Path integral - Scholarpedia

Path integrals are given by sum over all paths satisfying some boundary conditions and can be understood as extensions to an infinite number ... Pathintegral FromScholarpedia JeanZinn-Justin(2009),Scholarpedia,4(2):8674. doi:10.4249/scholarpedia.8674 revision#147600[linkto/citethisarticle] Jumpto:navigation, search Post-publicationactivityCurator:JeanZinn-Justin Contributors: 0.12-NickOrbeck 0.12-RiccardoGuida 0.06-LeoTrottier MasudChaichian AndreiA.Slavnov Prof.JeanZinn-Justin,CEA,IRFUandInstitutdePhysiqueThéorique,CentredeSaclay,F-91191Gif-sur-Yvette,France Asizablefractionofthetheoreticaldevelopmentsinphysicsofthelastsixtyyearswouldnotbeunderstandablewithouttheuseofpathor,moregenerally,fieldintegrals. Inthisarticlewewillfocusontheuseofpathintegralsandfieldintegralsindifferentbranchesoftheoreticalphysics.Arigorousstudyofthemathematicalpropertiesofpathandfieldintegralsisanopensubtopicoffunctionalanalysisandwillnotbedealtwithhere. Pathintegralsaregivenbysumoverallpathssatisfyingsomeboundaryconditionsandcanbeunderstoodasextensionstoaninfinitenumberofintegrationvariablesofusualmulti-dimensionalintegrals.Pathintegralsarepowerfultoolsforthestudyofquantummechanics.Indeed,inquantummechanics,physicalquantitiescanbeexpressedasaveragesoverallpossiblepathsweightedbytheexponentialofatermproportionaltotheratiooftheclassicalaction\(\mathcalS\)associatedtoeachpath,dividedbythePlanck'sconstant\(\hbar\.\)Thus,pathintegralsemphasizeveryexplicitlythecorrespondencebetweenclassicalandquantummechanics.Inparticular,inthesemi-classicallimit\(\mathcal{S}/\hbar\rightarrow\infty\,\)theleadingcontributionsintheaveragecomefrompathsclosetoclassicalpaths,whicharestationarypointsoftheaction.Thus,pathintegralsleadtoanintuitiveunderstandingandsimplecalculationsofphysicalquantitiesinthesemi-classicallimit. Theformulationofquantummechanicsbasedonpathintegralsiswelladaptedtosystemswithmanydegreesoffreedom,whereaformalismofSchrödingertypeismuchlessuseful.Therefore,itallowsaneasytransitionfromquantummechanicstoquantumfieldtheoryorstatisticalphysics.Inparticular,generalizedpathintegrals(functionalintegralsand,moreprecisely,fieldintegrals)leadtoanunderstandingofthedeeprelationsbetweenquantumfieldtheoryandthetheoryofcriticalphenomenaincontinuousphasetransitions. WefirstdescribeBrownianmotionandEuclidean-time(i.e,imaginarytime)pathintegrals.Thismeansthatweconsiderthepathintegralrepresentationofthematrixelementsofthequantumstatisticaloperator,ordensitymatrixatthermalequilibrium\(\mathrm{e}^{-\beta\hatH},\)\(\hatH\)beingthequantumHamiltonianand\(\beta\)theinversetemperature(measuredinaunitwheretheBoltzmannconstant\(k_B\)is1).Inthisway,weareabletodescribequantumstatisticalphysicsintermsofpathintegrals,butalso,perhapsmoresurprisingly,toexhibitarelationbetweenclassicalandquantumstatisticalmechanics.Moreover,forawholeclassofHamiltonians,theEuclidean-timepathintegralcorrespondstoapositivemeasure.Wethendefinethereal-time(inrelativisticfieldtheoryMinkowskian-time)pathintegral,whichdescribesthetimeevolutionofquantumsystemsandcorrespondsfortime-translationinvariantsystemstotheevolutionoperator\(\mathrm{e}^{-it\hatH/\hbar}\)(\(t\)beingtherealtime).Finally,webrieflylistafewgeneralizations:pathintegralsintheHamiltonianformulation,pathintegralsintheholomorphicrepresentationrelatedtobosonsystemsand,correspondingly,Grassmannianpathintegralsforfermions. Anumberofimportantapplicationstophysicsofthepathintegralideainvolveinfactintegralsoverfields.Inparticular,fieldintegralsareindispensableforthestudyofquantumgaugeinvarianttheorieswhichconstitutethebasisofthedescriptionoffundamentalinteractionsatthemicroscopicscale,aswellasforunderstandingofthecriticalpropertiesofphasetransitions.Theyrelyonapragmaticapproach,focusingmoreondevelopingcalculationaltoolsthanonestablishingrigorousproperties.Indeed,eventhoughanumberofinterestingrigorousresultshavebeenproved,onefacesextremelydifficultmathematicalproblemsinrealisticsituations(e.g.,infourdimensionalspace-time). Contents 1Randomwalk,Brownianmotionandpathintegral 1.1Discussion 1.2Explicitcalculation 2ApplicationoftheWienermeasuretostatisticalphysics 2.1Classicalstatisticalphysics 2.2Quantumstatisticalphysics 3Generalization 3.1PathintegralsandlocalMarkovprocesses 3.2Pathintegralsandclassicalstatisticalphysics 3.3Pathintegralsandquantumstatisticalphysics 4Gaussianpathintegrals:Thequantumharmonicoscillator 5Perturbativeexpansionforthepathintegral 5.1Correlationfunctions 5.2GaussianexpectationvaluesandWick'stheorem 5.3Perturbativeexpansion 6Quantumtimeevolution 7Barrierpenetrationinthesemi-classicallimit 8Pathintegrals:Generalizations 8.1Thequantumparticleinastaticmagneticfield 8.2Hamiltonianformulationandphasespaceintegration 8.3Holomorphicformalismandbosons 8.4Grassmannpathintegralsandfermions 9Ageneralization:Thefieldintegral 10References 11Furtherreading 12Seealso Randomwalk,Brownianmotionandpathintegral Asafirstexample,weconsiderarandomwalkonthereallinewithdiscretetimes\(k=0,1,2,\dots,n.