The Inuene of an External Magneti Field
on the Fermioni Casimir Eet
M.V. Cougo-Pinto
, C. Farina y
,A.C. Tort z
InstitutodeFsia,Universidade FederaldoRiodeJaneiro
CP68528,Riode Janeiro, RJ21945-970,Brazil
Reeivedon3July,2000. Revisedversionreeivedon8September,2000
TheinueneofanexternalonstantuniformmagnetieldontheCasimirenergyassoiatedwith
a Diraeld underantiperiodi boundaryonditionis omputedusingShwinger'smethod. The
obtained resultshows thatthemagnetieldenhanesthefermioniCasimirenergy,inoposition
tothebosoniCasimirenergywhihisinhibitedbythemagnetield.
H. B.G.Casimir showedin 1948 [1℄thatthe
pres-eneoftwoloselyspaedparallelmetalli plateswith
no harge on them would shift the vauum energy of
theeletromagnetieldbyanamountE
(a) givenby:
E
(a)
` 2
=
2
720a 3
; (1)
where` 2
istheareaofeahplate,aistheseparation
be-tweenthemandtherelationa`isassumedinorder
to implement the onditionof small separation. As a
onsequeneofthisshiftthereisanattrativeforeon
theplateswhihwasmeasuredbySparnaayin1958[2℄
andmorereentlywithhighauraybyLamoreaux[3℄
andbyMohideenandRoy[4,5℄. Thisshiftinenergyis
knownastheCasimirenergyandbelongstogetherwith
its onsequenes and related phenomena to the realm
of thesoalled Casimireet [6, ?, 8℄. Generally,the
Casimireetanbedenedastheeetofnon-trivial
spae topologyonthe vauum utuationsof any
rel-ativistiquantum eld (f., e.g.,[6, 7℄). Thesoureof
thenon-trivialityofthespae topologyisprovidedby
severalkindsofboundaryonditions,bakgroundelds
andonstraints. Underthisgeneraldenitiontheeet
disoveredbyCasimirandgivenbytheenergy(1)an
be viewed as a onsequene of passing from the
triv-ialtopologyof lR 3
to the topology oflR 2
[0;a℄, due
totheboundaryonditionsimposedonthe
eletromag-netieldbythemetalliplateswithseparationa. The
fermioniCasimireetisofpartiularimportanedue
tothefundamentalroleplayedbytheeletroninQED
andthequarksin QCD.Intheaseofquarkswehave
aboundaryonditionof onnementgivenby nature,
whihmakestheCasimirenergyanaturalingredientin
thehadronstruture.
ThefermioniCasimirenergywasrstomputedby
Johnson [9℄ in the ontext of the MIT-bag model [10℄
for a massless Dira quantum eld onned between
parallel planeswithseparationa. Amorerealisti
de-sriptionofquarksandgluonsinsideahadronrequires
theonning boundaryonditionsto beimposed ona
spherial surfae. The Casimir eet in spherial
ge-ometry for massive elds is amuh moreompliated
problem and hasonlyreentlybeenompletelysolved
formassivefermioni [11℄and salar[12℄ elds. Inthe
ase of onning planes the fermioni Casimir energy
E(a) obtainedbyJohnson[9℄isgivenby:
E(a)
` 2
=
2
720a 3
; (2)
where =7=4. As in the original Casimir eet this
energyomes fromashiftfrom theusual spaelR 3
to
the spae lR 2
[0;a℄. If instead of ompatifying one
dimension lR into [0;a℄ weompatify it into airle
S 1
[13, 14℄ of radius a=2 we obtain for the Casimir
energyassoiatedwiththemasslessDiraeld the
ex-pression(2), where now is equalto74or 84,
aordingto ahoieoftwistedoruntwistedspin
on-netion, whih orresponds to antiperiodi orperiodi
boundary onditions with period a, respetively. We
should notie the similarityof those three results, for
the Casimir energy of the Dira massless eld under
MIT, periodi and antiperiodi boundary onditions.
Theyshowthatalltheseboundaryonditionsgiverise
tothesamedependeneonaanddieronlyonthe
mul-tipliativenumerialfator. Wemaytakeadvantage
of this fat by hoosing the simplest boundary
ondi-tion in arstinvestigation ofaCasimir eet. In the
aseofafermionield theompatiationinto[0;a℄
e-mail:marusif.ufrj.br
y
e-mail:farinaif.ufrj.br
providedbytheMITboundaryondition[10℄givesrise
to themostompliatedalulations,espeiallyin the
massivease [15℄. The periodi and antiperiodi
on-ditions are muh simplerin the masslessand massive
ase. Let us also notie that, as shown by Ford [14℄,
intheaseoftheDiraeldtheantiperiodiboundary
ondition avoids the ausality problems whih ours
forperiodiboundaryondition.
