peskin量子场论课后答案(芝加哥大学版)

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Physics443Homework#1

DueThursday,October7,2010

1.)Peskin&Schroeder,problem2.1

2.)Peskin&Schroeder,problem2.2

3.)Peskin&Schroeder,problem2.3

4.)Theclassicallimitofaharmonicoscillatorcanbedescribedintermsofcoherentstates

|a)=exp[aat]|0).

Whenaislarge,theoscillatorstateissemiclassical.ProceedingsimilarlyfortheFouriermodesofthequantumKlein-Gordonfield,

(a)Evaluatetheexpectationvalueofthefieldoperator

(f|φ(x)|f),

andshowthatitsatisfiestheKlein-Gordonequation.

(b)Evaluatetherelativemeansquarefluctuationoftheoccupationnumberofthemodewithmomentump,andtherelativemeansquarefluctuationinthetotalenergy

Iseitheroftheseagoodmeasureofthedegreetowhichthefieldisclassical?Justifyyour

answer.

(c)Take△(x-y)=(0|φ(x)φ(y)|0)(equaltimes)asameasureofthefuctuationsorcorrelationsofthefieldamplitude.Useyourresultforproblem2.3(P&S)toevaluatethisquantity.Whatisthemeaningofthedivergenceasx→y?

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QFT1:ProblemSet1

1.)Peskin&Schroeder2.1

Webeginwiththeactionfortheclassicalelectromagneticfield:

whereF=0A,-0,Aμ

(a)

homogeneousMaxwellequations

ToderivethehomogeneousMaxwellequationsweformthePoincaredualoftheFaraday

tensor.

Gaβ=eaBμF

GisdivergencelessfromthedefinitionofF:

O₀Gaβ=eBμ↓O₀δA,=0

Fromtheexpressionsforthefields(E,B)intermsofthepotentialsA⁰=(重，A)“:

B=▽×A

Wefind(usingLatinlettersforspatialindicesandGreekforspacetimeindices):

Ei=-0⁰A¹+QA⁰=Fi⁰

Thus.0₀GaO=0,Gi⁰=0;Bi=0

OaGa=O₀G+0;ei⁰kFok=-O₀B¹-eijk0;EK=0

Or,

▽·B=0

inhomogeneousMaxwellequations

WebeginwiththeEuler-Lagrangeequation:

Now:

Thus.

and

0。FaO=0;FiO=0;Ei=0

0Fai=O₀F⁰i-0;eii⁰kGo=-O₀E¹+eijk0;B^=0

Or,

▽·E=0

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(b)

WederivetheNoethercurrentassociatedwithaninfinitesimaltranslationx→x+a.Using

theequationsofmotion:

FromδL=a*0aC

Or,

andδA,=a*0aA,wefindOaT⁴g=0where:

Notethatthistensorisnotsymmetric.Itisalsoneithergaugeinvariantnortraceless.Toremedytheseproblemsweconstruct:

“g=T⁴g+θ(F>Ag)

Usingtheequationsofmotion:

Or,

Wecomputetheenergyandmomentumdensitiesintermsofthefields.Now,

FFM=2F₀;F⁰j+F;;Fii

Fromabove:

F⁰j=-Foj=-E)andFii=Fi;=-eij⁰kGok=-EijkB^

UsingeoijkOijg=286q,wehave:

Weconsider,

Or,

Also,

Or,

FF”=-2(E²-B²)

S¹=0i=FiFK²nk=Eeii⁰kGo=iEiB

S

=E×B

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alternatederivation

Wemayalsoderivethesymmetric,tracelessandgaugeinvariantenergy-momentumtensorfromtheactiononacurvedspacetime.Theactionis:

where

√σ=√|det[gw]

Herethematrixofthecomponentsofatensorinaparticularcoordinatesystemisrepre-sentedbybraces.Forexample

[gw]-¹=[g'”]

Tocomputetheenergy-momentumtensorwevarytheactionwithrespecttothemetric

withthedefinition:

Where,

Now:

det[g]=exp[tr(ln[gμn])]

Thus.

det[9ow+δg]=det[g]]det(1+[gvw][og

≈det[g](1+tr(fgow)-[ōgw1))

Since,r(lw)-¹[ōgl)=g*δg=-9óg!"

