Path Integrals Path integrals were invented by Feynman while a graduate student as an - PDF document

Eq.(1.4)canbecheckedbydierentiatingbothsideswithrespecttotandverifyingthatj (t)iasdenedbyEq.(1.4)satisestheSchrodingerequation,andalsothecorrectboundaryconditionlimt!tij (t)i=j 0i:(1.6)WecallthesolutionEq.(1.4)`formal'onlybecauseisnoeasiertoevaluatetheexponentialoftheHamiltonianthanitistosolvetheSchrodingerequation.TherealuseofEq.(1.4)isforprovinggeneralresults.Thequantity^U(t;ti)def=e�i^H
(t�ti)(1.7)thatappearsinEq.(1.4)iscalledthetimeevolutionoperator.Ifweknowthisoperator,itisclearthatweknoweverythingthereistoknowaboutthewaythesystemevolvesintime.Itisalsosucienttondthematrixelementof^Ubetweenarbitrarypositioneigenstateshqfj^U(tf;ti)jqii:(1.8)Thisquantityissometimescalledthetimeevolutionkernel.(Notethatthenalstateappearsontheleft.Wewillalwayswriteourexpressionssothat`latertimesareontheleft.')WebeginourderivationofthepathintegralbydividingthetimeintervalfromtitotfintoNequalintervalsoflength
t=tf�ti N:(1.9)(WewilleventuallytakethelimitN!1,
t!0.)Wewritethetimeevolutionoperatorase�i^H
(t�ti)=e�i^H
te�i^H
t


e�i^H
t| {z }Nfactors:(1.10)Wetheninsertacompletesetofstatesbetweeneachofthefactorsaboveusingthecompletenessrelation1=Zdqjqihqj:(1.11)Inthisway,weobtainhqfj^U(tf;ti)jqii=ZdqN�1


Zdq1hqfje�i^H
tjqN�1ihqN�1je�i^H
tjqN�2i


hq1je�i^H
tjqii:(1.12)2 andtheidentitiesf(^q)jqi=f(q)jqi;hpjf(^p)=hpjf(p);(1.20)whichholdforanarbitraryfunctionf.NoticewhathashappenedinEq.(1.18):wehaveusedthecompletenessrelationtoreplacetheoperators^pand^qwithintegralsoverclassicalquantitiespandq.SubstitutingEq.(1.18)intoEq.(1.14),weobtainhqfj^U(tf;ti)jqii=ZdqN�1dpN�1 2


Zdq1dp1 2Zdp0 2exp(iN�1Xn=0
tpnqn+1�qn
t�H(pn;qn)):(1.21)InthecontinuumlimitN!1,
t!0,wecanidentifyqn+1�qn
t!_q(t);N�1Xn=0
tf(tn)!Ztftidtf(t):(1.22)WecanthenwriteEq.(1.21)inthecompactform hqfj^U(tf;ti)jqii=q(tf)=qfZq(ti)=qid[p]d[q]eiSH[p;q]; (1.23)where SH[p;q]=Ztftidthp(t)_q(t)�H(p(t);q(t))i; (1.24)andwehaveusedtheabbreviationsd[p]def=N�1Yn=0dpn 2;d[q]def=N�1Yn=1dqn:(1.25)Eq.(1.23)iscalledapathintegral(orfunctionalintegral)becausetheintegralisoverall`phase-spacepaths'(p(t);q(t)).Thepathq(t)mustsatisfytheboundaryconditionsq(ti)=qi,q(tf)=qf,whilethepathp(t)iscompletelyunconstrainedandisnotrelatedtoq(t)(or_q(t))inanyway.Weemphasizethatp(t)andq(t)aredenedbyintegratingoverthevaluesofp(t)andq(t)independentlyateachvalueoft,sothepathsthatcontributetothefunctionalintegralareingeneralhighlydiscontinuous.Itisoftenusefultohaveanintuitivepictureofthepathintegralasstatingthata4 neededtomakeexpressionswell-dened.Forexample,inthisapproachthetransfermatrixbecomesTq0;q0=Zdp 2exp(i(q0�q)p�i
t(1�i)"p2 2m+V(q)#):(1.30)Thisintegraliswell-denedaslongas�0becausethecoecientofp2intheexponenthasanegativerealpartthatsuppressestheintegrandasp!1.WenowperformtheintegraloverpinEq.(1.30).TheintegralhastheformofageneralizedGaussianintegralZ1�1dpe�1 2Ap2+Bp;Re(A)�0:(1.31)Wecanevaluatethisintegralbycompletingthesquareintheexponent�1 2Ap2+Bp=�1 2Ap�B A2+B2 2A(1.32)andshiftingthevariableofintegrationtop0=p�B=A:Z1�1dpe�1 2Ap2+Bp=eB2=(2A)Z1�1dp0e�1 2Ap02=eB2=(2A)2 A1=2:(1.33)ThistrickfordoingGaussianintegralswillbeusedrepeatedly.Applyingthisformula,weobtainTq0;q=m 2i
t1=2exp(i
t(1�i)24m 2 q0�q
t(1�i)!2�V(q)35):(1.34)Notethatthetimealwaysappearswithasmallnegativeimaginarypart.Omittingtheifactorsforbrevity,thetime-evolutionkernelishqfj^U(tf;ti)jqii=m 2i
tN=2ZdqN�1