The results that we shall presenthere stems from
theideathatvauumutuationsofahargedquantum
eld are aetednotonlyby boundaryonditionsbut
alsobyexternalelds. Therefore,intheaseofharged
quantumeldsitisnaturalandimportanttoaskwhat
kindofinterplayoursbetweentheCasimireetand
thevauumpolarizationeets,whenboundary
ondi-tions and external eld are both present. This
ques-tionanbeexaminedfromtwophysiallyverydistint
pointsof view. Fromonepointofviewweaskwhatis
theinueneofboundaryonditionsonthepolarization
eets of an external eld and from the other weask
whatistheinueneofanexternaleldontheCasimir
energyofahargedeld. Weshouldexpetonphysial
groundstheexisteneofsuhinuenesanditis
nees-sarytoalulatetheirfeaturesandmagnitudesto
lar-ifytheirrole onandto obtainadeeperunderstanding
of the Casimirand vaumpolarization eets. For a
Diraeldtherstpointofviewisonvenientlytreated
byalulatinganEuler-HeisenbergeetiveLagrangian
[16℄withboundaryonditions[17℄. Wepresentherethe
seond pointof view,in whih welookfor thepreise
inuene of anexternaleld on theCasimirenergyof
a Dira eld. The obtained results omplement the
ones obtained for the bosoni Casimir eet in
exter-nal magneti eld [18℄. We ompute the inuene of
an external magnetield onthe Casimir energyof a
harged Dira eld under antiperiodi boundary
on-ditions and nd that the energy is enhaned by the
magneti eld. This result appears in opposition to
the behaviourof ahargedsalareld under Dirihlet
boundaryonditions,whihhasits Casimirenergy
in-hibited by theexternal magneti eld. It is tempting
to advaneanexplanationofthisoppositebehaviorin
terms ofthespinorialharaterof theelds. After all
the permanent magneti dipoles of spin-half quantum
eld utuations should tend to paramagneti
align-mentwith theappliedexternaleld whiletheindued
diamagnetidipolesofthesalarquantumeldtendto
antialignment. However, ithas beenveried [19℄ that
themagnetipropertiesofquantumvauumdependnot
onlyonthespinorialharaterofthequantumeldbut
also onthe kindofboundaryonditionsto whih itis
submitted. Therefore,further investigationsare
nees-saryinordertoformulateasoundphysialexplanation
of the harater of the hange in the Casimir energy
due toappliedexternalmagnetields.
We will take as external eld a onstant uniform
magnetieldandasboundaryonditionontheDira
eld the antiperiodiity along the diretion of the
ex-ternal magneti eld. The hoie of a pure magneti
eldexludesthepossibilityofpairreationatanyeld
strength. Thesimpliityofantiperiodiboundary
on-dition was remarked above and the other hoies are
obvious simplifying assumptions. These assumptions
leadustoaonvenientformalismtostudythephysial
inueneofanexternaleld ontheCasimireet.
The inuene of external eld on vaum
u-tuations of quantum elds have been onsidered by
AmbjrnandWolfram[20℄andbyElizaldeandRomeo
[21℄ for the ase of quantum salar eld in (1+
1)-dimensional spae-time. Ambjrn and Wolfram have
onsideredtheaseofahargedsalareldinthe
pres-ene of an external eletri eld while Elizalde and
Romeo onsider the ase of a neutral salar eld in
a stati external eld with the aim of addressing the
problem of the gravitational inuene on the Casimir
eet. Letus also note that in the Sharnhorsteet
[22℄wehavetheinterationofaneletromagneti
exter-naleldwith theeletromagneti vauumutuations
aetedbyboundaryonditions. However,inthisase
theboundaryonditionsare imposed onthe quantum
eletromagnetieldandnotontheDiraeld. The
ef-fetisthenatwo-loopeet,sinetheouplingbetween
theexternaleldandthequantumeletromagneti
va-uumeldrequirestheintermediationofafermionloop.
Heretheboundaryonditionis ontheDiraeld,the
quantum eletromagneti eld need not to be
onsid-eredand theexternaleletromagneti eld isnot
sub-jetedto boundary onditions. Inthis waytheeets
thatwedesribeappearattheonelooplevel,although
higher orders orretions an be obtained with more
loops.