\) Suchastochasticprocessisspecifiedbyaprobabilitydistribution\(P_0(x)\)fortheposition\(x\)atinitialtime\(k=0\)andatime-independentdensity \(\rho(x,x')\)describingtheprobabilityoftransitionfromthepoint\(x'\)tothepoint\(x\,\)meaningthat theprobabilitydistribution\(P_k(x)\)attime\(k\)satisfiestherecursionrelationormasterequation \[P_k(x)=\int\mathrm{d}x'\,\rho(x,x')P_{k-1}(x'),\quad\int\mathrm{d}x\,\rho(x,x')=1\,.\] Thisrandomwalkisa(homogeneousorstationary)Markovchain,thatis,a(time-translationinvariant)stochasticprocesswiththeMarkovproperty(i.e.,thepropertythattheprobabilitydistributionattime\(k\)dependsonlyontheprobabilityattime\(k-1\,\)butnotontheprobabilitydistributionsatpriortimes). Inthefollowingwealsoassumethatthetransitionprobabilityistranslationinvariantandeven\[\rho(x,x')=R(x-x')=R(x'-x)\.\] Underrathergeneralconditions,themostimportantbeingthat\(R(x)\)decreasesfastenoughfor\(|x|\)large(wecallthisalocalMarkovprocess),onecanprove(aconsequenceofthecentrallimittheoremofprobabilities)thatthedistribution\(P_n(x)\)convergesasymptoticallyforlargetimes\(n\)towardaGaussiandistributionthatisindependentofthetransitionprobability\(R(x)\.\)Therefore,ifoneisinterestedonlyinlargetimeproperties,onecanstartdirectlyfromaGaussiantransitionprobabilityoftheform \[\tag{1} R(x)={1\over\sqrt{2\pi\xi}}\mathrm{e}^{-x^2/(2\xi)},\] where\(\xi>0\)characterizesthewidthofthedistribution. Itistheneasytocalculate\(P_n(x)\)explicitlybysuccessiveGaussianintegrations.However,forourpurposeitismoreinstructivetojustapplytherecursionrelation. Ifoneassumes,forexample,thattheinitialdistributionisconcentratedatthepoint\(x=x_0\)(i.e.,\(P_0(x)=\delta(x-x_0)\)where\(\delta(x)\)isDirac'sgeneralizedfunctionalsoknownasDiracfunctionor\(\delta-\)function)oneobtainsattime\(n\)theprobabilitydistribution \[\tag{2} P_n(x,x_0)=\int\mathrm{d}x_{n-1}\mathrm{d}x_{n-2}\ldots\mathrm{d}x_{1}\,R(x-x_{n-1})R(x_{n-1}-x_{n-2})\ldotsR(x_1-x_0).\] IntheGaussianexample(1),theexpressionbecomes \[\tag{3}P_n(x,x_0)=(2\pi\xi)^{(1-n)/2}\int\mathrm{d}x_{n-1}\mathrm{d}x_{n-2}\ldots\mathrm{d}x_{1}\,\mathrm{e}^{-\mathcal{S}(\mathbf{x})/\xi}\], where,defining\(\mathbf{x}\equiv(x_0,x_1,\cdots,x_n)\)and\(x\equivx_n\,\) \[\mathcal{S}(\mathbf{x})={1\over2}\sum_{k=1}^{n}\left(x_k-x_{k-1}\right)^2\,\]. Figure1:Piecewiselinearpath. Wethenintroduceatimestep\(\varepsilon>0\,\)themacroscopictimevariables \[\tau_k=t'+k\varepsilon\\mathrm{with}\0\lek\len\,,\] (suchthat\(\tau_0=t'\,\)\(\tau_n=t'+n\varepsilon\equivt''\))andacontinuous,piecewiselinearpath(seeFigure1) \[\tag{4} q(\tau)=\sqrt{\varepsilon}\left[x_{k-1}+{\tau-\tau_{k-1} \over\tau_k-\tau_{k-1}}\left(x_k-x_{k-1}\right)\right]\quad \mathrm{for}\\tau_{k-1}\le\tau\le\tau_k\quad\mathrm{and}\;k\ge1,\] withtheboundaryconditions \[\tag{5} q(t')=\sqrt{\varepsilon}x_0\equivq'\,,\quadq(t'')=\sqrt{\varepsilon}x\equivq''\,.\] Oneverifiesthat\(\mathcal{S}(\mathbf{x})\)canberewrittenas \[\mathcal{S}(\mathbf{x})=\mathcal{S}_\varepsilon(\mathbf{q})\equiv{1\over2}\int_{t'}^{t''}\,\dot{q}^2(\tau)\mathrm{d}\tau\] where\(\dot{q}(\tau)\equiv\mathrm{d}q/\mathrm{d}\tau\.\) Onecanthenstudythelargediscretetimeasymptoticbehaviour,bytakingthelarge\(n\)limitat\(t''-t'\)fixedand,thus,\(\varepsilon=(t''-t')/n\to0\.\)Onealsospeaksofatemporalcontinuumlimitsincethetimestepgoestozero.Inthislimit,thenormalizedprobabilitydistributioninthenewvariables\(\Pi_0(t'',t';q'',q')\)isgivenbyaEuclidean-timepathintegral(Wiener1923)thatwedenoteby \[\tag{6} \Pi_0(t'',t';q'',q')=\lim_{n\to\infty}{1\over\sqrt{\varepsilon}}P_n(x,x_0)=\int[\mathrm{d}q(\tau)]\mathrm{e}^{-\mathcal{S}_0(\mathbf{q})/\xi},\quad\mathcal{S}_0(\mathbf{q})={1\over2}\int_{t'}^{t''}\,\dot{q}^2(\tau)\mathrm{d}\tau\] (thefactor\(1/\sqrt{\varepsilon}\)comesfromthechangeofvariablesfrom\(x\)to\(q\))wherethesymbol\([\mathrm{d}q(\tau)]\)(alsodenotedby\(\mathcal{D}q(\tau)\)intheliterature)meanssumoverall(trajectories)\(q(\tau)\)satisfyingtheboundaryconditions(5). Discussion Afewsimpleremarksareinorder.First,theintegrandinthepathintegralispositiveand\([\mathrm{d}q(\tau)]\mathrm{e}^{-\mathcal{S}_0(\mathbf{q})/\xi}\)thusdefinesapositivemeasureonpaths,theso-calledWienermeasure.Second,itisdifficulttokeeptrackoftheabsolutenormalizationinthecontinuumpathintegrallimit.Therefore,onemostlyusespathintegralstocalculateexpectationvalues.If\(\mathcal{F}(\mathbf{q})\)isafunctionalofthepath\(\mathbf{q}\equivq(\cdot),\)itsexpectationvalueisdefinedby \[\tag{7} \langle\mathcal{F}(\mathbf{q})\rangle_0=\mathcal{Z}^{-1}\int[\mathrm{d}q(\tau)]\mathcal{F}(\mathbf{q})\mathrm{e}^{-\mathcal{S}_0(\mathbf{q})/\xi}\] with \[\mathcal{Z}\equiv\int[\mathrm{d}q(\tau)]\mathrm{e}^{-\mathcal{S}_0(\mathbf{q})/\xi}.