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QuantizationofGaugeFields

Wewi llnowturntot heproble mofthequanti zation ofgauget heories.We willbegin withthesimplestg augetheor y,thefre eelectromagneticfield. Thisisanabeliangaugethe ory.Afterthatwewilld iscussatlengthth e quantizationofnon-abeliangaugefiel ds.Un likeabeliantheories,suchasthe freeelect romagneticfield,evenintheabsenceofmatterfieldsnon- abelian gaugetheor iesarenotfreefieldsandha vehighl ynon-tr ivialdynamics.

9.1Canonic alquantizationofthefre eelectromagneticfield

TheMaxw elltheorywasthefirstfie ldtheorytobequant ized.Thequa nti- zationprocedureof agaugetheory,evenforafree field,i nvo lvesanumbe rof subtletiesnotsharedbytheother problems thatwehaveconsideredsofar. Theissu eisthefactthatt histh eoryhasal ocalgaugeinv ariance.Unlike systemswhichonlyh aveglobalsymmet ries,notallth eclassicalconfigu - rationsofvectorpot ential srepresentphysicall ydistinctstates.Itcouldbe arguedthatonesh ouldabandonth epictu rebasedonthevect orpotential andgoback toa picturebas edonel ect ricandmagneticfieldsinstead.How- ever,therei snolocalLagra ngiant hatcan describethetim eevolutionofthe systeminthatre present ation.Furth ermore,isnotclearwhichfield s,Eor B(orsome otherfield)pla ystheroleofc oordinatesandwhi chcanp laythe roleofmom entum. Forthatreason,andothers,ones tickstotheLa grangian formulationwiththevectorpotent ialA asit sindepend entcoordinate-like variable.

TheLagr angianfortheMaxwelltheory

L=! 1 4 F F (9.1)

262QuantizationofGaugeFields

whereF A A ,canbewrittenintheform L= 1 2 (E 2 !B 2 )(9.2) where E j 0 A j j A 0 ,B j jk k A (9.3)

Theelec tricfieldE

j andthes pacecomponen tsofthevector potentialA j formacano nic alpairsince,bydefinition,th emomentum j conjugateto A j is j (x)= #L 0 A j (x) 0 A j j A 0 =!E j (9.4) Noticethatsinc eLdoesnotcont ainany termswhichinclude! 0 A 0 ,the momentum 0 ,conjugatetoA 0 ,vanishes 0 #L 0 A 0 =0(9.5)

AconsequenceofthisresultisthatA

0 ises sentiallyarbitraryanditplays therol eofaLagrange multipl ier .Indeed,itisalwayspossi bletofind agauge transformation$ A 0 =A 0 0 A j =A j j $(9.6) suchthatA 0 =0.Th esolutionis 0 $=!A 0 (9.7) whichisconsi stentp rovidedthatA 0 vanishesbothintheremot epartand inth eremotefu ture,x 0 Thecano nicalformalismcanbeapplied toMaxwellelectrodynamicsif weno ticethatthefieldsA j (x)and! j "(x )obeytheequ al-timePois son

Brackets

{A j (x),! j "(x PB jj 3 (x!x )(9.8) or,int ermsoft heelectricfieldE, {A j (x),E j "(x PB jj 3 (x!x )(9.9) Thus,thespat ialcompon entsofthevectorpoten tialandthecomponents ofth eelectric fieldarecanonicalpairs.Howeve r,thetimecomponentof thevecto rfield,A 0 proceduretreatsitsepar ately,asaLagrangem ultiplie rfieldth atisimposing level,thecondit ion 0

9.1Can onicalquantizationofthefre eelectromagneticfield263

ledDirac toformulateth etheor yofquantizationofsystemswithconstr aints (Dirac,1966).Thereis ,however,anothe rapproach,alsoi nitiatedbyDirac, consistinginsettingA 0 =0andtoimposetheGaussLawasaconstraint ont hespaceofqu antumstates.As wew illsee,thisamountst ofixingthe gaugefirst (atthepriceofm anifes tLorentzin variance). Theclas sicalHamiltoniandensityisd efinedintheusualmanner H=! j 0 A j !L(9.10)

Wefi nd

H(x)= 1 2 (E 2 +B 2 )!A 0 (x)!$E(x)(9.11) Exceptforthelastt erm,thisi stheusual answer.Itis easytoseethatthe lasttermis aconstantofmotion.Indeedtheequal-timePoissonBracket betweentheHamiltoni andensityH(x)and!$E(y)isze ro.Byexplicit calculation,weget {H(x),!$E(y)} PB =!d 3 z!! #H(x) A j (z) #!$E(y) E j (z) #H(x) E j (z) #!$E(y) A j (z) (9.12) But #H(x) A j (z) =!d 3 w #H(x) B k (w) B k (w) A j (z) =!d 3 wB k (w)#(x!w)" k j w #(w!z) k j z !d 3 wB k (w)#(x!w)#(w!z) (9.13) Hence #H(x) A j (z) j k z (B k (x)#(x!z))=" j k B k (x)% x #(x!z)(9.14)

