THE CASIMIR EFFECT Günter PLUNIEN, Berndt MULLER and ... · THE CASIMIR EFFECT Günter PLUNIEN, Berndt MULLER and Walter GREINER ... THE CASIMIR EFFECT Günter PLUNIEN, Berndt MULLER

PHYSICSREPORTS(Review Sectionof PhysicsLetters)134, Nos.2 & 3 (1986) 87—193. North-Holland,Amsterdam

THE CASIMIR EFFECT

GünterPLUNIEN, Berndt MULLER andWalterGREINER

Institutfür TheoretischePhysikderI. W. Goethe UniversitütFrankfurt; 60(X)Frankfurt/Main, Fed. Rep.Germany

Received8 August 1985

Contents:

1. Introduction 89 4.3. Quarksandgluons in abag 1392. Energyof thevacuum state 91 5. The Dirac vacuumin externalelectromagneticfields 147

2.1. Zero-point energiesin field quantization and the 5.1. Vacuumenergyandvacuumpolarization 147definition of thephysicalvacuumenergy 91 5.2. Thesupercriticalvacuum 159

2.2. Implicationsof thevacuum energy 102 5.3. Thevacuumenergyin nuclearscattering 1673. Mathematical formulation and evaluation methods for 6. Casimir energyat finite temperature 172

vacuumenergies 106 6.1. Partitionfunctionsandfree energy 1723.1. Methodof modeSummation 106 6.2. Finite temperaturepropagators 1743.2. The local formulationand Greenfunction methods 110 6.3. Casimirenergy 17833. The multiple-scatteringexpansionfor Greenfunctions 7. Applications 180

andtheCasimirenergy 121 7.1. Casimir energyin dielectricmediaandtherelationto3.4. Phase-shiftrepresentation 130 thebagmodel of hadronicparticles 180

4. Casimireffect 132 7.2. Boundary problems and Casimir energy in gauge4.1. The Casimireffect betweenperfectlyconductingplates 132 theory 1864.2. The Casimirenergyof a massivescalarfield in afinite 8. Concludingremarks 189

cavity 137 References 190

Abstract:This report gives an introductionto theCasimir effect in quantumfield theoryand its applications.The interactionbetweenthevacuumof a

quantizedfield and anexternalboundaryor aclassicalexternalfield is investigatedlaying particularemphasison Casimir’sconceptof measurablechangein thevacuumenergy.The variousmethodsfor evaluationandregularizationof theCasiinu- energyarediscussedin detailfor specificfieldconfigurations.Recentapplicationsof theCasimir effect in supercriticalfields, QCD bagmodels andelectromagneticmediaarereviewed.

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THE CASIMIR EFFECT

Günter PLUNIEN, Berndt MULLER and Walter GRE1NER

Institutfür TheoretischePhysikderJ. W. GoetheUniversitâtFrankfurt, 6000 Frankfurt/Main,Fed. Rep. Germany

NORTH-HOLLAND - AMSTERDAM

G. Plunieneta!., TheCasimireffect 89

1. Introduction

The occurrenceof divergentzero-pointenergiesis an inherent featureof quantumfield theory. Itappears as a direct consequenceof canonical field quantization, which permits to establish thecorrespondencebetweenclassical(observable)quantitiesandquantumoperators.Zero-point energiesarisebecausethe canonicalquantizationschemedoesnot fix the orderingof non-commutingoperatorsin the field Hamiltonian.Thus,quantumfield theory basedon the operatorconcept,in principle,needsadditionalprescriptionsin order to becomea well-definedtheory.

Normally, this ambiguity is removed by requiring a certainordering of operatorproducts,Wick’snormal ordering.For the canonicalfield Hamiltonianthis prescriptionimplies a formal subtractionofthe infinite zero-pointenergyand,per definition, the vacuumexpectationvalueof the normal-orderedfield Hamiltonian is then equalto zero. Accordingly, this formally introduced(mathematical)vacuumstatehaspeculiarproperties,namely,it carriesneitherenergynor momentumnor angularmomentum.The standardargumentsfor sucha subtractionlay particularemphasison the fact that, in practice,it isnot possibleto measurethe absolutevalueof theenergy,but only differencesin energy,allowingfor anarbitrarychoiceof the origin on the energyscale.This statementis certainlyvalid for the microscopicSystems.However,the questionremainswhetherthe zero-pointfluctuationsof quantumfields must betakenseriouslyor not. For systemswith a finite numberof degreesof freedomzero-pointenergieshaveneverbeen envisagedas problematic,becausethey are finite and measurable(e.g., the zero-pointoscillationsof crystalsat zerotemperatureareobservable)and,thus, theyrepresenta real propertyofthe groundstate.That the situationshould be different in the caseof quantizedfields is not a priorievident. The absolutevalueof the vacuum energyis, in principle, a measurablequantity, becauseitgravitates.The energydensityof the vacuummust thereforebe expectedto contributeto someextentto the total energydensityof the universe.Howeverthat may be, onewould like to know how to dealwith infinite zero-pointenergiesin a meaningfulway within a field theory basedon the canonicalfieldHamiltonian and, in close connectionwith that, whetherone can conclusively show possible con-sequenceswhich indicatethe presenceof vacuumfluctuations.

Such questionsimmediately lead to the Casimir effect. In 1948 Casimir showed [11that neutralperfectly conductingparallelplatesplacedin the vacuumattracteachother.The attractiveforce can beconsideredas arising dueto the changein the zero-pointenergyof the electromagneticfield whentheplates are brought into position. The experimentalverification [2—8]of this attraction has put thediscussionaboutzero-point energiesin field theory on firm ground.The general importanceof thetreatmentof zero-point energy in the derivation of the original Casimir effect lies in the fact that itimplies that the energyof the vacuumstateof quantizedfields cannotbe correctly definedby normalordering.The basicideaof what will later on be called “Casimir’s conceptof vacuumenergy”can bedescribedlike the following: A meaningfuldefinition of the physical vacuum energymust take intoaccountthat in areal situationquantumfields alwaysexistin thepresenceof externalconstraints,i.e. ininteraction with matter or other externalfields. An idealized descriptionof such circumstancesisobtainedby forcing the field to satisfycertainboundaryconditions.The zero-pointmodesareaffectedby the presenceof theseexternalconstraints,andtherefore,the zero-pointenergyis modified.

Consequently,one is led to calculatethe physicalvacuumenergyor Casimirenergyof a quantizedfield with respectto its interactionwith externalconstraints.It is generallydefinedas the differencebetweenthe zero-point energy referring to the distortedvacuumconfiguration and the free vacuumconfiguration, respectively.This formal definition only makessense,if it is combinedwith appropriateregularizationmethodswhich guaranteea finite expressionfor this energydifference.Any variation of

90 G. Plunien etaL, The Casimir effect

theexternalboundaryconfigurationinducesachangein thevacuumenergy,which is now consideredasameasurablequantity. Themain consequenceof this conceptis that the physicalvacuumof a quantumfield no longer representsa statewith trivial global properties.Instead,the Casimir vacuummust beconsideredas a physical object reacting against distortions, i.e., possible vacuum effects can beinterpretedas the “response”of the vacuumdue to the presenceof externalconstraints.This line ofreasoningis spelledout in detail in the first part of section2 of the presentreview article, whereit isshown how zero-pointenergiesarisewhen quantizingthe scalarfield, the electromagneticfield andtheDirac field. This introductory sectioncontinueswith a more specific definition of the vacuumenergy.

Dependingon the particular kind of externalconstraintsdistorting the free vacuumconfiguration,thevacuumenergymaybe understoodas a contribution to the self-energyor to the interactionamongthe given boundaryitself. Possiblyit may causeboundaryeffectsdue to surfaceenergiescontributing,e.g., to surfacetensionor stressesinside the boundary.Basic investigationshavebeen madeof theCasimir energyof the electromagneticfield under constraints,which demonstratethe physical im-plication of the zero-point energy.As already shown by Casimir [301the zero-pointenergyof theelectromagneticfield can be usefully applied to explain van der Waals attraction. However, theexistenceof repulsive Casimir forces [37] of electromagneticorigin seemsto contradictthis inter-pretation.This examplealready revealsthat the physical contentof the concept of Casimir energyembracesmorethan its useas an alternativeway to understandcertainknown effectsthat can alsobederived by conventionalapproaches.Presently,Casimir energiesof quantizedfields are studied inconnectionwith a variety of problems,ranging from applicationsin particle physics,e.g. in QCD bagmodels,to gravitationalphysics,where its possible influenceon the structureof space-timeis studied.Examplesfor physical implicationsof Casimirenergiesarereviewedin the secondpart of section2.

In practice,the evaluationof vacuumenergiesremainsa problematicexercise,becausethe availablemethods,in most cases,only allow an approximatecalculation. Mainly two methods can be dis-tinguished:The mode-summationmethodis basedon the direct evaluationof infinite sumsoverenergyeigenvaluesof thezero-pointfield modes.Within thelocal formulationon the otherhand,oneexaminestheconstrainedpropagationof virtual field quantaandconsidersthe vacuumstresstensor,which can beexpressedin terms of propagators.Both treatmentscan be shown to be formally equivalent. Still,ambiguitiesbetweenthe resultsobtainedby thesemethodscannotbe excludeda priori sincethey implyinherently different regularization schemes,but correct results for the Casimir energyshould beindependentof the applied methods.

Specific difficulties have to besolvedin a successfulapplicationof the modesummationaswell as thelocal Greenfunction method. In the mode-summationmethod,which requiresthe knowledgeof thetotal energyspectrumof the free andthe constrainedfield modesparticularcutoff proceduresmustbeintroducedin order to deal with divergencies.Whethera finite result for thevacuumenergyis obtainedafter performing the involved infinite summations and cutoff removal crucially dependson theconsideredboundaryproblem. In the case of highly symmetric boundariesthis method has beensuccessfullycarriedout.

The local Greenfunction methodrepresentsa formally elegantway in order to derive the vacuumenergyor the vacuumpressure.The main difficulty within this method is the determinationof exactGreenfunctionsdescribingpropagationin the presenceof externalboundaries.This can be solvedbymeansof image sourceconstructions[35] for plane geometriesor, perturbativelyby the multiple-scatteringexpansion[41,42] in the caseof generalconstraints.In section3 we presentthe evaluationmethodsas theyhavebeeninvestigatedin connectionwith the Casimir energyof the electromagneticvacuum.

G. PlunienelaL, The Casimir effect 91

Section 4 dealswith selectedcasesof Casimirenergiesof quantumfields in the presenceof simplegeometricboundaries,in particular, the original Casimir effect, a massiveparticle constrainedto acavity, andthe Casimirenergyin the bagmodelsof elementaryparticles.

The extendedclassof Casimir problemsinvolvesall kindsof situationswherea given quantumfieldinteractswith an externalfield, which maybe createdby an arbitraryconfigurationof externalsources.Correspondingly,the vacuumenergyis definedas the differencebetweenthe zero-pointenergyin thepresenceof the externalfield andthat of the free vacuum.This definition of the vacuumenergyof thefermion field in an externalelectromagneticfield goesback to the work of Weiskopf, Schwingerandothers(97, 100, 101]. As typical examplesfor suchproblems,the vacuumenergyof a scalarfield in thepresenceof externalelectromagneticfields [78] or thevacuumenergyof Higgs field configurations[62]haverecently attractedinterest. In section5 we considerthe vacuumenergyof the electron—positronfield in the presenceof arbitrarily strong externalelectromagneticfields [115].It is commonfor bothtypesof Casimirproblemsto derive interactionsbetweenboundaryconfigurationsdueto thezeropointoscillationsof the constrainedfield.

The formalismasit hasbeendevelopedto evaluatethe Casimirenergyof the vacuumstateis directlygeneralizableto constrainedquantumfields in thermodynamicequilibrium at finite temperature.Themode-summationmethod may be directly applied for calculating the Casimir free energy [81]. Thecorrespondinglocal treatment is basedon thermal Green functions, combined with the multiple-scatteringexpansion [41]. Temperaturecorrectionsmay be of considerableinterest in experimentalmeasurements[81].The Casimireffect at finite temperaturewill be discussedin section6.

At the presenttime Casimirenergiesarediscussedwith increasinginterestin connectionwith variousphysical problems.This makesit impossibleto give a completesurvey of applications,and we have,instead,concentratedon providinga lucid introductioninto the field. Accordingly this reviewarticle onthe Casimir effect representsa particular selection of problemswhich to the authorsseemedmostlyconvenientin order to show the basicmethodsandunderlyingideas.We only give a few representativeexamplesof applications.The investigation of the Casimir effect in dielectric media placed in theelectromagneticfield [39,60,61] hasrecentlyfoundrenewedinterest,becauseit has a direct applicationin QCD bagmodels,wherethe “true” vacuumis regardedas a perfectcolour-magneticconductor[128].The Casimir energydensity on external boundariesmay be also useful in order to decide whethercertaingaugebreaking terms in a given field Lagrangian[134] have to be considered.The analogybetweenthe electromagnetictheory of Casimir energy in material media and QCD, as well as theinfluenceof a non-vanishingphoton mass will be discussedin section 7. The review ends with asummaryandspeculativeremarks.

Throughoutthe review, naturalunits (h = c = 1) and the signature(+, —‘ —, —) for the Minkowskimetric areused.

2. Energy of the vacuumstate

2.1. Zero-pointenergiesin field quantizationand the definition of thephysicalvacuumenergy

In order to approachthe problemof vacuum(ground-state)energiesin quantumfield theory let usstartwith ashort recapitulationof the canonicalfield quantizationscheme.

If ço~°(x)denotesthe dynamicalfields, from the classicalLagrangian

92 G. Plunieneta!., The Casimireffect

L(t) = Jd3x ~~[(p~°(x,t), ~o~°(x, t)] , (2.1)

onederivesthe conjugatefields

~L(t)H~(x,t) = ~(a~~°(x,t))~ (2.2)

For boson fields, quantizationproceedsby replacingthe c-number fields by operatorsthat satisfycanonicalequal-timecommutationrules:

[~~(x, t), .~(k)(x1 t)] = [I7~’~(xt), #~~(x’t)] = 0 , (2.3a)

[ç1~(x,t), fl’(k)(xP t)] = i~Ik45(X— x’) . (2.3b)

Accordingly, anticommutationrules are requiredfor fermion fields. In terms of ~ and 11 the fieldHamiltonianH, formally identicalwith a classicalHamilton function, is obtainedas

i-Ti = JcPx (~Hwc9~~t)— ~ t9~(z)) (2.4)

At thispoint onehasto keepin mind that to ensurethe hermiticity of i-Ti in generalone hasto (anti-)symmetrizewith respect to the adjoint fields 1i~and ~. Only the fundamentalpostulatefor thequantizationof wave fields, i.e., the replacementof canonicalfield variablesby operatorstogetherwiththerequirementof appropriatecommutationrelations,hasbeenusedso far.

Unfortunately, this postulatealone is not sufficient to determine the field Hamiltonian in asatisfactoryway. For instance,the order of non-commutingfactorsin the Hamiltonian is a prioriundefined.The difficulty is madeworseby the singularitiesarisingfrom taking field operatorsat equalspace-timepointswhich leadsto divergentvacuumexpectationvaluesof operatorslike the totalenergy.To ensuretheir elimination one requiresrenormalizabilityas a fundamentalpropertyof everyphysicalfield theory.

In order to make the first point mentionedabove more transparentand to see how zero-pointenergiesappearin different theories,let us now considerthe free scalarfield, the electromagneticfieldandthe electron—positronfield, in sequence.We then go on to definetheir physical vacuumenergies.

2.1.1. Thezero-pointenergyof themassivescalarfieldWe startwith the neutral (real) scalarfield characterizedby the Lagrangian

= — m242, (2.5)

which leadsto the field equation

(L1+m2)~=0. (2.6)

G. Plunieneta!., The Casimireffect 93

For the Heisenbergoperators4 and i~= d4 the following equal-timecommutationrulesarevalid:

[~(x, t), ~(x’, t)] = [fl(x, t), ñ(x’, t)] = 0, (2.7a)

[t~(x,t), JTi(x’, t)] = iô(x — x’). (2.7b)

To discussthe vacuumenergyit is more convenientto adopt the momentum representation.Inside afinite cubic box of volume 11, at anygiven time, the operatorsq5(x, t) and H(x’, t) can beexpandedintermsof the Fourierseries:

~(x, t) = ~ exp(ik . x)4a(t), (2.8)

J~(x,t) = exp(ik X)fi_k(t). (2.9)

The neutral fields ~ and iTi areHermitian, i.e. t~= ~, so consequentlywehave

4~(t)= 4_~(t), ~ö~(t)= J.3k(t). (2.10)

Inserting the aboveexpansionsinto eq. (2.7a)and (2.7b) we derive the commutationrules for theoperators~lk and j3~.So, for example,from

[t~(x,t), 1Ti(x’, t)] = ~ -~ exp(ik . x) exp(ik’ . x’)[4a(t), P_k’(t)]

= exp(ik’ . x) exp(—ik’ x’)[c~k(t),13k.(t)]

=ic5(x—x’)

one can identify

[4~(t),p~’(t)]= ~ (2.11)

since

exp[ik. (x — x’)} = 8(x — x’). (2.12)

Correspondinglythe othercommutatorsare

[4k(t), 4k.(t)] = [j3~(t),pk’(t)I = 0. (2.13)

94 G. Plunieneta!., The Casimireffect

Now weconsiderthe field Hamiltonian.Accordingto eq. (2.4) it reads

iTi=~Jd3x{I~+(V~)2+m2~2}, (2.14)

or after partial integrationof the gradientterm one obtainsi-Ti as aquadraticform of the fields ~ andH:

i-Ti = ~J d3x {fl2 + ~~(— V2+ m~)t~}, (2.15)

which can be expressedin termsof the operators/3k and4~: - -

i-Ti = ~ J d3x exp[i(k + k’) x}{ I-al-a’ + (k’2 + m2)4~4~.}

~ (uk=Vk+m. (2.1-6)

This is exactlythe Hamiltonianof a systemof uncoupledone-dimensi9nalharmonicoscillators,which isnot surprisingbecausewe havejust decomposedthe field qS into nOrmalmodes4~. Note that j3~and4~correspondto the canonicalvariablesof the classicaloscillatorproblem.To obtain the more compactenergyrepresentationof H, oneusuallytransformsto a newbasis ~ a~}of the Fock-space:

Ilk = V~w~(4a +-~--p~), (2.17a)

ak=V~cvk(qk__pk). (2.17b)

Thesecreationandannihilationoperatorssatisfythe commutationrule

[ak, Ilk’] = ‘5kk’, (2.18)

while all other commutatorsvanish.Expressedin terms of the ~k andá~the Hamiltonian takesthesymmetricform

(2.19)

where iI~= akak is the number operatorthat haseigenvaluesii,. = 0, 1, 2 The last step showsexplicitly howthe zero-pointenergyof the free massivescalarfield appears.The vacuum 0) is definedby the equationâ~I0)= 0. Obviously the vacuum expectationvalue E

0 = ~0IH~0)divergessince eachoscillatorcontributeswith ~Wk andonehasanunboundedsumover all momenta.

G. Plunienetal., The Casimireffect 95

2.1.2. The zero-pointenergyof the electromagneticfieldWe turn now to the discussionof the zero-pointenergyof the electromagneticfield.Fromthe free field Lagrangian

2= ~ F,~,.= — ~ (2.20)

onederivesthe conjugatefields as

= 92/9(90A~). (2.21)

In particular,the time-like componentH°vanishes,whereasthe spatial componentscoincidewith the

electric fieldH°=0, (2.22a)

= 3°A” — 3kAO = Ek. (2.22b)

Difficulties in quantizationarising from the relation (2.22a) can be removed by the Gupta—Bleulermethod [9, 10] in a Lorentz-covariantmanner.Sufficing for our purposehere, the manifestLorentzcovariancecan be abandonedby choosingthe transverse(Coulomb) gaugefor quantization.In thisgaugethe free electromagneticfield satisfiesthe conditions

A°=0, V•A=0, (2.23)

andMaxwell’s equationsreduceto

LIIA=0. (2.24)

Now quantizationis achievedby the following commutationrules:

[A~(x,t), A1(x’, t)] = [1TI~(1,t), 1Ti1(x’, t)] = 0, (2.25a)

[A1(x,t), JZ(x’, t)] = iô(s.r). (x — x’), (2.25b)

wherein the last relationthe usual 5-functionis replacedby the divergencefree transverse6-function

ô(tr) (x — x’) (ôq — z1~9,ô~)6(x— x’) (2.26)

to reconcileeq. [2.25b]with the gaugecondition V~A= 0 and Gauss’ law V~H = V~E = 0. Relation(2.25b)guaranteesthe absenceof photonswith longitudinalpolarization.

To representA andH in termsof Fourierseries,oneconvenientlyintroducesfor given ka particularorthogonalbasis of polarizationvectors{e~(A= 1, 2); k/!kP}, which satisfy the conditions

4~•k=0, E~~”ôAA’, A=1,2. (2.27)

96 G. Plunienet aL, TheCasimireffect

The completenessrelationfor thebasisrequires

~ E~E~+= 6~. (2.28)

The Fourierrepresentationof the fields A and 1? can then be written as

A(x, t) = ~ ~~exp(ik ‘x)e~4~(t) (2.29)

and

fl(x, t) = ~ ~ç~exP(ik . x)r~4~(t). (2.30)

As in the caseof the neutralscalarfield the hermiticity of A andH yields the relations

(A) ‘(A) — (A) “(A)± (A) -‘(A)±— (A) -‘(A)— ~ q, , ~-aPa — ~a P—a,

which allow oneto derive the commutationrules for the operators4~ andj3~

F4~A)(t)j3~’~(t)]= [j~~(t), j3~’~(t)]= 0 , (2.32a)

[4~(t), j3~’~(t)]= i6AA’6 kk’~ (2.32b)

We can now considerthe Hamiltonianof the electromagneticfield, which is derivedaccordingto eq.

(2.4) in its canonicalform

i4=~Jd3x{ft2+A.(_v2A)}. (2.33)

In the derivationusehasbeenmadeof the identity f~d3x (V x A)2= f0 d

3xA (— V2A), which can beprovedby applyingthe gaugecondition (2.23) andintegratingby parts. Expressedin termsof j3~and4~the Hamiltoriian becomesagainan infinite sumof uncoupledharmonicoscillators:

i-Ti = ~ {j5~j3~+ ~ (2.34)

The transformationto new creationand annihilationoperators~ andê~definedby

ê(A) = VT~ (4~ + 1 j3(A)+) (2.35)

G. PlunienetaL, The Casimireffect 97

satisfiesthe commutationrule

[-‘(A) ‘(A’)±l —, c~. — ~-‘AA’~’kk’

leadsto the representationin termsof photonnumberoperatorsñj~=

~ Wk(fl~+~). (2.37)

The vacuumof thequantizedelectromagneticfield is definedby ê~I0)= 0. Consequently,the quantizedfree electromagneticfield also carries an infinite zero-point energy, E

0 —(OIHIO). = ~l~A Wk. In thiscontextwe note that in the frameworkof the Gupta—Bleulerquantizationmethodonewould obtainaHamiltonian of similar form, but with additional contributions from the unphysicallongitudinal orscalarphotons[11].However,it can beshown[12]that thezero-pointenergyof the unphysicalphotonsis exactlycancelledby the introductionof Fadeev—Popovghost fields [13, 14], which carry a negativezero-pointenergy[15—17].In this sense,the covarianttreatmentof zero-pointenergyis equivalenttothat in the transversegauge.

2.1.3. Zero-pointenergyoffermionfieldsAfter having treatedtwo boson fields which exhibit divergent zero-point energiesbecausethey

correspondto an infinite collection of harmonicoscillators,let usnow turn to the zero-pointenergyoffermion fields taking as an examplethe electron—positronfield. This particular casewill be of furtherinterestin later sections.

Before taking chargeconjugationinvarianceinto account,we discussthe quantizationof the freeDirac field basedon the Lagrangian

(2.38)

which gives the field equation

(iJ—m)~P=0. (2.39)

The conjugatefields follow as

H~=i~P~. (2.40)

For fermion fields quantizationis achievedby anticommutationrelations

{#,,(x, t), #~(x’,t)} = {1?a(x, t), .ñ~~(xl,t)} = 0, (2.41a)

{1P,,(x, t), 1Ti,3(x’, t)} = ~5ajjS(X — x’). (2.41b)

According to eqs. (2.4) and(2.39) the field Hamiltonianreads

H= Jd3xi~i~ôo!P=Jd3xt~(—iy°y”~k + y°m)~P. (2.42)

98 G. Plunienet a!., The Casimireffect

In orderto derive the moreconvenientenergyrepresentation,the field operatorsare expandedin terms

of a completeset of single-particlestates

~P(x,t) = ~ i/i~(x)exp(—i~~t)l~+ ~ ~1i~(x)exp(—iE~t)l~. (2.43)

The first sum containselectronstatesreferringto a positiveenergye,> 0, whereasthe negativeenergystateswith ~. <0 are included in the second sum. With respect to eqs. (2.41a) and (2.41b) thecorrespondinganticommutationrelationsfor the creation-andannihilationoperatorsb, b~andb~,b~areobtained

{l~,,1;} = {l~,l~,}= {l~,,l~,}= 0, (2.44a)

{6.,,, l~.}= ~ {l~,/~.}= ~ (2.44b)

The expansion(2.43) allows the expressionof the Hamiltonianin the form

H = ~ l~l~+ ~ ~ (2.45)

In orderto ensureastablegroundstateonerequiresthatall electronstatesreferringto anegativeenergy~(the negativeenergycontinuum)are occupied.

Accordingly, the Dirac vacuumis definedin termsof the operatorsb~andb~as

= 0, ~ 0, - (2.46a)

l~j0)=0, ~,,<0. (2.46b)

Making use of the relations(2.44a)and(2.44b) the Hamiltoniancan be rewrittenas

i-Ti = ~ — ~ 1J~+ Eo. (2.47)

The last term of H representsthe infinite zero-pointenergyof the Dirac vacuumE0 = (OIHIO) ~. ~,which arisesfrom the occupiednegative energycontinuum.Insteadof dealingwith electron statesofnegativeenergy,Dirac’s holepictureimplies that holes in the negativeenergyspectrumb~~0)n) arereinterpretedas positron (antiparticle)stateswith a positive energyE~= (—e,~)and positive charge.Formallythe conceptof antiparticlesis introducedby the changeto positroncreationoperatorsd~= b~.Then, in termsof the operatorsb2 andd~,the vacuumis definedby the equations

l~0)=0, ~>0, (2.48a)

d~I0)=0, s11<0. (2.48b)

The completeset of single-particlestatescan be divided into a subsetof statesdescribingparticles


(e,> E~)and one describing antiparticles(e~< CF). Therefore the boundarybetween particle andantiparticlestateshasthe quality of a generalized“Fermi surface” (indicatedby the symbolF). For thedistinction betweenelectron- @article) and positron- (antiparticle) statesone introducesthe FermienergyCF. In the caseof thenon-interactingDirac field the conventionalchoicefor the Fermi energyisCF = 0. As we shall discussin more details in section5, the mostgeneraland consistentchoiceof theFermi energy,also in the casewhen the Dirac field interactswith an externalelectromagneticfield ofarbitrary strength and therefore bound states are present, turns out to be at a Fermi energy—m�eF~m[18].

We can rewrite the expansionof the field operator~Pin terms of single-particleand antiparticlestates

~P(x,t) = ~ t/Jk(X) exp(—iEkt)15 + ~ t/i~(x)exp(iEkt)d~, (2.49)

k>F k<F

wherebythe Ek are relatedto the r1, andE~by the relation: Ek = C~,for k >F, andEk = —E~fork <F.Now we expressthe Hamiltonianin aHermitianandchargeconjugationinvariant form [11]

~=~Jd3x{[~i~, (_iyoykak+yom)~fI]+[~fr+(_iyoykak + y°m),~I~]}. (2.50)

This leadsto the final expressionin termsof particleandantiparticleoperators:

H= ~ Ekb~bk+~ Ekc2~dk+EO. (2.51)k>F k<F

Accordingly one obtains the result that the symmetrizednon-interactingDirac vacuum carries theinfinite zero-pointenergy

E0=_~(~Ek+ ~ Ek). (2.52)k>F k<F

2.1.4. Zero-pointenergyand normalorderingSo far we have shown that the occurrenceof infinite zero-point energiesis inherentwithin the

canonicalfield quantizationschemebecauseit doesnot fix the orderingof quantumoperatorsin thefield Hamiltonian.To removethis ambiguity onehasseveraloptions. In the most acceptedprocedureonearbitrarily imposesacertainorderof the non-commutingcreationandannihilationoperatorsin theHamiltonian.It is basedon the observationthat zero-pointenergiesalwaysappearin connectionwith arearrangementof termscontaininga productof annihilationand creationoperators.One circumventsthis by introducing the so-called“normal ordering” of operatorproductsand replacingall relevantexpressionsby their normal-orderedcounterparts.Normal orderingof a product (indicatedby embrac-ing the expressionbetweencolons) is definedas rearrangementof its factorsin such a way that allannihilationoperatorsstandto the right. For fermionstheproductmustbe multiplied by the sign of therequiredpermutation.For instance,for bosonoperators

~:(â~ák+ akak): = a,~ak, - (2.53)

100 G. Plunieneta!., The Casi,nireffect

andfor fermionoperators

(!~l~— bkbk): = b~bk. (2.54)

Applying thisprocedureto the field Hamiltoniansconsideredabove,one obtains

:i-Ti: = i-Ti — (Oh-Tb)an i-Ti’. (2.55)

One observesthat replacingthe original Hamiltonian i-Ti by the normal-orderedHamiltonian i-Ti’ is

equivalentto a formal subtractionof zero-point energies.This has the effect that the energyof thecorrespondingfree vacuumstatebecomeszero.

It must be stressedthat the normalorderingcannotbe justified by referring to the correspondenceprinciple which can be envisagedas the underlyingguide in canonicalfield quantization.One hastoacceptnormalordering in this context as an additional postulaterequiredto renderthe energyof thevacuumstatefinite. It becomesacceptableby the fact that it bringsaboutconsistencywith the heuristicnotion of a non-interactingvacuumas a stateof zero energy.However, the generalargumentthat thesubtractionof the zero-pointenergycan be performedarbitrarily sinceit correspondsto a redefinitionof the energyscale,and since only energydifferencesare in reality measurablequantities,must beregardedwith caution. In relativistic quantum field theory the Hamiltonian, which is one of thegeneratorsof the Poincarégroup, is determinedin a uniquemannerfor a given set of field operators[19]. Takahashiand Shimodaira[19,20] havein fact shownthat its expectationvalue must vanish inorder that thegeneratorssatisfythecorrectcommutationrules. Thisprovidesa convincingargumentforthe usual notion of the vacuumstate,but not a uniqueone for introducingH’. It doesnot excludealternativeapproachestowardsasatisfactorydefinition of the vacuumstate.Thisproblemis particularlyimportant in the context of gravitationalphysicswherethe ad hocsubtractionof zero-pointenergiesseemsdubiouson formal andphysicalgrounds,becausetheexpectationvalueof theenergy—momentumtensordeterminesthe geometryof space-time.In principle, quantumfluctuationsmay influencethelocal space-timestructure in microscopicdomains which are of the order of the Planck length(hG/c3)1”2 = 1.6x 1O~cm.