\] Notethatintheratioof(7)thenormalizationcancels.Inthecaseoftherandomwalk,typicalexpectationvaluescorrespondtocorrelationsbetweenpositionsatdifferenttimes,whichcanalsobeconsideredasgeneralizedmomentsoftheprobabilitydistribution.Forexample, \[\mathcal{F}(\mathbf{q})\equivq(\tau_1)q(\tau_2)\ldotsq(\tau_{2n}).\] Expectationvaluesofthiskindarealsocalledcorrelationfunctions. Theformof\(\mathcal{S}_0(\mathbf{q})\)determinestheclassofpathsthatcontributetothepathintegral,whicharecalledinthiscaseBrownianpaths.Asthefactor\(\sqrt{\varepsilon}\)in(4)suggests,BrownianpathssatisfyaHölderconditionoforder1/2,thatis,for\(\tau-\tau'\to0\,\) \[|q(\tau)-q(\tau')|=O\left(\left|\tau-\tau'\right|^{1/2}\right);\] inparticularBrownianpathsarecontinuousbutnotdifferentiableand,thus,\(\dot{q}(\tau)\)isnotdefined. Inthissense,thenotation\(\dot{q}^2(\tau)\)hastobeconsideredasasymbolandshouldnottakenliterally.Nevertheless,itisausefulnotationsince,for\(\xi\to0\,\)thepathintegralisdominatedbypathsclosetotheclassicalpathsthatleave\(\mathcal{S}_0(\mathbf{q})\)stationary,andwhicharedifferentiable(seethecalculationbelow).Finally,thecontinuityofthepathsallowsunderstandingwhyitispossiblethatanintegrationoveranincreasinglydensesetofpoints(orsumoverallpossiblepiecewiselinearpaths)eventuallyconvergestowardanintegrationoverallcontinuouspaths. Explicitcalculation Gaussianpathintegrals,likefinitedimensionalGaussianintegrals,areexplicitlycalculable.LetusillustratethispropertywiththesimpleexampleoftheBrownianmotion.Wenowexplicitlyevaluatetheintegral(6)byamethodthatgeneralizestootherkindsofGaussianintegrals.Varyingthequantity\(S_0(\mathbf{q})\)withrespecttothepath\(q(\tau),\)oneobtainstheclassicalequationofmotion\(\ddotq(\tau)=0\.\)Imposingtheboundaryconditions(5)ofthepathintegraltotheclassicalsolution,oneobtains \[q_c(\tau)=q'+\frac{(\tau-t')}{(t''-t')}(q''-q').\] Onethenchangesvariables,\(q(\tau)\mapstor(\tau)\)with\(q(\tau)=q_c(\tau)+r(\tau).\)Ateachtime\(\tau\)thisisasimpletranslationandtheassociatedJacobianis\(1\.\)Theboundaryconditionsonthenewpath\(r(\tau)\)are\(r(t'')=r(t')=0\)andonefinds \[S_0(\mathbf{q})=\frac{1}{2}{(q''-q')^2\over(t''-t')}+S_0(\mathbf{r}).\] Thepathintegral(6)becomes \[\Pi_0(t'',t';q'',q')=\mathrm{e}^{-\frac{1}{2\xi}{(q''-q')^2\over(t''-t')}}\int[\mathrm{d}r(\tau)]\mathrm{e}^{-\mathcal{S}_0(\mathbf{r})/\xi}.\] Theremainingpathintegralis\(q'',q'\)-independentandgivesthenasimplenormalizationfactor.Here,itcanbedeterminedbyimposingtheconditionofprobabilityconservation \[\tag{8} \int\mathrm{d}q''\,\Pi_0(t'',t';q'',q')=1\\Rightarrow\\Pi_0(t'',t';q'',q')={1\over\sqrt{2\pi\xi(t''-t')}}\mathrm{e}^{-\frac{(q''-q')^2}{2(t''-t')\xi}}.\] Aspointedoutabove,thisnormalizationcancelsinexpectationvalues. ApplicationoftheWienermeasuretostatisticalphysics ThesamepathintegraldescribingtheBrownianmotionhasaninterpretationintheframeworkofstatisticalphysics. Classicalstatisticalphysics Theexpression(3)intheexample(1)mayalsobephysicallyinterpretedastheclassicalpartitionfunctionof\(n+1\)particlesonaone-dimensionallatticewithspatialsites\(k=0,1,\cdots,n\.\)Particlesdeviatefromtheirequilibriumpositionsbythevalue\(x_k\)andhavenearest-neighbourharmonicinteractions: \[\mathcal{Z}(x_n,x_0;\xi)=(2\pi\xi)^{(1-n)/2}\;\int\mathrm{d}x_{n-1}\mathrm{d}x_{n-2}\ldots\mathrm{d}x_{1} \,\exp\left(-{1\over2\xi}\sum_{k=1}^{n}\left(x_k-x_{k-1}\right)^2\right).\] (Theextremitiesofthechainbeingfixedatdeviations\(x_0\)and\(x_n\,\)respectively.) Here,theparameter\(\xi\)hastheinterpretationofatemperature.Thepathintegral(6)thencorrespondstothecontinuumlimitwherethelatticespacing\(\varepsilon\)betweentwoadjacentsitesgoestozeroatfixedtotalmacroscopiclengthofthechain\(L=n\varepsilon\.\) Quantumstatisticalphysics Remarkablyenough,thepathintegraloftheBrownianmotionyieldsalsothedensitymatrixofafreenon-relativisticquantumparticle. Thecontinuumdistribution\(\Pi_0(t,t';q,q')\)givenbyequation(8)satisfiesthediffusionequation \[{\partial\Pi_0\over\partialt}={\xi\over2}{\partial^2\Pi_0\over(\partialq)^2}\;,\] withinitialcondition: \[\lim_{t\rightarrowt'}\Pi_0(t,t';q,q')=\delta(q-q')\;.