature effectisimplementedinthedoubled Fockspacebya Bogoliubov transformationwhich introduces a rotationof thetilde and non-tildevariables.Thisformalismisusefulforsystemsinequilibrium. For such systems the Bogoliubov transformation is unitary. Here wechoosetousetheTFDformalism.

This paper is organized as follows. In section 2, the energy-momentum tensorfor fermions ofthe SME is calculated.In sec-tion3,abriefintroductiontoTFDispresented.Insection 4,some applications are developed. The Stefan–Boltzmann law and the Casimireffect atzero andfinitetemperature arederived. In sec-tion5,someconcludingremarksarepresented.

2. TheenergymomentumtensorfortheDiracfieldoftheSME TheLagrangianforthefermionsectoroftheextendedquantum electrodynamicsoftheSMEis

L

= ¯ψ



i



μ

∂

↔μ

−

M



ψ,

(2) where



μ

=

γ

μ

+ (

cμν

+

dμν

γ

5

)γ

ν

+

eμ

+

i fμ

γ

5

+

1 2g κμν

_σ

κν

,

(3) M

=

m

+ (

aμ

+

bμ

γ

5

)γ

μ

+

1 2H μν

_σ

μν

.

(4)

Theparametersin



μ aredimensionlesswhiletheonesinM have

dimension of mass.

γ

μ,

γ

5 and

σ

κν denote the Dirac matrices.

ThecoefficientsforLorentzviolationareaμ

,

bμ

,

cμν

,

dμν

,

eμ

,

f μ

,

gκμν andHμν .

The hamiltonian associated with the lagrangian (2) is non-hermitianandcorrespondstononunitarytimeevolution.This dif-ficulty can be resolved by a spinor redefinition

ψ

=

A

χ

in the lagrangian.Thequantity A ischosensuchthatthetime-derivative

isthat oftheusual Diraclagrangian

[10]

. Thisspinorredefinition leavesunchangedthephysics.Thusthelagrangianbecomes

L

= ¯

χ



iA

¯



μA

∂

↔μ

− ¯

AM A



χ.

(5)

Using A

¯

=

γ

0_A†

_γ

0 _{and A}†

_γ

0

_

0_A

₌

_{I,where I istheunitmatrix,}

thislagrangiancontainsonlytimederivativeastheusualterm,i.e.,

i

χ γ

¯

0 ↔

∂

0

χ

.

ThemodifiedDiracequationisobtainedas



iA

¯



μA

∂

μ

− ¯

AM A



χ

=

0

.

(6)

Using this field equation, the energy-momentum tensor for fermionsisgivenas

Tμν

=

i

χ

¯

A

¯



μA

∂

ν

χ.

(7)

In order to get the Casimir effect the energy-momentum tensor iswritten soastoavoidaproduct offieldoperators atthesame space–timepoint.Then

Tμν

(

x

)

=

iA

¯



μA

∂

ν lim

x→x

τ



¯

χ

(

x

)χ(

x

)



,

(8)

where

τ

isthetimeorderingoperator.

Thevacuumaverageoftheenergy-momentumtensoris



Tμν

(

x

)

=



0

|

Tμν

(

x

)

|

0

= −

lim

x→x

¯

A



μ_A

_∂

ν_S

₍

_x

₋

_x

₎



_,

₍₉₎

wheretheFeynmanpropagatorfortheDiracfield

[29]

is S

(

x

−

x

)

= −

i



0

|

τ



χ

¯

(

x

)χ

(

x

)



|

0

= (

i

γ

· ∂ +

m

)

G0

(

x

−

x

),

(10) with G0

(

x

−

x

)

=

−

i

(

2

π

)

2 1

(

x

−

x

)

2

₊

_i

_ξ

,

(11)

being the propagator of the massless scalar field. To obtain lin-ear order inparameters forLorentz violation thechoice A

=

1

−

1

2

γ

0

(

0

−

γ

0

)

and A

¯

=

1

−

1

2

(

0

−

γ

0

)

γ

0 is considered.Thus for

a massless fermionic field the average ofthe energy momentum tensorbecomes



Tμν

(

x

)

= −

i lim x→x

{ ∂

μ

_∂

ν_G 0

(

x

−

x

)

},

(12) where



=



1

+

9₄



i

γ

i

, with i

=

1,2,3. The Minkowski metric withsignature

(

+

− −−)

isused.