2.1.5. TheCasimir energySearchingfor a general conceptto define the propertiesof the vacuum stateof quantizedfields,

attentionis drawn to Casimir’swork on the zero-pointenergyof the electromagneticfield [1]. Theunderlying idea can begeneralizedto the caseof otherfields, leadingto variousversionsof the Casimireffect. It is essentialfor the understandingof thiseffect to realizethat quantizedfreefields, as theyhavebeen understoodabove, are abstractmathematicalconstructions.In reality fields exist and can bemeasuredonly in the presenceof sourcesor as fields interacting with eachother. A specialcaseofparticularinterest is a field that is confinedto a finite spatialvolume.Generallyspeakingaphysicalfieldmaybe forced to satisfy certainboundaryconditions,an important fact which hasto be consideredinthe mathematicaldescription.The presenceof boundariesfor the field inducesa changein the energyspectrumandthereforemodifies alsothe zero-pointenergy.Consequently,it is reasonableto definethephysical vacuumenergyof quantizedfields as a differencein zero-pointenergy.Let aT’ standfor anarbitrary boundaryrequired for the consideredfield. If E

0[aF] denotesthe zero-point energyin thepresenceof boundaries,and E0[0] that without such boundaries,then the vacuum energy (Casimirenergy)is formally definedas

G. PlunjenetaL, TheCasimireffect 101

Evac[t91’] = E0[aF] — E0[O]. (2.56)

In order to makethis formal definition more transparent,a few remarksare necessaryto specifywhat possiblesituationsor systemscan be envisagedandwhat kind of “boundaryconditions” mustbedealtwith. In a typical situationa field is given in the presenceof macroscopicobjectsor it is confinedto a finite cavity. Typical examples for this case are conductorsor dielectrics constraining theelectromagneticfield or the QCD vacuumconfining quarksandgluonsin a hadronbag. The field thenhasto satisfycertaingeometricalboundaryconditions,e.g.,to vanishon the surfaceor in theoutsideofa cavity. In such casesit is convenientto treat the vacuumenergyas a function of a suitableset ofparameters,hereabbreviatedby A, which characterizethe given geometricalconfiguration (fig. 2.1).This meansthat we write

Evac[A] = E0[A] — E0[A0], (2.57)

whereA0 denotesthe parameterset correspondingto the situationwithout boundaries.Any variationintheseparametersmayinducea changein vacuumenergy.

Boundaryconditionscan beconsideredas idealizationsof therealconditionswhereconfigurationsofmatteror externalforcesacton a given field. We are thus led to a generalizedconceptof the Casimirenergyby consideringthe changein the energyof the vacuumof, for instance,the electron—positronfield in thepresenceof an externalelectromagneticfield A,~.The externalfield now playstherole of theparametersA, which representthe boundarycondition.The analogousdefinition of the vacuumenergythen is

Evac[A] = E0[A] — E0[A = 0]. (2.58)

Alternatively, the vacuum energymay be treatedas a function of parameterswhich determinethegeometryof the given externalsourceconfiguration.More detailedconsiderationswill be necessarywhen the vacuumenergyis explicitly evaluated(seesection5).

To summarize,wenotethat(1) the ambiguity of divergentzero-point energiesmaybe removedby retaining the original field

Hamiltonianbut introducingthe definition, eq. (2.56), for the physicalvacuumenergy;(2) the heuristicnotion of a non-interactingvacuumis not affected,since Eyac[0] is, per definition,

equalto zero;(3) changesin zero-pointenergyarereintroducedin quantumfield theory as in principle measurable

corrections.

Fig. 2.1. The parameterA may describetherelativepositionor thegeometryof certainobjectsin an externalfield.

102 G. Plunieneta!., TheCasimireffect

Thispoint of view implies that propertiesof quantizedfieldsmaybe determinedfrom the “response”of the vacuumstateto externalconstraintsor fields. Since it can be experimentallytested(at leastinprinciple)whethersuchchangesin the vacuumenergyas a functionof certainparametersA do exist ornot, it is a questionof physics (and not only of formalism) whetherthe zero-point energymust beincludedin the field Hamiltonianor not.

2.2. implications of the vacuum energy

After introducingthe vacuumenergy,and havingmadeits formal definition plausible,we will nowoutline some implications of the vacuum energy. To do so we have to examine the meaningofzero-pointenergiesin the contextof variousphysicaleffects.

The forcesactingbetweenneutralbut polarizableobjects,often generallyclassifiedas vander Waalsforces,historically gavea stimulusto Casimir’swork on the zero-pointenergyof the electromagneticfield, which was motivatedby the searchfor an explanationof the simplicity of the resultsfor vanderWaals attraction obtainedin perturbationtheory [28]. It is principally possible to distinguishthreeclassesof van der Waalsforces[29],viz., orientation,inductionanddispersionforces.Thelatter classofforcesindicates that two polarizableparticleswill attracteachotherbecausequantumchargefluctua-tions in one particlewill producea dipole moment,which will in turn induce a dipole moment in thesecondparticle. Becauseof the relation of theseforcesto the phenomenonof optical dispersiontheyare termed“dispersion forces”.The conceptof electromagneticzero-pointenergyallows to calculateaparticular subclassof the van der Waals forces, the so-called“long-rangeretardeddispersionforces”.

For example,calculatingtheattractionof apolarizableparticleto a perfectconductingwall (fig. 2.2),onecan usethevacuumenergyproperlydefinedfor this situation[30,31]. Let E0 = ~,, w~denotethezero-point energyof the electromagneticfield in a large cavity with conductingwalls. The normalfrequencies~ aredeterminedby requiring periodic boundaryconditions.The presenceof a particleinside the cavity, with dielectricpolarizability a, separatedby a distanceR from the wall, will modifythe frequency of each mode and consequentlychange the zero-point energyby the R-dependentamount

~w~(R)~~ [w~(R)—w~],

which dependson R. For the combined system “particle plus conducting wall” the distance R

____________ /Fig. 2.2. Boundaryconfigurationassumedin thederivationof thevan derWaalsattractionbetweenapolarizableparticleandaperfectlyconductingwall.

G. PlunienetaL, The Casimireffect 103

representsthe significant parameter.One obtains the correspondingvacuum energyas the energydifferencebetweena configurationwherethe particle is placedat the distanceR andthat obtainedinthe limit R —~~ when~ —~0 by definition. For the consideredsystemthevacuumenergyrepresentsacontributionto the potentialenergydescribingthe attractionbetweenthe particle andthe wall

Evac(R) = ~ (w~°~+&o~(R))—~~ (w~+~wn(co)). (2.59)

Explicit evaluationof the expressionEvac(R) leadsto the result

3a 1Evac(R) = — ~, (2.60)

which agreeswith that first obtainedby Casimir and Polder [28] from perturbationtheory.The samemethod which leads to the above result can also be employed to derive the long-rangeretardedpotentialbetweentwo neutralpolarizableparticlesseparatedby a distanceR [30]:

Evac(R) = —23a1a2/41TR7. (2.61)

Again, the result is identicalwith that found by CasimirandPolder [28]from perturbationtheory.Thesuccessfuluse of the zero-point energy in this context is further supportedby Boyer’s [32, 33]recalculationof long-rangevan derWaals potentials,wherehealsoincludedthe magneticpolarizabilityof the particle.The full expressionscorrespondingto eqs. (2.60) and(2.61) coincidewith thoseobtainedby FeinbergandSucher[34] from dispersion-theorytechniques.

The original effect [1] consideredby Casimirwas the attractionof two unchargedparallelconductingplatesat zerotemperature.His work concerningthis simple configurationshowsmost impressivelytheexistenceof forces which may be ascribedentirely to the changein the zero-point energy of theelectromagneticvacuumbrought about by the presenceof boundaries.It was not until 30 years later,however,that the local form of Casimir’s result, i.e., the formulationin termsof vacuumenergydensityand vacuum pressure, was exploited by Brown and Maclay [351.They showed that the vacuumexpectationvalue of the regularizedenergy—momentumtensoris spatially constant,and derived theexpression

= —(n-2/180a4)(~g~”+ z’~z~), (2.62)

wherez~= (0, 0, 0, 1) denotesa space-likefour-vectororthogonalto the platesseparatedby a distancea.

The form of this resultbecomesplausibleby the following generalarguments.First of all, the onlytwo symmetrictensorsof rank 2 that can be formedareg~” andz”z~becauseof the symmetry of theconfiguration. @~‘ is invariant with respectto arbitrary Lorentzboostsparallelto the plates,but theperpendiculardirection is favoured. Since the electromagneticfield is masslessand conformallyinvariant, the energy—momentumtensor,eq. (2.62), must be tracelessand divergenceless.These factsrequire t9’”’ to be independentof position and only allow the combination(~g~’+ z~’z’~).Finally, ondimensionalgrounds it follows that Ø’~’must be proportional to the inverse fourth power of theseparation.

104 G. Plunienet aL, The Casimireffect

The numericalfactor (~i~2/18O),including thesign, can only be obtainedby explicit calculation.Thisis an importantpoint in that it showsthat the result (2.62) is unlikely to be representativefor thegeneral situation. Whetherthe zero-pointpressureon surfaceconfigurationstends to contract or toexpandthem,whetherthe vacuumenergyis negativeor positive, dependscrucially on the particulargeometry.Unfortunatelythe geometryentersin anon-trivial way. Thus,for example,the hopethat theparallel-plateresult may serve to stabilize the classicalmodel of the electron as a chargedshell [36]turnedout to bea fallacy. Boyer’s result for the vacuumenergyon perfectly conductingsphericalshells[37,38] is positiveand consequentlytendsto expandthis configuration:

E 0.093/2a. (2.63)

Thereforeit cannotplay the role of the Poincaréstressin electrodynamics.Nevertheless,over theyearsthis idea instigatedfurtherefforts to evaluatethe Casimireffect [39,40] for solid ballswith certaindielectricity andpermeability. Here in fact attractiveenergycontributionsare obtained,but they arequantitativelytoo small to explain the observedchargequantization.In this context one hasto provewhether the repulsive Coulomb energyof a chargedspheremay be balancedby an appropriateattractiveCasimir energy. If this would be the case,then one would obtain an eigenvaluefor thefine-structureconstant.From an approximatecalculation it had also beensuggestedthat the Casimirenergyfor perfectlyconductingcylindrical shellsmaybe zero[42].This would beintuitively reasonablefrom a geometricalpoint of view, classifying the cylindrical configuration somewherein betweentheparallel-plateand the spherical-shellconfiguration. Explicit calculationsfor this geometry,however,haveyieldeda negativevacuumenergy[43].

Theobviouscomplexityof therelationbetweenthemagnitudeof theelectromagneticCasimirenergyand the shapeof the surfacehas provided a stimulus for studying vacuumenergyin the presenceofarbitrarily shapedobjects in the framework of the local representation.Balian and Duplantier [42]studied the Casimir effect (including temperature)concerningthe electromagneticfield in regionsboundedby thin perfectconductorswith arbitrarysmoothshapes.In this contextafurtheraspectof thevacuumenergybecameobvious, namely that it can representa contribution to the surfacetension.Conceptually,thisis easyto understandwhenonestartsfrom an infinite planeconductorplacedin theelectromagneticvacuumas a situationof reference.When a smoothdeformationof the conductorisintroduced,this inducesa changein the zero-pointenergy.Suitableparametersfor measuringthelocaldeviationfrom a planargeometryare the principal curvaturesof the conductorsurface.Treatedas afunction of theseparameters,the vacuumenergycontributing to surfacetensionmay be rigorouslydefined as the difference between the zero-point energy at a certain deformationand that whichcorrespondsto the planargeometry.

This interpretation of the Casimir effect can also be applied to more realistic objects in theelectromagneticfield characterizedby a dielectric constantand magneticpermeability. An essentialproperty of such surfacecontributions of the vacuum energyis that they generallyremain cutoff-dependent[44, 45]. This fact, however,makesit possibleto obtain reasonablevalues for measurablequantitiesas, for example,the surfacetension of liquid helium [44] and many metals [46, 47] by asuitablechoiceof the high-frequencycutoff. Having themeansof deriving suchgeometricalexpansionsfor the vacuumenergyin the presenceof arbitrarily shapedobjects,one is close to the capability ofconsideringvacuum energyin curved space-time[49, 50]. Accordingly, as a parallel development,remarkableprogresshasbeenmadein revealingfurtheraspectsof the Casimirenergyin gravitationalphysics [21—27].In the frameworkof such studiesonecan try to investigatecosmologicalconsequences

G. P!unien etaL, The Casirnir effect 105

of changesin vacuumenergydue to deviationsfrom Minkowskian geometry, to mention only oneapplication.

Reviewingat first only the variouswork that hasbeendoneconcerningtheelectromagneticvacuum,one is led to the following judgementof the zero-pointenergyproblem: Casimir’s work stimulatedstudiesabout the zero-pointenergyproblem in field theory. Such studiesare generallyimportant inorder to derive an accuratefield quantizationschemeas a fully satisfactory“first principle theory”.Questionsabout the physical reality of the zero-pointenergycan only be settled by demonstratingobservableeffects that reveal the influenceof the vacuum. The electromagneticfield representsaconvenientexampleto examinethe physicalmeaningof thezero-pointenergy,becauseits fundamentalinteraction is well known. In this way it hasbeenshown that the conceptof Casimirenergycan besuccessfullyappliedto alternativecalculationsof the vander Waalsattraction[32,33]. But in contrastto that, the Casimir energycan alsocauserepulsiveforces as in the caseof the conductingsphericalshell [37,48]. This alreadyindicatestwo possibleimplicationsof the vacuumenergy:On the one handthe concept of a Casimir energy can lead to a different point of view or allows an alternativeinterpretationof knowneffects. On the otherhand,however,this conceptbrings to light certaineffectswhich may not be necessarilyunderstandablein the framework of conventionalfield theory whichneglectsthe zero-pointenergy.Directly connectedwith that is the clarification of thequestionof how todeterminethe “true” field Hamiltonian.

Having realized that the conceptof Casimir energy togetherwith its evaluationmethodscan beappliedsuccessfullyin the caseof the electromagneticfield, it seemslegitimate to look for the possiblerole played by zero-pointenergiesconsideringvarious other constrainedquantumfields. Let usnowmentionsomefieldsof applicationfor the useof zero-pointenergyconcerningbosonandfermion fields.First we haveto note that Wentzel [51,52], even before Casimir, used the zero-pointenergyof theKlein—Gordon field to calculate the forces in the meson-pairtheory of fixed sources.Wentzel’s pairmodel recently reemergedin the context of the self-energy calculationsfor masslessscalar fieldsconfinedto a sphericalcavity [53].

The conceptof the Casimireffect has recentlybecomea heuristicguide in particle physics [54—72].The idea that hadronscan be describedas confined fermion and (vector) bosonfields, the fields ofquarks and gluons, hasbecomegenerallyaccepted,although the precise nature of the mechanismresponsible for confinement remains unknown. The confinement itself gives rise to new effectsconnectedwith the Casimirenergydueto quantumfluctuationsin the fields. In the framework of theestablishedphenomenologicalbag model [73—75],which representsa widely usedapproximationto theconfinementmechanismexisting in quantumchromodynamics(QCD), one has exactly the situationwhere fields are forced to satisfy certainboundaryconditionson the bagsurfacein order to guaranteecolour andquark confinement.Much work hasbeendonein approachingandunderstandingthe role ofthe zero-pointenergyin the internal structureof hadronsin the bag model [54—60].Originally it wassuggested[76, 69] that the Casimir energiesof quark and gluon fields could explain a part of thepotential (—zia, where a is the bag radius and z is a pure positive number) necessaryfor thephenomenology.Goodfits wereobtainedfor the valuez = 1.84. In order to find an explanationfor thisempirical valueone hasto examineall the energycontributionswhich areof the form 1/a. BesidetheCasimir energyof confined quarksand gluons (their 1/a behaviouralready follows on dimensionalgrounds) the center-of-masscorrection hasto be takeninto account.The latter turns out to be thedominant1/a contribution[139]to the staticbagenergyandleadsto the value ZCM 1.95. The use ofthe Casimir energy in the framework of the bag model to explain the empirical value of z failsquantitatively. Both the zero-point energiesof confined quarks and gluons give rise to a repulsive


Casimirenergy[58,61]. Taking all theseeffectstogetheroneis left with avalue which is approximatelyabout z 1.5. The evaluationof the Casimir energiestreating quark and gluon fields as free andperfectly confinedwithin the bagis plaguedwith divergencies(cutoff-dependentterms)when restrictingto field modesin the interior of the bag[55,57]. Thesecutoff termscancelwhenconsideringalsotheexterior field modes[58,61].

The situationbecomesmoredifficult consideringmassivequarks.The result for the fermion stresscontainsnew ultraviolet-divergentterms in addition to those occurring in the masslesscase [59].

Concerning the Casimir energy in the hadron model, only further studies will allow a deeperunderstandinganda satisfactoryexplanationof their role. It remainsto be said that Casimirenergiesgive rise to interestingconsiderationsin connectionwith the structureof the QCD vacuum[66—72]aswell as the propertiesof thevacuumof scalarfields [77].

The investigationof the responseof the vacuum to the presenceof external fields representsafurther field of applicationof Casimirenergies.A problem of this typeconsidersthe zero-pointenergyof the electron—positronfield in the presenceof classicalelectromagneticfields, createdby an arbitraryconfigurationof externalsources.ThevacuumenergyEvac[A] = E0[A] — EO[A = 0] maybe treatedas afunctionof parameterswhich characterizethe geometricalshapeor the relativepositionsof the externalsourcesfor a given configuration.Which propertiesof the vacuumenergycan be expectedundersuchconditions?One could at first argue that the induction of polarizationcurrentsrepresentsthe mainreactionof the Dirac vacuumto the presenceof externalelectromagneticfields. Any variation of theparameterswhich determinethe external chargeconfiguration will change the external field and,correspondingly,cause a displacementof the polarization charge density. Accordingly, such “dis-tortions” of the vacuum will lead to a vacuum pressure.Consequentlyone would expect that thevacuumenergy representsa contributionto the interactionpotential. In addition, the correspondingvacuumenergymay containself-energycontributions.Whethertheseexpectationsprove to betrue hasto be shownby explicit calculations.An investigationalongthis line hasbeencarriedout by AmbjornandWolfram [78] whodiscussedthe vacuumenergyof a chargedscalarfield in an externalelectricfield.

Another extensionof the concept of Casimir energiesis obtainedby consideringthe combinedsystem,field plus boundaries,at finite temperature,assumingthermodynamicalequilibrium. Applyingthe standardformalism, the quantity which must be calculatedunder suchconditionsis the Helmholzfree energy

F=—-~--ln2, (2.64)

where ~ is the partition function of the system.The temperature-dependentCasimir energy forconductingplatesin the electromagneticvacuumhasbeencalculatedby severalauthors[44,35, 79—81].Correctionsarising from the fact that real metalsarenot perfectconductorshavealsobeenconsidered[82]. Theseinvestigationsare important in order to derive a betterapproximationto the measurableeffectsin real experiments.

3. Mathematicalformulationandevaluationmethodsfor vacuumenergies

3.1. Method of mode summation

The most direct methodto evaluatevacuumenergiesis that of modesummation.Startingfrom itsdefinition as a difference in zero-point energiesthis method involves the performanceof infinite

G. P!unieneta!., TheCasimireffect 107

summationsovereigenenergiesof correspondingfield normalmodesin thepresenceof externalconstraintsandfor the unperturbedvacuumconfiguration.

In orderto makethis kind of approachin principle applicable,onehasfirst of all to introducea largebut finite quantizationvolume which is boundedby a surface.~ (spacecutoff) of suitablegeometryadaptedto the constraintspresentin the consideredfield. This suggests,for example, in caseof theparallel-plateconfigurationto usea rectangularbox as quantizationvolume.Quantizingthe field underthe usual assumptionof periodic boundaryconditionsdeterminesthe free eigenmodefrequencies,andthe zero-point energy inside this volume becomesa discrete sum. Let us illustrate the situation forboson fields (scalar or vector fields), where the correspondingzero-point energy takes the formE0 = ~k Wk. The free field zero-pointenergyinsidethe volume dependsonly on the spacecutoff Z

E0[.~,0] = ~ Wk[I, 0], (3.1)

where the symbol 0 indicates that the volume is otherwiseempty. The vacuum inside the volumebecomesdisturbedby thepresenceof certainobjects.As an idealizationthoseobjectscan be treatedasadditional arbitrarysurfacesS, on which the physical fields are constrainedto fulfill certainboundaryconditions.Many possibleconfigurationscan bestudiedin thisway. The surfacemaybedisconnectedorsimply connectedconfining a finite subspaceor a combinationof the two.

Also different typesof boundaryconditionscan berequired,dependingupon the physicalsituation.The simplestonesareDirichlet conditions,wherethe field (or someof its components)must vanishonS:

Dirichlet: ço(x)!s = 0. (3.2)

Sucha condition is applicablewhen perfectconductorsareplacedin the electromagneticvacuum.TheNeumannboundarycondition,which statesthat the normalderivativeof the field hasto vanishon 5, isanotherpossibility:

Neumann: n~Vç(x)js= 0. (3.3)

This condition is analogousto the linear bag condition applied to quarksinside a hadron,wherethenormalcomponentof the quark currentmust vanishon the bagsurface.In certaincasesonemay alsoassumean arbitrary linear combination of these, the Robin boundary condition. The eigenmodeenergiesare obtainedby solving the field equationwith respectto theseconstraints.Accordingly, for agiven surfaceconfigurationthe zero-pointenergydependsalsoon S. In deriving the vacuumenergy,itseemsconvenientto considerthe energydifference

~ S] = wk [1’ 5] — Wk [.~,0]. (3.4)

This expressionis still indefinite, which indicates that in fact such “perfect conductor” boundaryconditionsaretoo rough an approximationto the real situation.Insteadof dealingwith those“sharp”boundaryconditionsit seemsto be more appropriateto “smooth” them out by taking into accountcertain microscopicdetails or physical surfaceeffects (for instance,the finite penetrationdepth ofelectromagneticwaves in real materials).On the other hand one can expect that a more realistic

108 G.P!unien eta!., The Casimireffect

treatmentbecomesrathercumbersome.It is possibleto circumvent theseproblemsby introducingasuitablehigh-frequencycutoff x(w), assumingthat it simulatesthe ignoredphysicaldetails.

Here an inherentproblemof the mode-summationmethodbecomesobvious,i.e., the embarrassingfact that it gives an indefinite expression(3.4), and requires the further introduction of a suitableregularization scheme in order to yield a finite result. This regularization should be, however,legitimizedon physicalgrounds.For example,the conductivity of physicalmaterialsactually decreasesat high frequencies,and the surfacesbecometransparentto electromagneticwaves of very shortwavelength.Thus onecan expectthat the high-frequencymodesarenot influencedby any boundarySand, consequently,they should not contribute to ~ S]. Transparencyof the surface can besimulated,for example,by choosingacutoff x(w) with the behaviourx(w) 1 for w~ w~andx(w) 0

for w ~‘ w~,wherethe cutoff frequencyw~is determinedby the propertiesof the material.Now one has~deaLwiththeregularizedquantity ~

~Ereg[.�~,5,x] = E (wk[I, S] — wk[.~:, O])X(wk), (3.5)

from which the Casimirenergyis obtainedby removingthe spacecutoff (movingI to infinity). At leastfor a conductingsurfaceS the Casimirenergyshouldbe the limit of 1~Ereg[I,S, x] whenboth the spacecutoff I and the high-frequencycutoff x are removed.Sucha limiting processhasbeenworkedout inseveralstudiesof theCasimireffect.However,only particularshapesfor I andx havebeenintroducedin connectionwith the consideredshapesfor S, for instancethe rectangularbox [1, 83], andthe sphere[37,48] for I, andan exponentialdecreasefor x. After detailedanalysisit turnsout in thesecasesthatthe resultsarecutoff-independent.

In order to makethe aboverepresentationmore specific let us considertheir application to theone-dimensionalversionof the conductingparallel-plateconfigurationand to the conductingsphericalshell configuration [37]. In the one-dimensionalCasimir configuration (see fig. 3.1) the systemcon-sideredconsistsof a large one-dimensionalbox of size L boundedby perfectly conducting“walls”which is divided into two regionsof length a and (L— a) by a perfectly conducting“plate”. Thisconfigurationenclosesthe electromagneticfield which is forced to satisfyperiodic boundaryconditionson the surfaces.The Casimirenergyis obtainedas the differencein zero-pointenergycorrespondingtotheconfigurationswherethe “plate” is placedat a given distancea ‘~ L andthat wherethedistancea islarge, say L/2. The zero-pointenergyof the electromagneticfield inside the region (I) is given byE1(a) = ~ mr/a. Obviously also the infinitesimal change ~E1(a)= ~ (nir/a

2) ~a is formallydivergent.Thereforeone has to introducea regularizationscheme.The simplestone employsa spacecutoff (box L) andan exponentialfrequencycutoff exp(—Anir/a). The Casimirenergyfollows accordingto the subtractionof zero-pointenergies(seefig. 3.1):

Ec(a)= lim lim {(E1(a, A)+ EIJ(L — a,A)) — 2E511(L/2,A)}. (3.6)

L-=~A—.O

E11 — ~ Em Epj

Q [-a Liq L-Lh1

Fig. 3.1. Subtractionof zero-pointenergiesin thecaseof theone-dimensionalversionof the Casimireffect.

G. P!unien eta!., TheCasimir effect 109

The zero-pointenergiesof thefield inside regions(II) and(III) areobtainedby a simple replacementof

the length parametera in E1(a, A). The regularizedzero-pointenergyinside the region (I) is given by

E~(a,A) = ±.-~-~exp(—A~)= —-f- [i_ exp(_-A ~)]= +—exp(—A—~)[1—exp(—A—~l . (3.7a)

a \ alL \ a/i

In the limit A —~0 thisenergycontributionbehaveslike

E1(a, A) = a/rrA2 — ii-/12a + C(A2). (3.7b)

Performing the cutoff removal A —+0 and L —~~, the Casimirenergyof the one-dimensionalversion ofthe parallel-plateconfigurationturnsout to be

E~(a) —ir/12a. (3.8)

The same techniquescan be applied in order to calculate the Casimir energyof the conductingsphericalshell configuration. In this casethe systemconsistsof a largeconductingsphereof radiusRenclosing the quantizationvolume, which concentricallysurroundsa small sphereof variable radiusa ~ R (see fig. 3.2). The correspondingenergydifference, eq. (3.4), is the changein the zero-pointenergyof the total systemwhen the radiusof the inner spheregrowsfrom radiusa to anotherradiusR/~(with a fixed value ~ > 1). The subtractionof zero-pointenergiesreferring to the different regionswill be performedas indicatedin fig. 3.2:

z~E(a,R)= (E1(a)+E11(a,R))—(E111(R/,~)+E1~(Rhj,R)). (3.9)

Sinceeverysingle contributionin eq. (3.9) diverges,one is forced to introducea suitablecutoff functionwhich vanishesfor large arguments,e.g., as before x(w) = exp(—Aw).The Casimirenergyis obtainedafter performingthe limit

E(a)= lim lim i~E(a, R, A), (3.10)A-.O R-+oo

andturns out to becutoff-independent(eq. (2.63)).

Fig.3.2. Subtractionof zero-pointenergiesfor theconductingsphericalshell configuration.

110 G. P!unien eta!., The Casimireffect

EvaluatingCasimirenergiesin similarcasesaccordingto this method— for instancethe parallel-plateconfiguration for a massivescalar field in a finite rectangularbox [77]— one also hasto introduceahigh-frequencycutoff or someotherregularizationscheme.Difficulties arise in understandingthe roleof the cutoff frequencywhen it would appearin the final result, becausein thiscasethe regularizationmethod is not well-motivated on physical groundsand must be consideredas a formal techniquedesignedto produce a finite quantity. Fortunately, the cutoff removal can be performed and theresultingCasimirenergyis cutoff-independentalso in this case.

We adda few commentsconcerningthe applicability of the modesummationmethod.It hasalreadybeenmentionedthat the eigenmodeenergiesWk areobtainedby solving the field equationwith respectto the requiredboundaryconditions.The caseswhere analytical expressionsfor 10k can be obtained,and the summationand regularizationcan be performed,are severely limited when more complexgeometriesareconsidered.Apart from the enormousnumericaleffort associatedwith the applicationofthis evaluationmethod in general, further technicaldifficulties occur in connectionwith the requiredboundaryconditions. Such problemsexist inherently in the caseof confined spinor fields, where theboundaryconditionhasto be satisfiedsimultaneouslyfor both the upperandlower component,or forthe electromagneticfield whereone hasto distinguishbetweentransverseelectric (TE) and transversemagnetic (TM) modes.For the highly symmetricalsphericalshell configuration it is possibleto dividethe total zero-point energy into the contributions from TE and TM modes,respectively,becauseparticularstepsof thewholecalculationcan be performedanalytically. Therebyone recognizesthat theCasimirenergybecomesfinite (in the limit R —~~ andA —~0) becauseof a delicatebalancebetweenthecontributionsof the TE and TM modes.The sphericalshell calculation already indicates that theapplicationof the modesummationmethodin the caseof moregeneralconfigurationscan be expectedto be very cumbersomeand evenunsatisfactory,when guidancefrom analyticalcalculationsis absent.Numericalconvergenceis not a priori guaranteed.Consequentlyonemayarguethat the successof themodesummationmethod is limited to problems,wherethe evaluationis accomplishedeitheranalytic-ally or, at least,semi-analyticallyas in the caseof the sphericalshell. On the otherhand,it shouldbeemphasizedthat this method always may be used, in principle, to define the Casimir energy.Nevertheless,due to the severeproblemsencounteredin solving non-trivial examplesusingthe directsummationmethod,it is necessaryto look for alternativeformulations.

3.2. The local formulationand Greenfunction methods

Anotherfield theoreticalapproachfor studyingthe propertiesof the vacuumstartsfrom an analysisof the behaviourof local field quantities.For ourpurpose,theenergy—momentumtensorT’~”representsthe appropriatequantity: the integral overT00 representsthe total energy, the componentsT°~’arerelatedto the flow of energyandmomentum,andthe stresscomponentsT~are usefulto deducethemechanicalpropertiesof the vacuum. A local formulation implies the introductionof the energy—momentumtensorof the vacuumf9~accordingto eq. (2.56) in the form:

~ (0hi’~’t0)ar(0hI’~’b0)o, - (3.11)

i.e., the measurableenergydensity of the vacuum is defined as the difference betweenthat in theconstrainedfield configurationandthe onecorrespondingto the unconstrainedfield. Since space-timeintegralsoverparticularcomponentsarerelatedto direct observablequantities,in addition to eq. (3.11)one has to require certaininvariancepropertiesunder fundamentalsymmetry transformationof the

G. P!unieneta!., The Casi,nir effect 111

system.For instance,the vacuumenergyof a relativistic chargedfield shouldbe invariant underchargeconjugation.@~shouldalso be symmetricin its indicesbecauseit acts,in principle, as a gravitationalsource.

Equation(3.11) hasto beregardedas aformal definition, i.e., ~ is not necessarilywell-determinedby thisequation,as we shall see.The advantageof thelocal definition is that it permitsa differentpointof view anda deeperunderstandingof the natureof vacuumenergyandvacuumstress.It turnsout that9~is expressablein termsof field propagators.The occurrenceof quantumfield fluctuationsand theassociatedobservablevacuumeffectscan thusbe understoodfrom the modificationsin the propagationof virtual field quantaunderexternalconstraints.

