On the action principle in quantum field theory

arXiv:hep-th/0204003v11Apr2002OntheactionprincipleinquantumfieldtheoryBozhidarZ.Iliev∗†‡Shorttitle:ActionprincipleinQFTBasicideas:→July21–22,2001Began:→August8,2001Ended:→August14,2001Initialtypeset:→August15–August20,2001Lastupdate:→March30,2002Produced:→February1,2008LANLxxxarchiveserverE-printNo.:hep-th/0204003BO/••HOrTMSubjectClasses:Quantumfieldtheory2000MSCnumbers:81P99,81Q99,81T992001PACSnumbers:02.90.+p,03.70.+k,11.10.EfKey-Words:Quantumfieldtheory,Actionprinciple,Schwinger’sactionprincipleActionprinciplesinquantumfieldtheory,ConservedquantitiesOperatorsofconservedquantitiesinquantumfieldtheoryEnergy-momentumoperator,CurrentoperatorSpinangularmomentumdensityoperatorEuler-Lagrangeequations,FieldequationsDerivativewithrespecttooperatorargument∗DepartmentofMathematicalModeling,InstituteforNuclearResearchandNuclearEnergy,Bul-garianAcademyofSciences,Boul.Tzarigradskochauss´ee72,1784Sofia,Bulgaria†E-mailaddress:[e-mail]‡URL:∼bozho/BozhidarZ.Iliev:ActionprincipleinQFT1Contents1Introduction12Problemswiththeequationsofmotionandwithconservedquantities13Schwinger’sactionprinciple(reviewandproblems)44Asolutionoftheproblems95Examples145.1Freeneutralscalarfield..............................145.2Freechargedscalarfield..............................155.3Self-interactingneutralscalarfield........................165.4Interactingneutralscalarfields..........................175.5FreeDirac(spinor)field1.StandardLagrangian................175.6FreeDirac(spinor)field2.ChargesymmetricLagrangian...........185.7GeneralquadraticLagrangian...........................206Conclusion21References22Thisarticleendsatpage..............................23AbstractAnanalysisoftheSchwinger’sactionprincipleinLagrangianquantumfieldtheoryispresented.Asolutionofaproblemcontainedinitisproposedviaasuitabledefinitionofaderivativewithrespecttooperatorvariables.ThisresultsinapreservationofEuler-Lagrangeequationsandachangeintheoperatorstructureofconservedquantities.Besides,itentailscertainrelationbetweenthefieldoperatorsandtheirvariations(whichisidenticallyvalidforsomefields,e.g.forthefreeones).Thegeneraltheoryisillustratedonanumberofparticularexamples.BozhidarZ.Iliev:ActionprincipleinQFT11.IntroductionThepaperdealswiththefollowingproblemsinLagrangianquantumfieldtheory:meaningofderivativeswithrespecttooperatorargument,orderoftheoperatorsinthestructureofconservedquantities,andcommutationofthevariationsofthefieldoperatorsinSchwinger’sactionprinciple.Theseproblemsarereviewed,analyzedandtheirsolutionisproposed.ThefirsttwooftheaboveproblemsarediscussedinSect.2.Sect.3reviewstheSchwinger’sactionprincipleandsomeitsconsequences.Specialattentionispaidtotheproblemwiththecommutativityofthefields’variationsandthefieldoperatorsor/andtheirpartialderivatives.IthasbeennoticeatfirstbySchwingerinhisoriginalwork[1]butlater,inseriousbookslike[2],ithasbeenforgotten.Asuitablesolutionofthatproblemispro-posedinSect.4bygivingarigorousmeaningofaderivativeofoperator-valuedfunctionofoperatorargumentswithrespecttosuchanargument.Itentailspreservationof(operator)Euler-Lagrangeequationsforthefieldoperatorsandauniquedefinitionoftheoperatorsofconservedquantities.Anewmomentisthatthevariationsofthefieldoperatorscannotbecompletelyarbitraryinthegeneralcase(e.g.forsomeinteractingfields)astheyshouldsatisfysomeconditionsderivedinthiswork.Sect5illustratesthegeneraltheoryofSect.4withparticularexamples(freeneutralorchargedscalarfield,(self-)interactingscalarfields,freespinorfield,andsystemoffieldsdescribedviaquadraticLagrangian).ItispresentedanexampleofaLagrangian,describingfree(orwithsomeself-interaction)spinorfield,forwhichthe(classicaloperator)Euler-Lagrangeequationsdonotexistinasensethattheyareidentities,like0=0.Regardlessofthatfact,thisLagrangianentailscompletelyreasonablefieldequations.ThemainresultsoftheworkaresummarizedinSect.6.IntheLagrangiansweconsiderisnotsupposednormalordering(oftheproductsofcreationandannihilationoperators).Besides,no(anti)commutation(orparacommutation)relationsaresupposedtobefulfilled.Buttheresultsobtainedare,ofcourse,validandifnormalorderingofproductsisusedsomekindof(anti)commutation(orparacommutation)relationsaretakenintoaccount.2.ProblemswiththeequationsofmotionandwithconservedquantitiesSupposeasystemofclassicalfieldsϕi(x),i=1,...,n∈N,overtheMinkowskispacetimeM,x∈M,isdescribedviaaLagrangianLdependingonthemandtheirfirstpartialderivatives∂μϕi(x)=∂ϕi(x)∂xμ,{xμ}beingthe(local)coordinatesofx∈M,i.e.L=L(ϕj(x),∂νϕi(x)).HereandhenceforththeGreekindicesμ,ν,...runfrom0todimM−1=3andtheLatinindicesi,j,...runfrom1tosomeintegern.Theequationsofmotionforϕi(x),knownastheEuler-Lagrangeequations,are1∂L∂ϕi(x)−∂∂xμ∂L∂(∂μϕi(x))=0(2.1)andarederivedfromthevariationalprincipleofstationaryaction,knownastheactionprinciple(see,e.g,[3,§1],[4,§67],[2,pp.19–20]).The(first)Noethertheorem[3,§2]saysthat,iftheaction’svariationisinvariantunderaC1transformationsx7→xω=xω(x)xω|ω=0=xω=(ω(1),...,ω(s))ϕi(x)7→ϕωi(xω)ϕωi(xω)|ω=0=ϕi(x)(2.2)1InthispapertheEinstein’ssummationconventionoverindicesappearingtwic