Practical task for exam in QFT(量子场论复习)

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1、1PracticaltaskforexaminQFT1.Operatoralgebra.Considerfreescalarfieldϕ(x)=ϕ+(x)+ϕ−(x),withϕ+(−)beingpositive(negative)-energypartoffieldoperator:⟨Ω

2、ϕ−=ϕ+

3、Ω⟩=0,and

4、Ω⟩isvacuumstate.-Showthat[]ϕ+(x),ϕ−(y)=⟨Ω

5、ϕ+(x)ϕ−(y)

6、Ω⟩I=⟨Ω

7、ϕ(x)ϕ(y)

8、Ω⟩I,whereIisunityoperato

9、r.-Showthat⟨Ω

10、Φ(x)Φ(y)

11、Ω⟩=⟨Ω

12、ϕ+(x)ϕ+(x)ϕ−(y)ϕ−(y)

13、Ω⟩,whereΦ(x)=:ϕ2(x):(normalordered).-Usethecommutationrelationandderive2⟨Ω

14、T(Φ(x)Φ(y))

15、Ω⟩=2(⟨Ω

16、T(ϕ(x)ϕ(y))

17、Ω⟩),(1)()*Comparethepreviousresult(1)to⟨Ω

18、Tϕ2(x)ϕ2(y)

19、Ω⟩givenbyapplicationofWick’stheorem.Discu

20、ssthediscrepancy.2.Mllerscattering.Electron-electronscatteringiscalledMøllerscattering.ConsidertheMøllerscatteringe−(p)e−(p′)→e−(k)e−(k′)attheleadingorder.Inparticular:-DrawFeynmandiagramsfortheleading(tree)orderamplitudeandwritedowntheirexpres-sions.

21、-Calculatethescatteringprobability(squaredamplitude)forunpolarizedfinalandinitialstates.(forsimplicitysetmassofelectronto0)-ExpressthescatteringprobabilityviatheMandelstamvariables,andviathescatteringangleinthecenter-massframeθ.*Explaintheoriginofsingul

22、arityatθ=0.23.Furry'stheorem.Vacuumexpectationofoddnumberofelectromagneticcurrentsiszero.Particulary,itmeans⟨Ω

23、T(J(x)J(x)J(x))

24、Ω⟩=0,(2)123whereJ(x)=ψ¯(x)γψ(x)iselectro-magneticcurrent.Checkexpression(2)atleadingorder.-DrawallFeynmandiagramscontrib

25、utingto(2)atleadingorder.-Showthatthediagramscanceleachother.*Givearguments,whyFurry’stheoremholdsatallorders.3Listoftopics1.Elementsofclassicalfieldtheory:•Leastactionprinciple,Euler-Lagrangeequation.•Hamiltonianformulation2.Transformationpropertiesoffi

26、elds•Lorentztransformationoffields,scalar,vector,spinorfields.•Conservedcurrents,Noether’stheorem.•Energy-momentumtensor,angular-momentumtensor.3.FreeKlein-Gordon(KG)field•KGequation,LagrangianforKGequation,Hamiltonianformulation•Quantization,fielddecompos

27、ition.•Creationandannihilationoperators,Hamiltonianinoperatorform,energyeigenstates.•Covariantcommutationrelation,KGpropagators(SchwingerandFeynman)•ComplexKGfield,chargeoperator.4.Interactingfields•ϕ4theory:Lagrangian,symmetries,equation