Zdq1exp(iN�1Xn=0
t"m 2qn+1�qn
t2�V(qn)#):(1.35)NotethatthereisonefactorC=m 2i
t1=2(1.36)foreachqintegral,withonefactorleftover.Wethereforedenethepathintegralmeasuretobed[q]def=N�1Yn=1Cdqn:(1.37)6 where�0istakentozeroattheendofthecalculation.Notethatthisisequivalenttotheiprescriptionthatwasusedtomakethepathintegralwell-denedabove.Insertingacompletesetofenergyeigenstates,wegethqfj^U(0;�T(1�i))jqii=Xne�iEnTe�EnThqfjnihnjqii:(1.44)TakingT!1,the-dependenttermsuppressesthecontributionofallexcitedstates,leavingonlythecontributionfromthegroundstaten=0:hqj^U(0;�T(1�i))jqii!eiE0T 0(qf) 0(qi):(1.45)Viewedasafunctionofqf,thisgivesthegroundstateofthesystemuptoa(singular)normalizationfactor.Therefore,wecanwrite 0(qf)=Nq(0)=qfZd[q]eiS[q]: (1.46)Here,theintegralisoverallpathsfromti!�1withtheiprescriptionisunder-stood,andNisa(singular)normalizationfactor.Thefactthattheiprescriptionprojectsoutthegroundstatewillbeusedfrequentlyinthefollowing.Thesingularnormalizationfactorsshouldnotbotheryoutoomuch.Conceptually,theyariseforthesamereasonasnon-normalizablestatesinquantummechanics,andwewillseehowtodealwiththemwhenwestartusingthepathintegraltocomputephysicalquantities.2FromPathIntegralstoQuantumMechanicsWenowreversetheprocedureaboveandshowhowtoreconstructtheoperatorformofquantummechanicsfromthepathintegral.Thatis,weattempttodeneaquantum-mechanicalevolutionoperatorbythepathintegralwithagivenaction.Thereisonegeneralizationoftheprevioussectionthatwillbeneededhere.Inthediscussionabove,thepathintegralmeasureisindependentofq(seeEq.(1.36)).Asimpleexampleofaquantum-mechanicalsystemwithanontrivialpathintegralmeasureisgivenbyaparticlewithaposition-dependentmass,denedbytheLa-grangianL=1 2m(q)_q2�V(q):(2.1)Thecanonicalmomentumisp=@L @_q=m(q)_q;(2.2)8 where~L(q;_q)=1 2m(q)_q2+i 2m0(q) m(q)_q�(m0(q))2 8m3(q)+V(q):(2.10)Again,the`extra'termscomefromtheoperatorordering.ThemeasurefactorisC(q)= m(q) 2i
t!1=2:(2.11)TheLagrangianappearinginthepathintegralisnotthesameastheclassicalLa-grangianwestartedwith.Thisshouldnotworryus,sinceweshouldaprioriallowallpossibletermsconsistentwithsymmetries(andrestrictedbyexperimentaldataifweareattemptingtodescribetherealworld).Anotherfeatureofthisexampleisthefactthatthemeasurefactordependsonq.Thepathintegralforthissystemcanthenbewritten hqfj^U(tf;ti)jqii=C(qi)q(tf)=qfZq(ti)=qid[q]eiS[q]: (2.12)Herethemeasureisdenedbyd[q]=limN!1N�1Yn=1C(qn)dqn;(2.13)wherethetimeintervalfromtitotfisdiscretizedintoNstepsasbefore.ThefactthatthemeasurefactorinEq.(2.12)involvesqiratherthanqforiginatesinthefactthatwechosetoorderthequantumHamiltoniansothat^pistotheleftof^q,ratherthantheotherwayaround.Itisclearthatphysicalquantitiesshouldnotdependonthischoice,andwewillseethatindeedthemeasurefactorC(q)cancelsoutwhenwecomputephysicalquantities.Wewanttoseeifwecandenethetimeevolutionoperatorusingtheright-handsideofthepathintegralEq.(2.12)withthegeneralizedmeasure.(Eq.(2.12)denesthematrixelementsofthetimeevolutionoperatorforacompletesetofstates,whichisthesameasdeningtheoperator.)Inordertodeneaconsistenttimeevolution,theoperator^U(tf;ti)denedbythepathintegralmustbeunitary,anditmustsatisfythe`timecompositionrule'^U(tf;ti)=^U(tf;t)^U(t;ti)(2.14)10 Tocheckthis,wecomputehq0j^Tjqi=Zd(
q)hq0je�i
q^pei
tL(^q;
q=
t)jqi=Zd(
q)hq0jq+
qiei
tL(q;
q=
t)=ei
tL(q;(q0�q)=
t):(2.22)Notethatthisdenesthematrixelementsof^Tinacompletesetofstates,sothesolutionisunique.Withthisresult,wecanwritethepathintegralasanintegral(sum)overinter-mediatestates:hqNj^U(tN;t0)jq0i=ZdqN�1