Wefind:

Thisleadsto:

Thuswehave:

Taβ=FaFβg¹”+49agFF

Thisclearlyreducestotheaboveresult(derivedviaNoethers'theorem)whenrestrictedtoflatspacetime.

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2.)Peskin&Schroeder2.2:Thecomplexscalarfield

Webeginwiththeactionforthecomplexscalarfield:

(a)

WecomputetheHamiltoniandensityassociatedwiththisaction:

H=πφ+πφd*-C

Where,

and

Thussince.

C=φ*φ-7p*·Vφ-m²φ*φ

Wefind:

H=π*π+▽φ*·Vφ+m²φ*φ

Wenowimposethecanonicalcommutationrelations:

[φ(x,t),π(y,t)]=i8³(x-y)

Allothercommutators(excepttheonegivenbyhermitianconjugation)vanish.WeusetheseandtheHeisenbergequationsofmotiontoverifythatφ=π*:

Wenowconsider方=*:

Integratingbypartswefind:

六(x)=▽·▽φ*(x)-m²φ

ThusosatisfiestheKlein-Gordonequation:

φ=π*=V²φ-m²φor0Q“φ+m²φ=0

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(b)

Byanalogywiththerealscalarfieldwepostulatethefollowingformforφ(x):

Thus,

AgainbyanalogywiththecaseoftherealKlein-Gordonfieldwepostulate:

and

Weassumeallothercommutatorsvanishandverifytheserelationsbycomputing

=i8³(x-y)

WenowshowthatHisdiagonalwhenwrittenintermsofthesecreationandannihilationoperators.Webeginwith:

WeconsidereachofthetermsthatmakeupHinturn.SinceHisaNoethercharge,wemayevaluateitatt=0)withoutlossofgenerality:

Now:

Thus.

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..

Also.

:

Combiningterms,takingp→-pforcrosstermsandusing

Thus.

Finally,wenormalordertoremovetheinfiniteenergyoftheso-calledDiracsea.

(c)

WenowexpresstheU(1)Noetherchargeintermsofcreationandannihilationoperators:

Againthechargeisconservedsowemayevaluateitatt=0

Since(πφ)+=φtπt:

Thus.

Uponnormalordering,weseethatthe(a,b)particleshavecharge

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(d)

WeconsidertheLagrangian:

Wewillfirstconsiderthegeneralcasea=1...NandthentakeN=2.WerewritetheLagrangianintermsofanNdimensionalcomplexvectorφanditshermitianconjugateφt

C=0φ²ə¹φ-m²φtφ

ThisisinvariantunderaglobalU(N)transformation:

φ→UφwhereU¹U=1

WemaydecomposeanyU(N)transformationintoaU(1)andanSU(N)transformation.GivenU∈U(N)suchthatdetU=eipwemayformM=e-iq/NU∈SU(N).Thuswemayconsidertheseinvariancesseparately.ForU(1)weconsiderφ→φ(a)=e-ia/2φ.SincetheLagrangianitself,ratherthanmerelytheaction,isinvariantundertheU(1),we

havetheconservedcurrent:

Now,

and

Thus.

J“=-言(a“φφ-φtaφ)

And,

Herewehave:

ForSU(N)weexpresseachelementofthegroup(hereweworkinthevectorrepresentationanditscomplexconjugate)intermsoftheexponentiationofelementsoftheLiealgebrasu(N).ThatisifM∈SU(N)thenitcanbeexpressedasM=e-ia'g'wheregj∈su(N)andai∈R.ThussinceUtU=1wehave(g))=giandsince,

weseethatgiistraceless.ThuswearelookingforasetoflinearlyindependenttracelesshermitianmatricesinNdimensions.Thedimensionalityofthisspaceis(N²-1).