Letusproeedtothealulationoftheinueneof
theexternalmagnetieldontheCasimirenergyofthe
Diraeld. We onsider aDira eld of massm and
hargee under antiperiodi boundaryondition along
the OZ axis. We implement the ondition on planes
perpendiular to OZ and separated by a distane a.
Weonsiderthoseplanesaslargesquaresofside`and
the limit ` ! 1 an be taken at the end of the
al-ulations. The onstant uniform magneti eld B is
takenalongtheOZaxiswithpositivediretionhosen
tomakepositivetheproduteB,where B isthe
om-ponentofBontheOZ axis. Inordertoalulate the
CasimirenergyoftheDiraeldinthepreseneofthe
magnetieldweuseamethodproposedbyShwinger
[23℄andbasedonhisproper-timerepresentationforthe
eetiveation [24℄. The method hasbeenapplied to
several situations (f. [18℄ for further referenes) and
herealso itwill leadusquiklyto theCasimirenergy.
We start with the proper-time representation for the
eetiveationW (1)
[24℄:
W (1)
= i
2 Z
1
s ds
s Tre
isH
wheres
o
isautointheproper-times,Tristhetotal
traeinludingsummationinoordinatesandspinor
in-diesand H is theproper-timeHamiltonian givenby:
H =(p eA) 2 (e=2) F +m 2
, wherephas
om-ponents p
= i
, A is the eletromagneti
poten-tialand F is theeletromagneti eld,whih is being
ontrated with the ombination of gamma matries
=i[ ;
℄=2. Theantiperiodiboundaryondition
gives for the omponent of p whih is along the OZ
axistheeingenvaluesn=a(n22lN 1),wherebylN
we denote the set of positive integers. The other two
spaeomponentsofpareonstrainedintotheLandau
levels generated by the magneti eld while the time
omponent p 0
has as eigenvalues any real number !.
Therefore, we obtain for the trae in (3) the
expres-sion:
Tre isH =e ism 2 X =1 2 X
n22lN 1 2e is(n=a) 2 X n 0
2lN 1 eB` 2 2 e iseB(2n 0 +1 ) Z dtd! 2 e is! 2 ; (4)
where therstsum takesareof thefour omponents
of theDira spinor, theseond sum is over the
eigen-valuesobtainedfromtheantiperiodiboundary
ondi-tion,thethirdsumisovertheLandaulevelswiththeir
degeneray fator eB` 2
=2, and the integral range of
t and ! are the measurement time T and the
ontin-uum of real numbers, respetively. Proeeding with
Shwinger's method we use Poisson's summation
for-mula [25℄ to invert the exponent in the seond sum
whihappearsin (4). We alsowrite thesum overthe
Landau levels n 0
whih appears in (4) in terms ofthe
Langevinfuntion L()=oth 1
and substitute
the trae obtained by these modiations into (3) to
obtain: W (1) =L (1) (B)a` 2
T E(a;B)T ; (5)
where L (1)
(B)isanexpressionwhih doesnotdepend
onaand
E(a;B)= a` 2 4 2 1 X n=1 ( 1) n Z 1 so ds s 3 e ism 2 +i(an=2) 2 =s
[1+iseBL(iseB)℄ (6)
istheutodependentexpressionwhihwillgiveusthe
Casimir energythat weare looking for. The quantity
L (1)
(B) given in (5) is atually the (unrenormalized)
Euler-HeisenbergLagrangian[16℄. In(5), itrepresents
adensityofenergyuniformthroughoutspaethatgives
no ontribution to theCasimir energy, whih by
de-nitionisset tozeroatinniteseparationoftheplates.