Theparametersei

_,

_fi_andgki j_in

_

i_{arenotextractabledirectly} fromSMEandaretakentobezeroorsuppressedduetothe renor-malizibiltyandgaugeinvariancerequirements.Theparameters ci j and di j are tracelessand symmetric. Forsimplicity we will con-siderthecasedi j

₌

_{0.Thentheaverageoftheenergy-momentum} tensorwithLorentzviolatingtermis



Tμν

(

x

)

= −

i

4xlim→x

{

c

∂

μ

_∂

ν_G

0

(

x

−

x

)

},

(13)

where



c

= (

31

+

9cii

),

with cii beingthe parameterthat violates Lorentz symmetry.Itisimportantnotethatthetermci_i isnotthe traceofci j_{,sinceeq.(12)yieldsatermproportionalto}



i

γ

i

=

ci j

γ

j

γ

i

=

cij

γ

j

γ

i

=

c1₁

γ

1

γ

1

+

c22

γ

2

γ

2

+

c33

γ

3

γ

3

=

cii

,

(14) where

γ

1

γ

1

=

γ

2

γ

2

=

γ

3

γ

3

=

1 isused.

TheTFDformalismisusedinordertointroducethefinite tem-peratureeffect.

3. BriefintroductiontoTFD

TFDconsistsinthegenerationofthermalstatesbydoublingthe degreesoffreedominaHilbertspaceaccompaniedbythe temper-aturedependentBogoliubovtransformation

[29,24–26,30,31]

.This doublingisdefinedby thetilde(∼) conjugationrules,associating each operatorin

S

to two operators in

S

T,where theexpanded space is

S

T

=

S ⊗ ˜

S

,with

S

beingthe standard Fockspaceand

˜

S

the fictitious space. For an arbitrary fermionic operator F the

standarddoubletnotationis Fa

=



F1 F2



=



F

˜

F†



,

(15)

wherethephysical variablesare described by nontilde operators. Thetilde operators areauxiliary degreesof freedom whichallow accommodationofthethermalpropertiesofthesystem.A Bogoli-ubovtransformation whichcorresponds to a rotationinthe tilde and non-tilde variables introduces thermal effects. For fermions andusingthedoubletnotationweget



b

(α)

˜

b†

(α)



=

B

(α)



b

(

k

)

˜

b†

(

k

)



,

(16)

where

(

b†

_,

_b

˜

†

₎

_{arecreationoperators,}

₍

_b

_,

_b

˜

₎

_{aredestruction}

opera-torsand

B(

α

)

istheBogoliubovtransformationgivenas

B

(α)

=



u

(α)

−

v

(α)

v

(α)

u

(α)



.

(17)

Thequantitiesu

(

α

)

andv

(

α

)

arerelatedtotheFermidistribution andaregivenas

v2

(α)

=

1

1

+

eαω

,

u

2

_(α)

₌

1

1

+

e−αω

,

(18)

suchthat v2

(

α

)

+

u2

(

α

)

=

1.Here

ω

=

ω

(

k

)

and

α

= β

.

Usingthisformalismthephysical

α

-dependent energy-momen-tumtensorisdefinedas

T

μν(ab)

₍

_x

_;

_α)

_{= }

_Tμν(ab)

₍

_x

_;

_α)

_

− 

Tμν(ab)

(

x

)

.

(19) Then

T

μν(ab)

₍

_x

_;

_α)

_{= −}

i 4xlim→x





c

∂

μ

∂

ν

×

×



G(ab)₀

(

x

−

x

;

α)

−

G(ab)₀

(

x

−

x

)

 

,

(20) wherea

,

b

=

1,2 and G(ab)₀

(

x

−

x

)

=



d4k

(

2

π

)

4

×

×

e−ik(x−x)G(ab)₀

(

k

),

(21) with G(ab)₀

(

k

)

=



G0

(

k

)

0 0 G∗₀

(

k

)



.

(22)

The

α

-dependentpartoftheGreenfunctionis G(ab)₀

(

x

−

x

;

α)

=



_d4_k

(

2

π

)

4

×

×

e−ik(x−x)G(ab)₀

(

k

;

α),

(23)

where G(ab)₀

(

k

;

α

)

=

B

−1(

α

)

G(ab)₀

(

k

)B(

α

).