Note:ThetrueunderlyinginvarianceofthelagrangianisO(2N)notU(N).Therearethus

actuallyN(2N-1),notN²,symmetrygenerators.

ForSU(N)wehavethecommutationrelations:

[g³,g^]=ifiklg'

Weusetheconventionalnormalizations:

andfimnfkmn=N8ik

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Weconsiderthesymmetryφ→φ(a)=e-ia'g'φ.Thisleadstotheconservedcurrents:

Now,

and

Thus,(J^)“=-i(O⁴φg^φ-φg^a*φ)

And.

WenowshowthatthechargessatisfythesamecommutationrelationsintheiractionontheHilbertspaceasthegeneratorsoftheLiealgebrasatisfyonCN.Asabove,sincethechargesareconserved,wemayevaluatethefieldsthatgointotheirconstructionatanytime.Wewillthereforesuppresstimelabelsonthefieldsinwhatfollows.WefirstrewritethechargeswithexplicitCNindices.

Wenowevaluatethecommutator:

+[πm(x)(g²)aφb(×),πe(y)(g⁴)aa(y)])

Now.

([φ(x)(g³)π+(x),φ*(y)(g⁴)π¹(y)])=-([π(x)(g³)φ(x),π(y)(g^)φ(y)])t

Thusweonlyneedtoevaluate:

[πa(x)(gì)。φo(x),π₀(y)(g^)a(y)]

=(gì)。(g^)c(πa(x)[φn(x),π₀(y)]φa(y)+πc(y)[π₀(x),φa(y)]φ(x))

Using;

[φa(x),πp(y)]=iôabδ³(x-y)

Wefind:

[πa(x)(gì)。φo(x),πc(y)(g^)a(y)]=iô³(x-y)(π(x)[g³,g^]φ(y))

Thus.

ForthecaseofSU(2)wemakethereplacements

and

fikl=cikl

WhereoJarethePaulimatricesandeiklisthecompletelyanti-symmetrictensorin3dimensions(e¹23=1)

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3.)Peskin&Schroeder2.3

Weevaluatethefunction

forspacelike(x-y),suchthat(x-y)²=-r²,explicitlyintermsofBesselfunctions.

SinceD(x)isinvariantunderLorentztransformations,D(x)=D(Ax)(A∈SO(3,1)),wemaychoosex⁰=y°.Thus,denotingx-y=r,

Wherewehaveintroducedaspericalcoordinatesystem(p,θ,φ)suchthatp·r=prcos0.

Thus.

From.

Definingu=p/m,

FromthepropertiesofKi(x),wefindthatfor(x,y)spacelikeseparated:

asx→y

ThefollowingisaplotofK₁(a)/xinredand1/x²inblue:

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4.)

WeconsidercoherentstatesfortherealKlein-Gordonfield:

|0)

Where,

(a)

Weevaluatetheexpectationvalueofthefieldoperator:

(f|φ(x)|f)

Where,

Since,φ_(x)=φ+(x)weneedonlyevaluate:

(f|φ+(x)|f)=|Nfl²(0|e-ir⁴φ+(x)eiT|0)

Where,

Thus,

Wenowshowthat|f)isnormalisedsothat:

(f|f)=|Nyl²(0|e-ir¹eiT|0)=1

Now,if[A,B]∈Cthen:

eAeB=eA+B+÷[A,B]=eβeAe[A,B]

Also,

Thus,

Now;

Thus,

r+|o)=0

and

[ap,T]=f(p)

since

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Thisleadsto:

Thustheexpectationvalueofthefieldoperatoris:

TheexpectationvaluetriviallysatisfiestheKlein-Gordonequationsincethefieldoperatorsatisfiesitand|f)isaHeisenbergstatevector.

(b)

Weevaluatetheexpectationvalueofthenumberdensityoperatorinmomentumspaceforthecoherentstate|f)

Also.