\] Thisequationcanbecomparedwiththeequationsatisfiedbytheelementsofthequantumdensitymatrix\(\mathrm{e}^{-\beta\hat{H}_0}\)(inthebasisinwhichthepositionoperator\(\hatq\)isdiagonal\(:\;\hat{q}|q\rangle=q|q\rangle\)),anoperatordescribingthethermalequilibriumofafreenon-relativisticquantumparticle, where\(\hat{H}_0={\hat{p}}^2/(2m)\)isthequantumHamiltonianinrealtime(alinearoperatoractingontheHilbertspaceofquantumstates),\(\hatp\)themomentumoperator(withthecommutationrelation\([\hatq,\hatp]=i\hbar\)),\(m\)istheparticle'smassand \(\beta\)theinversetemperature(inunitsoftheBoltzmann'sconstant\(k_\mathrm{B}\)).Thematrixelements\(\langleq|\mathrm{e}^{-\beta\hat{H}_0}|q'\rangle\)satisfythepartialdifferentialequation: \[{\partial\over\partial\beta}\langleq|\mathrm{e}^{-\beta\hat{H}_0}|q'\rangle={\hbar^2\over2m}{\partial^2\over(\partialq)^2}\langleq|\mathrm{e}^{-\beta\hat{H}_0}|q'\rangle,\] \(\hbar\)beingPlanck'sconstant. Since\(\Pi_0(t,t';q,q')\)and\(\langleq|\mathrm{e}^{-\beta\hat{H}_0}|q'\rangle\)alsosatisfythesameboundaryconditionsatinitialtime\(t-t'\equiv\hbar\beta=0\,\)itfollowsthat\(\langleq|\mathrm{e}^{-\beta\hat{H}_0}|q'\rangle\)isequalto\(\Pi_0(t=\hbar\beta,0;q,q')\)when\(\xi=\hbar/m\)and,therefore,isgivenbythepathintegral \[\tag{9} \langleq|\mathrm{e}^{-\beta\hat{H}_0}|q'\rangle=\int[\mathrm{d}q(\tau)]\mathrm{e}^{-\mathcal{S}_0(\mathbf{q})/\hbar},\quad\mathcal{S}_0(\mathbf{q})={1\over2}\int_{0}^{\hbar\beta}\,m\dot{q}^2(\tau)\mathrm{d}\tau\] withtheboundaryconditions\(q(0)=q',\)\(q(\hbar\beta)=q.\) Generalization Asimplegeneralizationofthepathintegral(9)relevantforquantumstatisticalphysicsisthepathintegral \[\tag{10} \Pi(t'',t';q'',q')=\int[\mathrm{d}q(\tau)]\mathrm{e}^{-\mathcal{S}(\mathbf{q})/\hbar}\quad\text{with}\;q(t')=q',q(t'')=q''\;,\] where \[\tag{11} \mathcal{S}(\mathbf{q})=\int_{t'}^{t''}\mathrm{d}\tau\,\mathcal{L}_{\mathrm{E}}(\dot{q}(\tau),q(\tau);\tau)\] andtheEuclideanLagrangianisdefinedas \[\tag{12} \mathcal{L}_{\mathrm{E}}(\dot{q},q;\tau)=\frac{1}{2}m\dot{q}^2+V(q,\tau)\;.\] NotethatintheEuclideanLagrangianthepotential\(V(q,\tau)\)isaddedtothekineticenergy,whileinthenormalLagrangianofclassicalmechanicsthepotentialissubtractedfromthekineticenergy. Theparameter\(m\)canbeidentifiedwiththemassofanon-relativisticquantumparticle.Wealsoassumethat\(V(q,t)\)isasmoothfunctionof\(q\)andthat \[\tag{13} \int\mathrm{d}q\,\mathrm{e}^{-\varepsilonV(q,\tau)}0\,.\] OnepossibledefinitionofthiskindofpathintegralsreferstotheWienermeasure: \[\tag{14} \Pi(t'',t';q'',q')=\Pi_0(t'',t';q'',q')\left\langle\exp\left[-{1\over\hbar}\int_{t'}^{t''}\mathrm{d}\tau\,V(q(\tau),\tau)\right]\right\rangle_0\,,\] wheretheexpectationvalueisdefinedin(7)with\(\xi=\hbar/m.\)Withthisnormalization,\(\Pi(t',t';q'',q')=\delta(q''-q')\,\)whichisthekernelassociatedwiththeidentityoperator. PathintegralsandlocalMarkovprocesses Weintroduceinsidethepathintegral(10)theidentity \[1=\int\mathrm{d}q\,\delta\bigl(q-q(t)\bigr)\\mathrm{with}\t'0\,.\] Weset \[\mathcal{S}(\mathbf{q})=\mathcal{S}_0(\mathbf{q})+\lambda\int\mathrm{d}\tau\,q^4(\tau)\] with \[\mathcal{S}_0(\mathbf{q})=\frac{1}{2}\int\mathrm{d}\tau\,\bigl(\dot{q}^2(\tau)+q^2(\tau)\bigr).\] Theexpansionofthecorrespondingpathintegralcanthenbewrittenas(hereweset\(\hbar=1\)) \[\mathcal{Z}(\lambda)=\int[\mathrm{d}q(\tau)]\mathrm{e}^{-\mathcal{S}_0(\mathbf{q})}\sum_{k=0}^\infty{(-\lambda)^k\overk!}\left[\int\mathrm{d}\tau\,q^4(\tau)\right]^k\sim\mathcal{Z}(0)\sum_{k=0}^\infty{(-\lambda)^k\overk!}\left\langle\left[\int\mathrm{d}\tau\,q^4(\tau)\right]^k\right\rangle_0,\] where\(\langle\bullet\rangle_0\)meansexpectationvaluewithrespecttotheGaussianmeasureassociatedwith\(\mathcal{S}_0\)andthesymbol\(\sim\)meansthattheperturbativeseriesisnotconvergent.EachtermintheseriescanthenbeevaluatedusingWick'stheoremandtheexplicitformoftheGaussiantwo-pointfunction.Forexample,atfirstorderin\(\lambda\,\) \[\langleq^4(\tau)\rangle_0=3(\langleq^2(\tau)\rangle_0)^2.\] Thenextorderinvolves \[\langleq^4(\tau)q^4(\tau')\rangle_0=9(\langleq^2(\tau)\rangle_0)^2(\langleq^2(\tau')\rangle_0)^2+72 \langleq^2(\tau)\rangle_0\langleq^2(\tau')\rangle_0(\langleq(\tau)q(\tau')\rangle_0)^2+24 (\langleq(\tau)q(\tau')\rangle_0)^4.\] ItisthenconvenienttorepresentindividualcontributionsgraphicallyintermsofFeynmandiagrams.Letuspointoutthatsuchanexpansionisdivergentforallvaluesoftheparameter\(\lambda\.\)Itisanasymptoticseries,usefulassuchonlyfor\(\lambda\)smallenough.Forlargervaluesoftheexpansionparameter,seriessummationmethodsarerequired. Quantumtimeevolution FollowingFeynman(Feynman1948),quantumtime-evolution(herewerefertorealphysicaltime)canbedescribedintermsof(oscillatory)pathintegrals. Inthisformalism,consideringasystemclassicallydescribedbytheCartesiancoordinates\(\mathbf{q}\equiv\{q^1,q^2\ldots\}\,\)thematrixelementsofthequantumevolutionoperator\(\mathbf{U}(t'',t')\)betweentimes\(t'\)and\(t''\)aregivenby asumoverallpossibletrajectories(paths)\(\mathbf{q}(\tau)\equiv\{q^1(\tau),q^2(\tau)\ldots\}\,\)whichinthesimplestcasescanbewrittenas \[\tag{29} \langle\mathbf{q}''\left|\mathbf{U}(t'',t')\right|\mathbf{q}' \rangle=\int\left[\mathrm{d}\mathbf{q}(\tau)\right]\exp\left({i \over\hbar}\mathcal{A}(\mathbf{q})\right)\] withtheboundaryconditions \[\tag{30} \mathbf{q}(t')=\mathbf{q}',\\mathbf{q}(t'')=\mathbf{q}'',\] wheretheclassicalaction\(\mathcal{A}(\mathbf{q})\)isthetime-integraloftheclassicalLagrangian: \[\tag{31} \mathcal{A}(\mathbf{q})=\int_{t'}^{t''}\mathrm{d}\tau\,\mathcal{L}\left(\mathbf{q}(\tau),\dot{\mathbf{q}}(\tau);\tau\right).