In order to elucidatethis connection we shall now derive the relation between the vacuumexpectationvalue of the energy—momentumtensorand the propagatorfor two conformally invariantfield theories,viz., the electromagneticfield and the masslessscalarfield. Besidethe dynamicalfieldsthemselves,the energy—momentumtensoralso containsfirst-order derivativesof the field. Suchtermsmaybe constructedby meansof a suitabledifferential operatoractingon the time-orderedproductofthe field operatorsT(~°(x)~’~(x’)).The operatorfor the energy—momentumtensor then formallyfollows after performingthe Lorentz-covariantlimit x’ —~x. By taking the vacuumexpectationvalue therelationshipto the propagator,which is generallydefinedas

iG~°~’~(x,x’) = (01 T~’~(x)~k)(xl))I0) (3.12)

becomesclear.For aspecific example,let usconsiderthe energy—momentumtensorof thefree electromagneticfield

with the Lagrangian2= ~ which leadsto the field equations:(g”Ll— a~a~)A~= 0. We wantto relateits symmetricalenergymomentumtensor

T~~A= —F’~F~+ ~g~AFsF~uJ (3.13)

to the propagator.The first step is to derive an equationbetweenthe time-orderedproduct of theelectromagneticfield tensors

t~AP(X x’) = T(P’~(x)F~’(x’)) (3.14)

andthat of the vectorfields as

= TX~,AP;~$T(A~(X)AP(X~))_(n~5(x~—x~)[A’~(x),ft~’(x’)] — n”6(x~—x~)[A~’(x),P~’(x’)]),

(3.15a)

wheren” = (1, 0, 0, 0) denotesa time-like vectorandthe differentialoperatorexplicitly reads

= (g”~a~’ — ~ 9~)(g1~9~A— gAl3a’~~) (3.15b)

The secondterm in eq. (3.15a)arisesfrom the differentiation of the function ~9(x0— x~)due to the

presence of time ordering. In the derivation of eq. (3.15a) terms of the form g°’~6(x0—x~,)[A,~(x),Al3(x’)] havebeenomittedsincefor equaltimes

t5(xo— X~)[Aa(X),Al3(x’)] = 6(x0—Xt)[Aa(X, x0), Al3(x’, x0)] = 0. (3.16)

112 G. P!unieneta!., The Casimir effect

According to eqs. (3.15a)and (3.15b) and replacingall commutatorswhich appeartogetherwith atemporaldeltafunction by the correspondingequal-time commutators,the energy—momentumtensorfollows as

j~~A(x)= —{~1~ T(A,,(x)A~(x’))+ 2i(n”n” — ~g~A)~5(x — x’)}I~~, (3.17a)

with the abbreviation

A~~A;~~l3_p~A ;a$l ~A ,‘p;— T~~’~. 4g ~

The delta-functionterm in eq. (3.17a) arisesfrom explicit evaluationof equal-timecommutators.Aftertaking the vacuumexpectationvaluethe desiredrelationbetweenthe unperturbedenergy—momentumtensor(Of T~~A(x)I0)o andthe free photonpropagator

iDas(x— x’) = (01T(Aa(X)A~(X’))f0) (3.18)

is found to be

(Of I’~(x)fO)o= —i{~i~ — x’) + 2(n~~nA— ~ig~A)6(x — x’)}I~.~. (3.19)

On the right-handside of eq. (3.19) D,,~(x— x’) can be expressedin termsof the scalarGreenfunction

G0(x— x’), which in the Feynman-gaugeis simply relatedto the photonpropagatoraccordingto

D~$(x—x’)=g~l3GO(x—x’)=——-’-~g~ . , (3.20)

4ir (x—x) +137

leadingto the result

(Of t~(x)f0)0 = —i{r~Go(x — x’) + 2(n~n”— ~g”jô(x — x’)}f ~ (3.21a)

= ~ = 2(3~’a —~g~”8”a.~)- (3.21b)

The first term diverges like 1/(x — x’)4 as x’ approachesx. In Ø~the delta-function term will be

cancelledby the correspondingcounterpartof (O!T~”(x)f0)~.That the spaceintegralover (Of T”~’(x)f0)~actually yields the zero-pointenergy E

0[O] becomesclear by inserting an appropriateeigenfunctionrepresentationof Go(x — x’) andintegratingover space.

Introducing boundaries,the propagationis modified due to the surface interactions, and con-sequentlythis perturbsthe homogeneityof the correspondingpropagator.In addition, the propagatorG(x,x’) must now satisfy the appropriateboundaryconditions, for instance Dirichlet or Neumannconditions as in the caseof perfect conductors(G(x, x’) = 0 or n~V G(x,x’) = 0 on the surfaceS).Under theseconditions the differential operator r~, which relates(OIT’~IO)sto G(x, x’), is thesame.Thus the energy—momentumtensorof the vacuumfor the constrainedfield configurationcan bewritten as [49]

G. Plunieneta!., The Casirnireffect 113

~9~(x)= —i{r~.(G(x,x’)— G0(x — x’))}j~’=~. (3.22)

The above expressionlooks formally quite simple. The vacuum subtractionappearsnow as thedifferencebetweentheconstrainedandthefree propagators.For explicit evaluationsall onehasto do isto constructG(x, x’) for the consideredconfiguration,which is, however,not a trivial exercise.This isnot surprising in view of the mathematicaldifficulties alreadyencounteredin the context of the modesummationmethod.One cannotexpectto circumvent theseproblemssimply by choosinga differentcalculationalmethod.Except for somespecialcaseswith simple geometrythe exact evaluationof ~for an arbitraryconfigurationis of comparableintricacy. On the otherhand, the local methodhastheadvantagethat allows one to apply systematicapproximationschemesfor the constrainedpropagator.For instance,the asymptoticbehaviourof ~ in the vicinity of a smoothboundarycan be obtainedexplicitly by this method [49].

Brown and Maclay [35] were the first to usethe Greenfunction methodin calculatingthe vacuumstresstensorfor the Casimirconfiguration.By virtue of the highly symmetricalparallel-plateconfigura-tion. Theexactexpressionfor theconstrainedGreenfunction G(x, x’) can beconstructedin termsof aninfinite sum of imagesourcefunctions(seefig. 3.3). If thereare no boundariesin the electromagneticfield, only “direct” propagation(emissionof a photonat a space-timepoint x’ and absorptionat x)occurs.Introducingthe parallelplatesseparatedby a distancea, severaladditionalcontributionsoccurcorrespondingto certainreflectionson theplates(mirrors). Thesecontributionscanbe treatedasif theyarisefrom an infinite sequenceof alternatingsourcesplacedalongthe directionz’~= (0, 0,0, 1) normalto the plates,propagatingfreely from the imagepointsx’ + 2lz andi’ + 2alz (1 = 0, ±1,..., ±c~)towardx, wherei” = (x’°,x”, x’

2, —x’3). Accordingly, the exactGreenfunction Dap(X,x’) can be expressedasa sum of free Greenfunctions. After the explicit calculatibn the energy—momentumtensorof thevacuumappearsas

~9~(x)= —i28~’8” ~‘ G~(x— x’ — 2alz)I~=~,

wherethe prime indicatesthat the I = 0 term (“direct” vacuumcontribution) is excluded.Performingthesummationandthe differentiationsleadsto the result,eq. (2.62),alreadymentionedin the previoussection.It is interestingto notethat only contributionscorrespondingto an evennumberof reflectionsenterin ~ Lukosz[84,85] showsin detailfor Casimir’soriginal parallel-plateconfigurationthat thiscancellation with an odd number of reflections arisesfrom the particular symmetry of the Greenfunctionscorrespondingto the electric andthe magneticfield modes,respectively.

0 • I I ~

Fig. 3.3. Illustration of the image-sourceconstructionof theconstrainedGreenfunctionfor theparallel-plateconfiguration.

114 G. Plunien et aL, The Casimir effect

3.2.1. Casimirstressbetweenperfectlyconductingparallel platesIn order to makethe previousconsiderationsmorespecific and to give an examplefor a successful

applicationof the Greenfunction method, let us now presentthe calculation of the Casimir effectbetweeninfinite platesin somedetail. It hasalreadybeenmentionedthat theenergy—momentumtensorcan be derivedfrom the Greenfunction

~ x’) = (O~F~’(x,x’)!O) = (OJTP~”’(x)E~(x’)JO). (3.23)

ThisGreenfunction itself can be constructedby meansof a suitabledifferential operatorwhich actsonthe photonpropagatorjDa~~(x, x’) = (O~T(Aa(x)A~(x’))jO). In Feynmangauge, where the photonpro-pagatoris simply relatedto the scalarGreenfunction, onederivesthe relations

G~~AP(X,x’)~.5~APG(X x’), (3.24a)

~5~~AP = g~’P~91~31A— g#~P3~9lA + g~’8~9”— g1~9~~a’~. (3.24b)

The notation = in eq. (3.24a) indicatesthat the equal-timecommutatorsareomitted (seeeq. (3.15a)).

Theenergy—momentumtensoris thengiven by

(Of k(x)IO)s~(i){G~,(x,~ ~ , (3.25)

omitting gauge-dependentdelta-functionterms, which arise from the evaluationof commutators,andwhichcancelin ~ The exactGreenfunction can be constructedin termsof an infinite sum over freeGreenfunctionscorrespondingto imagesourcesof alternatingsign. As illustrated in fig. 3.3, onehastodeal with two typesof images.Thosewhich carry a positivesign areplacedat positionsx’ + 2alz. Theycorrespondto those signalspropagatingfrom space-timepoint x’ to x, where an even numberofreflectionsbetweenthe plates has takenplace. The Greenfunction referring to an evennumber ofreflections(omittingdelta-functionterms) is given by

G~(x, x’) ~ ~ G(x - x’- 2alz). (3.26)

In orderto derive the contributionfrom an odd numberof reflections~ which refersto negativesourcesplaced at i’ + 2alz, one hasto considerthat in eachsingle propagationfunction Da~(X— X’ —

2alz) the normalcomponentof the potentialchangessign, i.e.,

+ 2alz)= (g~+ 2z0~z‘~)A13(f+ 2alz) ~‘

3A13(f+ 2alz), (3.27a)

with

(3.27b)

It wasconvenientto introducethemodified tensor~ which generatesthe transformationfrom normalvariablesto thereflectedones:x’~—* ~ andA’~—~N’. In view of the property ~ = â’~the field A”remainsunchangedfor an evennumberof reflectionsand has to be replacedby the transformedfield


(eq. (3.27a))if an odd numberof reflectionshastakenplace.Thecorrespondingpart of the total Greenfunction G”” Ap takesthe form

G~°(x,x’) — r~’~’~~ ~ (i) <OfT(Aa(X)A13(1’ + 2alz))fO)

= — ô~AP G(x-1’— 2alz), (3.28)

where the tilde indicatesthat instead of the normal Minkowski metric the modified tensor~“~‘ nowenters in the differential operator(3.24b). The total Green function describingphoton propagationbetweenparallelplatesfollows aftersubtractionof the free vacuumcontribution,given by the term with/=0:

W”~’AP(x, x’) ~ 6~”~G(x— x’ — 2alz)— ~ ~ G(x- x’ - 2alz). (3.29)

The boundaryconditionsfor the electromagneticfield require that the tangentialcomponentsof theelectric field and the normal componentof the magneticfield must be zeroat the conductorsurface;J~

11= J~12= F~2= 0 on the plates.One can prove that the correspondingcomponentsof w””~AP(x, x’)

vani~hi~iaccordanc~WiThTheSOrid tions.The eflfgy—momentumeIi~ö11~flOWObfãi1iëdãS - -- -

,n~”i \—~ns” I \.~a~”

‘-‘ vac~,Xj — ‘—‘ vac; event,X) ‘—‘ vac; odd X

= (_i){r~ ~ G(x- x’ — 2alz)} — (~ {~“~‘~ G(x— i’ - 2alz)} . (3.30)

Explicit evaluationshowsthat eachterm contributingto i9~,odd vanishes:

{r~”~~G(x— i’— 2alz)}f~...~,= {[~g””s9~c9’+g~Afza~aF

+ 2z”z”3~8,—2Z~Z~0V8IA— 2z”z~a”a,]G(x — i’ — 2alz)}f,,~

= — ~ (2x3 — 2al)4(—32z”z”z~z,,.z”z~

+ ~ + 16z’~z~z”z,,~)= 0. (3.31)

Accordingly, the energy—momentumtensorof the constrainedelectromagneticvacuumonly containsthe contributionfrom an evennumberof reflections.Oneobtainsexplicitly,

@~(x)=—-~ j ~ ~ _4(x— xl_2a1zY(x_x~_2a1z)A}~~=_~ (x — x — 2alz) (x — x — 2alz)

= — 24{~g~A + z”z~}~ -~. (3.32)2ira I~1l

116 G.P!unienetaL, TheCasimireffect

Performing-the-infinite-summation,oneis ledto the finairesult,eq~(262) -Thusfortheparalle[-pIateconfiguration one is able to derive the energy—momentumtensorof -the- constrained-vacuum—in-analyticalform. This is feasiblebecauseof the simplicity of theboundarygeometry,whichallows onetoconstruct the exact Greenfunction in terms of free image functions, since no retardationeffectsappear.

3.2.2. Relationshipbetweenthe local formulation and themethodof modesummationIn the previoussection we have shown that the definition of the vacuum energyin terms of a

differencebetweeninfinite summationsof eigenmodeenergiesreferring to the constrainedandthe freevacuumconfiguration,respectively,ina incaLversionflndsJts~counterpart,in terms of a differencebetweenthe energymomentumtensorsfor correspondingfield configurations.For the electromagneticfield we show nowexplicitly thatboth formaldefinitionsof thevacuumenergyareequivalent.We assumethat the boundaryconditionsdo not dependexplicitly on time. Such a caseis realized, e.g., for theelectromagneticfield inside a static cavity of volume V enclosedby a perfectly conductingsurfaceS.Accordingly, the photon propagatoris homogeneousin time, i.e., D4~13(x,x’) = DaJ~(X,x’, i-) where

= — 4. The sameis alsovalid for the scalarGreenfunction.In orderto obtaintheeigenfrequencies~

0k[S] of the constrainedelectromagneticfield, theboundaryvalueproblemwith respectto the boundaryconditionsn HI

5 = 0 and ii X El5=0 hastube~ol-sred.Atcordirrgtu D1 ,i1iwch~nd13or~iiis[86,871 this -

problem reducesto two scalar boundaryvalue problemsfor electric [E) andmagnetic (H) modes,respectively,which can be treatedseparately:

~~E.H)(x x0) = 0. (3.33)

With respectto theboundaryconditions(seeeqs.(3.35b) and(3.35c))onedeterminesthenormalmodes

~ ~. H)(~ x0) = ~ ~,E.H)(x) exp(—iw~’H)[S]Xo), (3.34)

wherethe p ~‘ H)(x) aresolutionsof the scalarHelmholtzequation

(zl + o~’~~’~(x) = 0, (3.35a)

andsatisfy the boundaryconditions:

= 0 (Dirichlet), (3.35b)

pj. V~t’’~(x)f~= 0 (von Neumann). (3.35c)

They alsoform a completeset of orthogonaleigenfunctions:

~ Q~ (x)p~E~~I)(x)~ 6(x— x’), (3.36a)k

J dsX 1~(~)E.1~(~)= 6kk’. (3.36b)


Obviouslyonecan definetwo scalarGreenfunctionswhich satisfy the equation

L~G~’H)(x x’, r) = 5(x — x’). (3.37)

For brevity the indices (E), (H) and S indicating the mode type and the boundarydependenceareomittedfrom now on. ThescalarGreenfunction can beexpressedin termsof eigenfunctions,eq. (3.34),obeyingthe Feynmancondition

* , exp(—iw~frf)G(x,x, T) = I ~, cc’k(X)cvk(X) . (3.38)k ZWk

Turning backto our purpose,we startfrom the equationfor the energy—momentumtensor

(Of ~I”‘~(x)f0) = —2i{(~9”3’~’— kg” 9”~9,~)G(x, x’, r) + (n”n~’ — ~g”~)ô(x— x’)}I~~. (3.39)

Owing to the homogeneityof the Greenfunction and eq. (3.37) the vacuumenergydensity can bewritten as

(Of i’°°(x)JO)= —i{(~ô’91— ~a19~)G(x,x’, ~ - (3.40)

Insertingthe eigenfunctionexpansion(3.38) andperformingthe limit x’ —s x oneobtainsthe expression

(Of i00(x)IO) = ~ w~(pk(x)(p~(x) — ~ —~— 8 [(ô,ço,,(x)q~~(x)]. (3.41)

The energyfollows after spatialintegration:

E = J d~x(OIT0°(x)IO)= — — J dI18~ok(x)co~(x). (3.42)

Oneseesthat the first term of eq. (3.42) is alreadyof the form which we haveexpected.The surfaceterm,in fact, vanishesin view of therequiredboundaryconditionseqs. (3.35b)and(3.35c). So what oneobtainsis in fact the mode sumfor the zero-pointenergyof the electromagneticfield, separatelyfor(E)- and (H)-type modes.For several technicalquestionsconnectedwith the proper identificationofthesemodeswe refer to references[84, 87]. Thus we haveexplicitly shown that in the caseof theelectromagneticfield the definition of vacuumenergyintroducedby the modesummationmethod isexactly reproducedby the local definition basedon ~ i.e., onecan conclude:

EvacES]= Jd3x (Of t°°(x)JO)5— f d

3x (Of i’°°(x)f0)0

(3.43)

= ~Wk[S]~ Wk[O].

118 G. P!unienet aL, The Casimireffect

The fact that the definitions are equivalentmust be takenwith caution, becauseboth formulationsrequire inherentlydifferent regularizationschemes.Thus, onehasin generalto prove whetherthe finalresults for Evac obtainedwithin theseformulations are in fact identical and independentfrom theparticularregularizationprocedureapplied.

The fact that ~ is finite for the Casimir configuration in the electromagneticfield does notrepresentits generalbehaviour.This becomesclear when one derivesthe vacuumstresson arbitrarysmooth boundariesor its contribution to the surface tension. In the vicinity of a plane perfectconductor,the energy—momentumtensorof the vacuumdivergeswith the inversefourth powerof thedistancee from the boundary, i.e., �7~-(1/e4)�”~4~+finiteterms. Near a slightly curved, smoothboundarythe asymptoticexpansionfor �~ is found to be of the form [49]ø~-~ ~i (1/e’)�~”’~°.Thecoefficients 9”~dependupon the shapeof the boundary,and they are purely local functionsof itsgeometry.Accordingly, the part of 9~contributingto the surfacetensionis also in generaldivergentnearthe boundary.This divergenceis not of the typewhich can beremovedby usualsubtractionsas intheparallel-plateconfiguration. It is real in the sensethat it originatesin the unphysicalidealizationofperfect conductorboundaryconditions. This gives reasonfor a careful analysisof how the physicalvacuumenergyshouldbe definedandhowit is relatedto both the modesum energyE~”~(eq. (3.4))andto theresultof the local formulationE~V’~~= .f d3xOC. The mode-sumenergycan be expressedasan integralover mode-numberdensities37(w):

~ = 1 Jdww(375(w)— flo(w)). (3.44)

First of all bothE~’~andE~~d)arein generalinfinite andthereforecannotdirectlybeidentified withthe physical vacuum energy. The divergenceof E~’~ arises from the high frequency modes.According to what we have already mentionedin section 3.1, when one takes into account thetransparencyof real conductors,this implies the introductionof a high-frequencycutoff x(w) thatmakes the expression(3.44) finite. Under thesecircumstancesE~”~should representthe physicalvacuumenergyand thusalsothe quantity, eq. (3.44), shouldin somesenserepresentthe limit of thisphysical vacuumenergywhen the cutoff frequencytends to infinity. Particularly in the caseof theCasimirconfigurationit turns out that the result for ~ is cutoff-independentand identical withEo0~~var

This fact alonedoesnot justify that both aresaidto be equalin general.DeutschandCandelas[49]havepointed out the fact that “renormalization”, as necessaryin the caseof deriving ~ in thepresenceof boundaries,doesnot generallycorrespondto theremovalof purely local divergenciesin theenergy—momentumtensor.This implies that performing the volume integral over 0~C to obtain thetotal energydoesnot generallylead to the sameresult as one would obtainby a “renormalization”,indicatedin the following relation:

~ = [Jd3x (Of T°°fO)~—Jd3x (Of T001o)o] ~ = J d3x ~ (3.45)

ren

The proof of the equivalenceof both energiesas well as the decision whether~ or E~::~~bestcorrespondsto the physical vacuumenergyeven at high conductivity can only be given after explicitcalculation.This implies that one has to be careful with the identification of quantitiescalculatedbydifferentmethods.

G. Plun!enetaL, The Casim!reffect 119

However,for particularconfigurationsof perfectconductors(parallelplates,sphericalshell) identicalresultsfor thevacuumenergyhavebeenobtainedby differentmethods.Thisgivesconfidencein thesetech-niquesandin the physicalreality of thequantitiesobtainedby them.In spiteof the inherentuncertaintyin identifying E~~O(asit stands)with the physicalvacuumenergy,it seemsreasonableto takeeq. (3.11)as a guidefor studyingvacuumenergiesof otherquantumfields underconstraints.For the electron—positronfield interactingwith a classicalelectric field, we will derivea relationanalogousto eq. (3.22)that leads to an expressionfor the correspondingvacuum energy which has a simple physicalinterpretation.

We now turn to explain furtherGreenfunction techniquesemployedfor calculatingvacuumenergiesandthe relationbetweenthem.

3.2.3. The local formulationin terms ofthe electricand magneticGreenfunctionsAt the beginning of this subsectionthe two-point function r””~AP(x x’) was introducedin eq. (3.14)

andusedto derive the relationbetweenthe energy—momentumtensorandthetime-orderedproductofthe electromagneticfield operators.In this way an expressionfor f9~of the constrainedelectromag-netic field was obtainedin terms of the difference between two-photonpropagators.The energy—momentumtensorof thevacuum,asit is definedby eq. (3.22),doesnot representtheonly possiblelocalformulation,as we will shownow. For this purposewecombineeqs. (3.14) and(3.15a),

iG””~°(x,x’) = (Of T(F”~(x)F5”(x’))f0)

= ~~i’;AP. aP(OfT(Aa(X)A13(x’))JO) — n”ô(x0 — x/)(Of[A”(x), P~”~(x’)]f0)

+ n”~(xo— x~)(Oj[A”(x), PAP(xl)]IO) (3.46)

andwe write explicitly the equationfor the photonpropagator

(ô~E — 8”o~)(OfT(A”(x)A~’(x’))fO)= ig”~(x— x’). (3.47)

We further makeuse of the fact that every commutatorwhich appearstogetherwith a temporaldeltafunction reducesto the correspondingequal-timecommutator.

The (dyadic) Greenfunctionsfor the physicalfields, i.e. the electricandthe magneticfield, can beintroducedand ~ maythen be expressedin termsof theseGreenfunctions.We define the electricandthe magneticGreenfunctionsaccordingto

iF”t(x, x’) = (Of T(E”(x)E’(x’))f 0), (3.48)

iJ~”(x,x’) = (OjT($k(x)fr(xF))fO). (3.49)

Consideringthe quantitiesG°”°”(x,x’) one recognizesthat only the spatialcomponentsarenon-zero,andtheycoincidewith the electricGreenfunction

GOk;OI(X, x’) = (Of T(PTh(x)ft01(xl))fO)

= i[~1 (x, x’). (3.50)

By looking at eq. (3.46) onerecognizesthat .1” can be expressedas

120 G. P!unienetal., The Casimireffect

iF”(x, x’) = ir°~7’~13D~13(x,x’)— i6~ô(x— x’). (3.51)

It can be provedthat the Greenfunction 1M is divergence-free

l9kFkl(x, x’) = 0, (3.52)

which is a direct consequenceof Gauss’law V~E = 0 for the source-freeelectromagneticfield. In order

to derive the magneticGreenfunction we considerthe purelyspatial componentsof G”~’ Ap which readi G~’“(x, x’) = 8~0I~ (Of T(A’(x)A”(x’))f 0) — 813,~~~(Of T(Ai (x)A~’(x’))fO)

+ a’a”(of T(Ai (x)A (x’))f0) — 8’a”(Of T(A1(x)Am(x’))IO). (3.53)

Contractingwith the totally antisymmetricalLevi—Civita tensorwe obtainthe identity

~ ‘\_i ij;mn(I’t’ ~X, X ~ — 4lEijkEmnl ~X, X

= EijkEmnIl9l9(OIT(A’(X)A(X))IO). (3.54)

The magneticGreenfunction alsohaszerodivergence

3kt~I3”(x,x’) = 0, (3.55)

due to the Maxwell equation V~B = 0. We now haveexplicitly obtainedthe electric and magneticcontributionsof the two-point Green function and,equivalently, their relationsto the normal photonpropagator.By comparisonwith the expressionfor the energy—momentumtensor,eq. (3.19), we findthe following equationsfor the energydensityandthe stresscomponents

(OJi’°°(x)JO)= —~i(F~(x,x)+ P~(x,x)), (3.56a)

(Of i~1k(x)fO) = ~i[F~ (x, x) + ~~J’Y”(x, x) — ~8°’(~(x,x) + f’~(x,x))] - (3.56b)

In the presenceof conductingsurfaces5, boundaryconditionsare requiredfor F”1 and ~‘. The normal

componentsof t~~”’and the tangentialcomponentsof I” must vanish on S. In order to derive thedesiredenergy—momentumtensorof the vacuumø~,onehasto changeto the GreenfunctionsF~and ‘~C~ where the correspondingfree vacuum counterparts are subtracted,i.e., F~C= F~’— TO”’.

Inserting this into eqs. (3.56a) and (3.56b)yields the componentsof ~ It should be notedthat it issufficient to know F”, since the quantities‘ii”’ are relatedto the electricGreenfunction accordingto

~o

8lO.pkl(~ x’) = Eij~Emnil9~49’

mF’~(x,x’)+ (ö”’3’3 + 8”8’)~(x — x’), (3.57)

which can be directly verified. A similar relationholds for the electricGreenfunction:

8°8’°F”(x, x’) = ~ijkEmnll9ö (P~”(x,x’) + (~“d~d~+0kaI),~(x — x’). (3.58)

Thestructureof theseequationsis derivedfrom the Maxwell equationsV x B = t9°Eand V X E = — 8°B.


To be complete,we notethat the GreenfunctionsF” and P” satisfythe samesecond-orderdifferentialequations:

LJFkt(x x’) = (3k13i3. + ~9”9’)5(x— x’), (3.59a)

L1~”(x,x’) = — (3”8~0,+ 8”i9’)3(x — x’). (3.59b)

If oneis only interestedin the Casimirforce, it is convenientto considerthe infinitesimal changein thevacuumenergy inducedby a changein the parameterswhich determinethe conductorconfiguration(energy-variationmethod):

~Evac = J d3x3(OfT°°(x)f0),

= -~iJ d3x (~T~k(x, x) + ~~k(X, x)). (3.60)

The Casimirforce perunit areacan alsobe obtainedfrom the spatialcomponents�1~J~.~Cwhich describethe mechanicalstressof the electromagneticvacuumon the conductingsurface(stress-tensormethod).This version of the local formulationhasbeenapplied by several authors[44,45, 88]. Although theydiffer slightly in their definition of the electric and magneticGreen functions, their resultsfor theCasimirforceareidentical.

SinceF1” and ~j.~kl satisfydifferentialequationsof the samestructureas Maxwell’s equationsfor theelectric and the magnetic field, the generalizationof the Casimir concept to caseswhere a homo-geneouslypolarizablemedium is placed in the vacuumis also possible.The vacuum energycan beformulated in terms of generalizedGreen functions iF” = (Of T(D”(x)E’(x’))JO) and i’P”(x, x’) =

(Of T(H”(x)B’(x’))f 0). The dielectricconstants~and the magneticpermeability~t aredeterminedby thegeometryor the relativepositionof polarizablemacroscopicobjects.In the caseof homogeneousmediathe infinitesimalchangein the vacuumenergythenis given by

~Evac = —~iJd~x(~st~+ 6i~~), (3.61)

wherecontinuityof the normalcomponentsof ~Ji”andfor the tangentialcomponentsof F” is required.This follows from the behaviourof the correspondingelectric and magneticfield componentsat theboundarybetweenthe mediumand the vacuum.

3.3. Themultiple-scatteringexpansionfor Greenfunctionsand the Casimirenergy

We turn now to explain the underlying idea of the multiple-scatteringexpansionwhich hasbeendevelopedby Balian andDuplantier [41, 42] (see also [89—92])in order to study the behaviourofelectromagneticwavesnearperfectconductors.It hasalreadybeenmentionedthat an exactevaluationof thevacuumenergyfails in generalconfigurations,sincethe constrainedGreenfunctionsF” and I~’are not analytically known.The multiple-scatteringexpansionallows for the generationof approximateexpressionsfor the constrainedmagneticandelectricGreenfunctionsby iterationof integralequations


for theseGreenfunctions[41].In principle, onecouldusethe expressionsinvolving F”’ and1” derivedabovein the contextof thestress-tensoror the energy-variationmethodin order to calculatetheCasimirforce. Balian and Duplantierwere inspired by the mode-summationmethodand obtain the Casimirenergyby deriving the eigenmodedensity in the presenceof boundaries.Via dispersionrelationstheeigenmodedensityis relatedto a so-called“mode generatingfunction” which can beexpressedin termsof the electricandmagneticGreenfunctions.

Let us first discussthe conceptof the multiple-scatteringexpansionfor the Greenfunctions.Weconsiderthe electromagneticfield in a region that is boundedby a perfectly conductingwall of arbitraryshape.In thepresenceof theconductingwall, free propagationof electromagneticwavesis disturbedbysurfaceinteractions. The exact Greenfunction which describesthe modified propagationcould beconstructedfrom termseachof which correspondsto a certainnumberof localizedsurfaceinteractions.In otherwords, a single termin this expansionrepresentsa processfor which the wavescattersseveraltimesoff the boundaryandpropagatesfreely in between(seefig. 3.4).The multiple-scatteringexpansioncan be applied for the magneticas well as the electric Green function. When the magneticGreenfunction is used,the multiple-scatteringexpansionhas a direct physical interpretation in terms ofinduced currents flowing on the boundary. The exact Green function can be divided into a partdescribingfree propagation,and the surfaceGreenfunction which containsthe interactionwith thesesurfacecurrents.The constructionof the surfacecontributionsthereforereducesto the determinationofthe inducedcurrents.For thesecurrentsan integralequationcan be derivedwhich maybe solvedbyiteration.Eachterm in this expansioncorrespondsto a certainnumberof scatteringprocesses,whichcanbe interpretedin the following way: The free magneticGreenfunction ~‘ representsthe magneticfield producedby an oscillating dipole of unit strengthJo. In the presenceof a boundarythis sourceinducesa primary surfacecurrentj~,which actsas the sourceof the scatteredwave (single-scatteringprocess).This in turn inducesa secondarysurfacecurrent which is thesourceof the nextscatteredwave(double-scatteringprocess),andso on. A formally analogousexpansionfor the electricGreenfunctionis also possible.The full geometryof conductingboundariesentersin the Greenfunction.

It is suggestiveto comparethe multiple-scatteringexpansionwith the semiclassicalapproximationofray optics, in which the ray propagatesfreely betweenany numberof reflectionson the surface.Thesetwo expansionscan be distinguishedin the way they describe a touching of the surface. As an

çxiçX2

Fig. 3.4. Illustration of themultiple-scatteringexpansion:(a) single-scatteringand (b) double-scatteringprocesses.Free propagationtakesplacebetweenthesourcepointx’, thescatteringpointsx

1 andx2, andthepoint x.

G. P!unien et aL, The Casimireffect 123

idealization, the interactions with the boundary are describedas pure mirror reflections in theframework of ray optics. The multiple-scatteringexpansion,in principle, representsan exacttreatmentof the full wave problem.Thesurfaceinteractionis treatedin sucha way that eachsurfacepoint actsasa sourceof an elementarywave (in analogyto Huygens’principle) andray optics is recoveredas thehigh-frequencylimit. In manycasesthe mirror reflection representsthe dominantcontribution in themultiple-scatteringexpansion,but additionally the ray may bescatteredoff any point on the surfaceinan arbitrarydirection.

It should be emphasizedthat the multiple-scatteringexpansiondevelopedby Balian, Bloch andDuplantier[89—92,41, 42] is similarly applicablefor deriving propagatorsof other constrainedquantumfields. HanssonandJaffe [93,941 haveapplied this methodto QCD in a cavity wheretheyused thesetechniquesto derive a systematicexpansionfor cavity propagatorsin order to calculate,e.g., theself-energyof quarksconfinedin a sphericalcavity.

3.3.1. Multiple-scatteringexpansionof the electricand magneticGreenfunctionWe turn now to the derivationof the integralequationfor magneticand electricGreenfunctions~J”

andF”’ given by Balian andDuplantier[41].As alreadymentionedabove(eqs.(3.59a)and(3.59b))thedefining equationsfor the Fourier-transformedcti”(x, x’, w) are

(LI + w2)~”= ~ (3.62a)

= 0. (3.62b)

For reasonsof brevity andtransparencyin thefollowing we suppressthetensorindicesandmakeuse ofthe dyadic notation, e.g. Cu” ~P.The required boundaryconditions for the normal and transversecomponentson the surface5, are (Cu)

0=0 and (Vx P~j,= 0 for x on S. Balian and Duplantier haveidentified j~’(x,x’) = ~mnri9t5(X— x’)6~’an Jo as the current densitydistribution at the position x of theunit magneticdipole sourcewith the direction indicated by the index r which is locatedat x’ andorientedalong the direction indicated by the index 1. The magnetic Green function describesthemagneticfield (the index k refers to its direction)producedby the dipole. The constrainedmagneticGreenfunction can be divided into the free and the surfacepart, i.e. 1i = Cu~+ Cu~,which satisfy theequations

(LI+w2)~orr_Vxj

0, (LI+-w2)Cu.~O.

