Feynman diagram - Wikipedia

In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic ... Feynmandiagram FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Pictorialrepresentationofthebehaviorofsubatomicparticles Foralesstechnicalversion,seethisarticleontheSimpleEnglishWikipedia. InthisFeynmandiagram,anelectron(e−)andapositron(e+)annihilate,producingaphoton(γ,representedbythebluesinewave)thatbecomesaquark–antiquarkpair(quarkq,antiquarkq̄),afterwhichtheantiquarkradiatesagluon(g,representedbythegreenhelix). RichardFeynmanin1984 Intheoreticalphysics,aFeynmandiagramisapictorialrepresentationofthemathematicalexpressionsdescribingthebehaviorandinteractionofsubatomicparticles.TheschemeisnamedafterAmericanphysicistRichardFeynman,whointroducedthediagramsin1948.Theinteractionofsubatomicparticlescanbecomplexanddifficulttounderstand;Feynmandiagramsgiveasimplevisualizationofwhatwouldotherwisebeanarcaneandabstractformula.AccordingtoDavidKaiser,"Sincethemiddleofthe20thcentury,theoreticalphysicistshaveincreasinglyturnedtothistooltohelpthemundertakecriticalcalculations.Feynmandiagramshaverevolutionizednearlyeveryaspectoftheoreticalphysics."[1]Whilethediagramsareappliedprimarilytoquantumfieldtheory,theycanalsobeusedinotherfields,suchassolid-statetheory.FrankWilczekwrotethatthecalculationswhichwonhimthe2004NobelPrizeinPhysics"wouldhavebeenliterallyunthinkablewithoutFeynmandiagrams,aswould[Wilczek's]calculationsthatestablishedaroutetoproductionandobservationoftheHiggsparticle."[2] FeynmanusedErnstStueckelberg'sinterpretationofthepositronasifitwereanelectronmovingbackwardintime.[3]Thus,antiparticlesarerepresentedasmovingbackwardalongthetimeaxisinFeynmandiagrams. Thecalculationofprobabilityamplitudesintheoreticalparticlephysicsrequirestheuseofratherlargeandcomplicatedintegralsoveralargenumberofvariables.Feynmandiagramscanrepresenttheseintegralsgraphically. AFeynmandiagramisagraphicalrepresentationofaperturbativecontributiontothetransitionamplitudeorcorrelationfunctionofaquantummechanicalorstatisticalfieldtheory.Withinthecanonicalformulationofquantumfieldtheory,aFeynmandiagramrepresentsatermintheWick'sexpansionoftheperturbativeS-matrix.Alternatively,thepathintegralformulationofquantumfieldtheoryrepresentsthetransitionamplitudeasaweightedsumofallpossiblehistoriesofthesystemfromtheinitialtothefinalstate,intermsofeitherparticlesorfields.ThetransitionamplitudeisthengivenasthematrixelementoftheS-matrixbetweentheinitialandfinalstatesofthequantumsystem. QuantumfieldtheoryFeynmandiagram History Background Fieldtheory Electromagnetism Weakforce Strongforce Quantummechanics Specialrelativity Generalrelativity Gaugetheory Yang–Millstheory Symmetries Symmetryinquantummechanics C-symmetry P-symmetry T-symmetry Lorentzsymmetry Poincarésymmetry Gaugesymmetry Explicitsymmetrybreaking Spontaneoussymmetrybreaking Noethercharge Topologicalcharge Tools Anomaly Backgroundfieldmethod BRSTquantization Correlationfunction Crossing Effectiveaction Effectivefieldtheory Expectationvalue Feynmandiagram Latticefieldtheory LSZreductionformula Partitionfunction Propagator Quantization Regularization Renormalization Vacuumstate Wick'stheorem Equations Diracequation Klein–Gordonequation Procaequations Wheeler–DeWittequation Bargmann–Wignerequations StandardModel Quantumelectrodynamics Electroweakinteraction Quantumchromodynamics Higgsmechanism Incompletetheories Stringtheory Supersymmetry Technicolor Theoryofeverything Quantumgravity Scientists Anderson Anselm Atiyah Bargmann Becchi Belavin Bender Bethe Bjorken Bleuer Bogoliubov Brodsky Brout Brown Buchholz Callan Caswell Coleman Connes Dashen DeWitt Dirac Doplicher Dyson Englert Faddeev Fadin Fermi Feynman Fierz Fock Fredenhagen Fritzsch Fröhlich Furry Glashow Glimm Gelfand Gell-Mann Goldstone Gribov Gross Gupta Guralnik Haag Heisenberg Hepp Higgs Hagen 'tHooft Iliopoulos Itzykson Ivanenko Jackiw Jaffe Jona-Lasinio Jordan Jost Källén Kaplan Kastler Kendall Kibble Klebanov Kontsevich Kuraev Lamb Landau Lee Lee Lehmann Lepage Lipatov Low Lüders Maiani Majorana Maldacena Migdal Mills Møller Naimark Nambu Neveu Nishijima Oehme Oppenheimer Osterwalder Parisi Pauli Politzer Polyakov Pomeranchuk Popov Proca Rubakov Ruelle Salam Schwarz Schwinger Segal Seiberg Semenoff Shirkov Skyrme Stora Stueckelberg Sudarshan Symanzik Thirring Tomonaga Tyutin Veltman Virasoro Ward Weinberg Weisskopf Wentzel Wess Wetterich Weyl Wick Wightman Wigner Wilczek Wilson Witten Yang Yukawa Zamolodchikov Zamolodchikov Zimmermann Zinn-Justin Zuber Zumino vteContents 1Motivationandhistory 1.1Alternativenames 2Representationofphysicalreality 3Particle-pathinterpretation 4Description 4.1Electron–positronannihilationexample 5Canonicalquantizationformulation 5.1Feynmanrules 5.2Example:secondorderprocessesinQED 5.2.1Scatteringoffermions 5.2.2Comptonscatteringandannihilation/generationofe−e+pairs 6Pathintegralformulation 6.1ScalarfieldLagrangian 6.2Onalattice 6.3MonteCarlo 6.4Scalarpropagator 6.5Equationofmotion 6.5.1Wicktheorem 6.5.2HigherGaussianmoments—completingWick'stheorem 6.5.3Interaction 6.5.4Feynmandiagrams 6.5.5Looporder 6.5.6Symmetryfactors 6.5.7Connecteddiagrams:linked-clustertheorem 6.5.8Vacuumbubbles 6.5.9Sources 6.6Spin1/2;"photons"and"ghosts" 6.6.1Spin1/2:Grassmannintegrals 6.6.2Spin1:photons 6.6.3Spin1:non-Abelianghosts 7Particle-pathrepresentation 7.1Schwingerrepresentation 7.2Combiningdenominators 7.3Scattering 8Nonperturbativeeffects 9Inpopularculture 10Seealso 11Notes 12References 13Sources 14Externallinks Motivationandhistory[edit] Inthisdiagram,akaon,madeofanupandstrangeantiquark,decaysbothweaklyandstronglyintothreepions,withintermediatestepsinvolvingaWbosonandagluon,representedbythebluesinewaveandgreenspiral,respectively. Whencalculatingscatteringcross-sectionsinparticlephysics,theinteractionbetweenparticlescanbedescribedbystartingfromafreefieldthatdescribestheincomingandoutgoingparticles,andincludinganinteractionHamiltoniantodescribehowtheparticlesdeflectoneanother.Theamplitudeforscatteringisthesumofeachpossibleinteractionhistoryoverallpossibleintermediateparticlestates.ThenumberoftimestheinteractionHamiltonianactsistheorderoftheperturbationexpansion,andthetime-dependentperturbationtheoryforfieldsisknownastheDysonseries.Whentheintermediatestatesatintermediatetimesareenergyeigenstates(collectionsofparticleswithadefinitemomentum)theseriesiscalledold-fashionedperturbationtheory(ortime-dependent/time-orderedperturbationtheory). TheDysonseriescanbealternativelyrewrittenasasumoverFeynmandiagrams,whereateachvertexboththeenergyandmomentumareconserved,butwherethelengthoftheenergy-momentumfour-vectorisnotnecessarilyequaltothemass,i.e.theintermediateparticlesareso-calledoff-shell.TheFeynmandiagramsaremucheasiertokeeptrackofthan"old-fashioned"terms,becausetheold-fashionedwaytreatstheparticleandantiparticlecontributionsasseparate.EachFeynmandiagramisthesumofexponentiallymanyold-fashionedterms,becauseeachinternallinecanseparatelyrepresenteitheraparticleoranantiparticle.Inanon-relativistictheory,therearenoantiparticlesandthereisnodoubling,soeachFeynmandiagramincludesonlyoneterm. Feynmangaveaprescriptionforcalculatingtheamplitude(theFeynmanrules,below)foranygivendiagramfromafieldtheoryLagrangian.Eachinternallinecorrespondstoafactorofthevirtualparticle'spropagator;eachvertexwherelinesmeetgivesafactorderivedfromaninteractiontermintheLagrangian,andincomingandoutgoinglinescarryanenergy,momentum,andspin. Inadditiontotheirvalueasamathematicaltool,Feynmandiagramsprovidedeepphysicalinsightintothenatureofparticleinteractions.Particlesinteractineverywayavailable;infact,intermediatevirtualparticlesareallowedtopropagatefasterthanlight.Theprobabilityofeachfinalstateisthenobtainedbysummingoverallsuchpossibilities.Thisiscloselytiedtothefunctionalintegralformulationofquantummechanics,alsoinventedbyFeynman—seepathintegralformulation. Thenaïveapplicationofsuchcalculationsoftenproducesdiagramswhoseamplitudesareinfinite,becausetheshort-distanceparticleinteractionsrequireacarefullimitingprocedure,toincludeparticleself-interactions.Thetechniqueofrenormalization,suggestedbyErnstStueckelbergandHansBetheandimplementedbyDyson,Feynman,Schwinger,andTomonagacompensatesforthiseffectandeliminatesthetroublesomeinfinities.Afterrenormalization,calculationsusingFeynmandiagramsmatchexperimentalresultswithveryhighaccuracy. Feynmandiagramandpathintegralmethodsarealsousedinstatisticalmechanicsandcanevenbeappliedtoclassicalmechanics.[4] Alternativenames[edit] MurrayGell-MannalwaysreferredtoFeynmandiagramsasStueckelbergdiagrams,afteraSwissphysicist,ErnstStueckelberg,whodevisedasimilarnotationmanyyearsearlier.Stueckelbergwasmotivatedbytheneedforamanifestlycovariantformalismforquantumfieldtheory,butdidnotprovideasautomatedawaytohandlesymmetryfactorsandloops,althoughhewasfirsttofindthecorrectphysicalinterpretationintermsofforwardandbackwardintimeparticlepaths,allwithoutthepath-integral.[5] Historically,asabook-keepingdeviceofcovariantperturbationtheory,thegraphswerecalledFeynman–DysondiagramsorDysongraphs,[6]becausethepathintegralwasunfamiliarwhentheywereintroduced,andFreemanDyson'sderivationfromold-fashionedperturbationtheorywaseasiertofollowforphysiciststrainedinearliermethods.[a]Feynmanhadtolobbyhardforthediagrams,whichconfusedtheestablishmentphysiciststrainedinequationsandgraphs.