\] Theexpression(29)isvalidwhenthekineticterm,thatis,thetermwithtwotime-derivativesintheLagrangianhastheform\(\textstyle{\frac{1}{2}\sum_i}m_i(\dot{q}^i)^2\,\)otherwisethemeasurehastobemodifiedandnewproblemsarise.Anexampleofthelattersituationisprovidedwhenthecoordinates\(q^i\)parametrizeaRiemannianmanifoldandthekineticterminvolves\(\textstyle{\sum_{i,j}}\dot{q}^ig_{ij}(\mathbf{q})\dot{q}^j\,\)where\(g_{ij}\)isthemetrictensor. Theformulationofquantummechanicsintermsofpathintegralsactuallyexplainswhyequationsofmotioninclassicalmechanicscanbederivedfromavariationalprinciple. Intheclassicallimit,thatis,whenthetypicalclassicalactionislargewithrespectto\(\hbar\,\)thepathintegralcanbeevaluatedbyusingthestationaryphasemethod.Thesumoverpathsisthusdominatedbypathsthatleavetheactionstationary:theclassicalpathsthatsatisfy \[\tag{32} \mathcal{A}\left(\mathbf{q}+\delta\mathbf{q}\right)-\mathcal{A}\left(\mathbf{q}\right)=O(\|\delta\mathbf{q}\|^2)\\Rightarrow\\frac{\delta\mathcal{A}}{\deltaq^i}=0\\Rightarrow\{\partial\mathcal{L}\over\partialq^i}-{\mathrm{d}\over\mathrm{d}t}{\partial\mathcal{L}\over\partial\dot{q}^i}=0\] withtheboundaryconditions(30).Theleadingordercontributionisthenobtainedbyexpandingthepatharoundtheclassicalpath,keepingonlythequadraticterminthedeviationandperformingthecorrespondingGaussianintegration. Thispropertygeneralizestorelativisticquantumfieldtheory. Fromthemathematicalpointofview,itismuchmoredifficulttodefinerigorouslythereal-timepathintegralthantheimaginary-timestatisticalpathintegral.Apossiblestrategyinvolves,whenapplicable,tocalculatephysicalobservablesforimaginarytimeandthentoproceedbyanalyticcontinuation. Barrierpenetrationinthesemi-classicallimit Thepurposeofthissectionistoillustratewithasimpleexampletheevaluationofstatistical(orimaginarytime)pathintegralsinthesemi-classicalapproximation.Itismoretechnicalandcanbeomittedinafirstreading. Thepathintegralassociatedwith\(\mathrm{tr}\,\mathrm{e}^{-t\hatH/\hbar}\,\) \[\tag{33} \mathcal{Z}=\int[\mathrm{d}q(\tau)]\mathrm{e}^{-\mathcal{S}(\mathbf{q})/\hbar}\quad\text{with}\;q(t/2)=q(-t/2),\] isespeciallywellsuitedtotheevaluation,inthesemi-classicallimit\(\hbar\to0\,\)ofspecificquantumphenomenacalledbarrierpenetrationortunnelling.Indeed,itcanbeshownthattheclassicallyforbiddenbarrierpenetrationappears,inthesemi-classicallimit,asformallyrelatedtoclassicalevolutioninimaginarytime. Toexplainthegeneralidea,weconsideranexampleoftheform \[\tag{34} \mathcal{S}(\mathbf{q})=\int_{-t/2}^{t/2}\mathrm{d}\tau\left[\frac{1}{2}m\dot{q}^2(\tau)+V\bigl(q(\tau)\bigr)\right],\] withthepotential \[\tag{35} V(q)=\frac{1}{2}q^2-\frac{1}{2}\lambdaq^3\,,\] whichhasonelocalminimumat\(q=0\)(\(V=0\)),alocalmaximumat\(q=2/(3\lambda)\)(\(V=4/(27\lambda^2)\))andgoesto\(\mp\infty\)when\(q\to\pm\infty\.\) Theproblemistoevaluatetheprobabilityperunittimeforaparticlelocalizedinitiallyinthewellofthepotentialat\(q=0\)toescapethewell.Sincethepotential(35)isnotboundedfrombelow,itisfirstnecessarytodefinethequantumHamiltonian.Inthisexample,onecanproceedbyanalyticcontinuationstartingfrom\(\lambda\)pureimaginary.AsconjecturedinitiallybyBessisandZinn-Justin(unpublished),thecorrespondingHamiltonian,thoughcomplex,hasadiscreterealspectrumasaconsequenceofthesymmetry\(q\mapsto-q\,\)\(\hatH\mapsto\hatH^*\,\)asymmetryalsocalledPTsymmetry(PbeingtheparitytransformationandTthetime-reversaltransformation).Returningbyanalyticcontinuationto\(\lambda\)real,onefindsinthiscaseacomplexenergyspectrum(quantumresonances),theimaginarypartoftheenergyeigenvaluesbeingdirectlyrelatedtotunnelling. Inthepathintegralframework,itcanbeshownthatbarrierpenetrationeffectscanbederivedfromanevaluationoftheintegral(33)for\(\hbar\to0\)and,therefore,bythesteepest-descentmethod,suitablygeneralizedtopathintegrals.Onelooksfornon-trivialsaddlepoints,herenon-constantsolutionsoftheclassicalequationsofmotionderivedfromtheEuclideanaction(34),whichcorrespondformallytoevolutioninimaginarytime.Moreover,ifoneisinterestedonlyinstateswithenergiesoforder\(\hbar\,\)thenonehastotakethelimit\(|t|\to\infty\.\)Therefore,onelooksforsolutionsthathaveafiniteactionontherealline.Thesesolutionsarecalledinstantons.Here, theequationofmotionobtainedbyvarying\(\mathcal{S}\)is \[-\ddotq(\tau)+q(\tau)-\frac{3}{2}\lambdaq^2(\tau)=0\,.\] Duetotimetranslationinvariance,onefindsaone-parameterfamilyofinstantonsolutions: \[q_c(\tau)={1\over\lambda\cosh^2((\tau-\tau_0)/2)}\\Rightarrow\\mathcal{S}(\mathbf{q}_c)=\int_{-\infty}^{+\infty}\mathrm{d}\tau\,\mathcal{L}_{\mathrm{E}}(\mathbf{q}_c)={8\over15\lambda^2}.