The current j0(x, x’) inducesa surfacecurrent Js(xi, x’) which dependsupon the surfacepoint Xi

andthe position x’ (fixed parameter)of the source.The total vectorpotentialA follows as the solutionof the defining equation

(LI+w2)A—(J

0+ Js), (3.63)

which is formally given by

A(x, x’) = J d3x” G

0(x, x”)j0(x”, x’)+ J do-1 G0(x, Xi) js(Xi, x’), (3.64)


whereG0(x, x’) denotesthe scalarGreenfunction of eq. (3.63). The magneticGreenfunction t~is givenby

i~P(x,x’)=VXA(x,x’)

= Cu)0(x, x’)+J do-1 Vx (G0(x, x1)j5(x1,x’)). (3.65)

The surfacecurrentcan bededucedfrom the discontinuityof ‘I acrossthe surfaceandits valueon thesurface[41]

j5(x1, x’) = 2n1 x cP(x1,x’). (3.66)

n1 denotesthe inwardly oriented,~nitnormalvectoron S at the surfacepointx1, andx’ denotesa pointin the interior of S. Expressing P(x1, x’) accordingto eq. (3.65) leadsto the integral equationfor thesurfacecurrent

Is(x1,x’) = j1(x1, x’) + 2J do-2n1 x (V1 x (G0(x,,x2)j5(x2, x’))), (3.67)

with J~ x’) = 2n1 x cb0(x1,x’). By iteration onegeneratesthe currentup to an arbitraryorder, i.e.,

J~(x1,x’) = j1(x1,x’)+ 2 J do-2 n, X (V1 X (G0(x1, x2)j1(x2,x’)))

+ 4Jdo-2 do-3 n1 X (V1 + (G0(x1, x2)n2 x (V2 x (G0(x2, X3)75(X3, x’))))) ~ (3.68)

As discussedabove,eachtermof this expansionhasa naturalinterpretation.Whenwe inserteq. (3.66)into eq. (3.65)we arrive at an integralequationfor the exactGreenfunction Cu~:

P(x, x’)= Cu(x, x’)+ 2J do-1 Vx (G0(x, x1)n1 X P(x1, x’)), (3.69)

which maybeiteratedaccordingto the sameschemeasthe current j~.In order to simplify thenotationit is convenientto introducethe following integralkernels:

Kan2n1X(V1G0(x1x2)X)and M= VG0(x,x,)x.

The expression

MJ5_=J do-,VG0(x,x,)X j5(x,)

involves the performanceof surfaceintegrations.In this short-handnotation the expansionsfor the


surfacecurrentand the magneticGreenfunction can be written as

j=j1+Kj1+K2j

1+”~ (3.70a)

and

u= ‘P+Mj1+MKj1+MK2j

1+~~~ (3.7Ob)

Note that terms which carry an odd power of K correspondto a processwith an evennumberofscatterings.

Equations(3.70a) and (3.7Ob)arevalid when both points, x andx’, arelocatedinside of S. Due tothe symmetryof the magneticGreenfunction Cu(x, x’) =~,Cu’(x’,x) and the structureof the discontinuityacrossthe surface,analogousexpressionsfor Cu~and J follow immediately for the casewhen x’ isoutsideS. Only the right-handsideof eq. (3.66)changessign and,consequently,successivetermsof theresultingexpansions(3.7Oa,b) areof alternatingsign.

For the electricGreenfunction similar equationscan be obtainedby an analogoustreatment.This ispossiblebecauseF(x, x’, cv) satisfiesthe samedifferentialequations(seealsoeq. (3.59a)):

(ii+w2)F=—VX Jo (3.71a)

V~~=O. (3.71b)

The boundaryconditionsare now (F)5 = 0 and (Vx F),, = 0 for x on the surface S. The resulting

equations,which arevalid for x and x’ both inside the boundedregion,are

F(x, x’) = P0(x, x’) + Jdo~V x (G0(x, x1) gs(xi,x’)) (3.72)

with the current

gs(Xi, x’) = — j1(x1, x’)— 2J do-2 n1 x (V1 x (G0(x1,x2) gs(x2,x’))). (3.73)

The expansionfor g obtainedby iterating eq. (3.73) differs from that for the current J (eq. (3.70a))only in sign. Thus the correspondingexpansionof the electricGreenfunction is also of oppositesign,i.e.,odd numberscatteringprocessescarry a negativesign:

i=~0-MJ1+MKJ1-MK2J

1+~~•. (3.74)

3.3.2. Relationbetweenthe Greenfunction, the eigenmodedensityand the modegeneratingfunctionWe now come to the relation betweenthe Green functions and the eigenmodedensity of the

constrainedelectromagneticfield. After this has been worked out, we shall turn directly to thecalculationof the Casimirenergy.

The intermediatequantity betweenthe Greenfunctionsandthe eigenmodedensity is the so-called

126 G. P!unien et a!., The Casimireffect

mode generatingfunction. The meaning of this function and its construction becomeobvious by

consideringthe spectralrepresentationof the magneticandthe electricGreenfunction.Equivalent to the knowledgeof the magneticGreen function cP is that of the eigenmodesof the

magneticfield inside the cavity. The eigenmodesB,,(x, t) = B~(x)exp(—iw,,t) with real amplitudeB,,(x)satisfythe equations

(LI + w~)B,,= 0, (3.75a)

V~B,,=O. (3.75b)

The B~should be normalizedaccordingto

J d3xBfl(x)Bk(x)=~,,k. (3.76)

In order to derivean expressionfor Cu(x, x’, cv) onecan makethe ansatz

‘~b(x,x’, cv) = ~ B~(x)®Z~(x’,cv) (3.77)

anddeterminethe unknownvectorsZ,, by insertion in the defining equation(3.62a).

~ (cvi — w~)B~(x)®Z~(x’,cv) = — Vx j0(x, x’)

= — Vx (Vxô(x — x’) 1). (3.78a)

Projectionwith Bk(x) leadsto

(w~_w2)Z1’(x1,w)Jd3xB1’(x).Vx(Vx~(x—x’)1)

= J d3x{(B

1’(x)~ V)Vô(x—x’)—B1’(x)z.1t5(x—x’)}

= cv~B1’(x’) = —LI’B1’(x’), (3.78b)

and one can identify the spectralrepresentationof the magnetic Greenfunction (Feynmanboundaryconditionsimply a small imaginarypart ie in the denominator)as

P(x, x’, cv) = ~ cv~—w2B,,(x)ØB,,(x’). (3.79)

Let us now define the mode generatingfunction !P(cv) as the trace of the magnetic Green functiontakenat coincidentspatialpoints

= Tr[Cu)(x x, cv)] anJd3xcP~(x,x, cv). (3.80)

G. P!unieneta!., The Casimireffect 127

Inserting eq. (3.79) and making use of the orthogonality relation (3.76) one obtainsthe dispersion

relation

Tr[~)(cv)]= ~ cv~-w~= J dcv’ (~3(w’- cv~))w~-cv2

= J dcv’ p(w’) ~ (3.81)

Herep(cv) is the eigenmodedensity inside a finite cavity which is given by a sumoverdeltafunctionsat

theeigenmodeenergiescv,,:

~ ô(cv—cv,,). (3.82)

Using the fact that p(cv) is an analytic function in the upper half-planeand shifting the pole by an

amountof is, eq. (3.81) can be solvedby contourintegration.Onefinds

p(cv) = —~-- Im{Tr[Cu(cv + iE)]} = —~-- Im{~P(w+ is)}. (3.83)

This equationrelatesthe magneticmode density to the imaginary part of the trace of the magneticGreenfunction. Now the role of çli(w) as the modegeneratingfunction becomesobvious.

In order to obtain the total modedensityof the electromagneticfield, the electricfield modesmustalsobe takeninto account.It turnsout thatthe electricGreenfunction F hasaspectralrepresentationofsimilar form as eq. (3.79) since it satisfiesthe samedifferential equations.An analogouscalculationleadsto the total modegeneratingfunction

~P(cv)= ~(Tr[Cu(cv)]+ Tr[cP(cv)]), (3.84)

or if oneinsertsthe multiple-scatteringexpansions(3.7Ob) and(3.74) oneobtains

= Tr[Cu~o]+ Tr[MK2’~’J1]. (3.85)

This explicit expression for !P reveals the quite remarkable result that, basedon the particularsymmetrybehaviourbetweenthe electric and the magneticGreenfunctionsin the multiple-scatteringexpansion,terms referring to an odd numberof scatteringscancel.Only evennumbersof scatteringscontributeto the distributionof eigenmodes.

In this context it should alreadybe noted that this property of the multiple-scatteringexpansionfor~J’ has its counterpart in the perturbationexpansionof the exact Feynmanpropagatorof theelectron—positronfield interactingwith an externalelectromagneticfield. Thereone obtainsthe resultthat, accordingto Furry’s theorem,only termsof evenpowerin the electromagneticfield A~,contributeto the expansion.This correspondsexactlyto the situationwherea fermionscattersat an evennumberof space-timepoints(seealso section5).

128 G. P!unieneta!., The Casimireffect

One further remark to the above relations: The equality (3.84) between the mode generatingfunction and the trace of the Green functions has to be understoodin the sensethat irrelevantdivergenciesarising from the limiting process x’ -~ x rnust be subtracted.Such divergenciesarecontained in the first two terms, Tr[p0] and Tr[MK j~] of the expansioneq. (3.85) which areproportionalto (cv

2/Ix — x’f) andto 1/Ix — x’j, respectively[41]. Besidethat, p alsocontainscontributionsbehavingas w2 and 1 (volumeand curvaturecontribution).The relationbetweent/i(w) andp(cv) via adispersion relation assumesthat the mode density satisfies the condition p(cv)—~O for cv—~x.Toguaranteethis behaviourwhen necessary,dispersionrelation mustbe subtracted.This meansthat theequation

1 r cv’2dcv’p(cv’) ~2 2 (3.86)

ITW cv —cv0

defines!P only up to a real additiveconstantwhich mustbedeterminedfrom theboundaryconditionatthe limit cv -~ 0.

3.3.3. TheCasimirenergyNow we are in possessionof all quantitiesand relations required for the discussionof Casimir

energieswithin the multiple-scatteringexpansionmethod [42]. We limit the following representationonly to the conceptionalpointsnecessaryto understandthe evaluationof Casimir energies.Balian andDuplantierconsidera perfectly conductingshell S in theelectromagneticfield enclosedin alargebox ~,

the walls of which areassumedto be perfectly conductingas well. Both the box andthe shell canbe ofarbitrarysmoothshape.The introductionof the spacecutoff ~ is convenientin order to obtaindiscretesums for the zero-pointenergies.The interior of the box I without the conductorS is taken as a“referencevacuum”.TheCasimirenergyfor theconsideredsystemis formally definedasthe differencein zero-pointenergiesreferring to internal, externaland referencevacuummodes:

Evac= cv~[int.]+ ~ cvm[ext.] — ~ cv,,[ref.]}. (3.87)

Accordingly, the first term dependsonly upon S while the vacuumterm dependsonly on I and thesecondterm containsboth. Figure 3.5 illustrates the situation,e.g., for a sphericalshell in a box ofidenticalshape.The expression(3.87) can be rewrittenin termsof the densitiesp(cv)= ~,. ~(cv— cv,,) asthe following integral

Evac~Jdcvcv~(cv), (3.88)

wherethe measurej5 describesthe changeof the eigenmodedensityin the box I whenthe conductingshell S is introduced

j5(cv) = p(cv; int.)+ p(cv; ext.)—p(cv; ref.). (3.89)


bext.

it-it.

—~ Q ez

Fig. 3.5. Various typesof pathscontributingto the Casimirenergyfor theconductingsphericalshell S andwith aspacecutoff £.

Sincetheseformal expressionsarestill divergent,onehas to introducea high-frequencycutoff x(w) inorderto dealwith a well-defined integral:

E[S, I, x] = ~J dcvcvt5(cv)~(cv). (3.90)

For calculatingE[S, I, x] usecan nowbe madeof the connectionbetweenmodedensityandthe modegeneratingfunction

~5(cv)= -~-- Im{ ~!‘(cv)} (3.91)

andits multiple-scatteringrepresentation.Accordingly, the modegeneratingfunction is given by

l~(~) ~P(cv;int.)+ !P’(cv;ext.)— V’(w;ref.). (3.92)

Expressing#(cv) in terms of the multiple-scatteringexpansionof the magneticand electric Greenfunctions (eq. (3.85)) one can analyze the scatteringprocesseswhich may contribute to ~ Themultiple-scatteringexpansionfor ~Pinv~lvesclosedpathswith scatteringseitheron S or on I. Fromtheparticularsymmetryrelatingof ~PandF oneknowsthat the contributionsto the Greenfunctionswhichare representedby pathswith an oddnumberof scatteringscancel.Only pathsof evenordercontribute.Figure3.4 illustratesthe varioustypesof pathscorrespondingto double-scatteringprocesses.Obviouslythe contributiondescribedby the first term of eq. (3.85), Tr[Cu

0], cancelsin ~P.In order to get animpressionof the various higher-orderscatteringprocesseswhich can take place in the constrainedconfiguration, and to see which of them may contribute to ~P,let us consider the caseof double-scatteringprocesses.

130 G.Plunienel aL, The Casimireffect

Two catagoriesof pathscontributingto ~P(cv,ref.) can be distinguishedaccordingto the positionx ofthe source,which maybe inside S or betweenS andI (path (a) and(b) of fig. 3.4). Thesepathsinvolvescatteringson I only. The term ~P(cv;mt.) containspathsreferring to the sourcepoint x inside S andwhich involve bothscatteringson S andI such as paths(a, c, d). Finally in the exteriorregion processesof type (b,e,f) can take place.Accordingly, the contributionscorrespondingto 1I’(cv; ref.), e.g. paths(a), (b) cancel completely with contributionsto 1P(cv;int) and ~P(cv;ext.). Consequently,the possibledouble-scatteringprocessescontributing to V’(cv) are representedby the paths(c, d, e,f). Higher-orderscatteringprocesseswhich contributeto ~P(cv)could beclassifiedin an analogousway. We thusseethat!P(cv) is representedby a multiple-scatteringexpansionwith an evennumberof scatteringswhere atleastoneoccurson the interior surfaceS.

In order to obtaintheCasimirenergyfromeqs.~3.90)and~3.91),Balian andDuplantierhaveremoved~both the high-frequencycutoff and the space cutoff. They have made rather general assumptionsabout the cutoff function x’ which are reasonablein the caseof perfectly conductingshells. The cvintegration can be performed after a rotation in the complex cv-plane and a suitable choice ofintegrationcontours.Thefinal result for the Casimirenergy,when I is pushedto infinity (undertheseconditionsthe only surviving contributionscomefrom pathsto type(c, e) in fig. 3.4.), is found to be

Evac[S] = dk [V’(ik)— ~P(i01)]. (3.93)

The secondterm of the integrandhasits origin in the high-frequencycutoff.Balian and Duplantier [42] have also applied the multiple-scatteringexpansionto the finite-

temperaturecasewherethey deriveda generalexpressionfor the Casimir free energy.By this methodthey recalculatedthe parallel-plateand the spherical-shellresult. The expansionfor ~Pup to secondorder (double-scatteringprocesses)turns out to be a rathergood approximation,which alreadyallowsthe investigation of the Casimir energy of the electromagneticfield in the presenceof perfectconductorswith arbitrary smooth shapes.

3.4. Phase-shiftrepresentation

Let usconsideroncemore the vacuumenergyof aquantumfield ~ insidea large but finite spatialregion enclosedby a surfaceI. For simplicity the field is chosento satisfyperiodic boundaryconditionson the surface.This field configurationrepresentsthe free referencevacuum,which is distortedby thepresenceof additionalboundariesinside I or by theinteractionwith localizedfields like, for instance,apotential V. The correspondingvacuumenergycan be formally expressedin termsof the phaseshiftsof thefield eigenmodes.The phase-shiftrepresentationwhichgoesbackto Friedel [150] andSchwinger[97], hasrecentlybecomea usefultechniquein the context of variousphysicalproblems[151—153].Ithas also beensuccessfullyapplied for the evaluationsof vacuumenergiesin the presenceof externalpotentials[154, 155, 62].

In order to explain the main points of this method we consider the following one-dimensionalproblem: Inside a largebut finite “box” of size ~L� fX1f a quantumfield ~ is interactingwith a static,localizedexternalpotential V of a finite rangewhich is assumedto be small comparedwith the size ofthe “box”. Dependingon whetherthe externalpotential is attractivea generalspectrumconsistsof aset of discreteboundstateswhich haveeigenenergies

5t, < ~minandcontinuumstateswith eigenenergies


> ~min. In the caseof a massiverelativistic particle field one hasfEmj,,J = m. The contributionof thevacuumenergydueto the changeof the continuumwhen the externalpotential is introducedcan bederived from the phaseshifts of each of thesestates.For this purposeone analyzesthe asymptoticbehaviour of the field eigenmodes,i.e. ço~(x

1 —* ±cx).For the free vacuumconfigurationonehas

~cos(kx1); for even~k,(Pk(X1)~ . (3.94a)lsin(kx1); for odd ~‘k,

wherethe wavenumbersk aredeterminedby the boundarycondition wk(L/2) = 0, i.e.,

kL—2rrn. (3.94b)

Eachof the field modes(3.94a)is modified by the presenceof the externalpotential.For distancesx1

much largerthanthe range of the potential the asymptoticform is given by

icos(kxi + z1~[V]/2); for even çok,~1’(x1)~ tsin(kx1+LI~[V]/2; for odd ~k, (3.95a)

wherethe boundayconditionimplies for evenandodd field modesrespectively:

kL+ii1’[V]=2irn. (3.95b)

The phaseshiftsLI~and LI~arefunctionalsof the potential V. Since the systemis restrictedto a largebox L, which inducesthe conditions(3.94b)and(3.95b)where L tendsto infinity the level densities areobtainedas

(3.96a)

In accordancewith (3.94b) the first term

= L/2rr (3.96b)

is nothing but the level density for vanishingexternal potential. Since the correspondingeigenmodeenergies are relatedto the wave numbers,i.e. s = s(k), one can define the zero-pointenergydensityaccordingto

w1’[V] = s(k)371’[V]. (3.97)

With respectto the definition of thevacuumenergyoneis led to the spectralrepresentation(the part of

the positiveenergycontinuum)

Evac[V] = J dks(k)(37~[V]—?7~[O]), (3.98)

132 G. PlunienetaL, The Cajimireffect

which is of a similar form as eq. (3.44) andformally consistentwith the expression(3.93) given in termsof the mode generatingfunction introduced within the treatmentof Balian and Duplantier [42]concerningthe electromagneticCasimirenergy. Insertingeqs. (3.96a,b) the vacuumenergy takestheform

Evac[V1 J ~e~(LI:[V]+z1~[V]), (3.99)21T deEmin

whichis thedefining equationfor the phase-shiftrepresentationof the vacuumenergy.Let us finish thissectionwith severalremarks.The fact that the high-energyfield modesdo not contributeto the vacuumenergyis ensuredby theproperty[d(zi~[V])Ids]j~...= 0. Forapplicationof thephase-shiftrepresentationonehasto calculatethe phaseshiftsexplicitly. Analytically expressionsmaybe derivedonly in specificcases.In order to evaluateEvac accordingto this methodappropriateregularizationproceduresmustbeapplied,e.g.,in thecaseof anexternalCoulombfield. In thiscontextwenotethatthevacuumenergyof theDirac field interactingwith aCoulombpotentialof arbitrarystrength,whichweshallderivebymeansof thelocal Greenfunction method(section 5), has beencalculatedearlier [154] by the phase-shiftmethod.

4. Casimireffect

4.1. TheCasimir effectbetweenperfectlyconductingplates

As an explicit examplefor calculatingCasimirenergieslet usconsiderthe parallelconductingplateconfiguration in the electromagneticfield. In this casethe modesummationmethodcan be appliedsuccessfullyfor an exact evaluationof the Casimir energy.Attention will be drawn to the involvedsubtractionsand the regularizationof infinite quantitieswhich are necessaryin order to derive a finiteresult for the Casimirenergy.

For thispurposeit is convenientto considera largecubiccavity of volume L3 boundedby perfectlyconductingwalls as quantizationbox, in which a perfectly conductingsquareplateof lengthL is placedat an adjustabledistanceparallelto the x, y face. To find the Casimirenergyonehasto considerthedifferencebetweenthe zero-pointenergycorrespondingto the situation in which the plate is placedat asmalldistancea from thewall andthe one whenthe distance is large, sayLI

37 ~ a with ~ > 1. The twoconfigurationsto be comparedare illustrated in fig. 4.1. Formally the Casimirenergyis definedas

Ec(a)= lim{(E,(a)+ E11(L—a))— (E111(L/’q)+E1~(L—L/’q))}. (4.1)

Each single term of eq. (4.1) representsthe zero-point energyas the sum of eigenmodesinside arectangularcavity. The correspondingfIeld eigeiiuiudesare-determined-~y--rcquiring4he-~Dirichlet .

boundaryconditionfor the electricandthe magneticfield, i.e. n - B = 0 andn X E = 0, on thewalls. It issufficient to derivethe energyE,(a), since it is the only contributionwhich survivesin the limit L—~cc.

The eigenmodesinside the cavity of volume L2a arecharacterizedby the eigenfrequencies

cv = Vk~+k~ k~=(nina)2 k~(n~in/L)2+(n~ir/L)2. (4.2)

G. P!unienetaL, TheCasimireffect 133

/~ / 44~ E~ - E~ ~

a (L-a) L/~ (L-L/~)Fig. 4.1. Subtractionof zero-pointenergiesin thecaseof theconductingparallel-plateconfiguration.

Since the condition a ‘~ L is satisfied, the componentsof k11 can be treatedas continuousvariables,whereasn still takesdiscretevalues.In this limit the eigenfrequenciesreferring to TE andTM modesare identical.To eachcomponentk, correspondtwo standingwaves (polarizations)and only a singleone,if one componentis equalto zero. The zero-pointenergyof the electromagneticfield inside thecavityL

2a (fig. 4.1) can be written as

Et(a)= (L/2ir)2 Jd2k1~{~fk~f+ ~ Vk~+ (niT/a)2}. (4.3)

As it stands this expressionas well as eq. (4.1) are ill-defined and require regularization.Variousproceduresof how to derivea finite result for the Casimirenergywill be discussedin the following.

In the caseof conductorspresentin the electromagneticfield regularizationcan be legitimized onphysicalgrounds.As mentionedearlier,real conductorsbecometransparentfor electromagneticwaveswith sufficiently high frequency.Accordingly, the high-frequencymodeswill remain unaffectedwhenthe platesarepresent,andeigenmodeswith frequencieslarger thana cutoff frequencycv~(in principledeterminedby the propertiesof the real material)shouldcancelin eq. (4.1). For the regularizationofexpression(4.3) this implies the introductionof a suitable (i.e. infinitely often continuouslydifferenti-able)cutoff functionf(k/k~) which satisfiesf(k/k~) 1 for k~ k~andtends to zerosufficiently rapidly forvalues k ~ k~.Here, k~denotesthe cutoff wave numberwhich is assumedto be of the order of theinverseatomicsizeandmaybedefinedaccordingtotheconditionf(1)= ~. Thesimplestchoiceisgivenby anexponential cutoff f(k) = exp(—k/k~).

In orderto calculatethe Casimir energyone hasto perform the energysubtraction,for instance,asindicatedin fig. 4.1. Doing this it hasto be takeninto accountthat, sincethe distances(L — a), LI37 and(L — LI37) are assumedto be large comparedwith a, the remaining discretesummationoccurring inE1(a) (eq. (4.3)) alsomaybe replacedby an integration.This leadsto the expression

L 2 2 1/2 2 1/2E(a, k~)=(i—) fd2k11{~Jki1ff(fki1f/k~)+~[k~+ (~)] f([k~+ (~)] /i~)

+ ~— ((L - a) - Li37 - (L - LI37)) f dk~kf(k/k~)}, (4.4)

134 G. P!unieneta!., TheCasimireffect

where the cutoff is already introduced and k = [k~+ k~]”2 in the third term. Obviously certaincontributionscorrespondingto the largercavitiescancel.The aboveequationrevealsexplicitly the factthat one hascertainfreedomdividing up the quantizationbox and in performing the energysubtrac-tions. For example,the subtractionaccording to E(a) = {E

5(a) + E55((L — a)I2 — E111(L)} is completelyequivalent.After performingthe angularintegrationeq. (4.4) takesthe form:

E(a,k~)= L2rr2 {~JdyVyf(—~-Vy)+ ~ J dyVyf(—~-Vy)- J dn J dyVyf(_~-Vy)~,

(4.5)

wherewe introducedthe dimensionlessvariabley= a2k2Rr2+ n2. Defining the function

F(n) = J dy\/yf(__ Vy) (4.6)2 ak~

theexpression(4.5) can bewritten as

E(a, k~)= {~F(o)+ ~ F(n) - J dnF(n)}. (4.7)a n~1

For the furtherevaluationonecan apply the Euler—MacLaurinformula.By making use of the assumedasymptoticbehaviourof the cutoff function, eq. (4.7) takesthe form

E(a, k~)= — L2ir2 ~ B2kF

t2”1~(0), (4.8)4a ~.

1(2k)!

whereB2~denotesthe Bernoulli numbers.Sincethe first two derivativesof F(n) vanishat n = 0, onlyhigher-derivativetermscontributeto the sum. For m � 3 the derivativesof the function F(n) explicitlyread

F(m)(n) = —2[(m — 1)(m — 2)f~m_3)(innIak~)+ 2(m — 1)nf(m~2)(innIak~)+ n2f(innIak~)]. (4.9)

Obviously only the first term doesnot vanish for n = 0 andthe first derivativecontributingin the sum(4.8) is F°’(O)= —4. Insertionof all non-vanishingtermsinto eq. (4.8) leadsto the total expressionforthe Casimirenergy

E(a, k~)= L2ir2 ~_2(2k)!B2k(2k— 2)(2k—3)(—~-)

2~4C2~_4. (4.10)

The coefficientsC2~_4aredeterminedas the valuesf(2k_4)(0), particularlyonehasC0 = 1.

G. Plunienet aL, The Casimireffect 135

The aboveexpressionfor the Casimirenergyrevealstwo interestingfacts: In the cutoff-dependentresult (4.10), which hasbeenderivedunder rathergeneraland reasonableassumptionsfor the cutofffunction, the leading term (—1/ad) turns out to be cutoff-independent.All the other termscontainpowers of (rrIak~)and vanish when the cutoff is removed (k~—*cc).In this limit one obtains thewell-known result for the Casimirenergy

L27r21Ec(a)~, (4.11)

720 a

which givesrise to an attractiveforce per unit area(vacuumpressure)

21

(4.12)240 a

A somewhatdifferentmethodfor calculatingthe Casimireffect betweenconductingplatesconsistsinusingthe Poisson’ssumformulatogetherwith aspecificchoiceof anexponentialcutoff function.We takethe exponentialcutoff yielding

F(n) J dyVy~ (4.13)

for the function F definedin (4.6) with the abbreviation~ = (rrIak~).In order to evaluateeq. (4.7) withthisspecific caseonecan usePoisson’ssum formula,which reads

c(a) = ~- J dn e~’F(n), (4.14a)

~ F(n) = 2~~ c(2rrn). (4.14b)

In the presentcasethe function F(n) is symmetricin n. Thus, theseequationssimplify:

c(a) = J dn cos(an)F(n), (4.14c)

~ F(n)+ ~F(O)= irc(O) + 2ir ~ c(2lTn). (4.14d)

Insertingeq. (4.14d) into eq. (4.7) we observethat irc(0) is cancelledby the term —f~°dnF(n),which

136 G.P!unien et aL, The Casimireffect

leadsto the following expressionfor the Casimirenergy:

Ec(a)= —i-- ~ c(2irn). (4.15)2a ~

Accordingto eq. (4.14c) we obtain the cutoff-dependent function

c(a, ji) = -~- J dn cos(an) J dy\/y e~”~

= —~- J dn sin(an)n2 e”~= - ~[~ ~ (4.16)ira

0

At this point it is permissible to take the limit /L —~0which gives c(a) = (—4Iira4). Nowone can perform theinfinite summation (4.15) with the result

2 2 L22E~(a)= — 8ir2a3 n4 = — 8ir2a3 = — 720a3~

The Poisson sum formula can also be applied in order to evaluate the temperature correction to the Casimireffect,as we shall see in section6.

Let us nowrepresent a third regularization method in order to evaluate the Casimir energy. It is based onthe analyticcontinuationin the numberof dimensionsd [77, 95]. Using the relation

2ir’~2

J d’~kf(k)= F(d/2)Jdkk”1f(k), (4.18)

the Casimir energy inside a d-dimensional cavity with one direction of finite length a reads

(4.19)

For evaluating the integral one can use the representation of the beta function [148]:

F(1+x)F(—y—x—i)j dttx(1+t)Y=B(1+x,_y_x_1)= F(—y) (4.20)

which formally yields

~~““2 F(—dI2)Ec(a,d) = (LI2)”~1 a’~ F(—112)~(—d). (4.21)

6. Plunieneta!., The Casimireffect 137

The gammafunction and the zeta function can be defined for values d >0 by analytic continuation.This is achieved by the reflection formulas [148]:

F(xI2)7r_z/2~(x)= [‘((1— x)I2)in~’~2~(1 — x), (4.22a)

F(1 — x)F(x)= inlsin(irx). (4.22b)

With these relations expression (4.21) can be rewritten and one obtains the Casimir energy also as afunction of the number of dimensions d:

E~(a,d) = — ~- (4ir)~~~~2E((l+ d)I2)~(1+ d). (4.23)

Obviously,the result (4.11) is recoveredfor d = 3.

With this we endour discussionof the variousmethodsusedto evaluatethe formal definition of theCasimirenergy.In particularthemethodof dimensionalregularization,whichis widely usedin the fieldtheoreticrenormalizationprogram,providesaveryefficient tool for thecalculationof Casimirenergyin allcaseswheremeaningfuldefinition can be given to the formal expression(2.57).

4.2. TheCasimir energyofa massivescalarfield in a finite cavity

In this section we describein somedetail the calculationof the Casimir energyin the caseof anon-interactingscalarfield p of massm insidea cavity of volume L2a assumingL ~‘ a. The field insidesatisfiesthe free Klein—Gordon equationwith Dirichlet boundaryconditionson thewalls. Thesituationconsideredhere is analogousto the electromagneticCasimir configuration discussed above. Thecorrespondingeigenfrequenciesof the field aregiven by

cvk- [k~+(~)2+m2]U2. (4.24)

Since ~ is a spin-0 field each of theseeigenmodescontributeonly with an amount ~cvkto the totalzero-pointenergy.The Casimirenergyof the scalarfield will be calculatedas a functionof the lengtha:

E(a, m)= (LI2rn)2 Jd2k11~ [k~+ (nirla)

2 + m2]~2— Jdn [k~+ (nrnla)2 + m2]1/2}. (4.25)

Again a certainregularizationschememust beapplied.Ambjørn andWolfram [77],who discussedthisconfiguration within a more general framework of quantumfields in finite cavities, performed theregularization by analyticalcontinuationin the numberof spacedimensions.In the following derivationwe simply useanexponentialcutoff andapply Poisson’ssum formula. In order to evaluatethe Casimirenergywe considerthe cutoff-dependentexpression

~ (4.26)

138 6. Plunien eta!., TheCasimireffect

with the abbreviations

b(n,~, A)= J dyVye~’~, ~ = amlir, A = irIak~. . (4~27).~‘

2±n2

Using Poisson’ssumformula (eqs. (4.14c)and (4.14d))onederivesthe cutoff-dependentexpression

E(a,~, A) = L2ir2 2ir E c(2irn, ~, A), (4.28)

where the tilde indicatesthat the term (L2ir2Il6a3)b(0, /L, A) hasbeenomitted, sinceit is independentfrom the separation a. Wenow have to evaluate the cutoff-dependent function

it- fc(a,~,A)=— I dncos(an) I dyVye~’~

in J0

2 d2(4.29)

ira dA

The remainingintegral in the aboveequationcan be analytically calculated[147]:

I(a,~,A)= Jdn sin(an)n(/.L2+n2)112exp[—A(j.c2+n2)~2]

= a/L(a2+ A2)112K1[~(a

2+A2)~2]. (4.30)

Insertingthis into eq. (4.29)andperformingthe limit A —~0 oneobtains

2/L2c(a, /1) = — —j K

2(~aa). (4.31)ira

Accordingto eq. (4.28) we obtain the cutoff-independent Casimir energyof the massivescalarfield:

Ec(a,m)~~ ~-~K2(2amn). (4.32)8ir a,,~.1n

This resultcoincideswith that obtainedby Ambjørn andWolfram [77]usingdimensionalregularization.The remaining summation cannot be performed analytically. Only in the limits of small and large massone can derive approximate expressions of the Casimir energy.