[7] Representationofphysicalreality[edit] Intheirpresentationsoffundamentalinteractions,[8][9]writtenfromtheparticlephysicsperspective,Gerard'tHooftandMartinusVeltmangavegoodargumentsfortakingtheoriginal,non-regularizedFeynmandiagramsasthemostsuccinctrepresentationofourpresentknowledgeaboutthephysicsofquantumscatteringoffundamentalparticles.TheirmotivationsareconsistentwiththeconvictionsofJamesDanielBjorkenandSidneyDrell:[10] TheFeynmangraphsandrulesofcalculationsummarizequantumfieldtheoryinaforminclosecontactwiththeexperimentalnumbersonewantstounderstand.Althoughthestatementofthetheoryintermsofgraphsmayimplyperturbationtheory,useofgraphicalmethodsinthemany-bodyproblemshowsthatthisformalismisflexibleenoughtodealwithphenomenaofnonperturbativecharacters...SomemodificationoftheFeynmanrulesofcalculationmaywelloutlivetheelaboratemathematicalstructureoflocalcanonicalquantumfieldtheory... Currently,therearenoopposingopinions.InquantumfieldtheoriestheFeynmandiagramsareobtainedfromaLagrangianbyFeynmanrules. DimensionalregularizationisamethodforregularizingintegralsintheevaluationofFeynmandiagrams;itassignsvaluestothemthataremeromorphicfunctionsofanauxiliarycomplexparameterd,calledthedimension.DimensionalregularizationwritesaFeynmanintegralasanintegraldependingonthespacetimedimensiondandspacetimepoints. Particle-pathinterpretation[edit] AFeynmandiagramisarepresentationofquantumfieldtheoryprocessesintermsofparticleinteractions.Theparticlesarerepresentedbythelinesofthediagram,whichcanbesquigglyorstraight,withanarroworwithout,dependingonthetypeofparticle.Apointwherelinesconnecttootherlinesisavertex,andthisiswheretheparticlesmeetandinteract:byemittingorabsorbingnewparticles,deflectingoneanother,orchangingtype. Therearethreedifferenttypesoflines:internallinesconnecttwovertices,incominglinesextendfrom"thepast"toavertexandrepresentaninitialstate,andoutgoinglinesextendfromavertexto"thefuture"andrepresentthefinalstate(thelattertwoarealsoknownasexternallines).Traditionally,thebottomofthediagramisthepastandthetopthefuture;othertimes,thepastistotheleftandthefuturetotheright.Whencalculatingcorrelationfunctionsinsteadofscatteringamplitudes,thereisnopastandfutureandallthelinesareinternal.Theparticlesthenbeginandendonlittlex's,whichrepresentthepositionsoftheoperatorswhosecorrelationisbeingcalculated. Feynmandiagramsareapictorialrepresentationofacontributiontothetotalamplitudeforaprocessthatcanhappeninseveraldifferentways.Whenagroupofincomingparticlesaretoscatteroffeachother,theprocesscanbethoughtofasonewheretheparticlestraveloverallpossiblepaths,includingpathsthatgobackwardintime. Feynmandiagramsareoftenconfusedwithspacetimediagramsandbubblechamberimagesbecausetheyalldescribeparticlescattering.Feynmandiagramsaregraphsthatrepresenttheinteractionofparticlesratherthanthephysicalpositionoftheparticleduringascatteringprocess.Unlikeabubblechamberpicture,onlythesumofalltheFeynmandiagramsrepresentanygivenparticleinteraction;particlesdonotchooseaparticulardiagrameachtimetheyinteract.Thelawofsummationisinaccordwiththeprincipleofsuperposition—everydiagramcontributestothetotalamplitudefortheprocess. Description[edit] GeneralfeaturesofthescatteringprocessA+B→C+D:•internallines(red)forintermediateparticlesandprocesses,whichhasapropagatorfactor("prop"),externallines(orange)forincoming/outgoingparticlesto/fromvertices(black),•ateachvertexthereis4-momentumconservationusingdeltafunctions,4-momentaenteringthevertexarepositivewhilethoseleavingarenegative,thefactorsateachvertexandinternallinearemultipliedintheamplitudeintegral,•spacexandtimetaxesarenotalwaysshown,directionsofexternallinescorrespondtopassageoftime. AFeynmandiagramrepresentsaperturbativecontributiontotheamplitudeofaquantumtransitionfromsomeinitialquantumstatetosomefinalquantumstate. Forexample,intheprocessofelectron-positronannihilationtheinitialstateisoneelectronandonepositron,thefinalstate:twophotons. Theinitialstateisoftenassumedtobeattheleftofthediagramandthefinalstateattheright(althoughotherconventionsarealsousedquiteoften). AFeynmandiagramconsistsofpoints,calledvertices,andlinesattachedtothevertices. Theparticlesintheinitialstatearedepictedbylinesstickingoutinthedirectionoftheinitialstate(e.g.,totheleft),theparticlesinthefinalstatearerepresentedbylinesstickingoutinthedirectionofthefinalstate(e.g.,totheright). InQEDtherearetwotypesofparticles:matterparticlessuchaselectronsorpositrons(calledfermions)andexchangeparticles(calledgaugebosons).TheyarerepresentedinFeynmandiagramsasfollows: Electronintheinitialstateisrepresentedbyasolidline,withanarrowindicatingthespinoftheparticlee.g.pointingtowardthevertex(→•). Electroninthefinalstateisrepresentedbyaline,withanarrowindicatingthespinoftheparticlee.g.pointingawayfromthevertex:(•→). Positronintheinitialstateisrepresentedbyasolidline,withanarrowindicatingthespinoftheparticlee.g.pointingawayfromthevertex:(←•). Positroninthefinalstateisrepresentedbyaline,withanarrowindicatingthespinoftheparticlee.g.pointingtowardthevertex:(•←). VirtualPhotonintheinitialandthefinalstateisrepresentedbyawavyline(~•and•~). InQEDavertexalwayshasthreelinesattachedtoit:onebosonicline,onefermioniclinewitharrowtowardthevertex,andonefermioniclinewitharrowawayfromthevertex. Theverticesmightbeconnectedbyabosonicorfermionicpropagator.Abosonicpropagatorisrepresentedbyawavylineconnectingtwovertices(•~•).Afermionicpropagatorisrepresentedbyasolidline(withanarrowinoneoranotherdirection)connectingtwovertices,(•←•). Thenumberofverticesgivestheorderofthetermintheperturbationseriesexpansionofthetransitionamplitude. Electron–positronannihilationexample[edit] Feynmandiagramofelectron/positronannihilation Theelectron–positronannihilationinteraction: e++e−→2γ hasacontributionfromthesecondorderFeynmandiagramshownadjacent: Intheinitialstate(atthebottom;earlytime)thereisoneelectron(e−)andonepositron(e+)andinthefinalstate(atthetop;latetime)therearetwophotons(γ). Canonicalquantizationformulation[edit] Theprobabilityamplitudeforatransitionofaquantumsystem(betweenasymptoticallyfreestates)fromtheinitialstate|i⟩tothefinalstate|f⟩isgivenbythematrixelement S f i = ⟨ f | S | i ⟩ , {\displaystyleS_{\rm{fi}}=\langle\mathrm{f}|S|\mathrm{i}\rangle\;,} whereSistheS-matrix.Intermsofthetime-evolutionoperatorU,itissimply S = lim t 2 → + ∞ lim t 1 → − ∞ U ( t 2 , t 1 ) . {\displaystyleS=\lim_{t_{2}\rightarrow+\infty}\lim_{t_{1}\rightarrow-\infty}U(t_{2},t_{1})\;.} Intheinteractionpicture,thisexpandsto S = T e − i ∫ − ∞ + ∞ d τ H V ( τ ) . {\displaystyleS={\mathcal{T}}e^{-i\int_{-\infty}^{+\infty}d\tauH_{V}(\tau)}.} whereHVistheinteractionHamiltonianandTsignifiesthetime-orderedproductofoperators.Dyson'sformulaexpandsthetime-orderedmatrixexponentialintoaperturbationseriesinthepowersoftheinteractionHamiltoniandensity, S = ∑ n = 0 ∞ ( − i ) n n ! ( ∏ j = 1 n ∫ d 4 x j ) T { ∏ j = 1 n H V ( x j ) } ≡ ∑ n = 0 ∞ S ( n ) . {\displaystyleS=\sum_{n=0}^{\infty}{\frac{(-i)^{n}}{n!}}\left(\prod_{j=1}^{n}\intd^{4}x_{j}\right){\mathcal{T}}\left\{\prod_{j=1}^{n}{\mathcal{H}}_{V}\left(x_{j}\right)\right\}\equiv\sum_{n=0}^{\infty}S^{(n)}\;.} Equivalently,withtheinteractionLagrangianLV,itis S = ∑ n = 0 ∞ i n n ! ( ∏ j = 1 n ∫ d 4 x j ) T { ∏ j = 1 n L V ( x j ) } ≡ ∑ n = 0 ∞ S ( n ) . {\displaystyleS=\sum_{n=0}^{\infty}{\frac{i^{n}}{n!}}\left(\prod_{j=1}^{n}\intd^{4}x_{j}\right){\mathcal{T}}\left\{\prod_{j=1}^{n}{\mathcal{L}}_{V}\left(x_{j}\right)\right\}\equiv\sum_{n=0}^{\infty}S^{(n)}\;.} AFeynmandiagramisagraphicalrepresentationofasinglesummandintheWick'sexpansionofthetime-orderedproductinthenth-ordertermS(n)oftheDysonseriesoftheS-matrix, T ∏ j = 1 n L V ( x j ) = ∑ A ( ± ) N ∏ j = 1 n L V ( x j ) , {\displaystyle{\mathcal{T}}\prod_{j=1}^{n}{\mathcal{L}}_{V}\left(x_{j}\right)=\sum_{\text{A}}(\pm){\mathcal{N}}\prod_{j=1}^{n}{\mathcal{L}}_{V}\left(x_{j}\right)\;,} whereNsignifiesthenormal-orderedproductoftheoperatorsand(±)takescareofthepossiblesignchangewhencommutingthefermionicoperatorstobringthemtogetherforacontraction(apropagator)andArepresentsallpossiblecontractions. Feynmanrules[edit] ThediagramsaredrawnaccordingtotheFeynmanrules,whichdependupontheinteractionLagrangian.FortheQEDinteractionLagrangian L v = − g ψ ¯ γ μ ψ A μ {\displaystyleL_{v}=-g{\bar{\psi}}\gamma^{\mu}\psiA_{\mu}} describingtheinteractionofafermionicfieldψwithabosonicgaugefieldAμ,theFeynmanrulescanbeformulatedincoordinatespaceasfollows: Eachintegrationcoordinatexjisrepresentedbyapoint(sometimescalledavertex); Abosonicpropagatorisrepresentedbyawigglylineconnectingtwopoints; Afermionicpropagatorisrepresentedbyasolidlineconnectingtwopoints; Abosonicfield A μ ( x i ) {\displaystyleA_{\mu}(x_{i})} isrepresentedbyawigglylineattachedtothepointxi; Afermionicfieldψ(xi)isrepresentedbyasolidlineattachedtothepointxiwithanarrowtowardthepoint; Ananti-fermionicfieldψ(xi)isrepresentedbyasolidlineattachedtothepointxiwithanarrowawayfromthepoint; Example:secondorderprocessesinQED[edit] ThesecondorderperturbationtermintheS-matrixis S ( 2 ) = ( i e ) 2 2 ! ∫ d 4 x d 4 x ′ T ψ ¯ ( x ) γ μ ψ ( x ) A μ ( x ) ψ ¯ ( x ′ ) γ ν ψ ( x ′ ) A ν ( x ′ ) . {\displaystyleS^{(2)}={\frac{(ie)^{2}}{2!}}\intd^{4}x\,d^{4}x'\,T{\bar{\psi}}(x)\,\gamma^{\mu}\,\psi(x)\,A_{\mu}(x)\,{\bar{\psi}}(x')\,\gamma^{\nu}\,\psi(x')\,A_{\nu}(x').\;} Scatteringoffermions[edit] TheFeynmandiagramoftheterm N ψ ¯ ( x ) i e γ μ ψ ( x ) ψ ¯ ( x ′ ) i e γ ν ψ ( x ′ ) A μ ( x ) A ν ( x ′ ) {\displaystyleN{\bar{\psi}}(x)ie\gamma^{\mu}\psi(x){\bar{\psi}}(x')ie\gamma^{\nu}\psi(x')A_{\mu}(x)A_{\nu}(x')} TheWick'sexpansionoftheintegrandgives(amongothers)thefollowingterm N ψ ¯ ( x ) γ μ ψ ( x ) ψ ¯ ( x ′ ) γ ν ψ ( x ′ ) A μ ( x ) A ν ( x ′ ) _ , {\displaystyleN{\bar{\psi}}(x)\gamma^{\mu}\psi(x){\bar{\psi}}(x')\gamma^{\nu}\psi(x'){\underline{A_{\mu}(x)A_{\nu}(x')}}\;,} where A μ ( x ) A ν ( x ′ ) _ = ∫ d 4 k ( 2 π ) 4 − i g μ ν k 2 + i 0 e − i k ( x − x ′ ) {\displaystyle{\underline{A_{\mu}(x)A_{\nu}(x')}}=\int{\frac{d^{4}k}{(2\pi)^{4}}}{\frac{-ig_{\mu\nu}}{k^{2}+i0}}e^{-ik(x-x')}} istheelectromagneticcontraction(propagator)intheFeynmangauge.ThistermisrepresentedbytheFeynmandiagramattheright.Thisdiagramgivescontributionstothefollowingprocesses: e−e−scattering(initialstateattheright,finalstateattheleftofthediagram); e+e+scattering(initialstateattheleft,finalstateattherightofthediagram); e−e+scattering(initialstateatthebottom/top,finalstateatthetop/bottomofthediagram). Comptonscatteringandannihilation/generationofe−e+pairs[edit] Anotherinterestingtermintheexpansionis N ψ ¯ ( x ) γ μ ψ ( x ) ψ ¯ ( x ′ ) _ γ ν ψ ( x ′ ) A μ ( x ) A ν ( x ′ ) , {\displaystyleN{\bar{\psi}}(x)\,\gamma^{\mu}\,{\underline{\psi(x)\,{\bar{\psi}}(x')}}\,\gamma^{\nu}\,\psi(x')\,A_{\mu}(x)\,A_{\nu}(x')\;,} where ψ ( x ) ψ ¯ ( x ′ ) _ = ∫ d 4 p ( 2 π ) 4 i γ p − m + i 0 e − i p ( x − x ′ ) {\displaystyle{\underline{\psi(x){\bar{\psi}}(x')}}=\int{\frac{d^{4}p}{(2\pi)^{4}}}{\frac{i}{\gammap-m+i0}}e^{-ip(x-x')}} isthefermioniccontraction(propagator). Pathintegralformulation[edit] Inapathintegral,thefieldLagrangian,integratedoverallpossiblefieldhistories,definestheprobabilityamplitudetogofromonefieldconfigurationtoanother.Inordertomakesense,thefieldtheoryshouldhaveawell-definedgroundstate,andtheintegralshouldbeperformedalittlebitrotatedintoimaginarytime,i.e.aWickrotation.Thepathintegralformalismiscompletelyequivalenttothecanonicaloperatorformalismabove. ScalarfieldLagrangian[edit] Asimpleexampleisthefreerelativisticscalarfieldinddimensions,whoseactionintegralis: S = ∫ 1 2 ∂ μ ϕ ∂ μ ϕ d d x . {\displaystyleS=\int{\tfrac{1}{2}}\partial_{\mu}\phi\partial^{\mu}\phi\,d^{d}x\,.