dq1hqNj^T^CjqN�1i


hq1j^T^Cjq0i;(2.23)where^C=C(^q).Thiscanbewrittenmuchmorecompactlyastheoperatorstatement^U(tN;t0)=(^T^C)N:(2.24)Thatis,^T^Cistheinnitesmaltimeevolutionoperator.Weseethat^Uisunitaryifandonlyiftheoperator^T^Cisunitary.Theoperator^TdenedbyEq.(2.21)isnotunitarybyitself,sinceitisanintegral(sum)ofunitaryoperators.Unitarityof^T^Cisequivalenttohq0j(^T^C)y(^T^C)jqi!hq0jqi=(q0�q):(2.25)AshortcalculationanalogoustoEq.(2.22)giveshq0j(C^T)y(C^T)jqi=C(q0)C(q)Zd(
q)exp(i
t"Lq;
q
t�L q0;q�q0+
q
t!#):Theintegralontheright-handsideisafunctionofqandq0thatissharplypeakedatq=q0forsmall
t.Thereasonisthatforq06=q,thephaseoftheintegrandoscillateswildly,suppressingthevalueoftheintegral.Theintegralbecomesmoreandmoresharplypeakedas
t!0,andwehavehq0j(C^T)y(C^T)jqi!jC(q)j2(sharplypeakedfunctionofq0�q):(2.26)WecanchooseC(q)(asafunctionof
t)sothatthishasunitareainthelimit
t!0,i.e.theintegralisequalto(q�q0).12 Notethatthereareindependentquantumoperators^(~x)and^(~x)ateachspatialpoint~x.Tomakethiswell-dened,wecanreplacethespatialcontinuumwithadiscretesquarelatticeofpointswithspacinga:~x=a
(n1;n2;n3);(3.4)wheren1;n2;n3areintegers.Wethenhaveindependentoperators^~xand^~xateachlatticesite.ThequantumHamiltonianthen^H=X~xa32641 2^2~x+1 2X~j0@^~x+~j�^~x a1A2+V(^~x)375;(3.5)where~jrunsovertheunitlatticevectors(a;0;0);(0;a;0);(0;0;a).Wenowretracethestepsleadinguptothepathintegral.Wewanttoevaluatethetimeevolutionkernelhfj^U(tf;ti)jii=hfje�i^H(tf�ti)jii;(3.6)wherejiisaneigenstateoftheeldoperator:^~xji=~xji:(3.7)(Thatis,jiisasimultaneouseigenstateofalloftheoperators^~x.)WeagainbreakthetimeintervalintoNintervalsoflength
t=(tf�ti)=N,andinsertacompletesetofstatesjibetweeneachinterval.Inthisway,weobtaintheHamiltonianpathintegralhfj^U(tf;ti)jii=(~x;tf)=f(~x)Z(~x;ti)=i(~x)d[]d[]eiSH[;];(3.8)withHamiltonianactionSH[;]=N�1Xn=0
tX~xa3"~x;n~x;n+1�~x;n
t�1 22~x;n�1 2X~j ~x+~j;n�~x;n a!2�V(~x;n)#(3.9)�!ZtftidtZd3x_�1 22�1 2~r2�V();(3.10)14 isnite,andtheintegralscanbeapproximatednumerically.Thisapproachiscalled`latticeeldtheory,'andthereiscurrentlyamajorresearcheortunderwaytoperformquantumeldtheorycalculationsusingthisapproach.16