A term proportionalto the area` 2
, whih is usual in
vauumenergyalulations,doesnotappearhere,due
to the alternatingharaterof the seriesin (6). After
the elimination of the uto in (6) we ontinue with
Shwinger's method [23℄ by using Cauhy theorem in
theomplexs planeto makea=2lokwise rotation
oftheintegrationpathin(6). Letusnotiethatin(3)
and(6)itisimpliitthattheintegrationpathisslightly
below the real axis, beause s must have a negative
imaginary partin order to render the trae
ontribu-tionsin(3),(4)and(6)welldened. Consequently,the
polesoftheLangevinfuntionin(6), whihareonthe
real axis,are not sweptby the=2 lokwise rotation
of theintegrationpath. Weareled bytherotationto
anexpression in whih the partoftheCasimir energy
whih exists in the absene of the external magneti
eld an be expressed in termsof the modiedBessel
funtion K
2
(formula3.471,9in [26℄). Inthis waywe
obtainfrom(6)theexpression:
E(a;B) ` 2 = 2(am) 2 2 a 3 1 X n=1 ( 1) n 1 n 2 K 2 (amn) eB 4 2 a 1 X n=1 ( 1) n 1 Z 1 0 de (n=2) 2 (am) 2 = L(eBa 2
=); (7)
d
whihgivestheexatexpressionfortheCasimirenergy
of theDiraeld in thepresene oftheexternal
mag-netield B. Whenthereisnoexternalmagnetield
the Casimir energy is given by the rst term on the
=74inthelimitofzeromass,asitshouldbe
ex-peted. Theseond termon ther.h.s. of equation(7)
measurestheinueneoftheexternalmagnetieldin
itive, dereases monotoniallyas ninreases and goes
to zero in the limit n ! 1. Consequently, we have
byLeibnitz riterionaonvergentalternatingseriesin
(7) and we may onlude that the external magneti
eldinreasesthefermioniCasimirenergy. Thisisthe
mainresultofthiswork,whiheluidatespartofthe
in-terplaybetweentwoofthemostfundamental
phenom-ena in relativisti quantum eld theory, namely: the
Casimir eet and thevauum polarization properties
due toanexternaleld. Theobtainedenhanementof
thefermioni Casimirenergy byanexternal magneti
eldmaybeompared withtheopposite behaviourof
thebosoni Casimirenergy ofansalareld,whih is
inhibited by the external magneti eld. To see this
weturnfromspinorialQEDtosalarQEDkeepingthe
sameboundaryonditions and externalelds that we
havebeenusing. Weobtain by alulationssimilar to
theonesperformedin[18℄thefollowingbosoniCasimir
energyin theexternalmagnetield:
E
s (a;B)
` 2
= +
(am) 2
2
a 3
1
X
n=1 ( 1)
n 1
n 2
K
2 (amn)
+ eB
8 2
a 1
X
n=1 ( 1)
n 1 Z
1
0 de
(n=2) 2
(am) 2
=
M(eBa 2
=); (8)
d
where thefuntion M()= ose() 1
was
intro-dued in [18℄ and plays in salar QED the samerole
playedbytheLangevinfuntioninspinorialQED.The
inhibitionofthebosoniCasimirenergybytheexternal
eld anthen beseenin (8)by just notingthat Mis
stritlynegative. Atually,thisbosoniCasimirenergy
isompletely suppressedinthelimitB !1.
For strong magneti elds regime hanges in the
harged vauum may be easier to our [27℄. In this
ase the integralin equation (7) is dominated by the
exponential funtion, whose maximum is exp( amn)
and ours at = 2am=n. Therefore, we are justied
insubstitutingtheLangevinfuntionby1 1
inthe
strongmagnetieldregime,whihintheasesam1
andam1isdesribed,respetively,byjBjj
o j=a
2
and jBj(j
o j=a
2
)(a=
),where
o
isthe
fundamen-tal ux 1=e and
is the Compton wavelength 1=m.
In the strong eld regime also the seond term in (7)
anbeexpressedintermsofamodiedBesselfuntion
(formula 3.471,9in [26℄), andthe Casimirenergyan
bewrittenas:
E(a;B)
` 2
= eBm
2
1
X
n=1 ( 1)
n 1
n K
1
(amn): (9)
Byusing in thisexpressiontheleadingtermin the
asending expansion and then in the assymptoti
ex-pansionoftheBesselfuntion(seeformulas8.446and
8.451,6in[26℄)weobtainthefollowingexpressionsfor
smalland largemasslimits,respetively:
E(a;B)
2 =
eB
(am1)
E(a;B)
` 2
=
(am=2) 1=2
eB
3=2
a e
2am
(am1): (10)
Wehaveobtained in (7) thegeneralexpression of the
fermioniCasimir energyunder the eet ofan
exter-nalmagnetield. Theresultshowsthat theexternal
eldinreasestheCasimirenergyandrevealsthe
inter-playbetweentwofundamentalagentswhihareknown
toaettheDiravauumutuations,namely:
exter-nal elds and boundary onditions. We havederived
expressionsfortheenergyintheregimeofstrong
mag-netieldandinthisregimewehavealsoobtainedthe
small and large mass limits. The approah we have
followedherehas anaturalextension tomore
ompli-atedgaugegroupsandonsequentlymaybeusefulalso
intheinvestigationoftheQCDvauum.
Theauthors areindebted to JanRafelskiand Ioav
Wagafor manyenlighteningonversationsonthe
sub-jetofthis work. M.V. C.-P.and C.F. would like to
aknowledgeCNPq(TheNationalResearhCounilof
Brazil)forpartialnanialsupport.
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