Fromthecommutationrelationsweseethatthisisadivergentquantity.

(f|npnp|f)=|f(p)|²(f(p)l²+(2π)³s²(O))

Thus

Thisdivergencearisessincenpisanoperator-valueddistributionandmustbeintegratedbeforeawell-definedproductwithanotheroperatorvalueddistributionmaybetaken.

WenowevaluatetheexpectationvalueoftheHamiltonianforthecoherentstate|f)

Also,

NoW.

(0|e-it*apake²T|0)=|Ny|-²(f(p)f(k)+(2π)³δ³(p-k),

Thus.

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Finally,

(c)

Aswefoundabove,for(x,y)spacelikeseparated:

as

x→y

Here,asabove,(x-y)²=-r².Again,asfornpabove,thisdivergencearisessinceφ(x)isanoperator-valueddistribution.Thisisalsoasignthatlocalquantitiesthatarequadraticinφ(x),suchastheenergy-momentumtensor,donothavewelldefinedvaluesandmustberenormalized.Notethatthedivergenceisindependentofthemassoftheparticle.Thisisanindicationthatallparticlesbehaveasmasslessparticlesatsufficientlyhighenergies(shortdistances).

Physics443Homework#2

DueThursday,October14,2010

1.)Considerthepathintegralforasinglepointparticle,withtheaction

Thisrepresentsthequantizationofthecoordinatesandmomentaoftheparticle,subjecttothemassshellconstraintp²=m²(togetherwiththeieprescription)imposedbytheLagrangemultiplierN.Thisactionadmitsthereparametrizationsymmetryδx=ap,δp=0,δN=-Oawherea(t)isanyfunction.ThissymmetryallowsustofixthegaugeconditionN(t)=T;theconstantTmuststillbeintegratedover,however.

a)Pathintegrateoverx(t),subjecttotheboundaryconditionsx/'(0)=x",x/(1)=y",yieldingadeltafunction8(p)alongthepath.Solvethisconstraint(findthesetoffunctionsthatsolveit)andpathintegrateoverthosep(t)tofindthequantummechanicalpropagationamplitude

wheredisthenumberofspacetimedimensions.

b)UsethisintegralrepresentationtoshowthatDrsatisfies

(ə²+m²)Dp=i8⁴)(x-y).

c)EvaluatetheTintegralintermsofBesselfunctions.

2.)PeskinandSchroeder9.2a-c

Hints:For9.2a,itissufficienttoformulatethepartitionfunctionintermsofapathintegral;youaregoingtoevaluateitinpart(b).For9.2c,firstshowthatthepartitionfunctioncanbeformulatedasapathintegraloverfieldsinEuclidean4-spacethatareperiodicintheimaginarytimedirection.Thespatialfieldmodesareharmonicoscillators;takethelogofthepartitionfunctiontogetthefreeenergyasasumovermodesofthefreeenergyofeachoscillator,anduseyourresultfor(b)toevaluateit.

Asecondapproachto(c)usesthemethodsofproblem(1).Usetherepresentation

log[Z]=log[det(-0b+m²)]=trlog(-0s+m²)

togetherwiththerepresentationofthematrixelementderivedinproblem(1),toevaluatethefunctionaldeterminantandhencethepartitionfunction.Youmaywanttotake0/dmoftheaboveexpressiontoremoveanm-independentdivergenceandrendertheintegralfinite.

3.)WriteafieldtheoryactiondescribingnonrelativisticscalarparticlesinteractingviaapotentialU(x-y)(thisaction-at-a-distanceformofinteractionispermissibleinanonrelativisticsetting,butnotinrelativisticfieldtheory,whereitwouldbreakLorentzinvariancebyselectingapreferredsurfaceofsimultaneity).FindthecorrespondingHamiltonianforthefield.UseyourexpressionfortheenergyintermsoffieldsandevaluateittoshowthattheexpectationvalueoftheHamiltonianinthenoninteractinggroundstateofasystemofNparticlesinavolumeVis,tofirstorderinperturbationtheory,

where

Useof'firstquantized'methodstoderivethisanswerisnotacceptable(thepointoftheexerciseistogainfamiliaritywithquantizedfields;youmayfindituseful,however,tocomparethetwoapproaches).