In the limit m ~a~the modified Besselfunction behaveslike [148]

K2(2man)= 2(man)2— + 0(m2). (4.33)

6. P!unienet aL, The Casimir effect 139

Performingthe n-summationthe Casimirenergyup to the order m2 turns out to be

L221 L2m21

Ec(a~m)j~~+ 96 (4.34)

The first term representsthe Casimir energyof a masslessscalar field. Its value is half of the resultfound in the electromagnetic case, due to the fact that the scalar field has only one polarization state.

In the opposite case,when the condition ma~ 1 is satisfied, the Casimir energyis found to beexponentiallysmall

L2 m2 i in \112Ec(a, m) = — —~j — e~2”~, (4.35)

16in a ma

reflecting the fact that the Casimirenergyvanishesin the classicallimit of particleswith largemass.

4.3. Quarks and gluons in a bag

Until nowwe havediscussedthe vacuumenergyof quantizedfields in rectangularcavities. For thisparticular geometry of variable external constraints it has been clearly shown that the zero-point energymustbe envisagedas a contributionto the potentialenergyof quantumfieldswhich representsas leastin principle a measurablequantity.

Particularly the investigation of Casimir energies of quantized fields confined in a spherical cavity hasrecentlybecomeof interestin the context of phenomenologicalbagmodelsin particlephysics[54—72].The bag model is basedon the belief that hadronsare built out of quarks(and gluons), and thatquantumchromodynamics(QCD) provides the appropriatedescriptionof hadronic matter. QCD isgiven by thefollowing Lagrangian,which is invariant underthecolour groupSU(3) and— in the absenceof massterms— underthe hadronicflavour symmetrygroup SU(N~):

-~OCD= ~ [~T’k(iy~a~— mk)~P’k— ~g~PkA,,y~PkA~,] — ~ (4.36a)

where Aa denotethe Gell—Mann matrices,g the couplingconstantandNf the numberof flavours.Theindex k will be suppressedlater on. The non-Abelianfield strengthtensoris given by

F~’= — a~A~+ gf~CA~A~. (4.36b)

Although the theory is not yet completelyunderstood,one expectsthat QCD containsquark andcolour confinementwhich would be in accordancewith the empiricalfinding that free quarksdo notexist andall known particlesarecolour singlets.

In a simplified, approximatemodel of hadronic structurebasedon QCD, confinementmust beintroducedby handassumingthat the fields are localizedinside finite regionsandrequiringappropriateboundaryconditions.Sucha treatmentrepresentsthe commonaspectin severalphenomenologicalbagmodels[75]. Now, evenin an “empty” bag, i.e., which containsno real quarks,therewill be non-zerofields due to quantumfluctuationsand thesegive rise to a Casimir energy.The Casimir energiescorrespondingto the quark gluon fields may be evaluatedseparately,consideringthe two simplified

140 G. P!unien etaL, TheCasimir effect

problems:Quarksconfinedin a finite (static)bagof volume V are describedby the free Lagrangian

= ~(iy~9,.— m)’I’, (4.37a)

togetherwith the linear boundarycondition

(iy~’n~.— 1)~PI~= 0, (4.37b)

where n~’= (0, n) is the outward normalvectorto the static bagsurfaceS. The correspondingCasimirenergyis the differencebetweenthe zero-pointenergydue to the cavity field modesof the seaquarksandtheoneof the unconstrainedfield modes.

Concerningthe colourgaugefieldsoneconsidersthe free Lagrangian(neglectingthenon-lineartermfabc~

4~4c’~in the field-strengthtensor):

= ~ (4.38a)

Within such an approachcolour confinementis achievedby treatingthe exteriorphysicalvacuumas aperfect colour magneticconductor, i.e., the vacuum inside the bag is characterizedby the colourmagneticpermeabilityp~= 1, while ~ is infinite in the exterior.Throughthe vacuumrelationep~= 1,this implies that the dielectricconstante~,= 0 (and ~ ci~)outsidethe bag(seesection7.1). The colourelectricand magneticfields must then satisfythe following boundaryconditionson the surface:

n.E’~I~=O, flXBajs=o. (4.38b)

The evaluationof the gluonic Casimirenergyis completelyanalogousto theprocedurefollowed for theelectromagneticCasimireffect. Note however,that the role of the colour-electricand colour-magneticfields is just the oppositeto that in normalelectrodynamics(seesection7.1).

The following discussionof Casimir energiesin a sphericalbagis intendedto give an outline of theappliedevaluationmethodwithout going too muchinto technicaldetails.We mention right-awaythatpresentresults are not completely satisfactory,since the final expressionsfor the Casimir energiescontaincutoff-dependentparts.The remainingfinite partsof the Casimirenergyof quarksandgluons -

areboth found to be proportionalto the inverseof the bag radiusa, as it must be sincea~ is thequantity settingthe energyscale.

4.3.1. Casimir energy in a sphericalbagIn orderto derivegeneralexpressionsfor Casimirenergiesof quantumfields inside a sphericalbag,it

is convenientto apply the local formulation. The stepsnecessaryto perform the derivationbecomemoretransparentif onefirst considersthecaseof aconfinedmasslessscalarfield. All methodwhich areapplied therecan be generalizedto the other casesof confined fermion or vector fields. Only thetechnicaldetailsaremoreinvolved.

The energy—momentumtensorof the vacuum of the masslessscalar field can be derived in ananalogousway as we haveshownin detailsfor the electromagneticfield. Sinceoneconsidersthe scalarfield in the presenceof staticboundariesthe exactGreenfunction is homogeneousin time. The vacuumstresstensorthus reads:

= {i(9~’3”~ — ~g 8~8”)(G(x,x’, x0 — x~)— G0(x— x’))}l~~..~. (4.39)

G. P!unien eta!., The Casimir effect 141

Both Greenfunctionssatisfy the equation

LIG(x, x’, x0 — x~)= — 3(x — x’), LIG0(x— x’) = —5(x— x’), (4.40)

where the constrained (or cavity) Green function may either fulfill Dirichlet or Neumann boundaryconditions on the surface, i.e., GD(x, x~,x0 — x~)j5= 0 or n~VGN(x, x’, x0 — x~)js= 0. Analogousconditionsmust be requiredwhenx’ is on the boundary.It is now of advantageto separatethe exactGreenfunction into the inhomogeneousfree vacuumpart G0(x — x’) anda remainder:

G(x, x’, x0 — x~)= G0(x— x’) + f~s(x,x’, x0 — x~). (4.41)

The remainingpart Fs(x, x’, x0 — x~),which describes the modified propagation due to the presence ofboundaries, consequentlysatisfiesthe homogeneousequation

L1Fs(x,x’, x0 — x~)= 0. (4.42)

Thus,one obtainsthe energy—momentumtensor(4.39) in termsof the boundarypart:

E~jx)= j{~j~I~s(xx’, x0— x~)}I~’~. (4.43)

Formallythe Casimirenergyfollows afterspatialintegrationof the energydensityover the bagvolume:

E~= J d3x �~~(x)=lirn i -~ J d3xP(x,x, r), (4.44)

wherethe limit T = — x~—~0 must beperformedat the end.For furtherevaluationone now needsanappropriateeigenfunctionexpansionfor the Greenfunctionswherethe boundaryconditionsareeasytotakeinto account.Concerningthe sphericalsymmetricboundaryproblemit is convenientto choosetheangularmomentumrepresentationof the Greenfunction. Introducingthe Fourier-transformedGreenfunction G(x, x, w) satisfying (A + w2)G(x, x’cv) = 6(x — x’), the partial wave expansionimplies:

G(x,x’, cv) = g1(r, r’) m~i Yim(12)Yi*m(flh). (4.45)

The radial part of the Greenfunction is determinedby the radial equation

1 d2 1(1+1) 1

(_~_~r_ ,.~ +cv2) g,(r, r’)=~6(r—r’), (4.46a)

subjectto Dirichiet or Neumannconditions(a denotesthe bagradius)

g~(a,r’) = 0, -~--g’~(r,r’)Pr.~a= 0. (4.46b)9r

The generalsolutionsof eq. (4.46a)which satisfiesthe boundaryconditions(4.46b)arewell known [93,


55] as certain combination of sphericalBesselfunctions.The Dirichlet and Neumannscalar Green

functionin the interior of the bag,i.e., r, r’ < a andwhich areregularat theorigin aregiven by (r’ < r):

GD(x, x’, w)= —ik ~ [j,(kr’)h~1~(kr)~h~(ka)j(kr~)j(kr)]~ Ytm(fl)Ytm(Q’), (4.47a)1.0 j

1(ka) m

GN(x,x’, cv) = —ik ~ [j~(kr’)h~1~(kr)—(i- h~1)(y)/~_ji(y)) y=ka 1i(kr’)Ji(kr)]

X~ Yim(Q)Y~m(f2’), (4.4Th)

k = Icv~.Thefirst terms,in eqs.(4.47a,b)correspondto thefreeparticalwave propagator,while thesecondtermsbelongto the surfacepart .P(x,x’, cv).

With theseinformationsone is able to derive closedexpressionsfor the Casimirenergyof confinedscalar fields satisfying either Dirichlet or Neumannboundaryconditions. In the case of Dirichletconditionsoneobtainsin accordancewith eq. (4.44)

cc a

Ec(a,T)= ~ (21+ 1)-~-~J ~e~’~k h~(ka)jdrr2j~(kr), (4.48)1=0 9r 2ir j

1(ka)—~ 0

wherethe sumformula for sphericalharmonics,

m~1 Yim(Q)12= 2~1, (4.49)

has beenused.The radial integral over- the interior of the-sphere- is -related-to the--sec-on4-Lommel---integral[149]:

J drrJ~(kr)= ~- [(-~-J(y))~ + (i - -A) J~(ka)]2 dy y=ka (ak)

0

= ~ [~~a) — J~i(ka)J~+i(ka)], p> -1. (4.50)

Inserting thisinto eq. (4.48)leadsto

Ec(a, r) = - ~(21+1) J ~ e’~i(ak)3h ~‘~(ak)[j2(ak) — j,1(ak)j,+1(ak)]. (4.51)

2ir j1(ak)

Performingnowa Wick rotation [88],

G. Plunienet aL, The Casimir effect 143

cv—*icv, r—*ir, k—*iIwj, (4.52)

andsubstituting

x=Iw~a, t~=rIa, (4.53)

the Casimirenergytakesthe form

E~(a)= lim~ { ~ (2/ + 1)1 dxx2 cos(ôx)K1÷112(x)[12() — I ,1,2(x)I

t=o 11+112(x)0

= lim ~ (4.54)~ ira

The cutoff-dependentconstantc(ö) abbreviatesthe remainingintegralincluding the infinite summationover 1. This formula hasbeenobtainedby Benderand Hays [54]. As could havebeen guessedongrounds of dimensionsthe Casimir energyof a masslessscalar field confined in a sphericalbag isproportional to lIa. The samebehaviouris also found in the caseof confined masslessquarksandgluons [54—56].The Casimirenergycontributionsreferringto thesefields differ from (4.54) only in thecoefficients.While the radiusdependenceof the Casimirenergiesis well determined,the calculationofthe exact coefficients is besetwith difficulties. As one sees,even in the caseof the scalarfield thecoefficient c(5) cannotbe obtainedanalytically. In addition, it turns out [54]that c(5) still contains atleastone termwhich divergesas 1I~in the limit ô—*0.

With the samemethodas appliedabove,one can derive the Casimirenergiesfor confined quark andgluon fields in a similar way. Concerningthe gluon field the evaluation reducesto solving two scalarGreenfunctionproblems[55].ThescalarGreen functioncorrespondingto transversecolour-magneticfieldmodes (TM) is related to the Green function problem (4.47a) by: GTht(x, x’, cv)= — G’~(x,x’, cv).Concerningtransversecolour-electricfield modes(TE) the Greenfunction is determinedas solutionof

(A + cv2)GTh(x,x’, cv) = ô(x — x’), (4.55a)