\] Completingthecalculationofthesaddlepointcontributionisanon-trivialexercisebecauseitrequiresfactorizingthepathintegralmeasureintoanintegrationover\(\tau_0\)(acollectivecoordinaterelatedtothebreakingoftime-translationsymmetrybythesolution)beforeusingasaddlepointapproximationfortheothermodesofthepath.Oneinfersthat,uptopowerlawcorrections,theprobabilityperunittimeofleavingthewell isoforder\(\exp{-\mathcal{S}(\mathbf{q}_c)/\hbar}=\exp{-8/(15\lambda^2\hbar)}.\) Pathintegrals:Generalizations Wehavepresentedonlythesimplestformofpathintegrals,whichforthepointofviewofquantummechanicsinvolveonlyaclassicalLagrangianwiththegeneralform(11).FormoregeneralLagrangiansorHamiltonians,oneencountersnewproblemsinthedefinitionofpathintegrals. Thequantumparticleinastaticmagneticfield WhentheLagrangianinvolvesatermlinearinthevelocity,asintheexampleofaquantumparticleinamagneticfield, \[\tag{36} \mathcal{L}(\mathbf{q},\dot{\mathbf{q}})=\textstyle{1\over2}\,m\,\dot{\mathbf{q}}^2-e\,\mathbf{A}(\mathbf{q})\cdot\dot{\mathbf{q}}\,,\] where\(\mathbf{A}(\mathbf{q})\)isagivenvectorpotential, anewproblemrelatedtoquantizationarises.TheclassicalLagrangiantogetherwiththecorrespondenceprinciple(replacingpositionandvelocitybythecorrespondingquantumoperators)doesnotdeterminethequantumtheorybecauseoperators\(\mathbf{A}(\hat{\mathbf{q}})\)and\(\dot{\hat{\mathbf{q}}}\)nolongercommute.Correspondingly,thenaivecontinuumformofthepathintegralisnotdefinedbecausethecontinuumlimitdependsexplicitlyonthetime-discretizedformofthepathintegralandleadstoaone-parameterfamilyofdifferenttheories.Thisreflects,forexample,intheappearanceofundefinedterms\(\operatorname{sgn}(0)\)incalculations. TheunderlyingquantumHamiltonianisthenuniquelydeterminedbydemandingeitheritshermiticityorequivalentlyitsgaugeinvariance.Todeterminethepathintegral,onecaneitherreturntoatime-discretizedformconsistentwiththequantumHamiltonian(whichimpliesthemidpointruleintheargumentofthevectorpotential),oraddatermwithhigherordertimederivativesintheaction,forexample, \[\mathcal{S}\mapsto\mathcal{S}+\eta\int_{t'}^{t''}\mathrm{d}\tau(\ddotq(\tau))^2,\\eta>0\,.\] ThishastheeffectofrestrictingtheintegrationtopathsthatsatisfyaHölderconditionoforder3/2andarethusdifferentiable,insuchawaythatexpectationsvalueswith\(\dot{\mathbf{q}}\)aredefined.ThisregularizationdoesnotviolategaugeinvariancebutviolateshermiticityoftheHamiltonian.Inthecaseofthemagneticfield,inthe\(\eta\to0\)limit,itfixestheambiguities(\(\operatorname{sgn}(0)=0\))inawaythatisconsistentwithagaugeinvariantHermitianquantumHamiltonian. Hamiltonianformulationandphasespaceintegration ForageneralclassicalHamiltonian,thequantumevolutionoperatorcanformallybeexpressedintermsofapathintegralinvolvinganintegrationoverphasespacevariables,position\(\mathbf{q}\)andconjugatemomentum\(\mathbf{p}\:\) \[\tag{37} \langle\mathbf{q}''\left|\mathbf{U}(t'',t')\right|\mathbf{q}' \rangle=\int\left[\mathrm{d}\mathbf{p}(\tau)\mathrm{d}\mathbf{q}(\tau)\right]\exp\left({i \over\hbar}\mathcal{A}(\mathbf{p},\mathbf{q})\right)\] withtheboundaryconditions \(\tag{38} \mathbf{q}(t')=\mathbf{q}',\\mathbf{q}(t'')=\mathbf{q}'',\) wheretheclassicalaction\(\mathcal{A}(\mathbf{p},\mathbf{q})\)isnowexpressedintermsoftheclassicalHamiltonian\(H(\mathbf{p},\mathbf{q};t):\) \[\tag{39} \mathcal{A}(\mathbf{p},\mathbf{q})=\int_{t'}^{t''}\mathrm{d}\tau\,\left[\mathbf{p}(\tau)\cdot\dot{\mathbf{q}}(\tau)-H\!\left(\mathbf{p}(\tau),\mathbf{q}(\tau);\tau\right)\right].\] WhentheHamiltonianisquadraticintheconjugatemomentum\(\mathbf{p},\)theintegralover\(\mathbf{p}(\tau)\)isGaussianandcanbeperformedexplicitly:firstoneshifts\(\mathbf{p}(\tau)\)intheexponentialbythesolutionoftheclassicalequation \(\dot{\mathbf{q}}(\tau)=\frac{\partialH\!\left(\mathbf{p}(\tau),\mathbf{q}(\tau);\tau\right)}{\partial\mathbf{p}(\tau)}\.\) OnethusrecoversintheexponentialtheclassicalLagrangian.Onethenintegratesover\(\mathbf{p}(\tau)\)andthismaymodifythe\(\mathbf{q}(\tau)\)-integrationmeasureifthecoefficientofthequadratictermin\(\mathbf{p}(\tau)\)isnotaconstant.Inthegeneralcase,theinterpretationofthispathintegralreflectstheproblemsofquantizingclassicalHamiltoniansandtheorderofoperatorsinproducts.TheHamiltonianpathintegralhasmainlyaheuristicvalue(exceptinthesemi-classicallimit). Holomorphicformalismandbosons Uptonow,wehavedescribedthepathintegralformalismrelevantfordistinctquantumparticles.Butquantumparticlesareeitherbosons,obeyingtheBose-Einsteinstatisticsorfermions,governedbyFermi-Diracstatistics.Todescribethequantumevolutionofseveralidentical(andthusindiscernible)quantumparticles,thepathintegralformulationhastobegeneralized. Inthecaseofbosons,itisbasedonthecoherentstatesholomorphicformalismandtheHilbertspaceofanalyticentirefunctions.Forbosonsoccupyingonlyafinitenumberofquantumstates,therelevantpathintegralcanformallybededucedfromthephasespaceintegralbyacomplexchangeofvariables,uptoboundarytermsandboundaryconditions.Intheexampleofonequantumstate,thechangeofvariablesissimply \[z=(p+iq)/i\sqrt{2}\,,\quad\barz=i(p-iq)/\sqrt{2}\,.