} Theprobabilityamplitudeforaprocessis: ∫ A B e i S D ϕ , {\displaystyle\int_{A}^{B}e^{iS}\,D\phi\,,} whereAandBarespace-likehypersurfacesthatdefinetheboundaryconditions.Thecollectionofalltheφ(A)onthestartinghypersurfacegivetheinitialvalueofthefield,analogoustothestartingpositionforapointparticle,andthefieldvaluesφ(B)ateachpointofthefinalhypersurfacedefinesthefinalfieldvalue,whichisallowedtovary,givingadifferentamplitudetoendupatdifferentvalues.Thisisthefield-to-fieldtransitionamplitude. Thepathintegralgivestheexpectationvalueofoperatorsbetweentheinitialandfinalstate: ∫ A B e i S ϕ ( x 1 ) ⋯ ϕ ( x n ) D ϕ = ⟨ A | ϕ ( x 1 ) ⋯ ϕ ( x n ) | B ⟩ , {\displaystyle\int_{A}^{B}e^{iS}\phi(x_{1})\cdots\phi(x_{n})\,D\phi=\left\langleA\left|\phi(x_{1})\cdots\phi(x_{n})\right|B\right\rangle\,,} andinthelimitthatAandBrecedetotheinfinitepastandtheinfinitefuture,theonlycontributionthatmattersisfromthegroundstate(thisisonlyrigorouslytrueifthepath-integralisdefinedslightlyrotatedintoimaginarytime).Thepathintegralcanbethoughtofasanalogoustoaprobabilitydistribution,anditisconvenienttodefineitsothatmultiplyingbyaconstantdoesn'tchangeanything: ∫ e i S ϕ ( x 1 ) ⋯ ϕ ( x n ) D ϕ ∫ e i S D ϕ = ⟨ 0 | ϕ ( x 1 ) ⋯ ϕ ( x n ) | 0 ⟩ . {\displaystyle{\frac{\displaystyle\inte^{iS}\phi(x_{1})\cdots\phi(x_{n})\,D\phi}{\displaystyle\inte^{iS}\,D\phi}}=\left\langle0\left|\phi(x_{1})\cdots\phi(x_{n})\right|0\right\rangle\,.} Thenormalizationfactoronthebottomiscalledthepartitionfunctionforthefield,anditcoincideswiththestatisticalmechanicalpartitionfunctionatzerotemperaturewhenrotatedintoimaginarytime. Theinitial-to-finalamplitudesareill-definedifonethinksofthecontinuumlimitrightfromthebeginning,becausethefluctuationsinthefieldcanbecomeunbounded.Sothepath-integralcanbethoughtofasonadiscretesquarelattice,withlatticespacingaandthelimita→0shouldbetakencarefully[clarificationneeded].Ifthefinalresultsdonotdependontheshapeofthelatticeorthevalueofa,thenthecontinuumlimitexists. Onalattice[edit] Onalattice,(i),thefieldcanbeexpandedinFouriermodes: ϕ ( x ) = ∫ d k ( 2 π ) d ϕ ( k ) e i k ⋅ x = ∫ k ϕ ( k ) e i k x . {\displaystyle\phi(x)=\int{\frac{dk}{(2\pi)^{d}}}\phi(k)e^{ik\cdotx}=\int_{k}\phi(k)e^{ikx}\,.} Heretheintegrationdomainisoverkrestrictedtoacubeofsidelength2π/a,sothatlargevaluesofkarenotallowed.Itisimportanttonotethatthek-measurecontainsthefactorsof2πfromFouriertransforms,thisisthebeststandardconventionfork-integralsinQFT.Thelatticemeansthatfluctuationsatlargekarenotallowedtocontributerightaway,theyonlystarttocontributeinthelimita→0.Sometimes,insteadofalattice,thefieldmodesarejustcutoffathighvaluesofkinstead. Itisalsoconvenientfromtimetotimetoconsiderthespace-timevolumetobefinite,sothatthekmodesarealsoalattice.Thisisnotstrictlyasnecessaryasthespace-latticelimit,becauseinteractionsinkarenotlocalized,butitisconvenientforkeepingtrackofthefactorsinfrontofthek-integralsandthemomentum-conservingdeltafunctionsthatwillarise. Onalattice,(ii),theactionneedstobediscretized: S = ∑ ⟨ x , y ⟩ 1 2 ( ϕ ( x ) − ϕ ( y ) ) 2 , {\displaystyleS=\sum_{\langlex,y\rangle}{\tfrac{1}{2}}{\big(}\phi(x)-\phi(y){\big)}^{2}\,,} where⟨x,y⟩isapairofnearestlatticeneighborsxandy.Thediscretizationshouldbethoughtofasdefiningwhatthederivative∂μφmeans. IntermsofthelatticeFouriermodes,theactioncanbewritten: S = ∫ k ( ( 1 − cos ⁡ ( k 1 ) ) + ( 1 − cos ⁡ ( k 2 ) ) + ⋯ + ( 1 − cos ⁡ ( k d ) ) ) ϕ k ∗ ϕ k . {\displaystyleS=\int_{k}{\Big(}{\big(}1-\cos(k_{1}){\big)}+{\big(}1-\cos(k_{2}){\big)}+\cdots+{\big(}1-\cos(k_{d}){\big)}{\Big)}\phi_{k}^{*}\phi^{k}\,.} Forknearzerothisis: S = ∫ k 1 2 k 2 | ϕ ( k ) | 2 . {\displaystyleS=\int_{k}{\tfrac{1}{2}}k^{2}\left|\phi(k)\right|^{2}\,.} NowwehavethecontinuumFouriertransformoftheoriginalaction.Infinitevolume,thequantityddkisnotinfinitesimal,butbecomesthevolumeofaboxmadebyneighboringFouriermodes,or(2π/V)d . Thefieldφisreal-valued,sotheFouriertransformobeys: ϕ ( k ) ∗ = ϕ ( − k ) . {\displaystyle\phi(k)^{*}=\phi(-k)\,.} Intermsofrealandimaginaryparts,therealpartofφ(k)isanevenfunctionofk,whiletheimaginarypartisodd.TheFouriertransformavoidsdouble-counting,sothatitcanbewritten: S = ∫ k 1 2 k 2 ϕ ( k ) ϕ ( − k ) {\displaystyleS=\int_{k}{\tfrac{1}{2}}k^{2}\phi(k)\phi(-k)} overanintegrationdomainthatintegratesovereachpair(k,−k)exactlyonce. Foracomplexscalarfieldwithaction S = ∫ 1 2 ∂ μ ϕ ∗ ∂ μ ϕ d d x {\displaystyleS=\int{\tfrac{1}{2}}\partial_{\mu}\phi^{*}\partial^{\mu}\phi\,d^{d}x} theFouriertransformisunconstrained: S = ∫ k 1 2 k 2 | ϕ ( k ) | 2 {\displaystyleS=\int_{k}{\tfrac{1}{2}}k^{2}\left|\phi(k)\right|^{2}} andtheintegralisoverallk. Integratingoveralldifferentvaluesofφ(x)isequivalenttointegratingoverallFouriermodes,becausetakingaFouriertransformisaunitarylineartransformationoffieldcoordinates.Whenyouchangecoordinatesinamultidimensionalintegralbyalineartransformation,thevalueofthenewintegralisgivenbythedeterminantofthetransformationmatrix.If y i = A i j x j , {\displaystyley_{i}=A_{ij}x_{j}\,,} then det ( A ) ∫ d x 1 d x 2 ⋯ d x n = ∫ d y 1 d y 2 ⋯ d y n . {\displaystyle\det(A)\intdx_{1}\,dx_{2}\cdots\,dx_{n}=\intdy_{1}\,dy_{2}\cdots\,dy_{n}\,.} IfAisarotation,then A T A = I {\displaystyleA^{\mathrm{T}}A=I} sothatdetA=±1,andthesigndependsonwhethertherotationincludesareflectionornot. Thematrixthatchangescoordinatesfromφ(x)toφ(k)canbereadofffromthedefinitionofaFouriertransform. A k x = e i k x {\displaystyleA_{kx}=e^{ikx}\,} andtheFourierinversiontheoremtellsyoutheinverse: A k x − 1 = e − i k x {\displaystyleA_{kx}^{-1}=e^{-ikx}\,} whichisthecomplexconjugate-transpose,uptofactorsof2π.Onafinitevolumelattice,thedeterminantisnonzeroandindependentofthefieldvalues. det A = 1 {\displaystyle\detA=1\,} andthepathintegralisaseparatefactorateachvalueofk. ∫ exp ⁡ ( i 2 ∑ k k 2 ϕ ∗ ( k ) ϕ ( k ) ) D ϕ = ∏ k ∫ ϕ k e i 2 k 2 | ϕ k | 2 d d k {\displaystyle\int\exp\left({\frac{i}{2}}\sum_{k}k^{2}\phi^{*}(k)\phi(k)\right)\,D\phi=\prod_{k}\int_{\phi_{k}}e^{{\frac{i}{2}}k^{2}\left|\phi_{k}\right|^{2}\,d^{d}k}\,} Thefactorddkistheinfinitesimalvolumeofadiscretecellink-space,inasquarelatticebox d d k = ( 1 L ) d , {\displaystyled^{d}k=\left({\frac{1}{L}}\right)^{d}\,,} whereListheside-lengthofthebox.EachseparatefactorisanoscillatoryGaussian,andthewidthoftheGaussiandivergesasthevolumegoestoinfinity. Inimaginarytime,theEuclideanactionbecomespositivedefinite,andcanbeinterpretedasaprobabilitydistribution.Theprobabilityofafieldhavingvaluesφkis e ∫ k − 1 2 k 2 ϕ k ∗ ϕ k = ∏ k e − k 2 | ϕ k | 2 d d k . {\displaystylee^{\int_{k}-{\tfrac{1}{2}}k^{2}\phi_{k}^{*}\phi_{k}}=\prod_{k}e^{-k^{2}\left|\phi_{k}\right|^{2}\,d^{d}k}\,.} Theexpectationvalueofthefieldisthestatisticalexpectationvalueofthefieldwhenchosenaccordingtotheprobabilitydistribution: ⟨ ϕ ( x 1 ) ⋯ ϕ ( x n ) ⟩ = ∫ e − S ϕ ( x 1 ) ⋯ ϕ ( x n ) D ϕ ∫ e − S D ϕ {\displaystyle\left\langle\phi(x_{1})\cdots\phi(x_{n})\right\rangle={\frac{\displaystyle\inte^{-S}\phi(x_{1})\cdots\phi(x_{n})\,D\phi}{\displaystyle\inte^{-S}\,D\phi}}} Sincetheprobabilityofφkisaproduct,thevalueofφkateachseparatevalueofkisindependentlyGaussiandistributed.ThevarianceoftheGaussianis1/k2ddk,whichisformallyinfinite,butthatjustmeansthatthefluctuationsareunboundedininfinitevolume.Inanyfinitevolume,theintegralisreplacedbyadiscretesum,andthevarianceoftheintegralisV/k2. MonteCarlo[edit] ThepathintegraldefinesaprobabilisticalgorithmtogenerateaEuclideanscalarfieldconfiguration.RandomlypicktherealandimaginarypartsofeachFouriermodeatwavenumberktobeaGaussianrandomvariablewithvariance1/k2.ThisgeneratesaconfigurationφC(k)atrandom,andtheFouriertransformgivesφC(x).Forrealscalarfields,thealgorithmmustgenerateonlyoneofeachpairφ(k),φ(−k),andmakethesecondthecomplexconjugateofthefirst. Tofindanycorrelationfunction,generateafieldagainandagainbythisprocedure,andfindthestatisticalaverage: ⟨ ϕ ( x 1 ) ⋯ ϕ ( x n ) ⟩ = lim | C | → ∞ ∑ C ϕ C ( x 1 ) ⋯ ϕ C ( x n ) | C | {\displaystyle\left\langle\phi(x_{1})\cdots\phi(x_{n})\right\rangle=\lim_{|C|\rightarrow\infty}{\frac{\displaystyle\sum_{C}\phi_{C}(x_{1})\cdots\phi_{C}(x_{n})}{|C|}}} where|C|isthenumberofconfigurations,andthesumisoftheproductofthefieldvaluesoneachconfiguration.TheEuclideancorrelationfunctionisjustthesameasthecorrelationfunctioninstatisticsorstatisticalmechanics.ThequantummechanicalcorrelationfunctionsareananalyticcontinuationoftheEuclideancorrelationfunctions. Forfreefieldswithaquadraticaction,theprobabilitydistributionisahigh-dimensionalGaussian,andthestatisticalaverageisgivenbyanexplicitformula.ButtheMonteCarlomethodalsoworkswellforbosonicinteractingfieldtheorieswherethereisnoclosedformforthecorrelationfunctions. Scalarpropagator[edit] EachmodeisindependentlyGaussiandistributed.Theexpectationoffieldmodesiseasytocalculate: ⟨ ϕ k ϕ k ′ ⟩ = 0 {\displaystyle\left\langle\phi_{k}\phi_{k'}\right\rangle=0\,} fork≠k′,sincethenthetwoGaussianrandomvariablesareindependentandbothhavezeromean. ⟨ ϕ k ϕ k ⟩ = V k 2 {\displaystyle\left\langle\phi_{k}\phi_{k}\right\rangle={\frac{V}{k^{2}}}} infinitevolumeV,whenthetwok-valuescoincide,sincethisisthevarianceoftheGaussian.Intheinfinitevolumelimit, ⟨ ϕ ( k ) ϕ ( k ′ ) ⟩ = δ ( k − k ′ ) 1 k 2 {\displaystyle\left\langle\phi(k)\phi(k')\right\rangle=\delta(k-k'){\frac{1}{k^{2}}}} Strictlyspeaking,thisisanapproximation:thelatticepropagatoris: ⟨ ϕ ( k ) ϕ ( k ′ ) ⟩ = δ ( k − k ′ ) 1 2 ( d − cos ⁡ ( k 1 ) + cos ⁡ ( k 2 ) ⋯ + cos ⁡ ( k d ) ) {\displaystyle\left\langle\phi(k)\phi(k')\right\rangle=\delta(k-k'){\frac{1}{2{\big(}d-\cos(k_{1})+\cos(k_{2})\cdots+\cos(k_{d}){\big)}}}} Butneark=0,forfieldfluctuationslongcomparedtothelatticespacing,thetwoformscoincide. Itisimportanttoemphasizethatthedeltafunctionscontainfactorsof2π,sothattheycanceloutthe2πfactorsinthemeasureforkintegrals. δ ( k ) = ( 2 π ) d δ D ( k 1 ) δ D ( k 2 ) ⋯ δ D ( k d ) {\displaystyle\delta(k)=(2\pi)^{d}\delta_{D}(k_{1})\delta_{D}(k_{2})\cdots\delta_{D}(k_{d})\,} whereδD(k)istheordinaryone-dimensionalDiracdeltafunction.Thisconventionfordelta-functionsisnotuniversal—someauthorskeepthefactorsof2πinthedeltafunctions(andinthek-integration)explicit. Equationofmotion[edit] Theformofthepropagatorcanbemoreeasilyfoundbyusingtheequationofmotionforthefield.FromtheLagrangian,theequationofmotionis: ∂ μ ∂ μ ϕ = 0 {\displaystyle\partial_{\mu}\partial^{\mu}\phi=0\,} andinanexpectationvalue,thissays: ∂ μ ∂ μ ⟨ ϕ ( x ) ϕ ( y ) ⟩ = 0 {\displaystyle\partial_{\mu}\partial^{\mu}\left\langle\phi(x)\phi(y)\right\rangle=0} Wherethederivativesactonx,andtheidentityistrueeverywhereexceptwhenxandycoincide,andtheoperatorordermatters.Theformofthesingularitycanbeunderstoodfromthecanonicalcommutationrelationstobeadelta-function.Definingthe(Euclidean)FeynmanpropagatorΔastheFouriertransformofthetime-orderedtwo-pointfunction(theonethatcomesfromthepath-integral): ∂ 2 Δ ( x ) = i δ ( x ) {\displaystyle\partial^{2}\Delta(x)=i\delta(x)\,} Sothat: Δ ( k ) = i k 2 {\displaystyle\Delta(k)={\frac{i}{k^{2}}}} Iftheequationsofmotionarelinear,thepropagatorwillalwaysbethereciprocalofthequadratic-formmatrixthatdefinesthefreeLagrangian,sincethisgivestheequationsofmotion.Thisisalsoeasytoseedirectlyfromthepathintegral.ThefactorofidisappearsintheEuclideantheory. Wicktheorem[edit] Mainarticle:Wick'stheorem BecauseeachfieldmodeisanindependentGaussian,theexpectationvaluesfortheproductofmanyfieldmodesobeysWick'stheorem: ⟨ ϕ ( k 1 ) ϕ ( k 2 ) ⋯ ϕ ( k n ) ⟩ {\displaystyle\left\langle\phi(k_{1})\phi(k_{2})\cdots\phi(k_{n})\right\rangle} iszerounlessthefieldmodescoincideinpairs.Thismeansthatitiszeroforanoddnumberofφ,andforanevennumberofφ,itisequaltoacontributionfromeachpairseparately,withadeltafunction. ⟨ ϕ ( k 1 ) ⋯ ϕ ( k 2 n ) ⟩ = ∑ ∏ i , j δ ( k i − k j ) k i 2 {\displaystyle\left\langle\phi(k_{1})\cdots\phi(k_{2n})\right\rangle=\sum\prod_{i,j}{\frac{\delta\left(k_{i}-k_{j}\right)}{k_{i}^{2}}}} wherethesumisovereachpartitionofthefieldmodesintopairs,andtheproductisoverthepairs.Forexample, ⟨ ϕ ( k 1 ) ϕ ( k 2 ) ϕ ( k 3 ) ϕ ( k 4 ) ⟩ = δ ( k 1 − k 2 ) k 1 2 δ ( k 3 − k 4 ) k 3 2 + δ ( k 1 − k 3 ) k 3 2 δ ( k 2 − k 4 ) k 2 2 + δ ( k 1 − k 4 ) k 1 2 δ ( k 2 − k 3 ) k 2 2 {\displaystyle\left\langle\phi(k_{1})\phi(k_{2})\phi(k_{3})\phi(k_{4})\right\rangle={\frac{\delta(k_{1}-k_{2})}{k_{1}^{2}}}{\frac{\delta(k_{3}-k_{4})}{k_{3}^{2}}}+{\frac{\delta(k_{1}-k_{3})}{k_{3}^{2}}}{\frac{\delta(k_{2}-k_{4})}{k_{2}^{2}}}+{\frac{\delta(k_{1}-k_{4})}{k_{1}^{2}}}{\frac{\delta(k_{2}-k_{3})}{k_{2}^{2}}}} AninterpretationofWick'stheoremisthateachfieldinsertioncanbethoughtofasadanglingline,andtheexpectationvalueiscalculatedbylinkingupthelinesinpairs,puttingadeltafunctionfactorthatensuresthatthemomentumofeachpartnerinthepairisequal,anddividingbythepropagator. HigherGaussianmoments—completingWick'stheorem[edit] ThereisasubtlepointleftbeforeWick'stheoremisproved—whatifmorethantwoofthe ϕ {\displaystyle\phi} shavethesamemomentum?Ifit'sanoddnumber,theintegraliszero;negativevaluescancelwiththepositivevalues.Butifthenumberiseven,theintegralispositive.Thepreviousdemonstrationassumedthatthe ϕ {\displaystyle\phi} swouldonlymatchupinpairs. Butthetheoremiscorrectevenwhenarbitrarilymanyofthe ϕ {\displaystyle\phi} areequal,andthisisanotablepropertyofGaussianintegration: I = ∫ e − a x 2 / 2 d x = 2 π a {\displaystyleI=\inte^{-ax^{2}/2}dx={\sqrt{\frac{2\pi}{a}}}} ∂ n ∂ a n I = ∫ x 2 n 2 n e − a x 2 / 2 d x = 1 ⋅ 3 ⋅ 5 … ⋅ ( 2 n − 1 ) 2 ⋅ 2 ⋅ 2 … ⋅ 2 2 π a − 2 n + 1 2 {\displaystyle{\frac{\partial^{n}}{\partiala^{n}}}I=\int{\frac{x^{2n}}{2^{n}}}e^{-ax^{2}/2}dx={\frac{1\cdot3\cdot5\ldots\cdot(2n-1)}{2\cdot2\cdot2\ldots\;\;\;\;\;\cdot2\;\;\;\;\;\;}}{\sqrt{2\pi}}\,a^{-{\frac{2n+1}{2}}}} DividingbyI, ⟨ x 2 n ⟩ = ∫ x 2 n e − a x 2 / 2 ∫ e − a x 2 / 2 = 1 ⋅ 3 ⋅ 5 … ⋅ ( 2 n − 1 ) 1 a n {\displaystyle\left\langlex^{2n}\right\rangle={\frac{\displaystyle\intx^{2n}e^{-ax^{2}/2}}{\displaystyle\inte^{-ax^{2}/2}}}=1\cdot3\cdot5\ldots\cdot(2n-1){\frac{1}{a^{n}}}} ⟨ x 2 ⟩ = 1 a {\displaystyle\left\langlex^{2}\right\rangle={\frac{1}{a}}} IfWick'stheoremwerecorrect,thehighermomentswouldbegivenbyallpossiblepairingsofalistof2ndifferentx: ⟨ x 1 x 2 x 3 ⋯ x 2 n ⟩ {\displaystyle\left\langlex_{1}x_{2}x_{3}\cdotsx_{2n}\right\rangle} wherethexareallthesamevariable,theindexisjusttokeeptrackofthenumberofwaystopairthem.Thefirstxcanbepairedwith2n−1others,leaving2n−2.Thenextunpairedxcanbepairedwith2n−3differentxleaving2n−4,andsoon.ThismeansthatWick'stheorem,uncorrected,saysthattheexpectationvalueofx2nshouldbe: ⟨ x 2 n ⟩ = ( 2 n − 1 ) ⋅ ( 2 n − 3 ) … ⋅ 5 ⋅ 3 ⋅ 1 ⟨ x 2 ⟩ n {\displaystyle\left\langlex^{2n}\right\rangle=(2n-1)\cdot(2n-3)\ldots\cdot5\cdot3\cdot1\left\langlex^{2}\right\rangle^{n}} andthisisinfactthecorrectanswer.SoWick'stheoremholdsnomatterhowmanyofthemomentaoftheinternalvariablescoincide. Interaction[edit] Interactionsarerepresentedbyhigherordercontributions,sincequadraticcontributionsarealwaysGaussian.Thesimplestinteractionisthequarticself-interaction,withanaction: S = ∫ ∂ μ ϕ ∂ μ ϕ + λ 4 ! ϕ 4 . {\displaystyleS=\int\partial^{\mu}\phi\partial_{\mu}\phi+{\frac{\lambda}{4!}}\phi^{4}.} Thereasonforthecombinatorialfactor4!willbeclearsoon.Writingtheactionintermsofthelattice(orcontinuum)Fouriermodes: S = ∫ k k 2 | ϕ ( k ) | 2 + λ 4 ! ∫ k 1 k 2 k 3 k 4 ϕ ( k 1 ) ϕ ( k 2 ) ϕ ( k 3 ) ϕ ( k 4 ) δ ( k 1 + k 2 + k 3 + k 4 ) = S F + X . {\displaystyleS=\int_{k}k^{2}\left|\phi(k)\right|^{2}+{\frac{\lambda}{4!}}\int_{k_{1}k_{2}k_{3}k_{4}}\phi(k_{1})\phi(k_{2})\phi(k_{3})\phi(k_{4})\delta(k_{1}+k_{2}+k_{3}+k_{4})=S_{F}+X.} WhereSFisthefreeaction,whosecorrelationfunctionsaregivenbyWick'stheorem.TheexponentialofSinthepathintegralcanbeexpandedinpowersofλ,givingaseriesofcorrectionstothefreeaction. e − S = e − S F ( 1 + X + 1 2 ! X X + 1 3 ! X X X + ⋯ ) {\displaystylee^{-S}=e^{-S_{F}}\left(1+X+{\frac{1}{2!}}XX+{\frac{1}{3!}}XXX+\cdots\right)} Thepathintegralfortheinteractingactionisthenapowerseriesofcorrectionstothefreeaction.ThetermrepresentedbyXshouldbethoughtofasfourhalf-lines,oneforeachfactorofφ(k).Thehalf-linesmeetatavertex,whichcontributesadelta-functionthatensuresthatthesumofthemomentaareallequal. Tocomputeacorrelationfunctionintheinteractingtheory,thereisacontributionfromtheXtermsnow.Forexample,thepath-integralforthefour-fieldcorrelator: ⟨ ϕ ( k 1 ) ϕ ( k 2 ) ϕ ( k 3 ) ϕ ( k 4 ) ⟩ = ∫ e − S ϕ ( k 1 ) ϕ ( k 2 ) ϕ ( k 3 ) ϕ ( k 4 ) D ϕ Z {\displaystyle\left\langle\phi(k_{1})\phi(k_{2})\phi(k_{3})\phi(k_{4})\right\rangle={\frac{\displaystyle\inte^{-S}\phi(k_{1})\phi(k_{2})\phi(k_{3})\phi(k_{4})D\phi}{Z}}} whichinthefreefieldwasonlynonzerowhenthemomentakwereequalinpairs,isnownonzeroforallvaluesofk.Themomentaoftheinsertionsφ(ki)cannowmatchupwiththemomentaoftheXsintheexpansion.Theinsertionsshouldalsobethoughtofashalf-lines,fourinthiscase,whichcarryamomentumk,butonethatisnotintegrated. Thelowest-ordercontributioncomesfromthefirstnontrivialterme−SFXintheTaylorexpansionoftheaction.Wick'stheoremrequiresthatthemomentaintheXhalf-lines,theφ(k)factorsinX,shouldmatchupwiththemomentaoftheexternalhalf-linesinpairs.Thenewcontributionisequalto: λ 1 k 1 2 1 k 2 2 1 k 3 2 1 k 4 2 . {\displaystyle\lambda{\frac{1}{k_{1}^{2}}}{\frac{1}{k_{2}^{2}}}{\frac{1}{k_{3}^{2}}}{\frac{1}{k_{4}^{2}}}\,.} The4!insideXiscanceledbecausethereareexactly4!waystomatchthehalf-linesinXtotheexternalhalf-lines.Eachofthesedifferentwaysofmatchingthehalf-linestogetherinpairscontributesexactlyonce,regardlessofthevaluesofk1,2,3,4,byWick'stheorem. Feynmandiagrams[edit] TheexpansionoftheactioninpowersofXgivesaseriesoftermswithprogressivelyhighernumberofXs.ThecontributionfromthetermwithexactlynXsiscallednthorder. Thenthordertermshas: 4ninternalhalf-lines,whicharethefactorsofφ(k)fromtheXs.Theseallendonavertex,andareintegratedoverallpossiblek. externalhalf-lines,whicharethecomefromtheφ(k)insertionsintheintegral. ByWick'stheorem,eachpairofhalf-linesmustbepairedtogethertomakealine,andthislinegivesafactorof δ ( k 1 + k 2 ) k 1 2 {\displaystyle{\frac{\delta(k_{1}+k_{2})}{k_{1}^{2}}}} whichmultipliesthecontribution.Thismeansthatthetwohalf-linesthatmakealineareforcedtohaveequalandoppositemomentum.Thelineitselfshouldbelabelledbyanarrow,drawnparalleltotheline,andlabeledbythemomentuminthelinek.Thehalf-lineatthetailendofthearrowcarriesmomentumk,whilethehalf-lineatthehead-endcarriesmomentum−k.Ifoneofthetwohalf-linesisexternal,thiskillstheintegralovertheinternalk,sinceitforcestheinternalktobeequaltotheexternalk.Ifbothareinternal,theintegraloverkremains. Thediagramsthatareformedbylinkingthehalf-linesintheXswiththeexternalhalf-lines,representinginsertions,aretheFeynmandiagramsofthistheory.Eachlinecarriesafactorof1/k2,thepropagator,andeithergoesfromvertextovertex,orendsataninsertion.Ifitisinternal,itisintegratedover.Ateachvertex,thetotalincomingkisequaltothetotaloutgoingk. Thenumberofwaysofmakingadiagrambyjoininghalf-linesintolinesalmostcompletelycancelsthefactorialfactorscomingfromtheTaylorseriesoftheexponentialandthe4!ateachvertex. Looporder[edit] Aforestdiagramisonewherealltheinternallineshavemomentumthatiscompletelydeterminedbytheexternallinesandtheconditionthattheincomingandoutgoingmomentumareequalateachvertex.Thecontributionofthesediagramsisaproductofpropagators,withoutanyintegration.Atreediagramisaconnectedforestdiagram. AnexampleofatreediagramistheonewhereeachoffourexternallinesendonanX.AnotheriswhenthreeexternallinesendonanX,andtheremaininghalf-linejoinsupwithanotherX,andtheremaininghalf-linesofthisXrunofftoexternallines.Theseareallalsoforestdiagrams(aseverytreeisaforest);anexampleofaforestthatisnotatreeiswheneightexternallinesendontwoXs. Itiseasytoverifythatinallthesecases,themomentaonalltheinternallinesisdeterminedbytheexternalmomentaandtheconditionofmomentumconservationineachvertex. Adiagramthatisnotaforestdiagramiscalledaloopdiagram,andanexampleisonewheretwolinesofanXarejoinedtoexternallines,whiletheremainingtwolinesarejoinedtoeachother.Thetwolinesjoinedtoeachothercanhaveanymomentumatall,sincetheybothenterandleavethesamevertex.AmorecomplicatedexampleisonewheretwoXsarejoinedtoeachotherbymatchingthelegsonetotheother.Thisdiagramhasnoexternallinesatall. Thereasonloopdiagramsarecalledloopdiagramsisbecausethenumberofk-integralsthatareleftundeterminedbymomentumconservationisequaltothenumberofindependentclosedloopsinthediagram,whereindependentloopsarecountedasinhomologytheory.Thehomologyisreal-valued(actuallyRdvalued),thevalueassociatedwitheachlineisthemomentum.Theboundaryoperatortakeseachlinetothesumoftheend-verticeswithapositivesignattheheadandanegativesignatthetail.Theconditionthatthemomentumisconservedisexactlytheconditionthattheboundaryofthek-valuedweightedgraphiszero. Asetofvalidk-valuescanbearbitrarilyredefinedwheneverthereisaclosedloop.Aclosedloopisacyclicalpathofadjacentverticesthatneverrevisitsthesamevertex.Suchacyclecanbethoughtofastheboundaryofahypothetical2-cell.Thek-labellingsofagraphthatconservemomentum(i.e.whichhaszeroboundary)uptoredefinitionsofk(i.e.uptoboundariesof2-cells)definethefirsthomologyofagraph.Thenumberofindependentmomentathatarenotdeterminedisthenequaltothenumberofindependenthomologyloops.Formanygraphs,thisisequaltothenumberofloopsascountedinthemostintuitiveway. Symmetryfactors[edit] ThenumberofwaystoformagivenFeynmandiagrambyjoiningtogetherhalf-linesislarge,andbyWick'stheorem,eachwayofpairingupthehalf-linescontributesequally.Often,thiscompletelycancelsthefactorialsinthedenominatorofeachterm,butthecancellationissometimesincomplete. Theuncancelleddenominatoriscalledthesymmetryfactorofthediagram.Thecontributionofeachdiagramtothecorrelationfunctionmustbedividedbyitssymmetryfactor. Forexample,considertheFeynmandiagramformedfromtwoexternallinesjoinedtooneX,andtheremainingtwohalf-linesintheXjoinedtoeachother.Thereare4 × 3waystojointheexternalhalf-linestotheX,andthenthereisonlyonewaytojointhetworemaininglinestoeachother.TheXcomesdividedby4!