with respectto the boundarycondition

~~~(rGTh(x,x’, cv))J~~~= a =0. (4.55b)8r

The angularmomentumrepresentationis given by [55] (r, r’ < a)

GTE(x, x’, cv) = ik ~ (Ji(kr’)h~’(kr)_[~_ (yh~(y))/~_(yj,(y~,)] y=ka)

Xm~:~~iYim([2)Y~m(fl’). (4.56)


Dueto theintrinsic spin oneof thegluon field thetermwith / = 0 isexcludedin bothGreenfunctions.TheCasimirenergyagainfollows from the homogeneouspart

Ec(a, T) = ~ J d3x (rFE(x x, r) + ~M(x, X, T)), (4.57)

andbecomesexplicitly

Ec(a, r)=~ (21+ i)J ~e~(ak)~ ~ [~(yh 0(y))/~(yje(y))j~yka)

x(j~(ka)— j,1(ka)j,±1(ka)). (4.58)

This formulawas derivedby Milton [55].After performing the Wick rotation (eqs. (4.52) and (4.53)) amoreconvenientexpressionis obtained:

Ec(a, ~) = - ~-~-— ~ (21+1)1 dx cos(5x)x{~-Sl(X)/s1(X) + ~2s,(x)/~ s1(x)

—2(~-e~(x)~- s~(x)— e,(x) th2 S~(X))}, (4.59)

wherethe functionss, ande, aredefinedas

s,(x)= (inxI2)U2I,±i,2(x), e,(x)= (2xIir)”

2K,÷112(x). (4.60)

In order to obtain an approximateexpressionfor (4.59) one can make useof uniform asymptoticexpansionsfor the Besselfunctions. Milton [55, 58] hasderivedthe following result for the Casimirenergyof confinedgluons:

Ec(a,~= I (_4/3~2+ ~). (4.61)

Apart from a quadratically divergent term (the leading divergence [57]) the cutoff-independentcontributionturns out as to bepositive andthusgives rise to a repulsivevacuumpressure.

Let us now continuewith a short considerationon the Casimir energyof the quark field inside asphericalbag. In order to calculateit from local quantities,we makeuse of the relation betweenthesymmetricalfermionicenergy—momentumtensorof the vacuumand the Feynmanpropagator(seeeq.(5.10)).The derivationof this relationwill be discussedin moredetails in section5 when we considerthevacuumenergyof the Dirac field interactingwith externalelectromagneticfields.

6. P!unienet aL, The Casimireffect 145

The energy—momentumtensorof the vacuumof confinedfermionsis definedas

EP’~’(x)= ~{tr[y’~(ô”— a’~)Ft(x,x’, r)] + (/2’~-t.~)}I~~ , (4.62)

where F’ denotesthe difference betweenthe cavity propagatorS(x, x’, r) and the free propagatorS

0(x— x’). The cavity propagator is defined as the solution of the equation

(iy~9~— m)S(x,x’, i-) = 6(x — x’), (4.63a)

with respectto the linear boundarycondition:

(1 + in~y)S(x, x’, r)I~= 0. (4.63b)

If one againseparatesout the inhomogeneouspart S0(x— x’) from the exactcavity propagator(4.63a),the boundarypart F~satisfies

(iy’~9~.— m)F’(x, x’, r) = 0. (4.64)

The Casimir energy is then obtainedas the spatial integral of the energydensity t9°°over the bag

volume:

E~(r)= -~- J d3x tr(y°f’~(x,x, r)), (4.65)

taking T —*0 at the end.For masslessquarksthe evaluationhasbeencarriedout by severalauthors[54,56] using the angularmomentumrepresentationof the fermion propagator,which can be similarlyconstructedas in the caseof confined scalaror vector fields, although the technicaldetailsare moreinvolved. The sphericalrepresentationof the propagatoris of the form

S(x, x’, cv) = ~ S111(r, r’, W)~jkm(I1)44l’m(fl’), (4.66)

j1l’m

where the ~jtm are two-component spinor spherical harmonics. For determining the radial partS11t-(r, r’, cv) one usesthe relation betweenthe fermion and the scalarpropagator:iy’~9~G(x,x’, T) =

S(x,x’, T). Accordingly the componentsof S~,t-are sphericalBesselfunctions[56, 93]. Explicit expres-sionsfor the (cutoff-dependent)Casimir energyhavebeenderivedby Benderand Hays [54] and by

Milton [56]:

Ec(a, r)= ~ (2j+ 1) J ~e~(ka)32{[(~jj÷i,2(y))~ j1112(ka)j=1/2 —cc 2ir dy yka

~‘ d . . 1 1h~i,2(ka)11÷ii2(ka)—h~2112(ka)j1112(ka)~~— I — 11—1/2(Y)1 j1±112(ka)I X 2 2 I - (4.67)\dy / y=ka -~ L [j1+112(ka)]— [j1112(ka)] ii

146 6. P!unien etaL, TheCasimir effect

Performingagaina Wick rotation and introducingthe notation (4.60) the expressionfor the Casimirenergytakesthe form [561

E~(a,~)= -~-- ~ (I + 1) J dxxcos(ôx)[-~-ln(s~±1(x)+ s~(x))ira =~ dx

t=0

—2 (~-s,±1(x)e,(x)+ ~- s,(x)ei±i(x))]. (4.68)

Using Debyeexpansionfor the BesselfunctionsMilton hasderivedthe approximateresult

Efa,8)=-~-(~=-th)-~ - - - -

which still containsa quadratic-allydivergentterm.The finite part carriesthe oppositesign as the onewhich hasbeenfound for gluonsand thusleadsto an attractivepressure.

This short review on the Casimir energyof confined massless,non-interactingquarks and gluonsclearly revealsthe similarities of their evaluationand the results. Both are proportional to lIa andcontain a leading quadraticdivergence.For the gluons E~doesalsocontain a logarithmic divergence[57]. RecentlyMilton [58]hasshownthat the quadraticdivergencein (4.69) appearsas a contacttermwhich cancelswhen exterior field modesare included.The correspondingcalculationconceptually isvery similar to Boyer’s treatmentof the sphericalshell configuration [37]. In the context of confinedquarks such an approach might have some physical relevancein the picture of K. Johnson andcollaboratorswherethe QCD vacuumis regardedas a foam of denselypackedbubblesof perturbativevacuum.The inclusion of the exteriormodesproceedssimilarly calculationsandeq. (4.68)changesto[58]

Ec(a,~)= ~ (/+1) J dxx cos(&) ln[(s~±1(x)+ s~(x))(e~+i(x)+ e~(x))]. (4.70)dx

0

In the asymptoticexpansionthequadraticdivergencecancelsandthe finite part of the Casimirenergyisnow found to be positive

Ec(a) = 0.02041a, (4.71)

which againgives rise to a repulsivepressure.In view of the results(4.61) and(4.71) it seemsunlikelythat the Casimirenergyof quarksand gluons can explain the “zero-point” contribution of the (static)MIT bagenergyE(a)= —zla (with z = 1.84) which occursin phenomenologicalbag model fits to thespectrum of hadronic particles. Calculating Casimir energies Bender and Hays [54]have also included finitequark masseswhich were recentlymorecompletelytreatedby Baackeand Igarashi[59].They obtainedmass-dependentcontributionswhich contain furtherdivergencies.Thesetermsrequiredetailedrenor-malizationprescriptionswhich havenot yet beenworkedout.


In the context of evaluatingCasimir energiesof confined quantumfields in the framework of thelocal Greenfunction methodsit shouldbenotedthat progresshasbeenmadein thederivationof exactcavity propagators.HansonandJaffe [93,94] haveinvestigatedthemultiple-reflectionexpansionfor thescalarandfermionic propagatorwhich is formally similar to techniquesused in the multiple-scatteringexpansionfor the photonpropagatorby Balian andDuplantier [41,42]. Applying suchtechniquesmaybe useful in order to calculatehigher-ordersurfaceand curvaturecorrectionsto the Casimir energyinside arbitrarily shapedbags.However, the fact that divergencesinherentlyoccur in the formalismmakesthe theoreticaltreatmentof zero-point energiesin finite cavities with sharpboundariesprob-lematic.

5. The Dirac vacuumin externalelectromagneticfields

5.1. Vacuumenergyand vacuumpolarization

In the previoussectionswe haveseen that the concept of Casimir energyallows a satisfactoryphysicalinterpretationof thezero-pointenergyof quantumfields. It permitsto regardphysicalvacuumenergiesas the responseof the vacuumto externalconstraints.As an idealizationof the real situation,fields undercertainconstraints(e.g.,macroscopicbodies)mustsatisfy appropriateboundaryconditions.We havealreadymentionedthat within a generalizedconceptof Casimir energysystemscan also beconsideredwhereexternalfields createdby an arbitrary sourceconfigurationplay the role of externalconstraints.

As an examplefor sucha situationwe will discussthevacuumenergyof the electron—positronfield inthe presenceof a classicalexternalelectromagneticfield. For this purposeonecan take the followingphysical picture as a guide: The zero-pointenergyof the Dirac field appearsas an infinite sum overeigenenergiesof the properly symmetrizedDirac Hamiltonian(eq. (2.48)).It is convenientto introducea large quantization box together with periodic boundary conditions, say, for the upper spinorcomponents,in order to obtainadiscreteenergyspectrum.The energyspectrumof the free Dirac fieldis visualizedin fig. 5.1. It consistsonly of thepossiblefree electronstatesandfree positronstates.Thisfree vacuumconfigurationis modified by externalelectromagneticfields. For instance,in the presenceof a Coulomb field createdby a positive externalchargedistribution, particularly that of a nucleus,additional boundstatesappearin the energyspectrum(fig. 5.1). Any changein the externalfield (e.g.displacementof the sources)producesa changeof the energylevelsandthus in the zero-pointenergy.Correspondingly,the physical vacuumenergyof the electron—positronfield hasto be definedin thiscaseas the differencein the zero-point energytaking the free vacuumconfiguration as the point ofreference(eq. (2.58)). The vacuum energycan be treatedas a function of suitableparameters,hereabbreviatedby A, which characterizethe relative positionsof the externalsourcesor their geometry.For instance,in the caseof severalchargesthe relativedistancesbetweeneach othercanbe takenas thevariables A (e.g., the two-centerdistanceR for two colliding nuclei). In the caseof a connectedchargedistributionthe vacuumenergymaybe analyzedas a function of someconvenientlychosenparametersdescribingthe shape or spatial extension(e.g., the radius a, when consideringa chargedsphere).Accordingly the vacuum energy is then obtained as the difference betweenthe zero-point energycorrespondingto a source-configurationcharacterizedby certainfixed valuesfor the parametersA andthat for such valuesA0, which correspondto the situationof a vanishingexternal field. Particularly inthe cases mentionedabove the free vacuum configuration is realized when either the two-center

148 G. P!unien et at, The Casimireffect

m m

0 0- -

-m _______ _______ -m _______ _______

a) b)

Fig. 5.1. (a)Energyspectrumof thefree Diracequation-and(b> thespectrumin-thepresence-ofan -externalCouIomb~fieId.- - -

distancebetweenthe nuclei or the radiusof the chargedspheretendsto infinity. The Casimirenergydefined in this context hasto be interpretedas the responseof the Dirac vacuumto “distortions”causedby the presenceof an externalsourceconfiguration.

One could expect that the vacuum energyenvisagedhere may be related to correctionsof thepotential energycarriedby an externalsourceconfiguration due to vacuumpolarization effects. Forinstance,whenconsideringthe Dirac vacuumin thepresenceof the staticelectromagneticfield A0(x, a)of a chargedsphere,a staticvacuumpolarizationp~~(x,a) will beinduced.Infinitesimal variationsFia ofthe radiuscause a change~A0(x,a) and thus also affect the vacuum energy. It seemsobvious tointerpretbEvac(a) asthe amountof energynecessaryto extendor to contracta sphereby ba againsttheaction of the polarizablevacuum.Thetotal vacuumenergycarriedby such a configurationis obtainedas the integral

I ,oEvac(a) = — da Evac(a ), (5.1)

.‘ i3aa

where the derivative F(a)= —aEyac(a)It9a representsthe generalizedCasimir force. A furthermechanicalquantity, the vacuumpressure,then follows accordingto

Pvac(a)= — Evac(a). (5.2)4ira 3a

In this examplethevacuumenergydefinedabovewould representtheself-energycorrectionof aspheredueto thevacuumpolarization.A furtherquestionthat can beraisedis whetherit is possibleto identifyEvac as a contributionto the interactionpotentialbetweenthe externalsources,as would be expected

6. P!unieneta!., The Casimireffect 149

when the separationbetweenthe sourcesis takenas the variable.Whetherthesephysicalimplicationsof thevacuumenergycanbe provento betrue hasto beshown by an explicit calculationof the vacuumenergy.

Consequentlyoneis led to the questionof which evaluationmethodfor the vacuumenergywould bemost convenientconcerningthe problemraised here. First of all the option of applying the modesummationmethodwill be discussed.The ingredientsnecessaryfor this method are the eigenmodeenergiesof the constrainedDirac field (assumingstatic external fields) and those of the free fieldconfiguration. In general, the eigenmodeenergiescan only be obtainedby numericalsolution of theDirac equationwith respectto the boundaryconditions requiredfor the spinor wavefunctionon thesurfaceenclosing the configuration. In this context one has to consider that for instance periodicboundaryconditionscannotbe fulfilled by the largeand the small spinorcomponentssimultaneously.Nevertheless,havingobtainedthe space-cutoffdependenteigenmodeenergiesand thusthe zero-pointenergiesE0[A, .~] andE0[0, .~], the nextsteptowardsthe evaluationof the vacuumenergywouldbe toapply an appropriateregularizationscheme(a high-energycutoff, for instance)in order to maketheformally divergentdifference of the zero-point energiesfinite. If this involved procedurewould becarriedout; the last stepwould be to attemptthe cutoff removal.Accordingto this method the wholecalculationis predominantlynumerical.Apart from very simple externalfield configurationsthe wholeprocedureseemsto be rathercumbersomeandnot very practical.

For this reasonandstimulatedby the Greenfunction methodssuccessfullyemployedto evaluatetheCasimirenergyin the caseof the electromagneticfield, we turn to investigatethe local formulation ofthe QED vacuumenergyin termsof the energy—momentumtensor.Accordingly, for this purposeit isnecessaryto deriveanexpressionfor thevacuumexpectationvalueof theenergy—momentumtensorin thefree Diracvacuum,(01 T~j0)0,andin the Diracvacuuminteractingwith theelectromagneticfield AM ofan externalsourceconfiguration, i.e., (01 TM~cI0)A.Basedon the analogousdefinition (eq. (3.11)), theenergy—momentumtensorof the QEDvacuumis formally definedby subtraction,

= (Oj T~I0)A— (01 iI0)~. (5.3)

ø~dependson the structureof the QED vacuum,since the exact Feynmanpropagatorappearsinas we shall see.Particularly in the caseof static externalfields, a generalexpressionfor the

vacuumenergywhich revealsthe dominantrole of vacuumpolarizationeffectscan be derived.

5.1.1. Theenergy—momentumtensorofthe interacting QED vacuumIn orderto derive the energy—momentumtensorof the interactingDirac vacuum,onefirst needsthe

relationshipbetween(01TM~~0)~and.the free propagatorS0(x— x’) andalso thatbetween(01 i’M~dI0)Aandthe propagatorin the externalfield SA(x, x’). t9~must beof a form which ensuresconsistencywith therequirementthat quantitiescharacterizingthe vacuum must be invariant under chargeconjugation(including the externalsource)and that ø~must be symmetricalin ~aand v. This meansthat thevacuumenergyof the electron—positronfield interactingwith an externalelectromagneticfield remainsthe sameevenwhen the role of electronsand positronsis interchangedwith a simultaneouschangeinsign of the externalsources.

The properlysymmetrizedenergy—momentumtensorof the free Dirac vacuumis given by

= ~{([~ yMa~’P]+ [~ yMô~t1])+(term ~-*v)}. (5.4)

150 6. P!unien etat, The Casimireffect

This bilinear form containsthe dynamicalfields andtheir first derivatives.Such a combinationcan beobtainedby meansof a suitabledifferential operatoractingon the time-orderedproductof the fermionfield operators

T(~P(x)!P(x’))= O(x~—x,)~i’(x)~i’(x’)—O(x~—xo)~i~(x’)~I’(x). (5.5)

The appropriatedifferential operatorfor thispurposereads(suppressingthe matrix indices)

D~= ~ + d~ç) (5.6a)

with the abbreviation

d~= ~iyM(8v — o’~). (5.6b)

When evaluatingthe quantities{tr D~[T(1I1(x)1~i(x’))]}onehasto be carefulwith expressionsarisingfrom derivatives of the stepfunctionsin the time-orderedproduct. Accordingly, terms containingatemporaldeltafunction appear,such as

~in”S(x0—x,) tr[y~’{#(x), 1I’(x’)}] = 2ig°MnP6(x— x’), (5.7)

wheren’~denotesthe time-like vectorn” = (1,0, 0, 0). The deltafunction on the right-handside of eq.(5.7) hasbeengeneratedby replacingthe anticommutatorby thecorrespondingequal-timecommutator.Performing the Lorentz covariant limit x’ —~x, we obtain the following expressionfor the energy—momentumtensorof the free Dirac field: -

T~(x)= {2i(g°Mn~’+ g°vnM)~(x— x’)— 2tr[D~çT(~t’(x)~1/(x’))]}Ix~cx. (5.8)

Substitutingin eq. (5.8) (Of T1I1(x)V~~(x’)l0)= iSo(x — x’), which satisfiesthe Dirac equation

(iYMoM — m)S0(x— x’) = 3(x — x’), (5.9)

leads to the desired equation linking the vacuum expectation value of T~ and the Feynmanpropagator:

(01fM~dI0)o= {2i(gOMn~’+ g°PnM)5(x— x’) + ~(tr[yM(9~’— 9’~)S0(x— x’)] + (term~a ~‘))}IX’cX. (5.10)

By making useof eq. (5.9) the energydensitycan be rewrittenin the form

(01 ~I’~(x)0)~= (—i) tr[y° h0(x)S0(x— x’)]I~’~, (5.lla)

where

ho(x)= ~,O~k jt~ + y°in (5.llb)

denotesthe free Dirac Hamiltonian. That the vacuum energy density (5.llb) is related to the

G. P!unien etat, TheCasimir effect 151

zero-point..,energyof the free Dirac field is simply shownby expressingagainthe propagatorin the form

(0jT~[’(x)V’(x’)I0) andintegratingover all space:

E0[0] = f d3x (0Ii’~(x)l)o= if d~x((0J[4’~,h

0lfr]l0)+ (0j[4’~h0,4110)). (5.12)

We now turn to derive an expressionsimilar to eq. (5.10) for the energy—momentumtensordescribingthe Dirac vacuumin the presenceof an electromagneticfield. Let usfirst considerthe totalsystem,the Dirac field coupledwith an electromagneticfield, which is characterizedby the Lagrangian

2?= ~i{[4~y~9~!P]+ [V~yM,9M~fl}_~m[4-~~P]— ~e[~PyM’I’]AM — ~FM,,FM~—J~xtAM, (5.13)

yielding the coupledfield equations:

(iyM9M — eyMAM — m)!1’ = 0, (5.14a)

tJt(jyM~9+ e7

MA,. + in) = 0, (5.14b)

3VFILZ, = ~ (5.14c)

The totalelectromagneticfield is asuperpositionof fields createdby the externalcurrentj~andthatofthe Dirac particlesj~= ~e[~P,y~P], i.e., ATM = ~ + Af~.The Lagrangian (5.13) is invariant underchargeconjugationtqgetherwith the changefrom j~,to (—j~J.The properly symmetrizedenergy—momentumtensorof the Dirac—Maxwell field readsexplicitly:

T~..M)= ~i{([~ yTM3~~1I/]+ [~1’ y’~’41)+ (term~ v)} — ~(j~A’~+ j~ATM)— (j~~1A”+ j~~1A

TM)

— FILaFV + gTM~~(~J~F~+ j~xtAa). (5.15)

In the following we are only interestedin describingthe responseof the Dirac vacuum to theelectromagneticfield of the externalsourcej~.For this reasonit is sufficient to consideronly the firstpart of the total energy—momentumtensor(5.15), i.e.,

T~)= ~i{([~ yTM~9~~~P]+ [1J~,yMt9~’11i})+(term1a~-*v)}. (5.16)

The index (A) indicatesthat the spinors11’ and ~[‘ aredeterminedby theDirac equations(5.14a,b), andthus T~)depends implicitly on the electromagneticfield. Such a division of the total energy—momentumtensoris basedon the following restrictions:Firstly, only a localizedexternalcurrentj~.,isassumedto bethe sourceof the electromagneticfield. Secondly,if thereis no chargedfermioncurrent,thenconsequentlythe total field As,. reducesto the externalfield A~’”(A~’is shorthandnotedby ATM inthe following). In this casethe energy—momentumtensor (5.15) consistsof the two dominantcon-tributions: T

TM~’= T~)+ TTM~(EM; ext), wherethe first term contains the spinorpart and the secondonedependsexclusivelyon the externalelectromagneticsources.A situationwherethis approximationholdsis whena single electronmovesin the localizedCoulombfield of a nucleus.Thus describingthedistortion of the Dirac vacuum due to the presenceof electromagneticsources,this external field

152 6. P!unien eta!., The Casimireffect

approximationis legitimate and the part T~) is the appropriatequantity for continuing with ourconsiderations.

In the next step we must derive the relation betweenvacuumexpectationvalue(01 TTM~~l0)Aof thequantizedexpression(5.16)andthe exactFeynmanpropagatorsatisfying the equation

(iyTM 3,. — eyTMA,. — m)SA(x,x’) = ô(x — x’). (5.17)

This is donein a similarway to the free field case,becauseT~)can be obtainedby meansof the samedifferentialoperatoractinguponthetime-orderedproductof thefield operators.Accordinglyweobtaintheexpression

(0ITI’~~(x)l0)A= {2i(go~n’~+ go~n~)ô(x— x’) + ~(tr[y~(3”— t9’~)SA(x,x’)] + (termjz~-*

(5.18)

Apart from the occurrenceof the exactpropagator,(Oj T~”l0)Ahas obviouslythe samestructureas eq.(5.10).The energydensitythen simply reads

(0li~0(x)I0)A= {4iö(x—_x~)+~tr[y°(c9°—1~’°)SA(x,x’)]}j~_ (5.19a

)

or by making useof eq. (5.17), .~- _______ ____________________________

(0ji’°O(x)l0)A= (—i)tr[y°hA(x)SA(x,x’)]I~~, (5.19b)

where

hA(x)= yoyk(i0+eA)+eA+ y°m. (5.19c)

Performingthe spatialintegrationthezero-pointenergyof the Diracfield in thepresenceof anexternalelectromagneticfield is recovered:

E0[A] = f d~x(O~i’°olO)A = — ~Jd

3x ((OlE~kF,hA4110)+ (OlE 4’hA, 4110). (5.20)

We nowhavederivedall quantitiesandrelationsin order to definethe energy—momentumtensorof theDirac vacuum. The equations(5.12) and (5.20) show that the correct local definition of the vacuumenergyis achievedby meansof the subtractedenergy—momentumtensor

9~(x)= ~{tr[y~(r9~— ô’v)S(x — x’)] + (term1a~-~v)}I~’~, (5.21)

where.~(x,x’) denotesthe differencebetweenthe exactandthefree Feynmanpropagator.Thus we areled to an expressionfor ø~which hasa similar structureas the onederivedfor theelectromagneticfield (eq. (3.22)). The expression(5.21)revealsthat all physicaleffectsdescribedby ø~originatefromthe modified propagationof electronsandpositronsin the vacuumunderthe action of externalfields.As it stands, ~ requiresregularization,since S(x — x’) containsdivergent contributions when x’approachesx. For explicit evaluationsit is convenientto deal with a differential form of eq. (5.21). The

6. Plunienetat, TheCasimireffect 153

correspondingexpressionis obtainedby consideringinfinitesimal changesof the externalfield A,..Rememberingthat we are treating the electromagneticfield as a function of suitableparametersAcharacterizingthe external sourceconfiguration, then, of course,any variation -in theseparameterscausesa change&~9~of the energy—momentumtensor.We assumethat A,.(x, A) varies continuouslywith A. Although any changeof the externalsourceconfigurationmodifiesthe propagation,we shall seethat the correspondingvariation ~SA(x,x’; A) is not necessarilya continuousfunction of A. Formally wewrite the infinitesimalchangeof theenergy—momentumtensoras

~9~:~(x,A) = ~{tr[y~(3” — 0’~)&SA(x,x’; A)] + (term j~-~~ (5.22)

andreobtaineq. (5.21)when integrating69~over A (A0 correspondsto the free field configuration)accordingto

@~(x,A) = J dA’ (01 ~TMv(x,A)JO)A. (5.23)

Thus we haveformulatedthe local form of the QED vacuumenergyin the context of a generalizedconcept of Casimir energies. It is satisfactory to note that the mathematicalformulation appliedsuccessfullyin the caseof the electromagneticandscalarfield alsocarriesoverto spinorfields.

5.1.2. Thevacuumenergyof the electron—positronfield in thepresenceofstaticelectromagneticfieldsWe now turn to discussthe energyof the QED vacuum. Having definedthe energy—momentum

tensor(5.21),Evac[A] can nowbe evaluatedby integrationof the energydensity �~C. We shall restrictourselvesto staticexternalelectromagneticfieldscreatedbyalocalizedsourceconfiguration,for whichananalyticalexpressioncan bederived.

In the caseof staticexternalfields A,.(x) the exactFeynmanpropagatoris homogeneousin time [96],i.e., SA(x, x’) = SA(X, x’, x0 — xi). Let us considerthe infinitesimal changeof the energydensity(sup-pressingthe A-dependencefor a moment),which undersuchconditionsreads

= tr[y°3°~SA(x, x’, x0— x~)]I~’=~. (5.24)

The right-handside of this equationcan be rewritten when consideringeq. (5.17) from which thefollowing differential equationfor ~SA(x,x’) canbe derivedby consideringinfinitesimalchangesin A,.:

(iyTM 3,. — eyTM A,.(x)— m)~SA(x,x’)= eyTM6A,.(x)SA(x,x’). (5.25)

Making useof this equationone is led to the expression

~9gvac(x) = —i tr[y°hA(x)~SA(x,x’, x0— x~)]l~~~— e~iA,.(x)itr[y~SA(x, x’, x0— x~)]I~’~,(5.26)

wherehA(x) is given by eq. (5.19c). In the secondterm of eq. (5.26) the vacuumpolarizationcurrent,generallydefinedas [96]

j~’~(x)= ei tr[yTM SA(x,~ = ~e(OI[~y~~]IO), (5.27)

occursexplicitly. Togetherwith the shorthandnotation

154 6. Plunienetat, The Casirnir effect

~WN(X) = —i tr[y°hA(x)~SA(x,x’, x0— x~)]l~~ (5.28)

for the first contributionin eq. (5.26),we obtainthe infinitesimal changeof the vacuumenergyof the

electron—positronfield in the presenceof staticexternalelectromagneticfields:

~Evac[A1= f d3x 6WN(X, A)_Jd3x~A,.(x,A)j~~(x,A)

= ~EN[A]+ ~EVP[A]. (5.29)

This is the basicequationfor our furtherdiscussion.Stimulatedby the conceptof Casimirenergywe haveanalogouslyderivedthe Dirac-vacuumenergy

basedon the same techniquesapplied for evaluatingvacuumenergiesof other quantumfields underconstraints. It is very satisfying that we have obtained a result which allows a direct physicalinterpretation.The secondterm in eq. (5.29) revealsthat onecontributionto the vacuumenergyrefersto the effect of vacuumpolarization.Converselywe could argue:Vacuum-polarizationeffectsin QEDcanbe understoodas a manifestationof a changein the zero-pointenergyinducedby the presenceofexternalelectromagneticsources.Thefirst term of eq. (5.29) is alsoof physicalmeaning.Its importantrole will becomeclearwhen we considerthe electron—positronfield in strong externalelectromagneticfields. As we shall discussin detail in section5.2 this term is closely relatedto the effect of the phasetransition from the neutral vacuum into a chargedvacuum as the new stable ground stateof theelectron—positronfield in strong electromagneticfields.

Let usnow show that the first term of eq. (5.29), i.e. ~EN = f d3x &WN’ vanishesin the caseof weakexternal fields A,.. For this purposewe considerthe usual eigenfunctionrepresentationof the exactpropagator:

iSA(x, x’, x0 — x~)= O(x0— x~)~ tfrk(x)IIJk(x ) exp[—irk(xo — x~)]

k>F

— O(x~—Xo) ~ ~1’k(x’)~’k(x)exp[—iEk(xO— xi)], (5.30)k<F

where the t//k form a complete set of single-particle solutions of the stationary Dirac equationhAt//k = ektfrk. Theeigenenergies

6k and the functions t/Jk are assumedto be continuousfunctionsof theparameterA. In the presentconsiderationthe externalfield A,. is assumedto be sufficiently weak thatall electronboundstateenergieslie abovetheFermi energy,i.e., Ek(A)> EF, whenA variescontinuously.We call such fields A,. subcriticalin contrastto so-calledsupercriticalfields, wherethis condition is nolonger fulfilled for someboundstateenergiesEk(A) after A hasreachedcertaincritical values A~.Theinfinitesimalchangeof the propagatoris given by

i~SA(x,x, x0— x~).=O(k0— x~)~ ~(Ifrk(x)4/k(x’))expHek(xO—x~)]

k>F

— O(x~— X~)~ ~i(t//k(x’)t/Jk(x)) exp[—irk(xo — x~)]

k<F+ (termsproportionalto i~ek(xo— x~)). (5.31)

6. P!unien etat, The Casimireffect 155

According to eqs. (5.28) and (5.29) only the first two termswill occur in SEN, while terms which areproportionalto ~Ek(x0—x~) vanish in the limit x’—*x. Insertingthe expansion(5.31) into eq. (5.28), theonly remainingtermsof ~WN arefound to be of the form

6WN(X) = —~{~ e~t5(ç1Jk(x)t/4(x))— ~ eko(t/4(x)t/Jk(x))}. (5.32)k>F k<F

Integrationoverall spacegives the result~EN= 0, becauseeachsingle term of eq. (5.32) is a variationof the norm integral andvanishes,i.e.,

J d3x~(c~k(x)c&~(x))= ~Jd3x t/4(x) c~k(x) 0.

Consequentlythe change of the vacuum energy of the electron—positronfield in the presenceofsubcritical,static,externalfields reducesto the part arisingfrom vacuumpolarization,i.e.,

~Evac(A)= ~E~~(A)= — f d3x~A,.(x,A)j~~(x,A). (5.33)

This result coincideswith that derivedby J. Schwinger[97],wherethe vacuumenergyis introducedinthe samemanneras hereby the conceptof Casimirenergy.Beforewe turn to investigatethe vacuumenergyin supercriticalexternal fields it is instructiveto give a morespecific explanationof the result(eq. (5.33)).

The evaluation of the vacuum polarization is now reducedto the calculation of the vacuumpolarizationcurrentj~,inducedby the externalelectromagneticfield. In the caseof subcritical fieldsthiscan bedoneby perturbationtheory,which is basedon theiterative solutionof the integralequationfor the exactpropagator:

SA(x, x’)= S0(x-x’)+ ef d

4yS0(x—y)ft4(y)SA(y,x). (5.34)

According to the definitionof thevacuumpolarizationcurrent(5.27)j~can be calculated,in principle,up to arbitrary order. In this way one obtains the perturbationexpansionfor j~,where eachcontributing term carriesan odd powerof the externalfield A,..This follows as a direct consequencefrom Furry’s theorem[98] which statesthat the seriesexpansionfor the vacuumenergycontainsonlytermsof evenpowersin A,.. This fact guaranteesthat ~Evac doesnot dependon the sign of the externalsourcesin consistencewith the requirementof chargeconjugationinvariance.

First-ordervacuumpolarization(one-loopcorrection)representsthe dominantcontribution to thevacuumenergy,and it is the only term which needsregularization[99]. For laterpurposeswe shortlydiscussthe first-orderterm ~E~~(A)= —f d

3x &A,.(x, A)j~1~(x,A) in the particularcaseof a staticchargedistribution Pext(X, A) as the externalsourceconfiguration. In general,the inducedvacuumpolarizationcurrent is relatedto the externalcurrentby

j~1~(x,A) = fd4yH~(x- y)j~(y), (5.35)

156 6. Plunieneta!., The Casimireffect

whereH~,,denotesthe renormalizedfirst-order (Uehling) polarization function. It hasthe Fourierrepresentation[96]

H~(p2)= - ~ f dzz(1- z) ln [i - p2z(1- z)] (5.36)IT m—1~

0

andits imaginary part reads[96]:

Im{H~(p2)}= —~a(1+ 2m2!p2)(1 — 4m2/p2)6(4m2/p2— 1). (5.37)

Fora staticexternalchargedistribution,Pext(’, A), the changeof the electrostaticpotential~iA0(x,A) is

simply relatedto the changeof thechargedistribution accordingto

~A0(x,A) = J d3x’ ~pext(X’,A) (5.38)Ix—xl

The first-orderterm of ~Eyacthen takesthe form

6E~(A)= — f d3x’ d3y ~pext(X’,A)pex

t(y, A) Jd3x H~(x~ (5.39)

— x

The integral over x can be evaluatedby inserting the Fourier representationof the polarizationfunction:

J d3x = f -~-~jexp[ip . (x’ — y)]H~(—p2)Jd3zexp(ip. z)(21T) zl

x(lx’—yI)= , = W(lx —yI), (5.40)

x

where the functionx(lx’ — ~I) is definedby (p = ~I)

x(Ix’ - ~I) = J dp sin(plx’ - yl)I1~(-p2). (5.41)

Inserting the expression(5.40) into eq. (5.39) one observesthat the integrandis symmetricalin thevariablesx’ andy. This property allowsoneto rewrite eq. (5.39) in the following form:

~E~~(A)= — J d3x’ d3y~pext(”, A )pext(y, A) W(jx’ — ~l]

= —~ f d3x’ d3ypext(X’, A)pext(y, A)w(jx’ — yl)

= ~E~(A). (5.42)

6. P!unienet at, The Casimireffect 157

The function W(jx’ — ~l)represents the effectiveCoulombinteractionbetweenthe externalchargesdueto the one-loopvacuumpolarizationcorrection.The correspondingFeynmandiagramis shown in fig.5.2. The aboveequationcan now be formally integratedover the parameterA. In accordancewith thedefinition of the vacuumenergyas being the differencebetweenzero-point energies,i.e., Evac(A) =

E0(A) — E0(A0) onedirectly obtainsE~~(A).The physicalmeaningof the first-ordervacuumenergydependson therole of the variableparameter

A chosento characterizethe chargeconfiguration. In the particular casethat A describesthe relativedistancebetweentwo externalchargedistributions,one easily verifies that ~ is identical with theUehling correction to the normal Coulomb interaction. Let us verify this point explicitly for theparticularconfigurationof two point chargesof oppositesign and taking the separationR = RI as theparameter:

pext(X, R) = e~(x — R). (5.43)

The vacuum energy we are interested to calculate is given by the difference E~~(R)=E~(R)— E~”(R—~cc). The configuration,wherethe chargesareinfinitely separatedcorrespondsto thesituationof vanishingexternalfield, i.e., Ao(x, R —~cc) = 0. Insertingthechargedistribution(5.43)into eq.(5.42) one obtainsthe expression

E~(R)= —e2Jd3x’ d3y~(x’)3(y)W(lx’ — ~I) + e2W(R). (5.44)

The first divergentand R-independentterm correspondsto the classicalself-energycorrection of thepoint chargesdue to the vacuumpolarization, which cancel in the vacuumenergy.As expected,thesecondterm representsa potential energycontribution. Using the fact that the function ~(R) (eq.(5.41) can be expectedin termsof the imaginarypart of the polarizationfunction (5.37) accordingto

2~ e’~~(R) = — j dq Im{H~(q2)}, (5.45)

IT q0

anddueto the fact that W(R—~cc) = 0, the vacuumenergybecomes

e22aE~j~(R)= — — — j d~ (1 + 1/2~2)(1— 1/~2)h/2. (5.46)

R3IT

This result is identical with the Uehling correction[102,103] to the attractivepotentialbetweentwopoint chargesof oppositesign. The fact that oneobtainsthe first radiative correction to the Coulomb

Fig. 5.2. One-loopvacuum-polarizationcorrectioncontributingto thevacuumenergy.

158 G. Plunienetat, TheCasimir effect

interactionis not in itself surprising,but its derivationis the interestingpoint. In thisexamplewe haveexplicitly usedthe zero-pointenergyof the Dirac field in the presenceof the static,externalCoulombfield of two chargesin orderto derive the interactioncorrectionbetweenthe sourcesas function of theseparation.This derivation follows proceduressuccessfullyapplied in derivationsof van der Waalsattractionbetweenpolarizableparticlesbasedon the zero-pointenergyof the electromagneticfield. Thefact that in the casediscussedabovethe conceptof vacuumenergyalsoleadsto reasonableandphysicalresults,givesconfidencein the underlying ideasandmethods,andit furthersupportsthe viewpoint thatzero-pointenergiesof quantum fields may be of general interestin the explanationof the origin ofinteractions.

To give also an example for a vacuum energy treated as a function of a parameterwhichcharacterizesthe geometryof an external source configuration, let us consider a homogeneouslychargedsphereof radiusa. In order to evaluatethe correspondingvacuumenergyit. is useful to startfrom its derivative

E~(a) = — f d3x’ d3y(~pext(X’, a)) pext(y, a)W(Ix’ — yI), (5.47)

wherethe externalchargedistribution is given by

3Ze 1pext(X’, a) = —~ O(a — jx’I). (5.48)

417 a

Insertingeq. (5.40) andperformingthe spatialintegrationsleadsto the remainingintegral:

E~(a)= - 18(Ze)2 j dPPH~(_P2)11(aP)12(ap) (5.49)IT ap ap

0

This expressioncan be evaluatedanalytically in the limits a —*0 and a —~cc~The first casemay beapplicablefor calculatingself-energycorrectionsof microscopicparticles,when theyareassumedto behomogeneouslycharged.Substituting-~ = ap andinsertingthe approximateexpressionof the-polariza-tion function in the casefor am 4 1, oneobtains

E~(a) = 6a~ Jd~(~-ln ( ~2 2)) J1(~)12(~) (5.50)ôa ~a (am)

0

Within this approximationit follows:

E~(a)= ~ a2 + 2 ln(am))c~+ 2c2], (5.51)

wherethe constantsc1 and c2 aredeterminedby thefollowing integrals:

6. P!unien et a!., The Casimireffect 159

= Jd~~Ji(~)J2(~) = ~, (5.52a)

C2= Jd ~ ~ (5.52b)

with the Euler constant y = 0.577 ... The vacuum energyof a chargedsphere is then obtainedaccordingto eq. (5.1):

2 ‘Z~ 1E~(a) = — ~‘ “ (In (—‘~ — y + i’). (5.53)

5 ira \ ~2ami /

This vacuumenergyis positive and supportsthe normal Coulombrepulsionwhich tries to expandthesphere.

We can summarizeso far: Within the local formulationone finds that the polarizationof the Diracvacuumis the importantphysical content of the vacuumenergy. Evac(A) can be understoodeither ascontributionto the interactionbetweenelectric chargesor as a correction to the Coulombenergyof agiven chargedistribution.

5.2. Thesupercritical vacuum

In the previoussectionwe haveshownthat the conceptof Casimirenergyis applicablein QED andallows for an interpretationof the zero-pointenergyof the Dirac vacuumin the field Hamiltonian.Defining the vacuumenergyin the presenceof externalelectromagneticfields in termsof a differencebetweenzero-pointenergiesno additionalformal difficulties occur.

We now turn to discuss the vacuum energyof the electron—positronfield in the presenceofarbitrarystrong,static electric fields. Let us first explain what is meantby “strong” fields. Considerapositiveexternalchargedistribution~ A) as the sourceof the static field A0(x, A). Correspondingly,the energyspectrumof the stationaryDirac equationcontainselectronboundstatesat energiesv~(A)betweenthe gap,which arealso functionsof a certainparameterA. Assumingsuch an externalchargeconfigurationthattheboundstateenergiesaremonotonicallydecreasingfunctions,i.e.,e~(A1)>e0(A2) forA1 < A2, we classify the externalfield in the following way: (a) A field is called “subcritical” if all boundstateslie abovethe Fermi energy