\] Theholomorphicpathintegralthentakestheform \[\tag{40} \langle\mathbf{z}''\left|\mathbf{U}(t'',t')\right|\bar{\mathbf{z}}' \rangle=\int\left[\mathrm{d}\bar{\mathbf{z}}(\tau)\mathrm{d}\mathbf{z}(\tau)\right]\mathrm{e}^{\bar{\mathbf{z}}(t')\cdot\mathbf{z}(t')}\exp\left({i \over\hbar}\mathcal{A}(\mathbf{z},\bar{\mathbf{z}})\right)\] withtheboundaryconditions \(\tag{41} \bar{\mathbf{z}}(t')=\bar{\mathbf{z}}',\\mathbf{z}(t'')=\mathbf{z}'',\) wheretheclassicalaction\(\mathcal{A}(\mathbf{z},\bar{\mathbf{z}})\)reads \[\tag{42} \mathcal{A}(\mathbf{z},\bar{\mathbf{z}})=\int_{t'}^{t''}\mathrm{d}\tau\,\left[-i\bar{\mathbf{z}}(\tau)\cdot\dot{\mathbf{z}}(\tau)-H\left(i(\mathbf{z}(\tau)-\bar{\mathbf{z}}(\tau))/\sqrt{2},(\mathbf{z}(\tau)+\bar{\mathbf{z}}(\tau))/\sqrt{2};\tau\right)\right].\] Moregenerally,to\(N\)quantumstatesareassociated\(N\)pairsofcomplexvariables\((z_i,\bar{z_i})\.\) EvenintheGaussianexample,thispathintegralsuffersfromthesameambiguitiesasintheexampleofaparticleinamagneticfield,andthisleadsalsototheappearanceof\(\operatorname{sgn}(0)\)incalculations. Grassmannpathintegralsandfermions TheunderstandingofthissectionnecessitatessomepriorknowledgeofGrassmannorexterioralgebras,includingthedefinitionandpropertiesofGrassmanndifferentiationandintegration. ThedescriptionofthestatisticalpropertiesorofthequantumevolutionoffermionsystemsrequirestheintroductionofelementsofaninfinitedimensionalGrassmannalgebraandtheintegrationoverGrassmannianpaths.Forexample,todescribeasystemwith\(N\)availablequantumstates,oneintroducesthegenerators\(\theta_i(\tau),\)\(\bar\theta_i(\tau),\)\(i=1\ldotsN\,\)ofaGrassmannalgebra.Theysatisfythecommutationrelations \[\theta_i(\tau)\theta_j(\tau')+\theta_j(\tau')\theta_i(\tau)=0\,,\quad \theta_i(\tau)\bar\theta_j(\tau')+\bar\theta_j(\tau')\theta_i(\tau)=0\,,\quad\bar\theta_i(\tau)\bar\theta_j(\tau')+\bar\theta_j(\tau')\bar\theta_i(\tau)=0\,.\] Then,rulesofGrassmanniandifferentiationandintegrationcanbeformulated.Itfollowsthattheelementsofthedensitymatrixatthermalequilibrium,ortheimaginary-timepathintegral,taketheform \[\left\langle\boldsymbol{\theta}''|U(t'',t')|\bar{\boldsymbol{\theta}}'\right\rangle=\int^{\boldsymbol{\theta}(t'')=\boldsymbol{\theta}''}_{\bar{\boldsymbol{\theta}}(t')=\bar{\boldsymbol{\theta}}'}\left[\mathrm{d} \boldsymbol{\theta}(\tau)\mathrm{d}\bar{\boldsymbol{\theta}}(\tau)\right]\mathrm{e}^{-\bar{\boldsymbol{\theta}}(t')\cdot\boldsymbol{\theta}(t')} \exp\left[-\mathcal{S}(\boldsymbol{\theta},\bar{\boldsymbol{\theta}})\right]\] with \[\mathcal{S}(\boldsymbol{\theta},\bar{\boldsymbol{\theta}})=\int^{t''}_{t'}\mathrm{d}\tau\, \left\{\bar{\boldsymbol{\theta}}(\tau)\cdot\dot{\boldsymbol{\theta}}(\tau)+H\left[\boldsymbol{\theta}(\tau),\bar{\boldsymbol{\theta}} (\tau)\right]\right\},\] where\(H\left[\boldsymbol{\theta}(\tau),\bar{\boldsymbol{\theta}} (\tau)\right]\)representstheHamiltonianactingonGrassmannfunctions. Ageneralization:Thefieldintegral Whilethepathintegralisaninterestingtopicforitsownsake,themostusefulphysicsapplicationsareprovidedbyageneralization:thefieldintegral,wheretheintegrationoverpathsisreplacedbyanintegrationoverfields.Forexample,inalocalfieldtheoryforaneutralscalarfield\(\phi(x),\)\(x\in\mathbb{R}^d\,\)thepartitionfunctionisgivenby \[\mathcal{Z}=\int[\mathrm{d}\phi(x)]\mathrm{e}^{-\mathcal{S}(\phi)/\hbar},\] wheretheEuclideanaction\(\mathcal{S}(\phi)\)isa\(d\)-dimensionalintegralofafunctionofthefieldanditsderivatives(\(\partial_\mu\equiv\partial/\partialx_\mu\)): \[\mathcal{S}(\phi)=\int\mathrm{d}^dx\,\mathcal{L}_{\mathrm{E}}(\partial_\mu\phi(x),\phi(x)).\] Whilethealgebraicpropertiesofthepathintegralgeneralizeeasily,thefieldintegralleadstonewproblemsrequiringnewconceptslikeregularizationandrenormalization.Indeed,anaturalphysicalchoice,forexample,wouldbe \[\tag{43} \mathcal{L}_{\mathrm{E}}(\partial_\mu\phi,\phi)=\frac{1}{2}\sum_\mu(\partial_\mu\phi)^2+\frac{1}{2}r\phi^2+{g\over4!}\phi^4,\] where\(r\)and\(g\ge0\)aretwoparameterscharacterizingthemodel. However,indimensions\(d>1\)thederivativetermnolongerselectsfieldsregularenough,asadiscreteorlatticeapproximationreveals,and,asaconsequence,fieldcorrelationfunctionsarenotdefinedatcoincidingpoints.Itisnecessarytomodify(inanunphysicalwayfromtheviewpointofquantumphysics)theactionatshortdistance,aprocedurecalledregularization. Onepossibilityistointroducequadratictermsinthefieldwithderivativesofhigherorder\(2n>d\,\)whichrestricttheintegrationtofieldssatisfyingaHölderconditionasin\(d=1\.