etingof p. mathematical notions of quantum field theory (lecture notes, web draft, 2002)(69s)

《etingof p. mathematical notions of quantum field theory (lecture notes, web draft, 2002)(69s)》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、MATHEMATICALIDEASANDNOTIONSOFQUANTUMFIELDTHEORY1.Generalitiesonquantumfieldtheory1.1.Classicalmechanics.Inclassicalmechanics,westudythemotionofaparticle.Thismotionisdescribedbya(vector)functionofonevariable,q=q(t),representingthepositionoftheparticleasafunctionoftime.T

2、hisfunctionmustsatisfytheNewtonequationofmotion,q¨=−U
(q),whereUthepotentialenergy,andthemassoftheparticleis1.Anotherwaytoexpressthislawofmotionistosaythatq(t)mustbeasolutionofacertainvariationalproblem.Namely,oneintroducestheLagrangianq˙2L(q)=−U(q)2(thedifferenceofkine

3、ticandpotentialenergy),andtheactionfunctional�bS(q)=L(q)dta(forsomefixeda

4、solutionofthevariationalproblemdefinedbyS.Remark1.Thename“leastactionprinciple”comesfromthefactthatinsomecases(forexamplewhenU

≤0)theactionisnotonlyextremizedbutalsominimizedatthesolutionq(t).Ingeneral,however,itisnotthecase,andthetrajectoryoftheparticlemaynotbeaminimu

5、m,butonlyasaddlepointoftheaction.Therefore,thelawofmotionisbetterformulatedasthe“extremal(orstationary)actionprinciple”;thisisthewaywewillthinkofitinthefuture.Remark2.PhysicistsoftenconsidersolutionsofNewton’sequationonthewholelineratherthanonafixedinterval[a,b].Inthisc

6、ase,thenaivedefinitionofanextremaldoesnotmakesense,sincethe�actionintegralS(q)=L(q)dtisimproperandingeneraldiverges.Instead,onemakesthefollowingR“correct”definition:afunctionq(t)onRisanextremalofSiftheexpression��d∂L∂L

7、s=0L(q+sε)dt:=(ε˙+ε¨+···),dsRR∂q∂q˙whereε(t)isanycom

8、pactlysupportedperturbation,isidenticallyzero.Withthisdefinition,theextremalsareexactlythesolutionsofNewton’sequation.1.2.Classicalfieldtheory.Inclassicalfieldtheory,thesituationissimilar.Inthiscase,weshouldthinknotofasingleparticle,butofa“continuumofparticles”(e.g.astrin

9、g,amembrane,ajetoffluid);sothemotionisdescribedbyaclassicalfield–a(vector)functionφ(x,t)dependingonbothspaceandtimecoor