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QFT1:ProblemSet2

1.)

Webeginbyattemptingtomotivatetheactionforarelativisticpointparticleappearinginthehomeworkset.Perhapsthemorefamiliaractionisthatgivenbytheinvariantlengthof

theworldline:

Where

i!=0sx¹=0x/0s.

InadditiontobeingPoincareinvariant,thisactionisinvariantunderarbitraryreparameter-izationsoftheworldlinecoordinates.Thecoordinatesx!ofcoursetransformlikescalars

underthistransformation.Theequationofmotionforx(s)isthefamiliar:

Wenowintroduceanewactionwhichincorporatesafieldwhichtransformslikeametricunderreparameterizationsoftheworldlinecoordinates.Wewillseethatitleadstothesameequationsofmotionforx(s).

Wheredsx·Osx=ii"andg⁵⁸gss=1.Theequationofmotionforgisfoundtobe:

9ss=0sx·0₉a

Thus,iftheequationsofmotionaresatisfied,gcoincideswiththemetricalongtheworldlineinherited(throughitsimbeddinginspacetime)fromη.Fromthiswefind:

S₂[x,O₈π·θ₅x]=S₁[x]

ThustheactionS₂leadstothesameclassicalequationsofmotionforxasdoesSi.Wenowmakeachangeoffieldvariablesbytakingadvantageofthefactthat,inonedimension,wemayreplacethemetricbyaone-form.ThuswechooseN=√9ss/m.Thisleadstothe

action:

Fromthisactionwe

derivetheHamiltonian:

Herepμ=aC/aiμ=-N-li,andthecanonicalWemaydefinethepathintegralassociated

momentumofthe

withSʒasfollows:

Nvariablevanishes.

Perhapsmorefundamentally,wemayconsiderthepathintegraltobedefinedthroughthe

useofthefollowingaction:

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Wewritethepathintegralas

ThismaybeseentogivethesameresultasthepathintegralinvolvingSʒsincetheactionS₄isquadraticinp.NotethatwedonotintegrateoverthecanonicalmomentumforNsinceitisidenticallyzero.Toensurethatthispathintegralconvergeswesubstitutem²→(m²-ie)andconstrainthepathintegraltopositivevaluesofN.

Inthediscussionofgaugefixingwearegoingtodivergeabitfromthestatementoftheproblem.TheproblempresentsasymmetryundercanonicaltransformationsinducedbytheHamiltonianwhereNistreatedasaLagrangeMultiplierforafirst-classconstraint.Thesymmetryisδx=ap,δp=0,δN=-0saforarbitrarya(s).Iammorecomfortablediscussingfixingthesymmetryoftheactionunderdiffeomorphisms;thatisreparameteriza-tionsofthetimeparameters.ThissymmetrytreatsxandpasscalarsandNasaone-formsothatδx=-β0₉x,δp=-β0₉p,8N=-0s(BN)forarbitraryβ(s).Istronglysuspectthatthesymmetriesareequivalentandcertainlyleadtothesameresult.ThissymmetryallowsustotransformNsubjecttotheconditionthatfdsN(s)ispreservedasitmustbeunderdiffeomorphisms.ThefinitetransformationofNisjustthetensortransformationlaw:

WemayusethisfreedomtotransformanyN(s)toN(S)=1.Thenwehave:

Wemaynowdoafurthertransformationtosetthelimitsoftheintegraltoso=0andS₁=1withN(s)=T.WecannotgaugeawayNentirelyandTmustbeintegratedoverinthepathintegral.TheprincipalreasonforavoidingthetranformationinthehomeworkisthatIamnotsurewhattheanalogoftheʃdsN(s)constraintis.Wearethusleadtothefollowinggaugefixedpathintegral:

(a)

Wewillrespecttheapparenttime-honoredtraditionintheoreticalphysicsoftreatingthesolutionofthepathintegralsomewhatloosely.Butfirstwepresentanexpressionthatmaybeworkedwithtoprovideaperhapsmorecarefulsolution(here△=1/n):

Wetreatthexpathintegral,followinganintegrationbypartsintheaction,asafunctional

Fouriertransform:

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Insertingordinary

thisinto

integral

thepathintegralweblithelyconvertthepfunctionalintegralintoansincep=0:

Notethattheintegraloversintheactionproducesitsintegrandsincep=0.Alsonotethatafactorλahasbeeninsertedtoprovidethenormalizationtobedeterminedbelow.Wenowusethefollowingformulafortheintegralofagaussian:

Settinga=-iT/2anda=(x-y)wefind:

(b)

WenowshowthatDrisaGreenfunctionfortheKlein-Gordonequation.Wefindthat:

(a²+m²-ie)Dr(x)

xexp(-i/2[T(m²-ie)+T-1x²])

Toseethatthedistributiongivenhereisadeltafunctionwemayintegrateitagainstatestfunction.WewillfindthattheintegraloscillateswildlyintheT→0limitexceptnearx=0.Weremovethetestfunction(evaluatedatx=0)andthedistributionintegratesto1sinceitisanormalizedgaussianforallT.Thisisofcourseprettylooselanguagebutisessentiallycorrect.Toverifythisresultwereturntothe(normalized)expressionforDppriortoperformingthemomentumintegral.ToconformwiththedefinitioninPeskinandSchroederwechoosethenormalizationλa=1/2andtakep→-pintheintegral:

WeperformtheTintegraltofind;

Thus.

(d²+m²-ie)Dr(x)=-i8⁴(x)

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(c)

ThemoststraightforwardwaytoapproachthisproblemistouseatableofintegralsorplugtheexpressionforDpasanintegraloverTintoaprogramlikemathematica.Theresultis:

Thefollowingaregraphsofx(1-d/2)Ka/z-1(x)(spacelike)ingreenandtherealandimagi-

narypartsof(-ir)(1-d/2)Ka/z-1(-ix)(timelike)inblueandredrespectively.

For

d=2:

Ford=3:

Ford=4:

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2.)Peskin&Schroeder9.2(a-c)

(a)

Wewanttoexpressthequantumstatisticalpartitionfunctionintermsintegral.Fornotationalclaritywewillconsideraone-dimensionalsinglesystem.Theextensiontoamorecomplicatedsystemistrivial.Insertingapositioneigenstateswehave:

ofafunctionalparticlequantumcompletesetof

Ratherthanevaluatingthepropagatorforcomplextimeandfacingrelativelydelicateissuesrelatedtoanalyticcontinuation,wederivethepathintegraldirectly.Defininge=β/Nandinsertingcompletesetsofpositionandmomentumeigenstates,wehave:

Now.

Thus,usingthedefinitionofthephasespacepathintegralappearinginP&S,wemaywritethepartitionfunctionas:

WhereweareusingthehybridEuclideanLagrangian:

m(q₃á,p)=-ipà+H(p,q)

Notethatthepathintegralisoverallperiodicpathsthathaveperiodβ.IfH(p,q)canbewrittenasH=p²/2m+V(q),wemayevaluatethepintegralsexplicitly:

Thus.

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Again,usingthedefinitionoftheconfigurationspacepathintegralappearinginP&S,we

maywritethepartitionfunctionas:

WhereweareusingtheEuclideanLagrangian:

ThemeasureintheconfigurationspacepathintegraliswrittenasDqtoreflecttheaddi-

tionalfactorsinthemeasurethatdonotappearinthephasespacepathintegral.