\)Anotherpossibilityistoconsideralatticeapproximationwithalatticespacing\(1/\Lambda\.\)Theexistenceofacontinuumlimitorlarge\(\Lambda\)limit,thenrequires,inaddition,tuningtheinitialparametersofthemodelasafunctionof\(\Lambda\,\)aprocedurecalledrenormalization.Renormalization,anditsconsequence,therenormalizationgroup,findanaturalinterpretationinthetheoryofcontinuousmacroscopicphasetransitions. Inarelativistic-covariantquantumfieldtheory,thereal-timeevolution(in3+1space-timedimensions)isthengivenby \[\int[\mathrm{d}\phi(x)]\exp\left[{i\over\hbar}\mathcal{A}(\phi)\right],\] where\(\mathcal{A}\)nowistheclassicalaction,space-timeintegraloftheclassicalLagrangiandensity \[\mathcal{A}(\phi)=\int\mathrm{d}^4x\,\mathcal{L}(\partial_\mu\phi(x),\phi(x)).\] IntheexampleoftheEuclideanLagrangian(43),thereal-timeLagrangianreads(\(t\equivx_0\)) \[\mathcal{L}=\frac{1}{2}(\partial_t\phi)^2-\frac{1}{2}(\nabla\phi)^2-\frac{1}{2}r\phi^2-{g\over4!}\phi^4,\] where\(\nabla\equiv\{\partial/\partialx^1,\partial/\partialx^2,\partial/\partialx^3\}\.\) Besidescalarbosonfields,ingeneralothertypesoffieldsarealsorequiredlikeGrassmannfieldswithspinforfermionmatter.Moreover,sinceintheStandardModelthatdescribesfundamentalinteractionsatthemicroscopicscale,interactionsaregeneratedbytheprincipleofgaugeinvariance,gaugefieldsalsoappear(andunphysicalspinlessfermionsafterquantization). 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Schweber,SS(1962).OnFeynmanQuantization.JournalofMathematicalPhysics3:831-842.doi:10.1063/1.1724296. Wentzel,G(1924).ZurQuantenoptik.ZeitschriftderPhysik22:193-199. Wick,GC(1950).TheEvaluationoftheCollisionMatrixPhysicsReview80:268-272. Wiener,N(1923).DifferentialSpace.JournalofMathematicsandPhysics2:132-174. Zinn-Justin,J(1981).Perturbationseriesatlargeordersinquantummechanicsandfieldtheories:Applicationtotheproblemofresummation.PhysicsReports70:109-167.doi:10.1016/0370-1573(81)90016-8. Internalreferences JamesMeiss(2007)Dynamicalsystems.Scholarpedia,2(2):1629. EugeneM.Izhikevich(2007)Equilibrium.Scholarpedia,2(10):2014. JeanZinn-JustinandRiccardoGuida(2008)Gaugeinvariance.Scholarpedia,3(12):8287. Gerard′tHooft(2008)Gaugetheories.Scholarpedia,3(12):7443. JamesMeiss(2007)Hamiltoniansystems.Scholarpedia,2(8):1943. AndreiD.Polyanin,WilliamE.Schiesser,AlexeiI.Zhurov(2008)Partialdifferentialequation.Scholarpedia,3(10):4605. 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Grosche,CandSteiner,F(1998).HandbookofFeynmanpathintegrals.Springer,Berlin,Heidelberg.ISBN978-3540571353 Kleinert,H(1995).PathIntegralsinQuantumMechanics,StatisticsandPolymerPhysics.WorldScientific,Singapore.ISBN978-981-270-009-4 Popov,VN(2002).FunctionalIntegralsinQuantumFieldTheoryandStatisticalPhysics,translatedfromRussian.Springer,Heildelberg.ISBN978-1-4020-0307-3 Schulman,LS(2005).Techniquesandapplicationsofpathintegration.DoverPublications,US.ISBN978-0486445281 Vasilev,AN(1998).FunctionalMethodsinQuantumFieldTheoryandStatisticalPhysics,translatedfromRussian.GordonandBreach,Amsterdam.ISBN90-5699-035-7 Zinn-Justin,J(2004).PathIntegralsinQuantumMechanics.OxfordUniversityPress,Oxford.ISBN978-0198566748 Frenchversion:Intégraledecheminenmécaniquequantique,EDPSciences2003ISBN978-2868836601,RussiantranslationFislitmat(Moscow2006). Zinn-Justin,J(2002).QuantumFieldTheoryandCriticalPhenomena(4thedition).OxfordUniversityPress,Oxford.ISBN0198509235 Seealso Coherentstate(quantummechanics),Densitymatrix,Gaugeinvariance,Gaugetheories,Instanton,Pathintegral(mathematicalphysics),Quantummechanics,Schrödingerequation, Sponsoredby:Dr.RiccardoGuida,InstitutdePhysiqueThéorique,CEA&CNRS,Gif-sur-Yvette,FranceReviewedby:AnonymousReviewedby:Dr.AndreiA.Slavnov,SteklovMathematicalInstitute,Moscow,RussiaReviewedby:Prof.MasudChaichian,DepartmentofPhysics,UniversityofHelsinki,Finland.Acceptedon:2009-02-0917:49:50GMT Retrievedfrom"http://www.scholarpedia.org/w/index.php?title=Path_integral&oldid=147600" Categories:PhysicsQuantummechanicsQuantumandstatisticalfieldtheoryQuantumfieldtheory(foundations) Personaltools Login/createaccount Namespaces Page Discussion Variants Views Read Viewsource Viewhistory Actions Search Navigation Mainpage About Proposeanewarticle InstructionsforAuthors Randomarticle FAQs Help Focalareas Astrophysics Celestialmechanics Computationalneuroscience Computationalintelligence Dynamicalsystems Physics Touch Moretopics Activity Recentlypublishedarticles Recentlysponsoredarticles Recentchanges Allarticles ListallCurators Listallusers ScholarpediaJournal Tools Whatlinkshere Relatedchanges Specialpages Printableversion Permanentlink Thispagewaslastmodifiedon11February2015,at09:21. 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