Explicitly, the physical componentofG(ab)₀

(

k

;

α

)

is

G(₀11)

(

k

;

α)

≡

G0

(

k

;

α)

=

G0

(

k

)

+

v2

(α)

[

G∗₀

(

k

)

−

G0

(

k

)

].

(24)

Forfermions theenergy-momentumtensor

(20)

is studiedfor somechoiceofthe

α

parameter.

4. Someapplications

Here three applicationswhich dependon the choiceof the

α

parameterarestudied.

4.1. Stefan–Boltzmannlaw

LetusconsiderthegeneralizedBogoliubovtransformation

[32]

whichiswrittenas v2

(

kα

;

α)

=

d



s=1



{σs} 2s−1

×

×

∞



lσ₁,...,lσs=1

(

−

η)

s+sr=1lσr

×

×

exp

⎡

⎣−



s j=1

α

σjlσjkσj

⎤

⎦ ,

(25)

whered isthenumberofcompactifieddimensions,

η

=

1(

−

1)for fermions (bosons) and

{

σ

s

}

denotes the set of all combinations withs elements.

Inthiscasethechoiceis

α

= (β,

0,0,0)andthen

v2

(β)

=

∞



l=1

(

−

1

)

l+1e−βk0l

.

(26)

Usingeq.(23)thethermalGreenfunctionbecomes

G(₀11)

(

x

−

x

; β) =

G0

(

x

−

x

)

+

∞



l=1

(

−

1

)

l+1

×



G∗₀

(

x

−

x

+

i

β

ln0

)

−

G0

(

x

−

x

−

i

β

ln0

)



,

(27)

wheren0

= (

1,0,0,0).Thentheenergy-momentumtensorisgiven

as

T

μν(11)

₍

_x

_;

_α)

₌

i 4xlim→x ∞



l=1

(

−

1

)

l



c

∂

μ

∂

ν

×

×



G∗₀

(

x

−

x

+

i

β

ln0

)

−

G0

(

x

−

x

−

i

β

ln0

)



.

(28)

For

μ

=

ν

=

0 weobtainthemodifiedStefan–Boltzmannlaw

T

00(11)

_(β)

₌

7

π

2 60 T 4



_a

₊

_{b c}i i

,

(29)

where ci_i is a Lorentz violating term. The constants a and b are

defined as a

=

31₁₆ and b

=

₁₆9. Thus the lowest-order prediction ofthefermionssector oftheSMEmodifiestheStefan–Boltzmann law.ThefieldredefinitionchangestheStefan–Boltzmannlawbya multiplicativefactor.

4.2. Casimireffectatzerotemperature

For parallel plates perpendicular to the z direction and sep-arated by a distance a the

α

parameter is chosen as

α

=

(0,

0,0,i2a

).

Then v2

(

a

)

=

∞



l=1

(

−

1

)

l+1e−i2ak3l

,

(30)

andtheenergy-momentumtensorbecomes

T

μν(11)

₍

_x

_;

_α)

₌

i 4xlim→x ∞



l=1

(

−

1

)

l



c

∂

μ

∂

ν

×

×



G∗₀

(

x

−

x

+

2alk3

)

−

G0

(

x

−

x

−

2alk3

)



.

(31)

FromthisequationtheCasimirenergyandpressureareobtained E

(

a

)

=

T

00(11)

(

a

)

= −

7

π

2 2880a4



a

+

b ci_i

,

P

(

a

)

=

T

33(11)

(

a

)

= −

7

π

2 960a4



a

+

b ci_i

,

(32)

whereci_i istheLorentzviolatingcoefficient.

4.3. Casimireffectatfinitetemperature

InordertoanalyzethetemperatureeffectintheCasimireffect

α

= (β,

0,0,i2a

)

isconsidered, wherethe temperatureeffectand spatialcompactificationarecombined.ThentheBogoliubov trans-formation,eq.(25),becomes

v2

(β,

a

)

=

v2

(

k0

; β) +

v2

(

k3

;

a

)

+

2v2

(

k0

; β)

v2

(

k3

;

a

),

(33)

=

∞



l0=1

(

−

1

)

l0+1e−βk0l0

+



∞ l3=1

(

−

1

)

l3+1e−i2ak3l3

+

2 ∞



l0,l3=1

(

−

1

)

l0+l3e−βk0l0−i2ak3l3

.