5F = —m, i.e., particularlyfor thelowestboundstater0(A)>

5F~(b) Avalue A~at which the energyof the lowest bound state reachesthe lower energycontinuum, i.e.,

SF, is called “critical”. (c) When the boundstate has joined the positron continuum, i.e.eo(A)< 5F, the externalfield is classifiedas “supercritical”.Figure5.3 illustratesthe energyspectrumofthe Dirac equation in an attractivestatic potential as a function of A. In order to discretize thecontinuumtheconsideredexternalchargeconfigurationmaybeenclosedin a largebut finite box. In thesubcriticalregion the eigenenergiesare obtainedfrom solutions of the Dirac equationfor localizedstates.Modified techniquesarerequiredin the caseof supercriticalfields, becausethe supercriticalstateappearsonly as a resonancein the positron continuum.The energyaround which the resonanceislocalized, r~(A),is identified as the continuation of the energyeigenvalueso(A) into the region ofsupercritical externalfields. In general, the distinction betweensubcritical and supercritical fields is


�

- --

C~(X) “

Fig. 5.3. Schematicdependenceof theenergyof lowestboundstateson thestrengthparameter.

closely connectedwith the fact, whetherthe deepestboundstateappearsin the gapbetween+ mand—m or whetherit is admixedto the lower continuumhasan immediateconsequencethat is mosteasilyunderstoodin the framework of Dirac’s hole theory. As mentionedbefore,the distinction betweenparticle and antiparticle states implies the definition of the Fermi energy, 5F = —m in potentialsattractiveto electrons.If, in supercriticalexternalfields an unoccupiedelectronboundstatecrossestheFermi energy,a hole is introducedinto the Diracsea.This holecan be filled without additionalsupplyof energyby a sea electronthat leavesa hole in the continuum. In terms of the hole picture thissituationcorrespondsto the processof spontaneouselectron—positronpair production.In the attractivepotentialthe electronbecomesstrongly boundwhile the positronescapesto infinity.

After this processhasterminated,the sourceconfigurationwill be surroundedby a strongly boundelectron.The former neutralvacuum(only the baresourcesand no other localizedreal charge)turnsinto the chargedvacuum (with the surroundingelectroncloud present)as a new stablegroundstate.The existenceof thedecayof theneutralvacuumin QED of strongexternalfieldswas predictedin theyears1970—73by two groupsatFrankfurt [104—109]andat Moscow [110—114].It hasbeenexperiment-ally studiedin connectionwith heavy-ioncollisions,wheresupercriticalfields can be realized.We shallcomebackto this topic in the nextsectionwhenwe discussthe vacuumenergyin the particularcaseofnuclearcollisions.

After these few introductory commentswe now enter a discussion of the vacuum energy insupercritical (static) external fields. Let us consider static external fields producedby a chargedistributionwhichis characterizedby aparameterA. We will not specifyA, but we assumethat the fieldis attractivefor electronsandthat bound-stateenergiesshouldvary with A as illustratedin fig. 5.3. Weagainstartfrom the infinitesimalchangeof the vacuumenergy:

~Evac(A) = ~EN(A)+ ~E~~(A), (5.54a)

6EN(A) = Jd3x (—i) tr[y°hA(x)~SA(x, x’, x — x~~ (5.54b)

6. P!unien eta!., The Casimir effect 161

~EVP(A) = — Jd3x bA0(x, A)p~~(x,A), (5.54c)

andcalculate~EN and~ separately.We show now that ~EN(A) contributesto the vacuumenergy in supercriticalfields and that it is

directlyconnectedwith the chargecontainedin thevacuum.For thispurposeit is usefulto expandtheexactpropagatorin termsof eigenfunctionstfrk(X IA) [97, 115]

iSA(x, x’, x0— x0 A) = O(x0—x~)~ O(s~— ~F)t/’k(’I A)t/Ik(x’IA)exp[—iEk(xO—x/~)]

— O(x~—xo)~O(EF— rk)t//k(xIA)t/Fk(xlA)exp[—isk(xO—xo)]. (5.55)

Both summationsrun overthe totalenergyspectrumof the Dirac Hamiltonian.The functionsO(~k — ~F)

and O(EF— ~k) guaranteethe correct distinction betweenparticle and antiparticle stateswhich areseparatedby the Fermi energy

5F = —m.The representationof the exactpropagator(5.55) is identicaltothai of eq. (5.30),but it hasthe advantagethat the crossingof the Fermi level by a boundstatecan beconsideredin a simple way. This becomesobviouswhen we considerthe infinitesimal changeof thepropagatorandcalculatethe variation of the energydensity~WN, which reads:

~WN(X, A)= ~ {O(ek(A)— CF)ek(A)~pk(x, A)— O(EF— Ek(A))5k(A)~Pk(X, A)}

— ~ ~(r~(A)— E)65k(A)pk(x, A)ek(A), (5.56)

where we have used the shorthandnotation pk(x, A) = l//k(X I A)ç14(x I A) for the normalized singleparticle densities.One recognizesthat the first term in (5.56) is the one alreadyshown to vanish forsubcritical fields (eq. (5.32)),whereasthe secondterm contributeswhenevera boundstatereachestheFermi level at a critical valueA~,counting the numberof stateswhich dive into the Dirac sea.Afterspatialintegrationof eq. (5.56) we find simply:

— ~ 5(Ek(A)— sF)rk(A)~ek(A). (5.57)

In order to obtainthe energyEN(A) contributingto Evac(A), the integrationover the parameterA mustbe performedaccordingto

E~(A)= — J dA’ ~ ~(ek(A’)— SF)Ek(A’)ÔA’Ek(A’)

= 5Ff dA’3A’ (~O(CF—

= ~F ~ (O(SF— sk(A))— 9(EF— ek(Ao))). (5.58)

162 6. P!unien et at, The Casimireffect

We mustnow recall that the k-summation(including spin degeneracy)only runsover boundstatesthatbecomecritical at values A ~ when the parametersvary continuouslybetweenA0 and A. The secondstep-functiontermin (5.58) vanisheswith respectto theconditionsk(Ao)>

5F~Consequentlywe find theresult

EN(A) = ~F ~ O(EF— Ek(A)) = EFN(A). (5.59)

This vacuumenergycontributionis negativeandit hasthe form of a stepfunction (seefig. 5.4). Eachdiving boundstatereducesthe vacuumenergyby an amountequalto the electronrest mass m. Thesuddenlowering of the vacuumenergyreflects thespontaneouscreationof an eIectron—positro~npair inasupercriticalfield. The vacuumenergyinside alargebox enclosingthe systemis reducedby the energycorrespondingto the rest massof the spontaneouslyemittedpositrons,while the vacuumbecomeschargeddueto the electroncloud surroundingthe origináib~à á1ëEài~ëebffgUfäIiOñT

This may be expressedin a different way by saying that a supercriticalfield must be treatedas anopen system,sinceit exchangesenergywith its surroundingsby particleemission.The changein particlenumbercausesa suddendrop in the vacuumenergy.The connectionbetweenparticle exchangeandvacuumchargecan be made moreexplicit. Consideringthe changein the vacuumpolarizationchargedensity

~ A) = ie tr[y°aSA(x,x’, x0— x~A)IIX.X (5.60)

andperformingthe samemanipulationsthat led to eq. (5.59)onefinds the following expressionfor thechangeof the vacuumcharge:

SQvac(A) = — e ~ 3(sk(A) — EF)~sk(A), (5.61)

and afterintegrationover A:

Qvac(i~t) = eN(A). (5.62)

The negativevacuum charge is carried by the electroncloud which is formed around the external

EN CXcr Acr

I IA

-2m-

-4m-

Fig. 5.4. Suddendecreaseof thevacuumenergywhenelectronboundstatesbecomesupercritical.

G. P!unien et a!., The Casimireffect 163

charges,when the electric field becomesovercritical. For reasonsof chargeconservationthis vacuumchargeis equalto the numberof emittedpositrons. Relation(5.62) allows oneto rewrite eq. (5.59) in aform which makesthe connectionbetweenvacuumchargeandenergymanifest:

EN(A) = ~ Qvac(A). (5.63)

Thus we haveobtaineda generalexpressionfor the vacuum energycontribution EN(A). We havedone this without explicitly specifying the external chargedistribution to underlinethat the phasetransitionto the supercriticalvacuumhasto be envisagedas a basicQED effect.Whenwe nowturn toevaluatethe part E~~(A)arising from vacuum polarization,we will sometimesmake referenceto theresults of theoretical investigations performedin the context of strong fields arising in heavy-ioncollision. However,the basicmethodswhich wereadoptedin this contextshouldalsobeusefulin moregeneralcases.

In the following evaluationof EVP(A) we restrict ourself to the simplest situationwhereonly thelowest bound state dives into the positron continuum. Due to spin degeneracythen the vacuumbecomeschargedtwice, i.e.,

Qvac(A) = J d3~pvp(X,A) = —2e0(A— A~). (5.64)

Undersuchconditionsit is legitimate to neglectthescreeningof theoriginal field of the externalchargeconfiguration by the vacuum charge. One now needs an appropriateexpressionfor the vacuumpolarizationchargedistribution. This point hasbeenstudied intensely[106, 107] and only the mainresultsnecessaryfor our purposewill be reportedhere.

Let us first considerthe Feynmanpropagatorin the externalfield SA(x, x’, x0 — x~A) which can be

expressedas a contour integral in the complexenergyplane:

SA(X,x’, x0 — x~A) = I ~ exp[ie(xo — x~)]G(x,x’, s; A). (5.65)

The Greenfunction G(x, x’, s; A) fulfils the equation

(hA(x, A)— s(A))G(x, x’, e; A)= 6(x—x’), (5.66a)

where hA(x, A) is again definedin eq. (5.19c). Its solutioncan be written in terms of a sum over the

spectrumof hA satisfying(hA — ek)’frk = 0, namely,t/fk(XIA)tI/k(xIA) (5.66b)

k 55k(A)

Each singularity of the Green function (5.66b) correspondsto an energyeigenvalueof the stationaryDirac equation.The Feynmanpropagatoris determinedby the choice of the integration contourillustratedin fig. 5.5, in the caseof a subcriticalconfiguration.

164 6. P!unien eta!., The Casimireffect

ImE

r ~I X X-m m Re~

Fig. 5.5.Theconventionalchoiceof the contourdetenniningthe Feynmanpropagator.The contour C crossesthereal axisat the Fermi energy= — m.

The two cuts beginning at e = ±mas well as the poles associatedwith the boundstatesbetween= —m and s = m are shown.The choiceof the contourC plays the samerole as the choice of the

Fermi energy.The contourcrossesthe real axis at the Fermi energy5F = —m and,thus, achievesthedistinction betweenparticle andantiparticlestates.

In accordancewith the definition of the vacuumpolarizationdensityoneis led to the representationin termsof contourintegralsafter WichmannandKroll [116]:

p~~(x,A)= £:~Idstr[y°G(x, x, e; A)]

= —f- { J ds tr[y°G(x, x, e; A)] + J dstr[y°G(x, x, e; A)]}. (5.67)4iri

C—

In fig. 5.6 thecontours,i.e. C÷andC_, areillustratedfor asubcriticalsituation.The contoursC±andC_enclosing the particle and antiparticle states,respectively,crossthe real s-axisat the Fermi energy5F= —m. The vacuum chargeis equal to zero in the subcriticalcase, i.e., the vacuum polarizationdensity(5.67) describesonly local chargedensityfluctuationsinducedby the externalelectric field. Forthis reasonit is calledvirtual vacuumpolarization.

Im�

-m m)( )C XXXI(1) (2) ReE

Fig. 5.6. Thecontourchosenby Wichmann andKroll to definevacuumpolarizationin thesubcriticalcase.

6. Plunieneta!., The Casimireffect 165

We now turn to the supercriticalvacuum,wherethe total vacuumpolarizationcan bedivided into avirtual part and the real part, which is responsiblefor the non-vanishingvacuum charge of thesupercriticalvacuum.It is bestto considerwhat happensin thecomplexs-planewhenthe externalfieldbecomessupercritical.When the field approachesthe critical strength,the polesassociatedwith certainbound statesapproachthe Fermi energy. When the field exceedsthe critical value, the pole (1)associatedwith the lowestboundstatemovesoff the real axis into the upperhalf planeon the secondRiemannsheet[117] (seefig. 5.7).The imaginarypart of thepoleenergy,~t = ~r + iF/2, reflectsthe factthat the formerboundstatehasbecomean unstableresonancein the lower continuum.Maintainingtheneutral vacuumas the referencestatewould imply choosinga contour that surroundsthe poles in thesameway asin asubcriticalfield. Consequently,the contourC in fig. 5.6mustbe deformedto a contourC’ as illustratedin fig. 5.7. It can be shownthat a choiceof sucha contourleadsto an unstablevacuumstate[106,107]. In order to definea new stablevacuumstatein a supercriticalfield, onehas to keepthecontourD unchanged(seefig. 5.8). Correspondinglythe Greenfunction G(x, x, s;A) is definedsuch asto includeonly the polesremainingon the real axis.

The vacuumpolarizationchargedensity is then calculatedaccordingto

p~~(x~A) = —f- Jdstr[y°G(x, x, s; A)]. (5.68)4171

D

This expressioncan be divided into the contributions of virtual and real vacuum polarization:Integrationalong contourC’ (see fig. 5.7) would representthe analyticalcontinuationof the virtualvacuumpolarization.In order to obtain the result (5.68), one hasto addthe contributionof the realvacuumpolarization,whichmaybe calculatedby integratingalongcontourR (fig. 5.7). In termsof realandvirtual vacuumpolarization(5.68)can be written as [107]

p~~(x,A) = p~(x, A)+ p~(x,A)

= —~- J de tr[y°G(x, x, s;A)] +—~- Jde tr[ y°G(x,x, s; A)]. (5.69)4iri 2iri

C~ R

Im CC’

--.~ ~

-

(~) ~ ~

Fig. 5.7. The contours determiningvirtual (C+;C_) and real(R) vacuumpolarizationin thesupercriticalcase.

166 G. Plunien eta!., The Casimir effect

ImE

x(1)

)( )C ~(2) Re~

Fig. 5.8. Thecorrect integrationcontourfor theFeynmanpropagatorin thecaseof supercriticalexternalfields.

The two chargedensitiessatisfy the conditions(including spin degeneracy)

f d3xp~(x,A)= 0, Jd3xp~(x~A)= —2e, (5.70)

which explicitly statesthat the discontinuousbehaviourin the vacuumchargeoriginatesfrom the realvacuumpolarizationwhich is presentonly in the supercriticalvacuum.

For the evaluationof the vacuumenergycontributionE~~(A)in supercriticalfields the representation(5.69) for p~is not very practical.In orderto obtaina moreconvenientexpressionfor it, one canmake-useof-the fact--that-the-exactresonancestateapproximately-behaveslike a-bound--state4ll8]-.--Therefore- -it is reasonableto construct a quasi-boundstate 1/Jr as an eigenstateof a slightly modified DiracHamiltonian (indicatedby a tilde) which hasan eigenvalue~r equalto the real part of the resonanceenergys~:

hA(x,A)çlir(x, A) = 5r(A)tfrr(X, A). (5.71)

Diagonalizingthe positronstateswith respectto this Hamiltonian,one achievesthat 1/er is orthogonaltoall of thesestates.This procedureis carried out by a projectionmethod [118] that was successfullyappliedin the caseof the supercriticalvacuumin the strong Coulombfield of superheavynuclei. Thereal vacuumpolarizationchargedensity is then approximatelygiven by

p~(x,A) = —2eI4’Jr(x,A)j2. (5.72)

It is important to note that in supercritical field the virtual vacuum polarization does not differsignificantly from that in subcritical fields. It has also been shown conclusively [119—121]that noanomalousbehaviouris found in higher-ordercontributions,andthat the dominantcontributionto thevirtual vacuumpolarizationis given by thefirst-order Uehling termevenin the supercriticalcase.

Consideringthesefacts,we calculatethe part E~~(A)of thevacuumenergyaccordingto the relation

3AEV~(A) = — J d3x 3AAO(x,A)p~(x,A) + 20(A — A~)Jd3x t/4(x, A)e3AAO(x,A)1/Ir(X, A)

= 3AE~(A)+ 3AE~(A). (5.73)

6. Plunien eta!., The Casimir effect 167

With the help of the Hellmann—Feynmantheorem [122] the part arising from the real vacuumpolarizationcan be rewritten:

aAEVP(A)= 20(A — A~)8Ae~(A). (5.74)

After integrationover A oneobtainsthe result

E~~(A)E~(A)+ 20(A — A~)(s~(A)— EF), (5.75)

wherethe relation e/(A~)= 5F hasbeenused.Oneobservesthat the real vacuumpolarizationlowersthe vacuumenergyby an amountequalto the diving depthof the supercriticalstate.It is also obviousthatE~~(A),is a continuousfunction of the parameterA.

Adding both contributions (eqs. (5.59) and (5.75)) oneobtains the total energy Evac(A) of thesupercriticalvacuum.If only the lowestboundstatedives into the positroncontinuumit explicitly reads

Evac(A)= E~(A)+ 20(A — A~)s~(A). (5.76)

Thus, one is led to the following conclusion:The vacuumenergyof the electron—positronfield in thepresenceof a staticexternalCoulombfield createdby a given chargeconfiguration, in general,consistsof two parts which correspondsto the effect of vacuum polarization and the effect of the phasetransitionof the neutral vacuum into the chargedvacuum as the new stablestate in supercriticalexternal fields. In the subcriticalcasethe vacuumenergycoincideswith the interactionenergyof thevirtual vacuum polarization with the external field. In the supercritical case the vacuum energy isabruptly lowered by the resonanceenergiesof supercritical states.Expression(5.76) can be easilygeneralizedto the casewhen several bound statesmay becomesupercritical. Then, of course,thescreeningeffect due to the real vacuumchargesurroundingthe original chargeconfigurationmust betakeninto accountwhenE~(A)andthe eigenvaluess~(A)of supercriticalstatesarecalculated.

Summarizingthis section,wehaveshownthat the zero-pointenergyof the electron—positronfield inthe presenceof externalelectric fieldsmaybe utilized to calculatevacuumpolarizationandthe effect ofthe phasetransition from the neutralto the chargedvacuum.This representsonemoreexamplefor asuccessfulapplicationof the conceptof a vacuumenergy.

5.3. Thevacuumenergyin nuclearscattering

In order to makeour generalconsiderationsconcerningthe vacuumenergyof the electron—positronfield more specific,we now turn to discussthe role of the vacuumenergyin nuclearscattering.Thisproblemis of considerableinterest,becausein heavy-nuclearcollisionsstrongelectromagneticfields canbecreatedwhich, in fact, permitexperimentaltestsof the predictedphasetransitionof the supercriticalvacuumto be carriedout.

The evaluationof the vacuumenergyin thepresenceof the electromagneticfield of scatteringnuclei,treatedas a time-dependentproblem, is no simple problem. In a semi-classicalapproach,wherethenuclearmotion is treatedclassically,onewould parametrizethe externalfield by the trajectoriesof thescatteringnuclei. Insteadof a full dynamical treatmentwe shall representthe nuclei by static chargedistributionsandtake their distanceR as the variableparameter.The geometryof the externalchargedistribution is illustrated in fig. 5.9. The two nuclei are representedby two homogeneouslycharged

168 6. P!unien et at. The Casimir effect

Fig. 5.9. Determinationof themodel chargedistribution for two scattering nuclei.

sphereswith nuclearchargesZ, andwith fixed radii a. The latter are determinedby the massnumbersA, of the nuclei accordingto the empiricalformulaa, = 1.2 A~’3.We takethe chargedistribution

Pexs(X, R) = ~ —~-~ O(a, — Ix — R1I). (5.77)

~=~4ira-

Treatingthevacuumenergyas a function of the nucleardistanceR = IRI = IR1 — R2I, it is definedas the

difference between the zero-point energies: Evac(R)= E0(R)— E0(R—* cc). The case of infinitelyseparatednuclei correspondsto the free vacuumconfiguration,since A0(x,R —* cc)= 0 in a finite boxlocatedat the center-betweenthe two nuclei. For scatteringnuclei the two-centerdistancecan takevaluesR > a1 + a2.

In the previoussection it has been shown that the vacuum energy (eq. (5.76)) consistsof thecontributionarisingfrom the virtual vacuumpolarization,which is suddenlylowered by the energyofthe supercritical statewhen the externalfield becomessupercritical. Both effects can be calculatedseparately.

The dominant contribution of E~j(R)is given by the first-order vacuum polarization, and oneexpectsthat it representsthe Uehling potentialfor extendednuclei. This is easilyverified by a simplecalculationstarting from eq. (5.42). Inserting the chargedistribution (5.77) one obtainsafter obvioussubstitutionsof the integrationvariables:

E~(R)= -~ Jd3z’d3zp

1(Iz’I)p,(IzI) W(Iz’ - zI)

- J d3z’d3zp

1(z’)p2(Jz— RI)W(Iz’ — zI). (5.78)

The first two terms in this equationcorrespondto the Coulomb energycorrectionof the individualnuclei dueto the vacuumpolarization.This part cancelsby the subtraction,sinceit is independentofthe separationR. The interaction term can be expressedas a Fourier integral over momentaandvanishesin the limit R —* cc~Consequentlyone obtainsthe Uehling correction

6. Plunien eta!., The Casimir effect 169

EUh(R) ~!Z1Z2e2Jdp sin(PR)[J(l)(~2)J1(”1P) j

1(a2p) (5.79)ir R p a1p a2p

0

The sphericalBessel-functionsj~arise from the Fourier transformednuclearchargedensities.Withsimilar argumentsasusedin the derivationof the Uehling potentialin the caseof pointchargesEuh(R)can be alsoexpressedin termsof the imaginary part of the polarizationfunction

EUh(R) = Z1Ze2~ Jd~e (1+ 1/2~2)(1— 1/~2)h/2i

1(2ma1~)i1(2ma2~) (5.80)R ir (2ma1~)(2ma2~)

where i1 aremodified sphericalBesselfunctions.The integral is well definedfor valuesR > a1 + a2.Thus we havederivedthe (repulsive)Uehling potentialbetweenextendednuclei characterizedby a

homogeneouschargedistribution. The result concerningpoint-like nuclei is regainedperforming thelimit a1, a2-~0. In figs. 5.10 and5.11 the Uehling potentialis shownfor two combinationsof scatteringnuclei, namely, Pb+ Cm and U + U. The behaviour of thesecurves is dominated by the factor

E [MeV]EUH [MeV] ((H

u+u\ Pb+ Cm (extended nuclei)

(extended reictei)

2.0- 2.0-

1.5— 1.5-

1.0— 1.0-

0.5— 0.5-

REfm] R[fm]a I I i 0 I I I i20 30 40 50 20 30 40 50

Fig. 5.10. The Uehling potential for the scatteringextended nuclei Fig. 5.11. Thesameas in fig. 5.10 for U+U.Pb + Cm as a function of the separation.

170 6. P!unien eta!., The Casimir effect

(Z1Z2e2/R).The first-orderradiationcorrectionto the interactionpotentialbetweenscatteringnuclei is

smallcomparedwith the bareCoulombinteractionandbecomesonly of the orderof 1 MeV in the caseof heavy systemslike U + U. The externalCoulomb potential createdby two such nuclei becomessupercriticalfor nucleardistancesR smaller than a certain critical value R~

1.Then, of course,thechangein the chargeof the vacuummust be takeninto account.

In order to determinefor which combination of scatteringnuclei the Coulomb field becomessupercritical,oneconsidersthefollowing problem[106]:Assumeabare“supernucleus”associatedwitha homogeneouslychargedspherewith total chargeZ. = Z1 + Z2 andwith a radiusa = 1.2(A1+ A2)Us.Solving the Diracequationwith the correspondingCoulombpotentialit turns out that theenergyof theis statedecreasesmonotonicallyas a function of Z. andjoins the positroncontinuumat a totalcriticalchargeZcr 173. According to this value we classify the subcritical (Z < Zcr) and the supercriticalsystems(Z> Zcr) of scatteringnuclei.

Concerningthe scatteringproblem it only dependsnow on the two-centerdistanceR whethertheCoulombpotentialremainssubcriticalor whetherit becomessupercritical.If only the lowest molecularboundstateis joins the positroncontinuum,the vacuumenergyin accordancewith eq. (5.76) reads

Evac(R) Euh(R) + 20(Rcr— R)s1~(R). (5.81)

In order to calculatethe energyof the molecular is stateas a function of R, onehasto solve thestationary two-centerDirac equation.The energyof the supercritical is state is determinedby theresonanceenergyof thesupercriticalstate[106].Quantitativeresultsfor thevacuumenergyareshowninfigs. 5.12 and5.13for the supercriticalsystemsPb+ Cm (Z11 = 178) andU + U(Z~= 184), respectively.Inthisadiabatictreatmentthe electricpotentialbecomescritical at certaindistancesRcr, which arefoundtobe: Rcr= 23.3fm for thesystemPb+ CmandRcr = 32.8fm for thesystemU + U. TheUehlingpotentialbecomessuddenlyloweredby theamountof twicetheenergyof thesupercriticalstate(spindegeneracy)forseparationsR <Rcr. In bothcasesthevacuumenergyrepresentsacorrectionof approximately1 MeVtothe normalCoulomb-interactionpotential,whereastheUehling potentialproducesmeasurablebut onlysmallmodifications.

It is appropriateto addone remarkconcerningthe effect of the phasetransitionof the supercritical

E [MeV] ~ 1MeV]10 — u+uPb~Cm . (extendedrsiclei)

____ I I

20\ 30~ 40 50R[fml N.~Q I Iii I I I I ~ -0.2 ~ I

20 30 40 - 50Fig. 5.12. The vacuum energy for the scattering extendednuclei Fig. 5.13. The sameasin fig. 5.12for U + U.Pb+ Cm asafunction of theseparation.

G. Plunienetat, TheCasimireffect 171

vacuumat critical distances.If the suddendrop in Evac(R) hasbeenderivedfrom an adiabaticmodelofa scatteringprocessthis jump would neverappearin such a pronouncedway, becausethe phasetransitionof the supercriticalvacuumis not an instantaneousprocess.The time-scale in the processofspontaneouselectron—positronpair production(the observableeffect of the phasetransition)is of theorder -=i019s, while in Rutherfordscatteringthe supercriticalfield may be presentfor only 1021s

[118].Nevertheless,this adiabaticpicture of nuclearscatteringallows a simple interpretationof thevacuumenergy.As alreadymentionedEvac(R) plays the role of a correctionto the static Coulombinteractionpotential. Two possible situationscan be distinguished:In the caseof subcritical systems,i.e., if no phasetransition from the neutral to the chargedvacuumcan take place during the nuclearscatteringprocess,the vacuum energy essentially representsthe Uehling correction to the staticinteractionpotentialV(R) betweenthe nuclei.Figure5.14aillustratesthescatteringpotentialfor such asituationasa function of the nucleardistanceR. The two nuclei feel the sameinteractionV(R) in bothdirections,i.e., whentheyapproacheachotherandwhentheyseparateagain.Thesituationis different,if thereis a changeto the chargedvacuumwhile the nuclei comeclose together.Then the vacuumenergyand, consequently,the interaction potential is abruptly lowered. When the distanceR againincreases,the two nucleifeel a potentialwhich is lowereddueto the bindingeffectsof the two electronsin thesupercritical is state(seefig. 5.i4b). In a plot of V versusR the systemthenfollows a differentpotential curve. Such a change in the interaction potential, causedby the vacuum energy,may beenvisagedas a kind of hysteresis effect in nuclearscattering.

Let us shortly summarize:The aim of this sectionwas to show that the conceptof Casimir energyallows oneto define a physical vacuumenergyof the electron—positronfield as a difference betweenzero-point energies.Applying argumentsused in connectionwith the Casimir energyof the elec-tromagneticfield we showedthat the vacuumenergyof the electron—positronfield is equivalentto theinteractionwith the vacuumpolarization,andthat the phasetransitionof the QED vacuumis reflectedin a decreaseof the vacuumenergyin the presenceof supercriticalexternalelectromagneticfields.

E,,~ I Evac

\\\\\

R~RCrR(a) (b)

Fig. 5.14. Schematicdiagramfor thevacuumenergyversustheseparationof scatteringnuclei: (a) for a subcriticalfield and(b) for asupercnticalfield.


6. Casimirenergyat finite temperature

6.1. Partitionfunctionsandfree energy

In the previoussectionswe have shown how one can define the energyof the vacuum stateofquantumfields basedon the correspondingfield Hamiltonian obtainedby canonicalquantizationandwe haveseenhow to deal with infinite zero-pointenergiesin ameaningfulway. The evaluationmethodsfor Casimir energiesdiscussedaboveallow one to extract their physical implications and observableconsequences.

So far we haveconsideredthe responseof thevacuumdueto the presenceof externalconstraints.Inreality fields exist at finite temperatureand contain real quanta, i.e., a large number of statesareoccupied.Suchconditionsrequirea quantumstatisticaldescription.In the following the assumptionismadethat the consideredquantumfields areto bedescribedassystemsin thermodynamicalequilibriumat finite temperatureT= 1//3 (BoltzmannconstantkB = 1), insteadof being in the vacuumstate.Fromstatisticalmechanicsone knows that an ensembleis characterizedby the statistical operatorj5. Letus considerthe canonicalensemble,wherethe functionalequation13(H1 + H2) = ~5(H1)~(H2)is valid asa consequenceof weakcouplingand statisticalindependenceof subsystems1 and2. This equationhasthewell-known solution

1~= ~exp(—/3$l, (6.1)

wherethe normalizationfactor ~ representsthe canonicalpartition function of the system,which isdefinedas tracein Fock space:

= tr(exp(—/3I~)). (6.2)

All relevantphysical quantitiesdescribingthe systemin equilibriumcan be derivedfrom the partitionfunction, e.g. the free energy

F = — T ln(2~), (6.3a)

andthe pressure(V denotesthe volume)

P = — 3F/S9V. (6.3b)

The free electromagneticfield inside a large resonantcavity can be treatedas a canonicalensembledetermined by the statistical operator(6.1) and the partition function (6.2). The samedescriptionisjustified in the caseof a Klein—Gordon field, when the particle numberis not conserved,i.e., forvanishingchemicalpotential/2.

The situationis different in the caseof the electron—positronfield in the presenceof strong externalelectromagneticfields. As we haveseenabove, the supercriticalelectron—positronvacuumhas to betreatedas an open system which can exchangeenergywith its surroundingsby particle emission.Accordingly, for a correctdescriptionof the supercriticalvacuumthe grand partitionfunction mustbe

6. P!unien eta!., The Casirnir effect 173

introduced,i.e.,

= tr(exp[—/3(I~— ,aN)]), (6.4)

where the Fermi energy SF = —m plays the role of the chemicalpotential: /2 =5F~The field Hamil-

tonianwhich enters in the aboveequationscontainsthe zero-pointenergyof the_particularquantumfield, i.e., H= :H: + E

0. Consequentlythe partition functionsfactorizesas = ~ where~ containsthezero-pointenergyand~ representsthe partwhich refersto real occupiedstatesforming the presentensemble.This meansthat quantumfluctuations(or zero-pointoscillations)and the real stateof thefield arestatisticallyindependentin the externalfield approximation.In accordancewith eq. (6.3a)thefree energy becomesa sum of the zero-point energyand the free energyof the quantumsystemreferring to real occupiedstates:

F=E0+F=E0—Tln(~t). (6.5)

This relation showsthat a correctgeneralizationof the conceptof Casimirenergyat finite temperatureis achievedby consideringthe free energyof a quantumfield and defining a so-calledCasimir freeenergyas the differencebetweenthe free energyof the field in the presenceof externalconstraintsandthe oneof the free field, i.e.,

F~[aF] = FEaT’] — F[0]. (6.6)

As in the caseof the previouslydefinedvacuumenergy(eq. (2.56))such a definition is, of course,onlymeaningfultogetherwith an appropriateregularizationschemeleadingto a finite result.The Casimirfree energyreducesto the vacuumenergyEvac[3F] whenthe temperaturetendsto zero.

In order to calculatethe thermalcorrectionto the Casimirenergyin principle one can apply bothmode-summationmethod and the local fonnulation using finite-temperatureGreen functions. Thedirect mode summationis possiblebecausethe free energyof a consideredfield can be expressedinterms of field eigenmodefrequenciesor energy eigenvalues.Considering, for instance, the freeelectromagneticfield inside a large, cubic cavity of volume L

3 (periodic boundaryconditionson thewalls) explicit expressionsfor thepartitionfunction andthe free energyareeasilyderived.The completeFock spaceof normalizedmultiple photon statesIX) with n~photonsof eachmomentumk andpolarizationA can be constructedaccordingto

IX) = [I (n~!Yu2(c~~+Y2~i0), (6.7a)

where the occupationnumberscan take values n~= 0, 1, 2 These statessatisfy the eigenvalueequation.

n~IX)=n~JX). (6.7b)

Since the Hamiltonian of the photon field (eq. (2.37)) is also diagonal in this representation,thepartitionfunction becomes


= fl (~exp[—f3w~(n+ 1/2)]) = H exp(—/3wk/2)[1— exp(—/3wk)]~, (6.8)

andthe free energyis given by

F = ~ {wk12+ T ln[i — exp(—wk/T)]}. (6.9)

The evaluation of the Casimir free energy according to the mode summationmethod implies asubtractionof infinite sums, and we have already seen that the first term of eq. (6.9) requiresregularization.Similar regularizationproceduresarenot necessaryfor thetemperature-dependentpart,which can be proveddirectly. In the caseof the electromagneticfield inside a largeempty cube(thisrepresentsthe free field configuration)the contributionFLU] of the free energyis finite. Replacingthesummationin eq. (6.9) by an integrationoverall momentak, oneobtainsthe free energycontribution:

P[0] = 2T (~)3 J d3k ln[i — exp(—wk/T)]= T4 (~)Jdxx2ln(i - e~)

= —2~-~(4)T~=—~IT2T4L3, (6.iOa)

which gives rise to the pressure

P[0]= —ôF[0I/8V=~IT2T4, (6.iob)

representingthe Stefan—Boltzmannlaw.The free thermalradiationchangeswhen externalboundaryconditions(e.g.perfectconductors)are

introduced.Thus,it seemsobviousto identify the thermalcorrectionto the Casimirenergyin termsof aderivation from free thermalradiationdue to externalconstraints.As we shall see,in the caseof theconductingparallel-plateconfigurationit turns out that in the high-temperaturelimit the effect due tothe zero-pointoscillationof the electromagneticfield becomescompletelycompensatedby the thermalradiation.

6.2. Finite temperaturepropagators

We haveseen that the thermalcorrection to the Casimir energycan be calculatedfrom the freeenergy F of the constrainedquantumfield by the direct mode-summationmethod,becauseF can beexpressedas a sumover thecorrespondingfield eigenmodeenergies.Besidethe directevaluationof theCasimirfree energyby meansof this method,one hasin principle the option to generalizethe localversion of Casimirenergiesin terms of the energy—momentumtensorto the finite-temperaturecase.The basicchangein the formalismconsistsin that insteadof consideringthe vacuumexpectationvaluesof relevantoperatorsone now has to dealwith their thermodynamicalexpectationvalues.Let usnowpointout the conceptualline of this treatment.

The thermodynamicalexpectationvalueof an operatorA referring to the quantumfield, which is

G. P!unien et at, The Casimireffect 175

characterizedby the densityoperator,5, is definedas tracein Fock space:

(A) = tr(15A). - (6.11)

In accordancewith suchan ensembleaveragingonedefinesthe energy—momentumtensorfor a field atfinite temperaturein the presenceof externalboundariesas the difference

ø~(x;/3) = (t~”’(x))8~— (i’~(x))~. (6.12)

The energy—momentumtensorof a quantumfield consistsof bilinearforms in the field operators.Suchforms may be derivedfrom the time-orderedproductof the field operatorsby meansof a suitabledifferentialoperator,as wehaveshownin detail before.Consideringfields wheresuch a constructionisfeasible,oneis thusled to the representationof &~“(x,/3) in termsof finite temperaturepropagators.Inprinciple,it is of the form

9(x, /3) = i{~ tXi)(~(~)(J)(x,x’)— ~(~)(J)(X — x’))}j~..,~~, (6.13)

wherethe latin indicescountthe dynamicalfield andstand,e.g., eitherfor Lorentzor spinor indices.Let us now considerthe thermalGreen function of a free massivescalar field. We note that the

Greenfunction obtainedin the limit m-+0 is directly relatedto the thermalphotonpropagator,whichrepresents,of course, the basic quantity in order to evaluate the temperaturecorrection to theelectromagneticCasimir energy.Assuming a canonicalensembleof bosonsat ~ = 0, but keepingthemassarbitrary,the thermalGreenfunction is definedas the average

iG(x - x’; /3) = ~tr[exp(-/3~)T(~(x)~(x’))] = (T(~(x)~(x’))). (6.14)

For calculatingthe explicit expressiononemayusethe discreteplanewaveexpansion(eq. (2.8))for theoperatorsç~(x)and ç~(x’).In terms of the creationand annihilationoperatorsâ~and ~k one obtains:

~(x) = ~—~ ,,.iL.. [exp(—iwkxo)exp(ik . x)âk(O)+ exp(iwkxo)exp(—ik . x)âi~ik(O)], (6.15)k vQv2wk

wherethe time evolution ~~(X0) = exp(—iwkxo)âk(O)hasalreadybeensubstituted.The operatorsâ~,~k

satisfy the Bose commutation relations, and considering the ensembleaverage([ak, a~~])=8kk’, it

follows directly:

(ak’ak) l5kk.n(wk), ~ = 8kk’(i + n(wk)). (6.16)

The function n(wk) representsthe averageoccupationnumberwhich is determinedfor Bosestatisticsas

n(wk) = (exp(13wk)— i)~. (6.17)

In accordancewith the commutationrelations[âk, ak’] = [a, a~.]= 0 all otheraveragesvanish, i.e.,

(~k~k.) = (a~á.)= 0. (6.18)

176 G. P!unien et at, The Casimir effect

With theseinformation,for instance,the average(~(x)c~(x’))can becalculated:

(çb(x)~(x’))= ~ ~ {exp[ik . (x — x’)] exp[—iwk(xo — x~)](âk(0)â~(0))

+ exp[—ik(x— x’)] exp[iwk(xo— x~)}(ái’ik(0)âk(0))}. (6.19)

Substitutingeq. (6.17) andperformingthe continuumlimit, eq. (6.19) takesthe form

rd3k i(~(x)c~(x’))= J —i — {exp[—iwk(xo — x~)]exp[ik . (x — x’)](i + fl(Wk))

(2IT) 2wk

+ exp[iwk(xO — xi)] exp[—ik . (x — X’)]fl(Wk)}. (6.20)

The thermalFeynmanpropagatorthen follows after incorporatingthe time ordering(k0 = = co_k):

(T(ç~(x)ç~(x’))) = J~ exp[—ik(x — x’)] { k2 - rn2 + ~ + 21ri[exp(/3wk) - i]15(k2 — m2)}. (6.21)