=4×3×2,butthenumberofwaystolinkuptheXhalflinestomakethediagramisonly4 × 3,sothecontributionofthisdiagramisdividedbytwo. Foranotherexample,considerthediagramformedbyjoiningallthehalf-linesofoneXtoallthehalf-linesofanotherX.Thisdiagramiscalledavacuumbubble,becauseitdoesnotlinkuptoanyexternallines.Thereare4!waystoformthisdiagram,butthedenominatorincludesa2!(fromtheexpansionoftheexponential,therearetwoXs)andtwofactorsof4!.Thecontributionismultipliedby4!/2×4!×4!= 1/48. AnotherexampleistheFeynmandiagramformedfromtwoXswhereeachXlinksuptotwoexternallines,andtheremainingtwohalf-linesofeachXarejoinedtoeachother.ThenumberofwaystolinkanXtotwoexternallinesis4 × 3,andeitherXcouldlinkuptoeitherpair,givinganadditionalfactorof2.Theremainingtwohalf-linesinthetwoXscanbelinkedtoeachotherintwoways,sothatthetotalnumberofwaystoformthediagramis4×3×4×3×2×2,whilethedenominatoris4!×4!×2!.Thetotalsymmetryfactoris2,andthecontributionofthisdiagramisdividedby2. Thesymmetryfactortheoremgivesthesymmetryfactorforageneraldiagram:thecontributionofeachFeynmandiagrammustbedividedbytheorderofitsgroupofautomorphisms,thenumberofsymmetriesthatithas. AnautomorphismofaFeynmangraphisapermutationMofthelinesandapermutationNoftheverticeswiththefollowingproperties: Ifalinelgoesfromvertexvtovertexv′,thenM(l)goesfromN(v)toN(v′).Ifthelineisundirected,asitisforarealscalarfield,thenM(l)cangofromN(v′)toN(v)too. Ifalinelendsonanexternalline,M(l)endsonthesameexternalline. Iftherearedifferenttypesoflines,M(l)shouldpreservethetype. Thistheoremhasaninterpretationintermsofparticle-paths:whenidenticalparticlesarepresent,theintegraloverallintermediateparticlesmustnotdouble-countstatesthatdifferonlybyinterchangingidenticalparticles. Proof:Toprovethistheorem,labelalltheinternalandexternallinesofadiagramwithauniquename.Thenformthediagrambylinkingahalf-linetoanameandthentotheotherhalfline. Nowcountthenumberofwaystoformthenameddiagram.EachpermutationoftheXsgivesadifferentpatternoflinkingnamestohalf-lines,andthisisafactorofn!.Eachpermutationofthehalf-linesinasingleXgivesafactorof4!.SoanameddiagramcanbeformedinexactlyasmanywaysasthedenominatoroftheFeynmanexpansion. Butthenumberofunnameddiagramsissmallerthanthenumberofnameddiagrambytheorderoftheautomorphismgroupofthegraph. Connecteddiagrams:linked-clustertheorem[edit] Roughlyspeaking,aFeynmandiagramiscalledconnectedifallverticesandpropagatorlinesarelinkedbyasequenceofverticesandpropagatorsofthediagramitself.Ifoneviewsitasanundirectedgraphitisconnected.TheremarkablerelevanceofsuchdiagramsinQFTsisduetothefactthattheyaresufficienttodeterminethequantumpartitionfunctionZ[J].Moreprecisely,connectedFeynmandiagramsdetermine i W [ J ] ≡ ln ⁡ Z [ J ] . {\displaystyleiW[J]\equiv\lnZ[J].} Toseethis,oneshouldrecallthat Z [ J ] ∝ ∑ k D k {\displaystyleZ[J]\propto\sum_{k}{D_{k}}} withDkconstructedfromsome(arbitrary)FeynmandiagramthatcanbethoughttoconsistofseveralconnectedcomponentsCi.Ifoneencountersni(identical)copiesofacomponentCiwithintheFeynmandiagramDkonehastoincludeasymmetryfactorni!.However,intheendeachcontributionofaFeynmandiagramDktothepartitionfunctionhasthegenericform ∏ i C i n i n i ! {\displaystyle\prod_{i}{\frac{C_{i}^{n_{i}}}{n_{i}!}}} whereilabelsthe(infinitely)manyconnectedFeynmandiagramspossible. AschemetosuccessivelycreatesuchcontributionsfromtheDktoZ[J]isobtainedby ( 1 0 ! + C 1 1 ! + C 1 2 2 ! + ⋯ ) ( 1 + C 2 + 1 2 C 2 2 + ⋯ ) ⋯ {\displaystyle\left({\frac{1}{0!}}+{\frac{C_{1}}{1!}}+{\frac{C_{1}^{2}}{2!}}+\cdots\right)\left(1+C_{2}+{\frac{1}{2}}C_{2}^{2}+\cdots\right)\cdots} andthereforeyields Z [ J ] ∝ ∏ i ∑ n i = 0 ∞ C i n i n i ! = exp ⁡ ∑ i C i ∝ exp ⁡ W [ J ] . {\displaystyleZ[J]\propto\prod_{i}{\sum_{n_{i}=0}^{\infty}{\frac{C_{i}^{n_{i}}}{n_{i}!}}}=\exp{\sum_{i}{C_{i}}}\propto\exp{W[J]}\,.} ToestablishthenormalizationZ0=expW[0]=1onesimplycalculatesallconnectedvacuumdiagrams,i.e.,thediagramswithoutanysourcesJ(sometimesreferredtoasexternallegsofaFeynmandiagram). Vacuumbubbles[edit] Animmediateconsequenceofthelinked-clustertheoremisthatallvacuumbubbles,diagramswithoutexternallines,cancelwhencalculatingcorrelationfunctions.Acorrelationfunctionisgivenbyaratioofpath-integrals: ⟨ ϕ 1 ( x 1 ) ⋯ ϕ n ( x n ) ⟩ = ∫ e − S ϕ 1 ( x 1 ) ⋯ ϕ n ( x n ) D ϕ ∫ e − S D ϕ . {\displaystyle\left\langle\phi_{1}(x_{1})\cdots\phi_{n}(x_{n})\right\rangle={\frac{\displaystyle\inte^{-S}\phi_{1}(x_{1})\cdots\phi_{n}(x_{n})\,D\phi}{\displaystyle\inte^{-S}\,D\phi}}\,.} ThetopisthesumoverallFeynmandiagrams,includingdisconnecteddiagramsthatdonotlinkuptoexternallinesatall.Intermsoftheconnecteddiagrams,thenumeratorincludesthesamecontributionsofvacuumbubblesasthedenominator: ∫ e − S ϕ 1 ( x 1 ) ⋯ ϕ n ( x n ) D ϕ = ( ∑ E i ) ( exp ⁡ ( ∑ i C i ) ) . {\displaystyle\inte^{-S}\phi_{1}(x_{1})\cdots\phi_{n}(x_{n})\,D\phi=\left(\sumE_{i}\right)\left(\exp\left(\sum_{i}C_{i}\right)\right)\,.} WherethesumoverEdiagramsincludesonlythosediagramseachofwhoseconnectedcomponentsendonatleastoneexternalline.Thevacuumbubblesarethesamewhatevertheexternallines,andgiveanoverallmultiplicativefactor.Thedenominatoristhesumoverallvacuumbubbles,anddividinggetsridofthesecondfactor. ThevacuumbubblesthenareonlyusefulfordeterminingZitself,whichfromthedefinitionofthepathintegralisequalto: Z = ∫ e − S D ϕ = e − H T = e − ρ V {\displaystyleZ=\inte^{-S}D\phi=e^{-HT}=e^{-\rhoV}} whereρistheenergydensityinthevacuum.Eachvacuumbubblecontainsafactorofδ(k)zeroingthetotalkateachvertex,andwhentherearenoexternallines,thiscontainsafactorofδ(0),becausethemomentumconservationisover-enforced.Infinitevolume,thisfactorcanbeidentifiedasthetotalvolumeofspacetime.Dividingbythevolume,theremainingintegralforthevacuumbubblehasaninterpretation:itisacontributiontotheenergydensityofthevacuum. Sources[edit] CorrelationfunctionsarethesumoftheconnectedFeynmandiagrams,buttheformalismtreatstheconnectedanddisconnecteddiagramsdifferently.Internallinesendonvertices,whileexternallinesgoofftoinsertions.Introducingsourcesunifiestheformalism,bymakingnewverticeswhereonelinecanend. Sourcesareexternalfields,fieldsthatcontributetotheaction,butarenotdynamicalvariables.Ascalarfieldsourceisanotherscalarfieldhthatcontributesatermtothe(Lorentz)Lagrangian: ∫ h ( x ) ϕ ( x ) d d x = ∫ h ( k ) ϕ ( k ) d d k {\displaystyle\inth(x)\phi(x)\,d^{d}x=\inth(k)\phi(k)\,d^{d}k\,} IntheFeynmanexpansion,thiscontributesHtermswithonehalf-lineendingonavertex.LinesinaFeynmandiagramcannowendeitheronanXvertex,oronanHvertex,andonlyonelineentersanHvertex.TheFeynmanruleforanHvertexisthatalinefromanHwithmomentumkgetsafactorofh(k). Thesumoftheconnecteddiagramsinthepresenceofsourcesincludesatermforeachconnecteddiagramintheabsenceofsources,exceptnowthediagramscanendonthesource.Traditionally,asourceisrepresentedbyalittle"×"withonelineextendingout,exactlyasaninsertion. log ⁡ ( Z [ h ] ) = ∑ n , C h ( k 1 ) h ( k 2 ) ⋯ h ( k n ) C ( k 1 , ⋯ , k n ) {\displaystyle\log{\big(}Z[h]{\big)}=\sum_{n,C}h(k_{1})h(k_{2})\cdotsh(k_{n})C(k_{1},\cdots,k_{n})\,} whereC(k1,...,kn)istheconnecteddiagramwithnexternallinescarryingmomentumasindicated.Thesumisoverallconnecteddiagrams,asbefore. Thefieldhisnotdynamical,whichmeansthatthereisnopathintegraloverh:hisjustaparameterintheLagrangian,whichvariesfrompointtopoint.Thepathintegralforthefieldis: Z [ h ] = ∫ e i S + i ∫ h ϕ D ϕ {\displaystyleZ[h]=\inte^{iS+i\inth\phi}\,D\phi\,} anditisafunctionofthevaluesofhateverypoint.OnewaytointerpretthisexpressionisthatitistakingtheFouriertransforminfieldspace.IfthereisaprobabilitydensityonRn,theFouriertransformoftheprobabilitydensityis: ∫ ρ ( y ) e i k y d n y = ⟨ e i k y ⟩ = ⟨ ∏ i = 1 n e i h i y i ⟩ {\displaystyle\int\rho(y)e^{iky}\,d^{n}y=\left\langlee^{iky}\right\rangle=\left\langle\prod_{i=1}^{n}e^{ih_{i}y_{i}}\right\rangle\,} TheFouriertransformistheexpectationofanoscillatoryexponential.Thepathintegralinthepresenceofasourceh(x)is: Z [ h ] = ∫ e i S e i ∫ x h ( x ) ϕ ( x ) D ϕ = ⟨ e i h ϕ ⟩ {\displaystyleZ[h]=\inte^{iS}e^{i\int_{x}h(x)\phi(x)}\,D\phi=\left\langlee^{ih\phi}\right\rangle} which,onalattice,istheproductofanoscillatoryexponentialforeachfieldvalue: ⟨ ∏ x e i h x ϕ x ⟩ {\displaystyle\left\langle\prod_{x}e^{ih_{x}\phi_{x}}\right\rangle} TheFouriertransformofadelta-functionisaconstant,whichgivesaformalexpressionforadeltafunction: δ ( x − y ) = ∫ e i k ( x − y ) d k {\displaystyle\delta(x-y)=\inte^{ik(x-y)}\,dk} Thistellsyouwhatafielddeltafunctionlookslikeinapath-integral.Fortwoscalarfieldsφandη, δ ( ϕ − η ) = ∫ e i h ( x ) ( ϕ ( x ) − η ( x ) ) d d x D h , {\displaystyle\delta(\phi-\eta)=\inte^{ih(x){\big(}\phi(x)-\eta(x){\big)}\,d^{d}x}\,Dh\,,} whichintegratesovertheFouriertransformcoordinate,overh.Thisexpressionisusefulforformallychangingfieldcoordinatesinthepathintegral,muchasadeltafunctionisusedtochangecoordinatesinanordinarymulti-dimensionalintegral. Thepartitionfunctionisnowafunctionofthefieldh,andthephysicalpartitionfunctionisthevaluewhenhisthezerofunction: Thecorrelationfunctionsarederivativesofthepathintegralwithrespecttothesource: ⟨ ϕ ( x ) ⟩ = 1 Z ∂ ∂ h ( x ) Z [ h ] = ∂ ∂ h ( x ) log ⁡ ( Z [ h ] ) . {\displaystyle\left\langle\phi(x)\right\rangle={\frac{1}{Z}}{\frac{\partial}{\partialh(x)}}Z[h]={\frac{\partial}{\partialh(x)}}\log{\big(}Z[h]{\big)}\,.