Thefirst termleads tothe StefanBoltzmannlawandthe second termtotheCasimireffectatzerotemperature. TheCasimireffect atfinitetemperatureis

T

μν(11)

_(β,

_a

₎

_{= −}

i 2xlim→x ∞



l0,l3=1

(

−

1

)

l0+l3

×

× 

c

∂

μ

∂

ν



G∗₀

(

x

−

x

+

i

β

l0n0

+

2alk3

)

−

G0

(

x

−

x

−

i

β

l0n0

−

2alk3

)



.

(34)

The Casimir energy,

T

00(11)

_(β,

_a

_),

_andpressure,

_T

33(11)

_(β,

_a

_),

re-spectively,aregivenas

T

00(11)

_(β,

_a

₎

_{= −}

8

π

2 ∞



l0,l3=1

(

−

1

)

l0+l3

×

(35)

×

(

2al3

)

2

−

3

(β

l0

)

2

[(β

l0

)

2

+ (

2al3

)

2

]

3

(

a

+

b ci_i

),

and

T

33(11)

_(β,

_a

₎

_{= −}

8

π

2 ∞



l0,l3=1

(

−

1

)

l0+l3

×

(36)

×

3

(

2al3

)

2

− (β

l0

)

2

[(β

l0

)

2

+ (

2al3

)

2

]

3

(

a

+

b ci_i

).

ThustheCasimirenergyis E

(β

;

a

)

=



₇

_π

2 60

β

4

−

7

π

2 2880a4 (37)

−

8

π

2 ∞



l0,l3=1

(

−

1

)

l0+l3

(

2al3

)

2

₋

₃

_(β

_l 0

)

2

[(β

l0

)

2

+ (

2al3

)

2

]

3



(

a

+

b ci_i

).

Note that at low temperatures this energy recovers the Casimir energy at zero temperature, while the high temperaturelimit is dominated by the positive contribution of the Stefan–Boltzmann term.TheLorentzviolatingtermsemergeatbothlimits.

TheCasimirPressureis P

(β

;

a

)

=



₇

_π

2 180

β

4

−

7

π

2 960a4 (38)

−

8

π

2 ∞



l0,l3=1

(

−

1

)

l0+l3 3

(

2al3

)

2

_{− (β}

_l 0

)

2

[(β

l0

)

2

+ (

2al3

)

2

]

3



(

a

+

b ci_i

).

Forlowtemperaturesthepressureisnegative.Whentemperature increases,atransitiontopositivepressurehappens.Itispossibleto determinethe criticalcurveofthetransition. Thepointof transi-tionoccurswhenthepressurevanishes.Thenanalyzingourresult wenotethattheLorentzviolatingtermdoesnotmodifythis tran-sitionvalue.

5. Conclusion

Symmetry,symmetrybreakingandphysicallawsareconnected tothedescriptionofnature.Instringtheoryitispossibletoviolate Lorentz andCPTsymmetries.Theextension oftheseideas forSM leads toSME where break down ofLorentz and CPTsymmetries ispossible.InthispaperweusethefermionsectoroftheSMEto calculatedtheCasimireffectatzeroandfinitetemperature.

The Casimirenergyfortheelectromagneticandfermionsfield within the SM at zero and finite temperature is considered and experimentallyestablished.HereourinterestistostudySMEwith LorentzandCPTviolatingtermsforfermionssystems.The energy-momentum tensor for the fermion sector of SME is calculated. Using the TFD formalism the Stefan–Boltzmannlaw and Casimir energy are obtained atfinite temperature. The Casimirenergy is foundtobe

(

a

+

bci

i

)

P ,where P isthestandardCasimirpressure, ci

i istheLorentz violating parameteranda

,

b areconstants.Final resultsaremultipliedbyaconstantfactorduetothefield redefini-tion.Thisisnecessarytoobtainatheorywherethehamiltonianis hermitian. Temperatureeffects contributeto constrainLorentz vi-olationparameters.OveralltheeffectofLorentzandCPTviolation onCasimirenergyissmall.