Thermal Green functionsfor other fields can be derived in a similar way. For fermions the thermal-Feynmanpropagatorat finite chemicalpotential/2 is found to be

d4k ( ~‘k +m)S(x - x’; ~3)= J~—3-~exp[—ik(x — x’)] {k~ rn

2 + ~ + 2ITi6(k2- m2)(y~’k~.+ m)

< F (9(k0) + ø(—k0) iT (6.22)

Lexp[/3(wk — /2)] + 1 exp[/3(wk + /2)] + iii

In an alternativeformulation the propagator(6.21) can be representedby the series[35, 123]

(T(ç~(x)ç&(x’))) i ~ G(x— — if3pn), (6.23)

wheren denotesthetime-like unit vectorn’~= (1, 0, 0, 0). In thisrepresentationthe thermalpropagatorappearsas an infinite sumoverfree propagatorscorrespondingto imagesourcesdisplacedin imaginarytime at i$v. The term with p = 0 is the free vacuumpropagatorG0(x— x’).

A representationof the Greenfunction of the form (6.23) is also valid for the thermal photonpropagator(m = 0) and is thususeful in order to derivethe thermalcorrection to the Casimireffectbetweenconductingparallelplates.The scalarphotonpropagatorreads

G(x - x’; /3) = (- -~-~)~ [x - x’ - i/3pn]2. (6.24)

G. P!unienetat, The Casimireffect 177

Basedon -this-explicit expressionBrown andMaclay[35] havederivedthe thermalenergy—momentumtensor.For the parallel-plate configuration all that is left to do is to consider the image sourcesdisplacedin space.According to eq. (3.23) the energy—momentumtensorof the electromagneticfieldbetweenthe conductingplatesat finite temperaturefollows as

(i~(a)) = - -~ ~ k~ ~ + (2a1)zz~-(k~in~n~} (6.25)

In order to obtain @~“‘(a, /3) the free vacuumcontributionat zerotemperaturemustbe subtracted,i.e.the termreferring to 1 = k= 0. (T~(a)) doesalsocontainthe free black-bodyradiationwhich appearsexplicitly in (6.25) whenconsideringthe limit of very largeseparation.This contributioncorrespondstothetermswith I = 0 andk� 0. It explicitly reads

-+ ca))= —41~1T2T4(g’~”’— 4n~’n~), (6.26)

and gives the well-known results [124] for the energydensityand pressureof a photongas inside aninfinite volume: -

(i’°°(a -+ cc) = 12T4, (6.27a)

(TI’~k(a-+ca))=18ik(j~~00(a_~cc)) (6.27b)

Excludingthesetermsas well as the zero-temperaturecontributionof the vacuumstress,i.e., I � 0 andk = 0, one obtainsthe thermalcorrectionto the energy—momentumtensorat finite temperatureandplateseparation:

ø~(a, /3) = — U(a,/3)(g~+4z~z~)+ g(a, $)(z~z~+ n~n~)}, (6.28)

wherethe functionsf andg aredefinedas

f(a, /3) = ~ [(2a1)2+ (/3k)2]2, g(a, /3) = ~ ~ ~. (6.29)I,k=1 a 3a

Approximate expressionscan be derivedin the low- and high-temperaturelimit. The result for theenergy density and pressurecoincideswith that obtainedby Mehra [81] who applied the modesummationmethod (seethe nextsection).

We concludethis sectionwith the remark that an alternativederivationof temperature-dependentGreen functions may be basedon an analytical continuation of the time evolution of Heisenbergoperatorsto imaginary times.This implies that the operatorexp(—/3H)is interpretedas the evolutionoperatorto imaginary times i/3 (restricting to the interval [0,i$]). For details concerningthis for-mulation of thermalGreenfunction we refer to the literature(e.g. [125,126]).

178 G. Plunien eta!., The Casimir effect

6.3. Casimir energy

For the parallel,conductingplateconfigurationin the electromagneticfield explicit results for theCasimir free energyhavebeen obtainedby the modesummation[79—81]as well as the local Greenfunction method[35,42]. For the caseof thissimple geometryof the boundarythe exacttemperature-dependentGreen function can be derived. Balian and Duplantier have investigatedthe Casimir freeenergyconcerningthe sphericalshell andin somedetailsits behaviourfor moregeneralgeometries[41,421. Calculationson the free energyof a massivescalarfield inside a rectangularbox, which is finite inonedirection,havealsobeencarriedout [77].

In order to give an explicit examplewe now turn to calculatethe thermalcorrectionto the originalCasimireffect. Applying the modesummationmethod,onehasto performthe energysubtractionin thesameway as in the zero-temperaturecase.In accordancewith the energysubtractioneq. (6.6) alreadyusedpreviously,weonly needto calculatethe temperature-dependentpartof the Casimirfree energy:

~

+-~-[(L-a)_(L_L/n)-L/n]Jdk2 ln[i_exp(_~Vk~+k~)]}

= 2 (~~)2 TJ d2kIl{ ~‘ln [i — exp(—~~Jk~+ (?2~)2)]

- Jdn ln[1_exp(_~~k~+ (~))]}. (6.30)

The prime on the summationsign indicates that the n = 0 term is taken with a factor ~. As alreadymentionedF~requiresno regularization.In termsof the function

b(a, T, n) = ~T J dy ln[i — exp(— ‘rr\/y/aT)], (6.3i)

eq. (6.30)reads

Ec(a, T)=~{~b(a, ~0)+~b(a, T,n)-Jdnb(a, T,n)}

= ~ {_~i~(aT)2~(

3)+ ~ b(a, T, n)+ 2T (~—~)3~(4)}~ (6.32)

Furtherexactevaluationfor arbitrary temperaturefails sincethe remainingsum cannotbe performedanalytically. Approximate expressionsfor Fc(a, T) can be obtainedin the low-temperatureandin the

G. Plunieneta!., TheCasimir effect 179

high-temperaturelimit. For low temperature(aT ~1) a seriesexpansionof the logarithmin eq. (6.31) isjustified yielding

~ b(a, T, n)=— j T(aT)2 ~ e’~”~~(~i~+i)n=l ~ P ITt’ n=1 aT

r/aT\2 aT]= — T{(~—_)+—_Je”~T+O(e2~T). (6.33)

IT IT

Insertingthis expressioninto eq. (6.32) leadsto the finite-temperaturecorrection

Pc(a, T) = L2air2{ [— ~ (.?1) ~_ (.?i)~(~.~.)e’~~] + 2 (.?i)~~(4)}. (6.34)

As is alreadyclear from its definition, eq. (6.30),one recognizesthat the first term of eq. (6.34)referstothe thermalradiationof a constrainedelectromagneticfield insidethe cavity of volume L2a with L ~ a.The last term is the free field contribution (Stefan—Boltzmannterm eq. (6.lOa)) correspondingto thesamevolume which was subtracted.Togetherwith the zero-temperaturepart the Casimirfree energybecomes

L2ir2 aT3 aT3 irFc(a, T) = — 720a3{ 1 + 360~(3)(_—)— (2aT)4+ 720 (__—) (i + —~)e_~n1aT}. (6.35)

This energygives rise to a temperaturedependentforceper unit area:

~c(a, T) = - 240a4{i + ~(aT)4—240(~~)e~T}. (6.36)

The thermalcorrectioncan also be obtainedfrom the definition of thermodynamicpressure

p(a,T)=—-c(V,T)=—-~--~--Pc(a,T), where V=L2a.av Lôa

In order to calculate the correction to the Casimir effect at high temperatures,i.e. aT>> 1, thesummationover n in eq. (6.32) mustbe performed.Again onecan usePoisson’ssumformulawhich inthe presentcasestatesthat if

c(a, T,a) = -~- J dn cos(an)b(a,T, n) (6.37a)

then

T, 0)+ ~ b(a, 1’, n) irc(a, T,0)+2ir c(a, T,2irn). (6.37b)


Due to the relation ITc(a, T, 0) J~°dn b(a, T, n) this term cancelsin accordancewith the energysubtractionand the free energycontributionF~becomes

T’c(a, T)= (~.)2 ~ c(a, T, 2ITn). (6.38)

The function c(a, T, a) follows as

c(a, T, a)= -~-Jdn sin(an)n ln(i —e’~T)

I(—’~---~——coth(aTa)l, (6.39a)2rradaL\aT!a a j

which, in thehigh-temperatureexpansion,up to termsof the order ~ leadsto the expression

1 T iT 2aT2\c(a, T, a)= —~——--~— ~ (6.39b)

aa 2cr a a /

Evaluatingthe sumover n, the resultup to termsof the order C~(e_4~~T)reads

1 ~(3)T T 1 4irTFc(a, T)= L21T2{

72003_ 8IT4a2~4Ir3(_~+_—_)e4~wT}. (6.40)

It is interestingto note that this expressioncontainsexactly thezero-temperatureCasimirenergywithoppositesign which, therefore,cancelsin the total Casimirfreeenergy.Thus,for hightemperaturestheCasimirforce becomes

~c(a, T) = — ____-—~--~ [1- 4rraT+ ~(2ITaT)2]e4’~T. (6.41)

4ira 2ira

It is possible to show that this cancellationbetweenthe zero-temperatureCasimir energyand thethermalcontributions in the high-temperaturelimit is not an artefactof the one-dimensionalboxproblem,but holds generally[156].

7. Applications

7.1. Casimirenergyin dielectric mediaand therelation to the bag modelof hadronicparticles

A main subjectof the previoussectionshas been the Casimireffect betweenperfectly conductingplates that are placedin the electromagneticvacuum. We restrictedour presentationto very specificconfigurations,where the basic ideasand calculationalmethodsthat are also applicablein treating

G. Plunienet at, The Casimireffect 181

zero-pointenergiesof otherquantumfields, could be most transparentlydemonstrated.In ageneralizedtheory of the Casimir effect one considersthe electromagneticCasimir energy in the presenceofdielectricandpermeablemedia[44,39, 40, 60, 61]. Thesestudieshavebecomeof considerableinterestin particle physics,becausethereare somestriking formal similarities betweenelectromagnetismofcontinuousmedia and the phenomenologicaltheory of bag models. As a basic assumptionin suchmodels,the QCD vacuumis modelledas a mediumwith infinite permeabilityandperfectdielectricityconcerningthe colour gaugefields [127,128]. In order to point out this connection,let usspenda fewremarkson the electromagneticCasimirenergyin materialmedia.

Investigationson this subjectwere startedby Schwinger,DeRaadand Milton [39,40, 44] based onsourcetheory [129]. A review of this methodhasbeengiven recentlyby DeRaad[130]. Particularly,theCasimireffect in a solid spherewith dielectricity r andpermeability,u hasattractedinterest.A generalclassof configurationsis basedon the following model assumption:A sphericalball of fixed radiusawith constantdielectricity r~andpermeability/2i (non-dispersivemedium)maybe embeddedin an alsohomogeneousmediumcharacterizedby certainvalues~2 and/22 (seefig. 7.1). Theconstitutiverelationsbetweenthe electromagneticfields in vacuum and the ones in the mediaare given by D = sE andB = /LH. Also well known from electrodynamics,the following continuity conditions for normal andtangentialfield componentshaveto be imposedon the sphericalsurfaceS which dividesthe interiorandexteriormedium:

f EEr, E~,B5,Brj = continuous. (7.1)/2 is

Theseboundaryconditionsdirectly carry over to the electricandmagneticGreenfunctions,as alreadymentionedin section3.2.2.

As a particularcaseMilton [39] hascalculatedthe Casimirstresson a dielectric solid ball placedinthe vacuum,i.e., for ~2 = /22 = = 1 and ~i = s. The specialcaseof a conductingball thenis obtainedwhen e tendsto infinity. With respectto the boundaryconditions, which imply the continuity of thefield components{SEr, E5, B = H} on the surface,the problemcan be solved.However, the final resultfor the Casimir stress (force per unit area) is found to be still cutoff-dependent,even after theperformanceof energysubtractions.The finite part of the stressturns out to be attractive. Miltonderives explicit expressionsfor two limiting cases: For weakly polarizablemedia, i.e., e — 1 ~ 1 the

- / .-/7~

�1

mt.

Fig. 7.1. Thesphereof radius a divides interiorandexteriormedium.

182 6. Plunieneta!., The Ca.simireffect

Casimirforce per unit areais found as

(s_1)2 16 1

= — 256’ira” ~ ~ (7.2)

where8 —*0 is the cutoff, andthe finite part of the pressurearisesfrom the Casimi-renergy

Ec(a)= —(s — 1)2/256a. (7.3)

For a perfectly conducting ball, i.e. in the limit s —~cc, the finite part of the Casimir energy isapproximatelygiven by [39]

Ec(a) —1/8ira. (7.4)

Taking theseresults for the Casimir energyone may be temptedto revive Casimir’s semiclassicalelectronmodel [36].We mentionthis pointbecause,historically, elementaryparticlephysicsbeganwiththe classicaltheory of the electron[131].Endowingthe electronwith an extendedchargedistributionremoveson onehand,the classicalself-energydivergencebut leads,on theotherhand,to instability oftheelectrondue to the Coulombrepulsion.In order to stabilizethe electronwithin sucha model,oneisforced to introduceadditionalattractiveforces(Poincaréstress).It seemsreasonablethat an attractiveCasimirforce mayplay the role of the Poincaréstress,regardingthe electronas a sphericalball withsomesort of effectivedielectricity s. Assumingthat the chargeon the electronis preferablylocalizedonthe surfacethe Casimir energyshould balancethe repulsiveCoulombenergyof a chargedsphericalshell:

E(a)= e2/8ira. (7.5)

Since the Casimir energy is also proportional to 1/a, such an electron model would allow one tocalculatethe fine-structureconstanta = (e2/4rr) from the energybalance.In the caseof s 1 onehastotakeeq. (7.3) as stabilizingCasimir energy.Accordingly it follows

a(s—1)2/128, (7.6)

wherethe valuee = 0.97 would reproducethe observedvaluea = 1/137 for the fine-structureconstant.If the electronis imaginedto be aconductingball, accordingto eq. (7.4) aparameter-freemodel resultis found to be

a—’1/4rr, (7.7)

which is about a factor 10 too large.Nevertheless,it must be said that one should not take such“explanations”of the fine-structureconstanttoo seriously,becausethey are essentiallymodel-depen-dent. For instance,Brevik’s [40] calculation of the Casimir force on a dielectric sphereincludingelectrostrictivecontributions,where the medium is also assumedas non-dispersiveand moreover tosatisfythe Claudius—Mossottirelation(r — 1)/(e+ 2) = const.p, leadsto the total Casimirpressure(finitepart) .

G. Plunienetat, The Casimireffect 183

= — 4(0.308— o.45V~). (7.8)24ira

If this pressureis assignedto balancethe Coulomb repulsion ~= a/(8ira~),one seesthat for therequiredvalue of a this can not be possiblefor any real valuesfor s. Apparently, thereis no evidentway of explainingthe fine-structureconstantin this manner.

A further notable calculation of the Casimir effect in a solid ball was carried out by Brevik andKolbenstvedt[60,61, 72]. They considereda particularclassof non-dispersivematerialmedia,assumingthat in both interior and exteriormediathe vacuumrelationship

E/.L = 1 (7.9)

is fulfilled. The boundaryconditionson the surfacedividing the two mediathen takethe form

ii 1 -~~ Er, E5, Br, — = continuous. (7.10)1/2 /2

An obviousconsequenceof (7.9) is that within such a modelthe electricandmagneticfieldsare treatedin a symmetricway, i.e. one has: E = /2D and B = /LH. In the presenceof condition (7.9) the explicitcalculationshowsthat the samedelicatecancellationsbetweenexteriorand interior contributionstakeplace, which are characteristicfor the caseof a conductingsphericalshell placed in the vacuum.Inparticular the cutoff problem normally presentin the theory of Casimir energyof dielectricmediadisappears.The Casimir energyis found to be cutoff-independentandpositive[61]:

Ec(a)=Eo(a)(/2121)[1+0.311 /212 2]. (7.lla)

/212+ 1 (/212+ 1)

Here

E0(a)= 0.0923/(2a) 3/Ma (7.llb)

representsthe well-known result for the idealized caseof a perfectly conductingsphericalshell [37].Correspondinglythe Casimir force is repulsive and dependson the propertiesof the interior andexteriormediaonly through theratio of their permeabilities/212 = /Ll/ic2. Oneobservesthat theCasimirenergy(7.lla) is also invariant when interchanginginterior and exteriormedium,i.e., the substitutioniL12--* 1//212 leadsto the sameresult. Let us mention somelimiting casesof eq. (7.lla). For /212~*0 or/L12—*cc (i.e. /2i—*0 or ic2--*O) one obtains the conducting-shellresult. The case /22 = 1 (and ~2 = 1)

correspondsto the situationof a compactsphereplacedin the vacuum,acasewhich maybe associatedwith somekind of semi-classicalelectronmodel. If in addition ~ —*0 (and g~—* cc) theelectronbecomesa perfectelectricconductorwherethe E andB fields vanishin the interior. Conversely,if 1a1 -+ cc (and

—*0) the electronbecomesaperfectmagneticconductorwith vanishingD andH fields in the interior.In both limiting casesthe Casimirenergyis equalto E0(a).

The electromagnetictheory consideredabove is formally directly applicableto bagmodelsof QCD.In section4.3. we havealreadymadereferenceto the duality betweenthe electromagneticfield andthecolour gaugefields. Thecorrespondenceof colour theory with the Maxwell theory of polarizablemedia

184 G. P!unien eta!., TheCasimir effect

is invoked, if oneconsiderstheso-calledperturbativevacuumin theinteriorof thebagandthe exterior,“true” vacuumas somekind of colour-dielectricmedia.The duality betweensuch a gaugetheory andthe electromagnetictheory describedabovemaybe summarizedas follows:

Electromagneticfield Colour field

fields: B = ~ = SCE’~ (7.12a)

D = rE ~ > B~= (7.12b)

dielectricproperty: = 1 < > = 1 (7.12c)

boundaryconditions:

{~Er, E5, Br, ~ B5}~ = continuous ( > B~,B~,E~,-~— E~’}= continuous. (7.12d)

The constraint(7.12c) in colour theory ensuresthat the gluonic modespropagatein the vacuumalwayswith the velocity of light. The continuityconditionson the surfaceformally remainthe same.TheQCDcaseis now obtainedas the limit p~—* cc ands~—*0 for the exterior vacuumand icc = s~= 1 for theinterior vacuum. The situation assumedin QCD, namely, that a sphericalbag is embeddedin thephysical vacuum may directly correspondto that of a sphericalsuperconductingball placed in theelectromagneticvacuum.The analogyis illustratedin fig. (7.2).

The conceptof consideringthe physical(exterior)QCD vacuumas a kind of materialmedium,whichis characterizedby the propertyp~s,= 1 in order to ensurerelativistic invariance,hasbeendiscussedindetailby Lee [128].The assumptionthat the physicalvacuumhaspropertiescomparablewith a perfectcolour magnetic conductorguaranteesconfinementin the sensethat the bag surfaceis impermeableagainstthe gluon field, i.e.,

D’~-+O, H”—10 (r>a). (7.13)

Accordingly, the continuity conditionsimply that the interior field componentsE~andB~vanishwhenapproachingthe bagsurface:

E~(r—*a—)=0, B~(r—*a—)=0. (7.14)

Theseconditionshavealreadybeenmentionedin section4.3, wherewe havediscussedvarious effortsto evaluatethe Casimir energyin a sphericalcavity. The first of the two conditions(7.14) implies no

colour electric flux through the bagsurface: dfn . E’~= 0, whereasthe secondone implies vanishing

energyflux of gluons throughthe surface,sinceit follows: n (Ea x B’~)I~= 0.As onesees,the conditions(7.13) areformally in accordancewith the onesvalid in the electromag-

netic case(interchanginginterior andexteriormedium)when /22 = 1/s2 tendsto zero.Thus, BrevikandKolbenstvedt[61]concludethat theelectrom-agnetic-Casimirenergy(7.lla) is identicalwithiheoneofa

6. Plunien etat, The Casimireffect 185

~21 Ec~O‘~ -V7~-2Z -

Z2 . -C

~2-~-’- ~zv2z~za) b)

Fig. 7.2. Duality betweenthesituation of a perfect conductingspherein the electromagneticvacuum (a) and theQCD vacuumassumedin thetheoryof bagmodels (b).

confined gluon field after performing the limit /L12—*O. Since each componentof the colour fieldcontributesto the energywith the amountof eq. (7.llb), the total Casimir energyof a gluonic bag isobtainedby multiplying thisresultby eight:

E~(a) = 0.73880/(2a) 3/8a. (7.15)

This energyis positive andgives rise to a repulsive Casimirpressure.The above“evaluation” of thegluonic Casimir energyreveals that the appearanceof cutoff-dependentterms (divergencies),formercalculations [39, 40] are plaguedwith, are in fact a consequenceof restricting exclusively to thecontributionof interior bag modes.As an alternativeto such calculations,Brevik and Kolbenstvedthaveshown that it is possibleto apply to the QCD casea limiting result, which originatesfrom theelectromagnetictheory of Casimir energy,where it is legitimate to take into account both thecontributions of interior and exterior field modes.In a recentpaperMilton [58] also considersthecontribution of exterior field modesassuminga certain vacuum structure(foam of denselypackedbubbles)which may legitimize such a treatmentalso in the caseof calculatingthe fermion stress.Itshouldbe emphasizedthat the basicassumption(7.12c)is of greatadvantagein the sensethat it allows ameaningfulevaluationof the global Casimirenergyreferring to interior andexterior field modes.Thesamedelicatecancellationof cutoff terms takesplace as in caseof the conductingsphericalshell, andthus leadsto a finite result.

The details of thesecancellationscan be clearly analyzedwhen consideringthe Casimir energydensityWc(T) in the interior andexterior region [71,72, 68]. The featureof finite global Casimirenergydoesnot imply the regularity of the energydensity,but this local quantity turns out to divergeat thebag surfacein a particular way (wc(r—* a)--* —cc from inside and wc(r—* a)—*cc from outside)whichmakesthe total Casimirenergyfinite [72]. The investigationof local quantitieslike the energydensityor the expectationvalue (0IF~F~0)may be of interestin order to elucidatefundamentalpropertiesand structuresof the physical vacuum (e.g. gluon condensate[71] or the formation of flux tubes[66—68]).

The evaluationof Casimir energiesof arbitrarily shapedbags is plaguedwith similar difficulties,which alsooccur in the electromagnetictheory of Casimirenergyin the presenceof arbitrarily shapedsmooth boundaries.An appropriate treatment of correspondingbag-boundaryconditions remains

186 G.P!unien eta!., The Casimir effect

problematic. However, the Casimir energy of bagsdeviating from sphericalgeometry may be ofconsiderable interest in connectionwith the theory of highly excitedmesonstates(Reggetrajectories[144—146],bagfission, etc.).The Casimirenergyof a perfectconductingcylindrical shell[132]mayhavesomeapplicationin connectionwith the formationof QCD strings.

A further considerableproblemwhich could be of relevancein QCD bag modelsis basedon thefollowing electromagnetic Casimir problem: Consider two perfectly conducting spherical shellsseparatedby adistanceR which are placedin the electromagneticvacuum.An approximateresult forthe Casimir force for this configuration (assuminga scalar “photon”) has recently been derived byDeRaad[130]. (The large-distancebehaviourfor spin-onephotonhasalreadybeendiscussedby Balianand Duplantier[42].)The correspondingCasimirenergygives rise to an attractiveinteractionbetweenthe spheres.One should also rememberthat Casimir alreadyused the zero-pointenergyin order toderive the retardedvander Waalsattractionbetweenstatic polarizableparticles. It may be interestingto generalizethesecalculationsand treating two sphericalsolid balls placedin an exterior medium,assumingBrevik’s condition p.s = 1 for both media. One can expect that from the generalizedexpressionfor the Casimirforce asimilar resultasgiven by DeRaadwould beobtainedin somelimitingcasesof the permeabilityof the spheresand the imbeddingmedium.Of course,onewould then like toknow whetherthe QCD limit as consideredby Brevik for a single sphereleadsto a non-vanishingCasimir energy,or not. A remainingCasimir energyas a function of separationwould representacontributionto the interactionpotentialof two bagsdueto fluctuatingcolour fields.

7.2. Boundaryproblemsand Casimirenergyin gaugetheory

It should be clear by now that zero-point energiesin quantum field theory can be given awell-definedmeaning.We now want to give onemore exampleof boundaryproblemsin field theorywhich haverecentlybeeninvestigatedin connectionwith the Casimireffect.

Let us considera massivevectorfield in the presenceof an infinite conductingplane.This problemhas attractedinterestfor two reasons[133, 134]: Firstly, onewould like to know whetherthe Maxwelltheory is smoothlyrecoveredasexpectedin the zero-masslimit [135],andsecondly,it is interestingtoknow whether a non-zero mass could causeobservableboundaryeffects. In addition, one likes toexamine the influenceof possible non-minimal coupling terms in the field Lagrangianwhich breakgaugeinvariancein the masslesslimit.

- The local treatment is based on the vacuum-stresstensor &~ (the indication “vac” will besuppressedlater on). Without loss of generality,the conductingplanemaybe phcéd~afthe pOsiTi~fl-x3 = 0. We first note that the correspondingsituation for the masslesselectromagneticfield leadsto avanishing energy momentum tensor (all componentsare equal to zero). This is easily proved byrepresenting9~”explicitly in termsof photonpropagatorsasdiscussedin section3.2. As aconsequenceof a finite photon mass, the energy—momentumtensorhas a non-vanishingtrace, i.e. ~ � 0. Thegeneralstructureof the vacuum-stresstensorcan be determinedon the basis of symmetryargumentsand conservationlaws. Since the geometryof the boundaryprefersthe x3 direction, 9~”’ can onlyconsistof the tensorsg~”andz~’z”which areweightedby arbitraryscalarfunctionsof the position x,i.e.:

~(x)=f(x)g~+g(x)z~z”, (7.16)

with the normal surfacevector z” = (0, 0, 0, 1). The absenceof externalsourcesrequiresthe con-

G. P!unien etat, TheCasimireffect 187

servationof energyand momentum.The condition t9~9~”~= 0 determinesthe scalar functionsto be

equaland to dependonly on the x3 coordinate:

~9~(x3)= f(x3)(g~’+ z”z~)+ ag”. (7.1-7)

Sincethe vacuumis expectedto remainunaffectedinfinitely far away from the plane, ~9~”mustvanishin the limit x31—*cc. This conditionis satisfiedif f(x3—*cc)--*0 anda = 0. Expressedin termsof the traceE~(x3)= 3f(x3), the vacuum-stresstensortakesthe generalform

= ~.(x3)(g~’ + z’~z”). (7.18)

This result implies that the vacuumpressure933 vanisheseverywhere,but the vacuumenergydensitymay be different from zero. Davies and Toms [134] haveinvestigatedthe field theory which is

determinedby the following classicalaction:

I[A] = Jd~x\/—g{—~F~F~+ ~m2A~+1RA~’A~+ R~”A~A~}. (7.19)

The massivevector field A” is consideredin a curvedbackgroundspace-timewith metric tensorg”.From this given metric the Ricci tensorRi”’ and the scalarcurvatureR are constructed.~ and ~2

denotedimensionlesscoupling constantswhich describea non-minimal interaction (i.e. ~, ~2 ~ 0)betweenthe vectorfield andthe backgroundgeometry.Although the Lagrangianin eq. (7.19) reducesto that of the Proca theory [136] in the Minkowski spacelimit, the energy—momentumtensorstill

retains an imprint of the non-minimal curvature-dependentterms. This turns out in the followinglimiting result for the trace[134]:

T~= —m2A’’A,~— (3~~+

2)LIJ(A’’A,~)— ~29.~3~A’’A”. (7.20)

Basedon this fact DaviesandToms [134]haveexaminedthe masslesslimit of ~9’”~for theProcatheory,i.e. ~ = ~2 = 0 (minimal coupling) as well as the casewhen non-minimalcouplingsare present.Theappropriateexpressionfor the renormalized trace ~ can be derived for both cases.Again oneexpressesthe vacuumstresstensorin termsof free propagatorsof the massivephotonfield:

= —im2{G~(x,x’) — G

0’t~(x— x’)}I~.~. (7.21)

Constructing the renormalized massivevector field propagator,one has to consider the physicalboundaryconditions for the Proca field on the conducting plane. The finite mass breaksgaugeinvarianceand in contrast to the Maxwell theory, the massivevector field has threeindependentpolarization states, two transverseand a longitudinal (scalar)mode. Requiring Dirichlet boundaryconditionson the conductingsurfaceis only meaningfulwith respectto the transversefield modes.Forthe third spin state,the longitudinalmodes,it is unrealisticto assumereflection on the boundary,too,sincein the masslesslimit thesemodesdecouplefrom matter andwill penetratethe boundaryevenatinfinite conductivity [137].As shown by Bass and Schrödinger[138]they will remainunabsorbedinconductingmediato adepth of the orderof lim. The appropriateboundaryconditionfor the massivephotonfield havebeenrecentlyexaminedby Barton andDombey,who classifiedthe modesfor a plane

188 G. Plunieneta!., TheCasimireffect

geometry.They havealso analyzedthe effect of a smallmasscorrectionto the Casimirforce betweenconducting parallel plates [137] and have shown explicitly that contributions from the penetratinglongitudinalmodesarenegligible.

The remaining transversemodes, which will be reflected between the plates, lead to twice theCasimirenergyof a massivescalarfield undersimilar conditions(seeeq. (4.34)). Consideringonly thenon-penetratingfield modes,DaviesandTomshaveexplicitly derivedtherenormalizedpropagator,andtheyobtainthe following result for the trace ~ in the minimal-couplingcase:

ø~(x3)= — K1(2mx

3). (7.22)

4IT (x)

Accordingly, when the massof the vectorfield tendsto zerothe vacuum-stresstensorbehaveslike

— ~.~(m/ITx3)2(g’”+ z’~z~), (7.23)

i.e., for arbitrary small but non-vanishingmassthe energydensitydivergeson the boundary. In themasslesslimit ~9’~is seen to vanish, a result, which is consistentwith the Maxwell case.Includingnon-minimalcouplingtermsonederivesa modified expressionfor the trace:

= ~j-~--~(~+ 6~)[3 (rn)2K2(2mx

3)+ 2- Ki(2mx3)] 423 K1(2mx

3). (7.24)

In this casethe expansionof 9’~for small m reads

_____— (1+ ~ + 6~~)](g~+ z~z~). (7.25)

The relations(7.23) and (7.25) revealthat evenfor an arbitrarily smallmassthereis an infinite vacuumenergyper unit areaatthe boundary.In addition,eq. (7.25) showsthat thenon-minimalgauge-breakingtermssurviveevenin thelimit of vanishingmassandgive alsorise to a divergentvacuumenergydensityon the plane.

The questionariseswhetherthissurfaceenergymaybe experimentallydetectable,which wouldbe atest for the masslessnessof the photon. The presentupper experimentallimit for the photonmassisabout10’~g. In the abovederivation,a perfectly conductingmediumwith a sharpboundaryhasbeenassumed.This is, of course,an idealizationsincereal materialmediawill be of finite conductivity, andthe surfacewill not actas a perfectmirror. Onemayhaveto consideran effectivepenetrationdepth8which will serveto damp the divergenceof the vacuumenergyon the plane.Daviesand Toms [134]concludecarefully that the surfaceenergyinducedby a non-zerophotonmass,and in the presenceofgauge-breakingterms in the electromagneticfield Lagrangian,could perhapsbe of measurablemag-nitude~ 1020eVcm2,assuming6 -~-10_8cm). However, onehasto considerthat even in the massless,gauge-invariantMaxwell case boundarieswith extrinsic curvatureproduce in addition a divergentsurface-energycontribution.

G. P!unien eta!., TheCasimir effect 189

8. Concluding remarks

During the last twentyyearsnotableprogresshasbeenmadein particlephysicstowarda unified fieldtheory.In closeconnectionwith theseefforts, our understandingof the natureof the vacuumstatehasbeendeepenedand is now restingon a much firmer basis. Generally the (true) vacuum state JO) isdefinedas the stateof lowestenergy.According to the observationthat space-time,on the smallandonthe very large scale,is isotropic andhomogeneous,the vacuumstateof a free quantumfield must beinvariant against rotationsand translations.These symmetry propertiesimply that the free vacuumcarriesno energy,momentumor angularmomentum,i.e., (OJP’’jO) = 0 and(0~M~”J0)= 0. On the otherhand, the canonicalfield quantizationleadsto quantumoperatorswith ill-defined vacuumexpectationvalueslike, for instance,thecanonicalfield Hamiltonianwhich gives rise to an infinite zero-pointenergy.Such ambiguitieswhich arise due to the existenceof quantumfluctuationsare removed by formalsubtractionarguingthat they do not produceobservableeffects.

Thesituationis differentin thecaseof fields interactingwithexternalsourcesorboundaries.Sincelong itis known that the presenceof external fields breaksfundamentalsymmetries and thus, quantumfluctuationsbecomeobservable.Alreadyin theearlydaysof quantumelectrodynamicsit wasdemonstratedthat fluctuationsin the vacuum chargedensity of the electron—positronfield induced by an externalelectromagneticfield — the vacuumpolarization— give rise to a correctionto the interactionbetweenelectromagneticsources(the Uehling potential).

More recently,it hasbeenrecognizedthat externalboundaryconditions(asan idealizationfor realsourcesinteractingwith a given field) play a similar role as externalfields, since theycan also breaksymmetriesandinduce observableeffects dueto quantumfluctuations.This hasrevivedthe discussionabout the vacuumenergyand the Casimir effect. The basic featureof Casimir’s conceptof vacuumenergyis that the physical vacuumstateof quantizedfields must be determinedwith respectto theconditionthat fields usually exist in the presenceof externalconstraints.In particular,this facilitatesameaningfultreatmentof zero-pointenergiesexaminingtheir measurableconsequencesas contributionto the self-energyor to the interactionbetweentheexternalconstraints.

Detailed studieson the Casimir energy of the constrainedelectromagneticfields haverepresen-tatively demonstratedthe role of zero-point energies. All evaluation methods which have beeninvestigatedin this context, particularly the modesummationmethod [1, 37, 81] and various localGreenfunction methods[35,42, 140], aresimilarly applicablein order to calculateCasimirenergiesofother constrainedquantumfields (e.g.quarksand gluons confinedin a bag).Both evaluationmethodsrequire inherentlydifferent regularizationproceduresin order to yield finite results. In the caseofmodesummationthis can be achieved,for instance,by introducinghigh-frequencycutoffs or by meansof dimensionalregularization.The local treatmentsdeal with the boundarypart of the exact Greenfunction wherethe free vacuumGreenfunction is alreadysubtracted(local regularization).Calculatingthe vacuumenergy in terms of Greenfunctions divergenceswhich arisefrom taking expressionsatequal space-timepoints, in addition, requiresappropriaterenormalization.Seriesexpansionmethodslike the multiple scatteringexpansionor the perturbationexpansionshowthat divergenciesariseonlyfor the lowest-order terms, as is typical for renormalizablefield theories. The vacuum energy isunambiguouslydeterminedin both formulations.For the original Casimir effect, they lead to identicalresults.It is important to note that the sameresult for the Casimir force betweenconductingparallelplatesis also derivablewithin a totally different approachdealingwith fluctuatingclassicalelectromag-netic fields [141—143].This, of coursegives confidencein the calculationalmethodsmentionedabove.

190 G. P!unien eta!., The Casimir effect

It is worth recalling that a fully analyticalevaluationof Casimirenergiesis only possiblefor planeboundaries(parallel plates, rectangularbox). In the caseof boundariesof genericshapethe Greenfunctiontechniquesfacilitateapproximateevaluations.As theseareusuallyquite involved, it maybe ofinterestto developalternativecalculationalmethods.Consideringthat oneis mainly interestedin globalquantitieslike the Casimir energyor Casimir pressure,it seemssomewhatinadequateto base theevaluationon quantities,such as a completeset of eigenmodesor the exact Green function, whichcontain more information about the constrainedfield configuration than necessary.It is thereforeofgreat practicalrelevanceto develop new approximationschemesfor the calculation of the Casimirenergy,e.g.variationalmethods.For example,it would be interestingto formulatethe Casimir energyas a functionalof the vacuumpolarizationchargedensity,which is stationaryat the correctvalue.

The conceptof Casimirenergyis furthersupportedby the resultsobtainedfor thezero-pointenergyof the electron—positronfield interactingwith external electromagneticfields. Here we saw that thevacuumpolarizationcurrent can be expressedas the functional derivativeof the vacuumenergywithrespectto the externalelectromagneticfield. This demonstratesthe role of the vacuumenergyas acontributionto the interactionpotentialbetweentheexternalsources.The transitionfrom aneutralto achargedvacuumstatein strongfields is reflectedin adiscontinuousloweringof the vacuumenergy.

In the final chapterwe discussedseveralapplicationsof the Casimirenergy,in particularthe vacuumcontribution to hadronicmassesin the context of the MIT bag model, and alternativederivationsofinteractionsin polarizablemedia,such as van der Waalsforces. In addition to theseapplications,theCasimir effect is an essential ingredient in recent developmentsin high-energy physics, namely,Kaluza—Klein theoriesof unified interactions.As Appelquist and Chodos[25] havepointed out, theCasimirenergyin a five-dimensionalKaluza—Klein theory with one compactdimensionis attractiveinthe sensethat it tendsto shrink the size of the compactdimension.While this doesnot prove thatspace-timein higher dimensionsmust be compact,it lends support to the conceptthat the motion inhigher dimensionsmay be frozen in due to their tiny size,if theseareindeedcompact.

The existing literature on this and other applicationsof the Casimir energy is too vast to becomprehensivelyreviewed here. However, we hope that the present review article will help theinterestedreaderto explorethesefascinatingdevelopmentsby himself.

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