} InEuclideanspace,sourcecontributionstotheactioncanstillappearwithafactorofi,sothattheystilldoaFouriertransform. Spin1/2;"photons"and"ghosts"[edit] Spin1/2:Grassmannintegrals[edit] ThefieldpathintegralcanbeextendedtotheFermicase,butonlyifthenotionofintegrationisexpanded.AGrassmannintegralofafreeFermifieldisahigh-dimensionaldeterminantorPfaffian,whichdefinesthenewtypeofGaussianintegrationappropriateforFermifields. ThetwofundamentalformulasofGrassmannintegrationare: ∫ e M i j ψ ¯ i ψ j D ψ ¯ D ψ = D e t ( M ) , {\displaystyle\inte^{M_{ij}{\bar{\psi}}^{i}\psi^{j}}\,D{\bar{\psi}}\,D\psi=\mathrm{Det}(M)\,,} whereMisanarbitrarymatrixandψ,ψareindependentGrassmannvariablesforeachindexi,and ∫ e 1 2 A i j ψ i ψ j D ψ = P f a f f ( A ) , {\displaystyle\inte^{{\frac{1}{2}}A_{ij}\psi^{i}\psi^{j}}\,D\psi=\mathrm{Pfaff}(A)\,,} whereAisanantisymmetricmatrix,ψisacollectionofGrassmannvariables,andthe1/2istopreventdouble-counting(sinceψiψj=−ψjψi). Inmatrixnotation,whereψandηareGrassmann-valuedrowvectors,ηandψareGrassmann-valuedcolumnvectors,andMisareal-valuedmatrix: Z = ∫ e ψ ¯ M ψ + η ¯ ψ + ψ ¯ η D ψ ¯ D ψ = ∫ e ( ψ ¯ + η ¯ M − 1 ) M ( ψ + M − 1 η ) − η ¯ M − 1 η D ψ ¯ D ψ = D e t ( M ) e − η ¯ M − 1 η , {\displaystyleZ=\inte^{{\bar{\psi}}M\psi+{\bar{\eta}}\psi+{\bar{\psi}}\eta}\,D{\bar{\psi}}\,D\psi=\inte^{\left({\bar{\psi}}+{\bar{\eta}}M^{-1}\right)M\left(\psi+M^{-1}\eta\right)-{\bar{\eta}}M^{-1}\eta}\,D{\bar{\psi}}\,D\psi=\mathrm{Det}(M)e^{-{\bar{\eta}}M^{-1}\eta}\,,} wherethelastequalityisaconsequenceofthetranslationinvarianceoftheGrassmannintegral.TheGrassmannvariablesηareexternalsourcesforψ,anddifferentiatingwithrespecttoηpullsdownfactorsofψ. ⟨ ψ ¯ ψ ⟩ = 1 Z ∂ ∂ η ∂ ∂ η ¯ Z | η = η ¯ = 0 = M − 1 {\displaystyle\left\langle{\bar{\psi}}\psi\right\rangle={\frac{1}{Z}}{\frac{\partial}{\partial\eta}}{\frac{\partial}{\partial{\bar{\eta}}}}Z|_{\eta={\bar{\eta}}=0}=M^{-1}} again,inaschematicmatrixnotation.ThemeaningoftheformulaaboveisthatthederivativewithrespecttotheappropriatecomponentofηandηgivesthematrixelementofM−1.ThisisexactlyanalogoustothebosonicpathintegrationformulaforaGaussianintegralofacomplexbosonicfield: ∫ e ϕ ∗ M ϕ + h ∗ ϕ + ϕ ∗ h D ϕ ∗ D ϕ = e h ∗ M − 1 h D e t ( M ) {\displaystyle\inte^{\phi^{*}M\phi+h^{*}\phi+\phi^{*}h}\,D\phi^{*}\,D\phi={\frac{e^{h^{*}M^{-1}h}}{\mathrm{Det}(M)}}} ⟨ ϕ ∗ ϕ ⟩ = 1 Z ∂ ∂ h ∂ ∂ h ∗ Z | h = h ∗ = 0 = M − 1 . {\displaystyle\left\langle\phi^{*}\phi\right\rangle={\frac{1}{Z}}{\frac{\partial}{\partialh}}{\frac{\partial}{\partialh^{*}}}Z|_{h=h^{*}=0}=M^{-1}\,.} SothatthepropagatoristheinverseofthematrixinthequadraticpartoftheactioninboththeBoseandFermicase. ForrealGrassmannfields,forMajoranafermions,thepathintegralisaPfaffiantimesasourcequadraticform,andtheformulasgivethesquarerootofthedeterminant,justastheydoforrealBosonicfields.Thepropagatorisstilltheinverseofthequadraticpart. ThefreeDiracLagrangian: ∫ ψ ¯ ( γ μ ∂ μ − m ) ψ {\displaystyle\int{\bar{\psi}}\left(\gamma^{\mu}\partial_{\mu}-m\right)\psi} formallygivestheequationsofmotionandtheanticommutationrelationsoftheDiracfield,justastheKleinGordonLagrangianinanordinarypathintegralgivestheequationsofmotionandcommutationrelationsofthescalarfield.ByusingthespatialFouriertransformoftheDiracfieldasanewbasisfortheGrassmannalgebra,thequadraticpartoftheDiracactionbecomessimpletoinvert: S = ∫ k ψ ¯ ( i γ μ k μ − m ) ψ . {\displaystyleS=\int_{k}{\bar{\psi}}\left(i\gamma^{\mu}k_{\mu}-m\right)\psi\,.} ThepropagatoristheinverseofthematrixMlinkingψ(k)andψ(k),sincedifferentvaluesofkdonotmixtogether. ⟨ ψ ¯ ( k ′ ) ψ ( k ) ⟩ = δ ( k + k ′ ) 1 γ ⋅ k − m = δ ( k + k ′ ) γ ⋅ k + m k 2 − m 2 {\displaystyle\left\langle{\bar{\psi}}(k')\psi(k)\right\rangle=\delta(k+k'){\frac{1}{\gamma\cdotk-m}}=\delta(k+k'){\frac{\gamma\cdotk+m}{k^{2}-m^{2}}}} TheanalogofWick'stheoremmatchesψandψinpairs: ⟨ ψ ¯ ( k 1 ) ψ ¯ ( k 2 ) ⋯ ψ ¯ ( k n ) ψ ( k 1 ′ ) ⋯ ψ ( k n ) ⟩ = ∑ p a i r i n g s ( − 1 ) S ∏ p a i r s i , j δ ( k i − k j ) 1 γ ⋅ k i − m {\displaystyle\left\langle{\bar{\psi}}(k_{1}){\bar{\psi}}(k_{2})\cdots{\bar{\psi}}(k_{n})\psi(k'_{1})\cdots\psi(k_{n})\right\rangle=\sum_{\mathrm{pairings}}(-1)^{S}\prod_{\mathrm{pairs}\;i,j}\delta\left(k_{i}-k_{j}\right){\frac{1}{\gamma\cdotk_{i}-m}}} whereSisthesignofthepermutationthatreordersthesequenceofψandψtoputtheonesthatarepaireduptomakethedelta-functionsnexttoeachother,withtheψcomingrightbeforetheψ.Sinceaψ,ψpairisacommutingelementoftheGrassmannalgebra,itdoesn'tmatterwhatorderthepairsarein.Ifmorethanoneψ,ψpairhavethesamek,theintegraliszero,anditiseasytocheckthatthesumoverpairingsgiveszerointhiscase(therearealwaysanevennumberofthem).ThisistheGrassmannanalogofthehigherGaussianmomentsthatcompletedtheBosonicWick'stheoremearlier. Therulesforspin-1/2Diracparticlesareasfollows:ThepropagatoristheinverseoftheDiracoperator,thelineshavearrowsjustasforacomplexscalarfield,andthediagramacquiresanoverallfactorof−1foreachclosedFermiloop.IfthereareanoddnumberofFermiloops,thediagramchangessign.Historically,the−1rulewasverydifficultforFeynmantodiscover.Hediscovereditafteralongprocessoftrialanderror,sincehelackedapropertheoryofGrassmannintegration. TherulefollowsfromtheobservationthatthenumberofFermilinesatavertexisalwayseven.EachtermintheLagrangianmustalwaysbeBosonic.AFermiloopiscountedbyfollowingFermioniclinesuntilonecomesbacktothestartingpoint,thenremovingthoselinesfromthediagram.RepeatingthisprocesseventuallyerasesalltheFermioniclines:thisistheEuleralgorithmto2-coloragraph,whichworkswhenevereachvertexhasevendegree.ThenumberofstepsintheEuleralgorithmisonlyequaltothenumberofindependentFermionichomologycyclesinthecommonspecialcasethatalltermsintheLagrangianareexactlyquadraticintheFermifields,sothateachvertexhasexactlytwoFermioniclines.Whentherearefour-Fermiinteractions(likeintheFermieffectivetheoryoftheweaknuclearinteractions)therearemorek-integralsthanFermiloops.Inthiscase,thecountingruleshouldapplytheEuleralgorithmbypairinguptheFermilinesateachvertexintopairsthattogetherformabosonicfactorofthetermintheLagrangian,andwhenenteringavertexbyoneline,thealgorithmshouldalwaysleavewiththepartnerline. Toclarifyandprovetherule,consideraFeynmandiagramformedfromvertices,termsintheLagrangian,withFermionfields.ThefulltermisBosonic,itisacommutingelementoftheGrassmannalgebra,sotheorderinwhichtheverticesappearisnotimportant.TheFermilinesarelinkedintoloops,andwhentraversingtheloop,onecanreorderthevertextermsoneaftertheotherasonegoesaroundwithoutanysigncost.Theexceptioniswhenyoureturntothestartingpoint,andthefinalhalf-linemustbejoinedwiththeunlinkedfirsthalf-line.Thisrequiresonepermutationtomovethelastψtogoinfrontofthefirstψ,andthisgivesthesign. Thisruleistheonlyvisibleeffectoftheexclusionprincipleininternallines.Whenthereareexternallines,theamplitudesareantisymmetricwhentwoFermiinsertionsforidenticalparticlesareinterchanged.Thisisautomaticinthesourceformalism,becausethesourcesforFermifieldsarethemselvesGrassmannvalued. Spin1:photons[edit] Thenaivepropagatorforphotonsisinfinite,sincetheLagrangianfortheA-fieldis: S = ∫ 1 4 F μ ν F μ ν = ∫ − 1 2 ( ∂ μ A ν ∂ μ A ν − ∂ μ A μ ∂ ν A ν ) . {\displaystyleS=\int{\tfrac{1}{4}}F^{\mu\nu}F_{\mu\nu}=\int-{\tfrac{1}{2}}\left(\partial^{\mu}A_{\nu}\partial_{\mu}A^{\nu}-\partial^{\mu}A_{\mu}\partial_{\nu}A^{\nu}\right)\,.} Thequadraticformdefiningthepropagatorisnon-invertible.Thereasonisthegaugeinvarianceofthefield;addingagradienttoAdoesnotchangethephysics. Tofixthisproblem,oneneedstofixagauge.ThemostconvenientwayistodemandthatthedivergenceofAissomefunctionf,whosevalueisrandomfrompointtopoint.Itdoesnoharmtointegrateoverthevaluesoff,sinceitonlydeterminesthechoiceofgauge.ThisprocedureinsertsthefollowingfactorintothepathintegralforA: ∫ δ ( ∂ μ A μ − f ) e − f 2 2 D f . {\displaystyle\int\delta\left(\partial_{\mu}A^{\mu}-f\right)e^{-{\frac{f^{2}}{2}}}\,Df\,.} Thefirstfactor,thedeltafunction,fixesthegauge.Thesecondfactorsumsoverdifferentvaluesoffthatareinequivalentgaugefixings.Thisissimply e − ( ∂ μ A μ ) 2 2 . {\displaystylee^{-{\frac{\left(\partial_{\mu}A_{\mu}\right)^{2}}{2}}}\,.} Theadditionalcontributionfromgauge-fixingcancelsthesecondhalfofthefreeLagrangian,givingtheFeynmanLagrangian: S = ∫ ∂ μ A ν ∂ μ A ν {\displaystyleS=\int\partial^{\mu}A^{\nu}\partial_{\mu}A_{\nu}} whichisjustlikefourindependentfreescalarfields,oneforeachcomponentofA.TheFeynmanpropagatoris: ⟨ A μ ( k ) A ν ( k ′ ) ⟩ = δ ( k + k ′ ) g μ ν k 2 . {\displaystyle\left\langleA_{\mu}(k)A_{\nu}(k')\right\rangle=\delta\left(k+k'\right){\frac{g_{\mu\nu}}{k^{2}}}.} TheonedifferenceisthatthesignofonepropagatoriswrongintheLorentzcase:thetimelikecomponenthasanoppositesignpropagator.Thismeansthattheseparticlestateshavenegativenorm—theyarenotphysicalstates.Inthecaseofphotons,itiseasytoshowbydiagrammethodsthatthesestatesarenotphysical—theircontributioncancelswithlongitudinalphotonstoonlyleavetwophysicalphotonpolarizationcontributionsforanyvalueofk. Iftheaveragingoverfisdonewithacoefficientdifferentfrom1/2,thetwotermsdon'tcancelcompletely.ThisgivesacovariantLagrangianwithacoefficient λ {\displaystyle\lambda} ,whichdoesnotaffectanything: S = ∫ 1 2 ( ∂ μ A ν ∂ μ A ν − λ ( ∂ μ A μ ) 2 ) {\displaystyleS=\int{\tfrac{1}{2}}\left(\partial^{\mu}A^{\nu}\partial_{\mu}A_{\nu}-\lambda\left(\partial_{\mu}A^{\mu}\right)^{2}\right)} andthecovariantpropagatorforQEDis: ⟨ A μ ( k ) A ν ( k ′ ) ⟩ = δ ( k + k ′ ) g μ ν − λ k μ k ν k 2 k 2 . {\displaystyle\left\langleA_{\mu}(k)A_{\nu}(k')\right\rangle=\delta\left(k+k'\right){\frac{g_{\mu\nu}-\lambda{\frac{k_{\mu}k_{\nu}}{k^{2}}}}{k^{2}}}.} Spin1:non-Abelianghosts[edit] TofindtheFeynmanrulesfornon-Abeliangaugefields,theprocedurethatperformsthegaugefixingmustbecarefullycorrectedtoaccountforachangeofvariablesinthepath-integral. Thegaugefixingfactorhasanextradeterminantfrompoppingthedeltafunction: δ ( ∂ μ A μ − f ) e − f 2 2 det M {\displaystyle\delta\left(\partial_{\mu}A_{\mu}-f\right)e^{-{\frac{f^{2}}{2}}}\detM} Tofindtheformofthedeterminant,considerfirstasimpletwo-dimensionalintegralofafunctionfthatdependsonlyonr,notontheangleθ.Insertinganintegraloverθ: ∫ f ( r ) d x d y = ∫ f ( r ) ∫ d θ δ ( y ) | d y d θ | d x d y {\displaystyle\intf(r)\,dx\,dy=\intf(r)\intd\theta\,\delta(y)\left|{\frac{dy}{d\theta}}\right|\,dx\,dy} Thederivative-factorensuresthatpoppingthedeltafunctioninθremovestheintegral.Exchangingtheorderofintegration, ∫ f ( r ) d x d y = ∫ d θ ∫ f ( r ) δ ( y ) | d y d θ | d x d y {\displaystyle\intf(r)\,dx\,dy=\intd\theta\,\intf(r)\delta(y)\left|{\frac{dy}{d\theta}}\right|\,dx\,dy} butnowthedelta-functioncanbepoppediny, ∫ f ( r ) d x d y = ∫ d θ 0 ∫ f ( x ) | d y d θ | d x . {\displaystyle\intf(r)\,dx\,dy=\intd\theta_{0}\,\intf(x)\left|{\frac{dy}{d\theta}}\right|\,dx\,.