Similarly,weget

#!$E(y) E j (z) y j #(y!z), #!$E(y) A j (z) =0(9.15)

Thus,thePoiss onBracket is

{H(x),!$E(y)} PB =!d 3 z[!" j k B k (x)% x #(x!z)% y j #(y!z)] j k B k (x)% x y j #(x!y) j k B k (x)% x x j #(x!y)=0 (9.16)

264QuantizationofGaugeFields

providedthatB(x)isno n-singular.Thus,!$E(x)isa constantofmotion. Itis easyto checkthat !$Egeneratesinfinitesimalgauget ransformations. Wewi llprovethis statementdirect lyinthequan tumtheory. Since!$E(x)isac ons tantofmotion,ifwepickavalue foritat some initialtimex 0 =t 0 !$E(x)=%(x)(9.17) whichwerecog nizeto beGauss'sLaw.Naturally,anex ternal chargedistri- butionmaybeex plicitlyti medepende ntandthen d dx 0 (!$E)= x 0 (!$E)= x 0 ext (x,x 0 )(9.18) Beforeturning tothequantizationofthi stheor y,wemustn oticethatA 0 playstheroleof aLagra ngemultiplierfie ldwhose var iationyieldstheGauss Law,!$E=0.He nce,theGaussLawshoul dberegarde dasaconstraint ratherthananequ ationofmotio n.This issuebecomesveryimportantin thequan tumtheory.Indeed,with outtheconstraint!$E=0,th etheory isab solutelytrivial,andwrong. Constraintsimposeverysevereres trictionsontheallo wedstatesofa quantumtheory.Cons iderforinstanceapart icleofmassmmovingfreelyin threedimensio nalspace.Itsstationarystateshaveplane wavewavef unctions p (r,x 0 ),withanenergyE(p)= p 2 2m .Ifweconstraintheparticletomove onlyonthesu rfac eofasphere ofradiusR,itbecomesequivalenttoarigid rotorofmoment ofinert iaI=mR 2 andener gyeigenvalues" m h 2 2I &(&+1) 2 =R 2 doeshavenon- triviale ects. Unlikethecas eofapartic leforcedtomo veonth esu rfaceofasphere,the constraintsthatwehavetoimposew henquantizing Maxwell electrodynam- icsdon otchan getheener gyspectrum.Thisisso becausewe canreduce thenumb erofdegreesoffreed omtobe quantizedbytakingadvantageof thegaug einvarianceofthe classicaltheory.Thisprocedureiscalledgauge 2 A A )=0(9.19) inth eCoulombga uge,A 0 =0and!$A=0,be comes 2 A j =0(9.20) HowevertheCoulombgau geisnotco mpatiblewiththePoisson Bracket $A j (x),! j ,(x PB jj "#(x!x )(9.21)

9.1Can onicalquantizationofthefre eelectromagneticfield265

sincethespatia ldivergen ceofthedeltafunct iondoesnotvanish.Itwill followthatthequa ntizationoft hetheory intheCoulombgaugeis achiev ed atth epriceofam odificationofthecom mut ation relations. Sincetheclas sicalt heoryisgauge-invariant,wecan alwaysfixthegauge withoutanylossofphy sicalcont ent.Theproce dureof gaugefixinghasthe attractivethatthenumberofin dependent variablesi sgreatlyredu ced.A standardapproachtotheq uantizationofagaugetheo ryisto fixthe gauge first,atthecla ssical level, andtoquantizelater. However,anumberofprob lems ariseimmediately.Forin stance,in most gauges,suchasthe Coulombgauge,Lore ntz invariancei slost,or atleas t itism anif estlyso.Thus,althoughtheC oulombgaug e,alsoknownasthe radiationortransversegauge, spoil sLorentzinvariance,itha stheattr active featurethatthenatur eofthephysi calstates (thephotons)isquitetrans- parent.Wewillseebel owthatt hequantizatio nofthetheor yinth isgauge hassom epeculiaritie s.

Anotherstandardcho iceistheLorentzgauge

A =0(9.22) whosemainappe alisitsmanif estcovariance.Theq uantiz ationoft hesyst em isth isgaugefollo wsthemethoddev elopedbyandGupta andBleuer.While highlysuccessf ul,itrequirestheintroductionofst atesw ithneg ativenorm (knownasghosts)wh ichcan celallthegauge-depe ndentcontributionsto physicalquantities.This approachisdescribedindetailinthebookby

ItzyksonandZuber(Itzy ksonandZub er,1980).

Moregener alcovariantgaugescana lsobedefined.Ageneralapproach consistsnotonimposing arigidrest rictio nonthedegreesoffreedom,but toad dnewtermsto theLagrang ianwhicheli min atethega ugefreedom.For instance,themodifiedLagran gian L=! 1 4 F 2 1quotesdbs_dbs14.pdfusesText_20

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