Acknowledgements

This work by A. F. S. is supported by CNPq projects 476166/ 2013-6 and 201273/2015-2. We thank Physics Department, Uni-versityofVictoriaforaccesstofacilities.

References

[1]T.D.Lee,C.N.Yang,Phys.Rev.104(1956)254. [2]C.S.Wu,etal.,Phys.Rev.105(1957)1413.

[3]J.H.Christenson,J.W.Cronin,V.L.Fitch,R.Turley,Phys.Rev.Lett.13(1964)138. [4]V.A.Kostelecky,S.Samuel,Phys.Rev.D39(1989)683;

V.A.Kostelecky,R.Potting,Nucl.Phys.B359(1991)545.

[5]M.Bojowald,H.A.Morales-Tecotl,H.Sahlmann,Phys.Rev.D71(2005)084012. [6]S.M.Carroll,J.A.Harvey,V.A.Kostelecky,C.D.Lane,T.Okamoto,Phys.Rev.Lett.

87(2001)141601.

[7]F.R.Klinkhamer,Nucl.Phys.B535(1998)233.

[8]D. Colladay,V.A. Kostelecky, Phys. Rev.D55 (1997)6760, Phys. Rev.D58 (1998)116002.

[9]V.A.Kostelecky,Phys.Rev.D69(2004)105009.

[10]V.A. Kostelecky, C.D. Lane, J. Math. Phys. 40 (1999) 6245, arXiv:hep-ph/ 9909542.

[11]H.G.B.Casimir,Proc.K.Ned.Akad.Wet.51(1948)793. [12]M.J.Sparnaay,Physica24(1958)751.

[13]S.K.Lamoreaux,Phys.Rev.Lett.28(1997)5. [14]U.Mohideen,A.Roy,Phys.Rev.Lett.81(1998)21.

[15]M.Bordag,U.Mohideen,V.M.Mostepanenko,Phys.Rep.353(2001)1. [16]S.K.Lamoreaux,Am.J.Phys.67(1999)850.

[17]M.Bordag,J.Phys.A39(2006)6173.

[18]M.T.D.Orlando,etal.,J.Phys.A,Math.Theor.42(2009)025502.

[19]H.Queiroz,J.C.daSilva,F.C.Khanna,J.M.C.Malbouisson,M.Revzen,A.E. San-tana,Ann.Phys.317(2005)220.

[20]E.Elizalde,F.C.Santos,A.C.Tort,Int.J.Mod.Phys.A18(2003)1761. [21]E.Elizalde,M.Bordag,K.Kirsten,J.Phys.A31(1998)1743. [22]T.Matsubara,Prog.Theor.Phys.14(1955)351.

[23]J.Schwinger,J.Math.Phys.2(1961)407;

J.Schwinger,LectureNotesofBrandeis,UniversitySummerInstitute,1960. [24]Y.Takahashi,H.Umezawa,Collect.Phenom.2(1975)55,Int.J.Mod.Phys.B10

(1996)1755.

[25]Y. Takahashi,H. Umezawa,H. Matsumoto, ThermofieldDynamics and Con-densedStates,North-Holland,Amsterdam,1982.

[26]H.Umezawa,AdvancedFieldTheory:Micro,MacroandThermalPhysics,AIP, NewYork,1993.

[27]A.E.Santana,F.C.Khanna,Phys.Lett.A203(1995)68.

[28]A.E.Santana,F.C.Khanna,H.Chu,C.Chang,Ann.Phys.249(1996)481. [29]F.C. Khanna, A.P.C. Malbouisson, J.M.C. Malboiusson, A.E. Santana, Thermal

Quantum FieldTheory:AlgebraicAspectsandApplications,WorldScientific, Singapore,2009.

[30]A.E.Santana,A.MatosNeto,J.D.M.Vianna,F.C.Khanna,PhysicaA280(2000) 405.

[31]F.C.Khanna,A.P.C. Malbouisson,J.M.C.Malbouisson,A.E.Santana,Ann.Phys. 324(2009)1931.

[32]F.C. Khanna,A.P.C. Malbouisson,J.M.C.Malbouisson,A.E.Santana,Ann.Phys. 326(2011)2634.