} Theintegraloverθjustgivesanoverallfactorof2π,whiletherateofchangeofywithachangeinθisjustx,sothisexercisereproducesthestandardformulaforpolarintegrationofaradialfunction: ∫ f ( r ) d x d y = 2 π ∫ f ( x ) x d x {\displaystyle\intf(r)\,dx\,dy=2\pi\intf(x)x\,dx} Inthepath-integralforanonabeliangaugefield,theanalogousmanipulationis: ∫ D A ∫ δ ( F ( A ) ) det ( ∂ F ∂ G ) D G e i S = ∫ D G ∫ δ ( F ( A ) ) det ( ∂ F ∂ G ) e i S {\displaystyle\intDA\int\delta{\big(}F(A){\big)}\det\left({\frac{\partialF}{\partialG}}\right)\,DGe^{iS}=\intDG\int\delta{\big(}F(A){\big)}\det\left({\frac{\partialF}{\partialG}}\right)e^{iS}\,} Thefactorinfrontisthevolumeofthegaugegroup,anditcontributesaconstant,whichcanbediscarded.Theremainingintegralisoverthegaugefixedaction. ∫ det ( ∂ F ∂ G ) e i S G F D A {\displaystyle\int\det\left({\frac{\partialF}{\partialG}}\right)e^{iS_{GF}}\,DA\,} Togetacovariantgauge,thegaugefixingconditionisthesameasintheAbeliancase: ∂ μ A μ = f , {\displaystyle\partial_{\mu}A^{\mu}=f\,,} Whosevariationunderaninfinitesimalgaugetransformationisgivenby: ∂ μ D μ α , {\displaystyle\partial_{\mu}\,D_{\mu}\alpha\,,} whereαistheadjointvaluedelementoftheLiealgebraateverypointthatperformstheinfinitesimalgaugetransformation.ThisaddstheFaddeevPopovdeterminanttotheaction: det ( ∂ μ D μ ) {\displaystyle\det\left(\partial_{\mu}\,D_{\mu}\right)\,} whichcanberewrittenasaGrassmannintegralbyintroducingghostfields: ∫ e η ¯ ∂ μ D μ η D η ¯ D η {\displaystyle\inte^{{\bar{\eta}}\partial_{\mu}\,D^{\mu}\eta}\,D{\bar{\eta}}\,D\eta\,} Thedeterminantisindependentoff,sothepath-integraloverfcangivetheFeynmanpropagator(oracovariantpropagator)bychoosingthemeasureforfasintheabeliancase.ThefullgaugefixedactionisthentheYangMillsactioninFeynmangaugewithanadditionalghostaction: S = ∫ Tr ⁡ ∂ μ A ν ∂ μ A ν + f j k i ∂ ν A i μ A μ j A ν k + f j r i f k l r A i A j A k A l + Tr ⁡ ∂ μ η ¯ ∂ μ η + η ¯ A j η {\displaystyleS=\int\operatorname{Tr}\partial_{\mu}A_{\nu}\partial^{\mu}A^{\nu}+f_{jk}^{i}\partial^{\nu}A_{i}^{\mu}A_{\mu}^{j}A_{\nu}^{k}+f_{jr}^{i}f_{kl}^{r}A_{i}A_{j}A^{k}A^{l}+\operatorname{Tr}\partial_{\mu}{\bar{\eta}}\partial^{\mu}\eta+{\bar{\eta}}A_{j}\eta\,} Thediagramsarederivedfromthisaction.Thepropagatorforthespin-1fieldshastheusualFeynmanform.Thereareverticesofdegree3withmomentumfactorswhosecouplingsarethestructureconstants,andverticesofdegree4whosecouplingsareproductsofstructureconstants.Thereareadditionalghostloops,whichcancelouttimelikeandlongitudinalstatesinAloops. IntheAbeliancase,thedeterminantforcovariantgaugesdoesnotdependonA,sotheghostsdonotcontributetotheconnecteddiagrams. Particle-pathrepresentation[edit] FeynmandiagramswereoriginallydiscoveredbyFeynman,bytrialanderror,asawaytorepresentthecontributiontotheS-matrixfromdifferentclassesofparticletrajectories. Schwingerrepresentation[edit] TheEuclideanscalarpropagatorhasasuggestiverepresentation: 1 p 2 + m 2 = ∫ 0 ∞ e − τ ( p 2 + m 2 ) d τ {\displaystyle{\frac{1}{p^{2}+m^{2}}}=\int_{0}^{\infty}e^{-\tau\left(p^{2}+m^{2}\right)}\,d\tau} Themeaningofthisidentity(whichisanelementaryintegration)ismadeclearerbyFouriertransformingtorealspace. Δ ( x ) = ∫ 0 ∞ d τ e − m 2 τ 1 ( 4 π τ ) d / 2 e − x 2 4 τ {\displaystyle\Delta(x)=\int_{0}^{\infty}d\taue^{-m^{2}\tau}{\frac{1}{({4\pi\tau})^{d/2}}}e^{\frac{-x^{2}}{4\tau}}} ThecontributionatanyonevalueofτtothepropagatorisaGaussianofwidth√τ.Thetotalpropagationfunctionfrom0toxisaweightedsumoverallpropertimesτofanormalizedGaussian,theprobabilityofendingupatxafterarandomwalkoftimeτ. Thepath-integralrepresentationforthepropagatoristhen: Δ ( x ) = ∫ 0 ∞ d τ ∫ D X e − ∫ 0 τ ( x ˙ 2 2 + m 2 ) d τ ′ {\displaystyle\Delta(x)=\int_{0}^{\infty}d\tau\intDX\,e^{-\int\limits_{0}^{\tau}\left({\frac{{\dot{x}}^{2}}{2}}+m^{2}\right)d\tau'}} whichisapath-integralrewriteoftheSchwingerrepresentation. TheSchwingerrepresentationisbothusefulformakingmanifesttheparticleaspectofthepropagator,andforsymmetrizingdenominatorsofloopdiagrams. Combiningdenominators[edit] TheSchwingerrepresentationhasanimmediatepracticalapplicationtoloopdiagrams.Forexample,forthediagramintheφ4theoryformedbyjoiningtwoxstogetherintwohalf-lines,andmakingtheremaininglinesexternal,theintegralovertheinternalpropagatorsintheloopis: ∫ k 1 k 2 + m 2 1 ( k + p ) 2 + m 2 . {\displaystyle\int_{k}{\frac{1}{k^{2}+m^{2}}}{\frac{1}{(k+p)^{2}+m^{2}}}\,.} Hereonelinecarriesmomentumkandtheotherk+p.TheasymmetrycanbefixedbyputtingeverythingintheSchwingerrepresentation. ∫ t , t ′ e − t ( k 2 + m 2 ) − t ′ ( ( k + p ) 2 + m 2 ) d t d t ′ . {\displaystyle\int_{t,t'}e^{-t(k^{2}+m^{2})-t'\left((k+p)^{2}+m^{2}\right)}\,dt\,dt'\,.} Nowtheexponentmostlydependsont+t′, ∫ t , t ′ e − ( t + t ′ ) ( k 2 + m 2 ) − t ′ 2 p ⋅ k − t ′ p 2 , {\displaystyle\int_{t,t'}e^{-(t+t')(k^{2}+m^{2})-t'2p\cdotk-t'p^{2}}\,,} exceptfortheasymmetricallittlebit.Definingthevariableu=t+t′andv=t′/u,thevariableugoesfrom0to∞,whilevgoesfrom0to1.Thevariableuisthetotalpropertimefortheloop,whilevparametrizesthefractionofthepropertimeonthetopoftheloopversusthebottom. TheJacobianforthistransformationofvariablesiseasytoworkoutfromtheidentities: d ( u v ) = d t ′ d u = d t + d t ′ , {\displaystyled(uv)=dt'\quaddu=dt+dt'\,,} and"wedging"gives u d u ∧ d v = d t ∧ d t ′ {\displaystyleu\,du\wedgedv=dt\wedgedt'\,} . Thisallowstheuintegraltobeevaluatedexplicitly: ∫ u , v u e − u ( k 2 + m 2 + v 2 p ⋅ k + v p 2 ) = ∫ 1 ( k 2 + m 2 + v 2 p ⋅ k − v p 2 ) 2 d v {\displaystyle\int_{u,v}ue^{-u\left(k^{2}+m^{2}+v2p\cdotk+vp^{2}\right)}=\int{\frac{1}{\left(k^{2}+m^{2}+v2p\cdotk-vp^{2}\right)^{2}}}\,dv} leavingonlythev-integral.Thismethod,inventedbySchwingerbutusuallyattributedtoFeynman,iscalledcombiningdenominator.Abstractly,itistheelementaryidentity: 1 A B = ∫ 0 1 1 ( v A + ( 1 − v ) B ) 2 d v {\displaystyle{\frac{1}{AB}}=\int_{0}^{1}{\frac{1}{{\big(}vA+(1-v)B{\big)}^{2}}}\,dv} Butthisformdoesnotprovidethephysicalmotivationforintroducingv;vistheproportionofpropertimeononeofthelegsoftheloop. Oncethedenominatorsarecombined,ashiftinktok′=k+vpsymmetrizeseverything: ∫ 0 1 ∫ 1 ( k 2 + m 2 + 2 v p ⋅ k + v p 2 ) 2 d k d v = ∫ 0 1 ∫ 1 ( k ′ 2 + m 2 + v ( 1 − v ) p 2 ) 2 d k ′ d v {\displaystyle\int_{0}^{1}\int{\frac{1}{\left(k^{2}+m^{2}+2vp\cdotk+vp^{2}\right)^{2}}}\,dk\,dv=\int_{0}^{1}\int{\frac{1}{\left(k'^{2}+m^{2}+v(1-v)p^{2}\right)^{2}}}\,dk'\,dv} Thisformshowsthatthemomentthatp2ismorenegativethanfourtimesthemassoftheparticleintheloop,whichhappensinaphysicalregionofLorentzspace,theintegralhasacut.Thisisexactlywhentheexternalmomentumcancreatephysicalparticles. Whentheloophasmorevertices,therearemoredenominatorstocombine: ∫ d k 1 k 2 + m 2 1 ( k + p 1 ) 2 + m 2 ⋯ 1 ( k + p n ) 2 + m 2 {\displaystyle\intdk\,{\frac{1}{k^{2}+m^{2}}}{\frac{1}{(k+p_{1})^{2}+m^{2}}}\cdots{\frac{1}{(k+p_{n})^{2}+m^{2}}}} ThegeneralrulefollowsfromtheSchwingerprescriptionforn+1denominators: 1 D 0 D 1 ⋯ D n = ∫ 0 ∞ ⋯ ∫ 0 ∞ e − u 0 D 0 ⋯ − u n D n d u 0 ⋯ d u n . {\displaystyle{\frac{1}{D_{0}D_{1}\cdotsD_{n}}}=\int_{0}^{\infty}\cdots\int_{0}^{\infty}e^{-u_{0}D_{0}\cdots-u_{n}D_{n}}\,du_{0}\cdotsdu_{n}\,.} TheintegralovertheSchwingerparametersuicanbesplitupasbeforeintoanintegraloverthetotalpropertimeu=u0+u1...+unandanintegraloverthefractionofthepropertimeinallbutthefirstsegmentoftheloopvi=ui/ufori∈{1,2,...,n}.Theviarepositiveandadduptolessthan1,sothatthevintegralisoverann-dimensionalsimplex. TheJacobianforthecoordinatetransformationcanbeworkedoutasbefore: d u = d u 0 + d u 1 ⋯ + d u n {\displaystyledu=du_{0}+du_{1}\cdots+du_{n}\,} d ( u v i ) = d u i . {\displaystyled(uv_{i})=du_{i}\,.} Wedgingalltheseequationstogether,oneobtains u n d u ∧ d v 1 ∧ d v 2 ⋯ ∧ d v n = d u 0 ∧ d u 1 ⋯ ∧ d u n . {\displaystyleu^{n}\,du\wedgedv_{1}\wedgedv_{2}\cdots\wedgedv_{n}=du_{0}\wedgedu_{1}\cdots\wedgedu_{n}\,.} Thisgivestheintegral: ∫ 0 ∞ ∫ s i m p l e x u n e − u ( v 0 D 0 + v 1 D 1 + v 2 D 2 ⋯ + v n D n ) d v 1 ⋯ d v n d u , {\displaystyle\int_{0}^{\infty}\int_{\mathrm{simplex}}u^{n}e^{-u\left(v_{0}D_{0}+v_{1}D_{1}+v_{2}D_{2}\cdots+v_{n}D_{n}\right)}\,dv_{1}\cdotsdv_{n}\,du\,,} wherethesimplexistheregiondefinedbytheconditions v i > 0 and ∑ i = 1 n v i < 1 {\displaystylev_{i}>0\quad{\mbox{and}}\quad\sum_{i=1}^{n}v_{i}<1} aswellas v 0 = 1 − ∑ i = 1 n v i . {\displaystylev_{0}=1-\sum_{i=1}^{n}v_{i}\,.} Performingtheuintegralgivesthegeneralprescriptionforcombiningdenominators: 1 D 0 ⋯ D n = n ! ∫ s i m p l e x 1 ( v 0 D 0 + v 1 D 1 ⋯ + v n D n ) n + 1 d v 1 d v 2 ⋯ d v n {\displaystyle{\frac{1}{D_{0}\cdotsD_{n}}}=n!\int_{\mathrm{simplex}}{\frac{1}{\left(v_{0}D_{0}+v_{1}D_{1}\cdots+v_{n}D_{n}\right)^{n+1}}}\,dv_{1}\,dv_{2}\cdotsdv_{n}} Sincethenumeratoroftheintegrandisnotinvolved,thesameprescriptionworksforanyloop,nomatterwhatthespinsarecarriedbythelegs.Theinterpretationoftheparametersviisthattheyarethefractionofthetotalpropertimespentoneachleg. Scattering[edit] Thecorrelationfunctionsofaquantumfieldtheorydescribethescatteringofparticles.Thedefinitionof"particle"inrelativisticfieldtheoryisnotself-evident,becauseifyoutrytodeterminethepositionsothattheuncertaintyislessthanthecomptonwavelength,theuncertaintyinenergyislargeenoughtoproducemoreparticlesandantiparticlesofthesametypefromthevacuum.Thismeansthatthenotionofasingle-particlestateistosomeextentincompatiblewiththenotionofanobjectlocalizedinspace. Inthe1930s,Wignergaveamathematicaldefinitionforsingle-particlestates:theyareacollectionofstatesthatformanirreduciblerepresentationofthePoincarégroup.Singleparticlestatesdescribeanobjectwithafinitemass,awelldefinedmomentum,andaspin.Thisdefinitionisfineforprotonsandneutrons,electronsandphotons,butitexcludesquarks,whicharepermanentlyconfined,sothemodernpointofviewismoreaccommodating:aparticleisanythingwhoseinteractioncanbedescribedintermsofFeynmandiagrams,whichhaveaninterpretationasasumoverparticletrajectories. Afieldoperatorcanacttoproduceaone-particlestatefromthevacuum,whichmeansthatthefieldoperatorφ(x)producesasuperpositionofWignerparticlestates.Inthefreefieldtheory,thefieldproducesoneparticlestatesonly.Butwhenthereareinteractions,thefieldoperatorcanalsoproduce3-particle,5-particle(ifthereisno+/−symmetryalso2,4,6particle)statestoo.Tocomputethescatteringamplitudeforsingleparticlestatesonlyrequiresacarefullimit,sendingthefieldstoinfinityandintegratingoverspacetogetridofthehigher-ordercorrections. TherelationbetweenscatteringandcorrelationfunctionsistheLSZ-theorem:ThescatteringamplitudefornparticlestogotomparticlesinascatteringeventisthegivenbythesumoftheFeynmandiagramsthatgointothecorrelationfunctionforn+mfieldinsertions,leavingoutthepropagatorsfortheexternallegs. Forexample,fortheλφ4interactionoftheprevioussection,theorderλcontributiontothe(Lorentz)correlationfunctionis: ⟨ ϕ ( k 1 ) ϕ ( k 2 ) ϕ ( k 3 ) ϕ ( k 4 ) ⟩ = i k 1 2 i k 2 2 i k 3 2 i k 4 2 i λ {\displaystyle\left\langle\phi(k_{1})\phi(k_{2})\phi(k_{3})\phi(k_{4})\right\rangle={\frac{i}{k_{1}^{2}}}{\frac{i}{k_{2}^{2}}}{\frac{i}{k_{3}^{2}}}{\frac{i}{k_{4}^{2}}}i\lambda\,} Strippingofftheexternalpropagators,thatis,removingthefactorsofi/k2,givestheinvariantscatteringamplitudeM: M = i λ {\displaystyleM=i\lambda\,} whichisaconstant,independentoftheincomingandoutgoingmomentum.Theinterpretationofthescatteringamplitudeisthatthesumof|M|2overallpossiblefinalstatesistheprobabilityforthescatteringevent.Thenormalizationofthesingle-particlestatesmustbechosencarefully,however,toensurethatMisarelativisticinvariant. Non-relativisticsingleparticlestatesarelabeledbythemomentumk,andtheyarechosentohavethesamenormateveryvalueofk.Thisisbecausethenonrelativisticunitoperatoronsingleparticlestatesis: ∫ d k | k ⟩ ⟨ k | . {\displaystyle\intdk\,|k\rangle\langlek|\,.