(b)

WeconsidertheEuclideanactionfortheunitmassharmonicoscillator:

Sinceweareconsideringapathintegraloverperiodicfunctions,weexpandx(t)inaFourierSeries;

and

Therealityofx(t)imposes.Wewillproceedinacavaliermannerandsimplydefinethepathintegralmeasuretobe:

Wherexn=an+ibn.Notethatboisabsentduetotherealitycondition.Wewillpaydearlybelowforthischoiceofmeasureintheformofinfiniteβ-dependentprefactors.Itispossibletoavoidtheseinfinitiesthroughamorecarefuldefinitionofthediscreteformofthepathintegral(seeItzyksonandZuber9-1).WewillproceedasP&Sintendsandneglectthedivergentpieces.WefirstcomputetheactionintermsoftheFouriermodes.

Now,

And,

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Thus.

Thisleadsto:

Or,

Wemaywritethisas:

Neglectingthew-independentfactorinbacketsandusingtheproductrepresentationforsinhappearinginP&Swefind:

Z(B)=(2sinh(βw/2))-1

Youareinvitedtofeeltroubledbythisderivation.

(c)

Weformulatethepartitionfunctionforarealscalarfieldbyfirstconsideringthefollowing

matrixelement.

U(φa,φo|-iγ)=〈φb|e-~H|φa)

WeareworkingintheSchroedingerpicturewithHamiltonian:

Ratherthantreatingtheproblemofrealandimaginarytimeseparately,withtimeortemper-atureasacontinuousparameterinthepathintegral,itismorestraightforwardtointroduceacontinuousparameterwhichindexestheinsertionofaninfinitenumberofcompletesetsofstates.Defininge=1/Nandinsertingcompletesetsoffieldandmomentumeigenstates:

Wherewehaveintroducedthefunctionaldeltafunction:

Fromthecanonicalcommutationrelations:

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Thus.

WenowintroduceacontinuousparameterowhichindexesthecompletesetofstatesanddefineahybridLagrangian:

E,[ó,φ,π]=F[π,]+iγH[π,where

Thisleadstothepathintegralformofthematrixelement:

Substitutingγ=itandφ(o)→φ(ot)andchangingvariablestos=otwefind:

Where,

E(ó,Vφ,φ,π)=πb-H(π,▽φ,φ)where

Thepartitionfunctionisdefinedas:

Substitutingγ=βandφ(o)→φ(oβ)andchangingvariablestos=σβwefind

Where,

EE(ó,Vo,φ,π)=-iπó+H(π,Vo,φ)where

WenowtakeadvantageofthefactthattheHamiltonianisquadraticinπandintegrateoutthemomentumvariables.Wewritethepathintegralas:

Performingthegaussianintegralandabsorbingγdependenttermsintothemeasure:

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Substitutingγ=itandφ(o)→φ(ot)andchangingvariablestos=otwefind:

Where,

where

Substitutingγ=βandφ(o)→φ(oβ)andchangingvariablestos=oβwefind:

Where

where

Integratingbypartswehave:

Or,

z(3)=(det(-0g+m²))-1/

Wewillcomputethispathintegralinamanneranalogoustothatusedforthepartitionfunctionfortheharmonicoscillator.WeintroduceperiodicboundaryconditionsonR³andFourierdecomposeφ(x,s)(V=L³):

Sinceφisreal,ifwedefineφ(n,n)=A(n,n)+iB(n,n),wefind

A(n,n)=A(-n,-n)andB(n,n)=-B(-n,-n)

Thisallowsustodefinethefunctionalmeasureas:

Withsomealgebrawefind:

Defining

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Wehave:

Defining;

Wefind:

Thus,

Droppingwn-independentfactorsasinpart(b)above,wemaywritethisas:

Werewritethisas:

Droppingthefirstterm,whichamountstothenormalorderingprescription,andwriting

thesumasanintegraloverkwefind:

Asfortheharmonicoscillator,thisderivationismuchsimplerusingoperatormethods.PleaseseeItzyksonandZuber3-1-5.ThefollowingisaplotoflnZ(β)/(Vm³)asafunctionofßmobtainedthroughnumericalintegration.

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3.)

Webeginwit