} Inrelativity,theintegraloverthek-statesforaparticleofmassmintegratesoverahyperbolainE,kspacedefinedbytheenergy–momentumrelation: E 2 − k 2 = m 2 . {\displaystyleE^{2}-k^{2}=m^{2}\,.} Iftheintegralweighseachkpointequally,themeasureisnotLorentz-invariant.TheinvariantmeasureintegratesoverallvaluesofkandE,restrictingtothehyperbolawithaLorentz-invariantdeltafunction: ∫ δ ( E 2 − k 2 − m 2 ) | E , k ⟩ ⟨ E , k | d E d k = ∫ d k 2 E | k ⟩ ⟨ k | . {\displaystyle\int\delta(E^{2}-k^{2}-m^{2})|E,k\rangle\langleE,k|\,dE\,dk=\int{dk\over2E}|k\rangle\langlek|\,.} Sothenormalizedk-statesaredifferentfromtherelativisticallynormalizedk-statesbyafactorof E = ( k 2 − m 2 ) 1 4 . {\displaystyle{\sqrt{E}}=\left(k^{2}-m^{2}\right)^{\frac{1}{4}}\,.} TheinvariantamplitudeMisthentheprobabilityamplitudeforrelativisticallynormalizedincomingstatestobecomerelativisticallynormalizedoutgoingstates. Fornonrelativisticvaluesofk,therelativisticnormalizationisthesameasthenonrelativisticnormalization(uptoaconstantfactor√m).Inthislimit,theφ4invariantscatteringamplitudeisstillconstant.Theparticlescreatedbythefieldφscatterinalldirectionswithequalamplitude. Thenonrelativisticpotential,whichscattersinalldirectionswithanequalamplitude(intheBornapproximation),isonewhoseFouriertransformisconstant—adelta-functionpotential.Thelowestorderscatteringofthetheoryrevealsthenon-relativisticinterpretationofthistheory—itdescribesacollectionofparticleswithadelta-functionrepulsion.Twosuchparticleshaveanaversiontooccupyingthesamepointatthesametime. Nonperturbativeeffects[edit] ThinkingofFeynmandiagramsasaperturbationseries,nonperturbativeeffectsliketunnelingdonotshowup,becauseanyeffectthatgoestozerofasterthananypolynomialdoesnotaffecttheTaylorseries.Evenboundstatesareabsent,sinceatanyfiniteorderparticlesareonlyexchangedafinitenumberoftimes,andtomakeaboundstate,thebindingforcemustlastforever. Butthispointofviewismisleading,becausethediagramsnotonlydescribescattering,buttheyalsoarearepresentationoftheshort-distancefieldtheorycorrelations.Theyencodenotonlyasymptoticprocesseslikeparticlescattering,theyalsodescribethemultiplicationrulesforfields,theoperatorproductexpansion.Nonperturbativetunnelingprocessesinvolvefieldconfigurationsthatonaveragegetbigwhenthecouplingconstantgetssmall,buteachconfigurationisacoherentsuperpositionofparticleswhoselocalinteractionsaredescribedbyFeynmandiagrams.Whenthecouplingissmall,thesebecomecollectiveprocessesthatinvolvelargenumbersofparticles,butwheretheinteractionsbetweeneachoftheparticlesissimple.[citationneeded](Theperturbationseriesofanyinteractingquantumfieldtheoryhaszeroradiusofconvergence,complicatingthelimitoftheinfiniteseriesofdiagramsneeded(inthelimitofvanishingcoupling)todescribesuchfieldconfigurations.) Thismeansthatnonperturbativeeffectsshowupasymptoticallyinresummationsofinfiniteclassesofdiagrams,andthesediagramscanbelocallysimple.Thegraphsdeterminethelocalequationsofmotion,whiletheallowedlarge-scaleconfigurationsdescribenon-perturbativephysics.ButbecauseFeynmanpropagatorsarenonlocalintime,translatingafieldprocesstoacoherentparticlelanguageisnotcompletelyintuitive,andhasonlybeenexplicitlyworkedoutincertainspecialcases.Inthecaseofnonrelativisticboundstates,theBethe–Salpeterequationdescribestheclassofdiagramstoincludetodescribearelativisticatom.Forquantumchromodynamics,theShifman–Vainshtein–Zakharovsumrulesdescribenon-perturbativelyexcitedlong-wavelengthfieldmodesinparticlelanguage,butonlyinaphenomenologicalway. ThenumberofFeynmandiagramsathighordersofperturbationtheoryisverylarge,becausethereareasmanydiagramsastherearegraphswithagivennumberofnodes.Nonperturbativeeffectsleaveasignatureonthewayinwhichthenumberofdiagramsandresummationsdivergeathighorder.Itisonlybecausenon-perturbativeeffectsappearinhiddenformindiagramsthatitwaspossibletoanalyzenonperturbativeeffectsinstringtheory,whereinmanycasesaFeynmandescriptionistheonlyoneavailable. Inpopularculture[edit] Theuseoftheabovediagramofthevirtualparticleproducingaquark–antiquarkpairwasfeaturedinthetelevisionsit-comTheBigBangTheory,intheepisode"TheBatJarConjecture". PhDComicsofJanuary11,2012,showsFeynmandiagramsthatvisualizeanddescribequantumacademicinteractions,i.e.thepathsfollowedbyPh.D.studentswheninteractingwiththeiradvisors.[11] VacuumDiagramsasciencefictionstorybyStephenBaxterfeaturesthetitularvacuumdiagram,aspecifictypeofFeynmandiagram. Feynmanandhiswife,GwenethHowarth,boughtaDodgeTradesmanMaxivanin1975,andhaditpaintedwithFeynmandiagrams.[12] Seealso[edit] One-loopFeynmandiagram JulianSchwinger#SchwingerandFeynman Stueckelberg–Feynmaninterpretation Penguindiagram Pathintegralformulation Propagator ListofFeynmandiagrams Angularmomentumdiagrams(quantummechanics) Notes[edit] ^"ItwasDyson'scontributiontoindicatehowFeynman'svisualinsightscouldbeused[...]HerealizedthatFeynmandiagrams[...]canalsobeviewedasarepresentationofthelogicalcontentoffieldtheories(asstatedintheirperturbativeexpansions)".Schweber,op.cit(1994) References[edit] ^Kaiser,David(2005)."PhysicsandFeynman'sDiagrams"(PDF).AmericanScientist.93(2):156.doi:10.1511/2005.52.957. ^"WhyFeynmanDiagramsAreSoImportant".QuantaMagazine.5July2016.Retrieved2020-06-16. ^Feynman,Richard(1949)."TheTheoryofPositrons".PhysicalReview.76(6):749–759.Bibcode:1949PhRv...76..749F.doi:10.1103/PhysRev.76.749.Inthissolution,the'negativeenergystates'appearinaformwhichmaybepictured(asbyStückelberg)inspace-timeaswavestravelingawayfromtheexternalpotentialbackwardsintime.Experimentally,suchawavecorrespondstoapositronapproachingthepotentialandannihilatingtheelectron. ^Penco,R.;Mauro,D.(2006)."PerturbationtheoryviaFeynmandiagramsinclassicalmechanics".EuropeanJournalofPhysics.27(5):1241–1250.arXiv:hep-th/0605061.Bibcode:2006EJPh...27.1241P.doi:10.1088/0143-0807/27/5/023.S2CID 2895311. ^GeorgeJohnson(July2000)."TheJaguarandtheFox".TheAtlantic.RetrievedFebruary26,2013. ^Gribbin,John;Gribbin,Mary(1997)."5".RichardFeynman:ALifeinScience.Penguin-Putnam. ^Mlodinow,Leonard(2011).Feynman'sRainbow.Vintage.p. 29. ^Gerardus'tHooft,MartinusVeltman,Diagrammar,CERNYellowReport1973,reprintedinG.'tHooft,UndertheSpellofGaugePrinciple(WorldScientific,Singapore,1994),Introductiononline ^MartinusVeltman,Diagrammatica:ThePathtoFeynmanDiagrams,CambridgeLectureNotesinPhysics,ISBN 0-521-45692-4 ^Bjorken,J.D.;Drell,S.D.(1965).RelativisticQuantumFields.NewYork:McGraw-Hill.p. viii.ISBN 978-0-07-005494-3. ^JorgeCham,AcademicInteraction–FeynmanDiagrams,January11,2012. ^Jepsen,Kathryn(2014-08-05)."SavingtheFeynmanvan".SymmetryMagazine.Retrieved2022-06-23. Sources[edit] 'tHooft,Gerardus;Veltman,Martinus(1973)."Diagrammar".CERNYellowReport.doi:10.5170/CERN-1973-009.{{citejournal}}:Citejournalrequires|journal=(help) Kaiser,David(2005).DrawingTheoriesApart:TheDispersionofFeynmanDiagramsinPostwarPhysics.Chicago,IL:UniversityofChicagoPress.ISBN 0-226-42266-6. Veltman,Martinus(1994-06-16).Diagrammatica:ThePathtoFeynmanDiagrams.CambridgeLectureNotesinPhysics.ISBN 0-521-45692-4.(expanded,updatedversionof'tHooft&Veltman,1973,citedabove) Srednicki,Mark(2006).QuantumFieldTheory.Script. Schweber,S.S.(1994).QEDandthemenwhomadeit:Dyson,Feynman,Schwinger,andTomonaga.PrincetonUniversityPress.ISBN 978-0691033273. Externallinks[edit] WikimediaCommonshasmediarelatedtoFeynmandiagrams. AMSarticle:"What'sNewinMathematics:Finite-dimensionalFeynmanDiagrams" DrawFeynmandiagramsexplainedbyFlipTanedoatQuantumdiaries.com DrawingFeynmandiagramswithFeynDiagramC++librarythatproducesPostScriptoutput. OnlineDiagramToolAgraphicalapplicationforcreatingpublicationreadydiagrams. JaxoDrawAJavaprogramfordrawingFeynmandiagrams. Bowley,Roger;Copeland,Ed(2010)."FeynmanDiagrams".SixtySymbols.BradyHaranfortheUniversityofNottingham. vteQuantumelectrodynamicsFormalism Euler–HeisenbergLagrangian Feynmandiagram Gupta–Bleulerformalism Pathintegralformulation Particles Dualphoton Electron Faddeev–Popovghost Photon Positron Positronium Virtualparticles Concepts Anomalousmagneticdipolemoment Furry'stheorem Klein–Nishinaformula Landaupole QEDvacuum Schwingerlimit Uehlingpotential Vacuumpolarization Vertexfunction Ward–Takahashiidentity Processes Bhabhascattering Breit–Wheelerprocess Bremsstrahlung Comptonscattering Delbrückscattering Lambshift Møllerscattering Schwingereffect Photon-photonscattering Seealso:Template:Quantummechanicstopics vteRichardFeynmanCareer Feynmandiagram Feynman–Kacformula Wheeler–Feynmanabsorbertheory Bethe–Feynmanformula Hellmann–Feynmantheorem Feynmanslashnotation Feynmanparametrization Pathintegralformulation Partonmodel Stickybeadargument One-electronuniverse Quantumcellularautomaton RogersCommissionReport Feynmancheckerboard Feynmansprinkler Works "There'sPlentyofRoomattheBottom" (1959) TheFeynmanLecturesonPhysics (1964) TheCharacterofPhysicalLaw (1965) QED:TheStrangeTheoryofLightandMatter (1985) SurelyYou'reJoking,Mr.Feynman! (1985) WhatDoYouCareWhatOtherPeopleThink? (1988) Feynman'sLostLecture:TheMotionofPlanetsAroundtheSun (1997) TheMeaningofItAll (1999) ThePleasureofFindingThingsOut (1999) PerfectlyReasonableDeviationsfromtheBeatenTrack (2005) Family JoanFeynman (sister) CharlesHirshberg (nephew) Related Namesakes Cargocultscience QuantumMan:RichardFeynman'sLifeinScience TuvaorBust! FeynmanPrizeinNanotechnology Infinity(1996film) QED (2001play) TheChallengerDisaster(2013film) Retrievedfrom"https://en.wikipedia.org/w/index.php?title=Feynman_diagram&oldid=1094537781" Categories:ConceptsinphysicsScatteringtheoryQuantumfieldtheoryDiagramsRichardFeynman1948introductionsHiddencategories:ArticleswithshortdescriptionShortdescriptionisdifferentfromWikidataWikipediaarticlesneedingclarificationfromMay2016AllarticleswithunsourcedstatementsArticleswithunsourcedstatementsfromDecember2012CS1errors:missingperiodicalCommonscategorylinkisonWikidata Navigationmenu Personaltools NotloggedinTalkContributionsCreateaccountLogin Namespaces ArticleTalk English Views ReadEditViewhistory More Search Navigation MainpageContentsCurrenteventsRandomarticleAboutWikipediaContactusDonate Contribute HelpLearntoeditCommunityportalRecentchangesUploadfile Tools WhatlinkshereRelatedchangesUploadfileSpecialpagesPermanentlinkPageinformationCitethispageWikidataitem Print/export DownloadasPDFPrintableversion Inotherprojects WikimediaCommons Languages العربيةAzərbaycancaবাংলাБеларускаяБългарскиCatalàČeštinaDeutschEestiΕλληνικάEspañolفارسیFrançais한국어HrvatskiBahasaIndonesiaItalianoעבריתLietuviųMagyarNederlands日本語Oʻzbekcha/ўзбекчаPolskiPortuguêsРусскийShqipSimpleEnglishSlovenčinaSlovenščinaСрпски/srpskiSrpskohrvatski/српскохрватскиSuomiSvenskaТатарча/tatarçaTürkçeУкраїнськаTiếngViệt粵語中文 Editlinks