Quantum Field Theory Example Sheet

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1、QuantumFieldTheory:ExampleSheet1DrDavidTong,October20071.Astringoflengtha,massperunitlengthσandundertensionTisfixedateachend.TheLagrangiangoverningthetimeevolutionofthetransversedisplacementy(x,t)isZ"22#aσ∂yT∂yL=dx−(1)02∂t2∂xwherexidentifiespositionalongthestringfromoneendp

2、oint.ByexpressingthedisplacementasasineseriesFourierexpansionintheformr2X∞nπxy(x,t)=sinqn(t)(2)aan=1showthattheLagrangianbecomesX∞σTnπ222L=q˙n−qn.(3)22an=1Derivetheequationsofmotion.Henceshowthatthestringisequivalenttoaninfinitesetofdecoupledharmonicoscillatorswithfreque

3、nciesrTnπωn=.(4)σa2.Showdirectlythatifφ(x)satisfiestheKlein-Gordonequation,thenφ(Λ−1x)alsosatisfiesthisequationforanyLorentztransformationΛ.3.Themotionofacomplexfieldψ(x)isgovernedbytheLagrangianλ∗µ2∗∗2L=∂µψ∂ψ−mψψ−(ψψ).(5)2WritedowntheEuler-Lagrangefieldequationsforthissystem.V

4、erifythattheLa-grangianisinvariantundertheinfinitesimaltransformation∗∗δψ=iαψ,δψ=−iαψ(6)DerivetheNoethercurrentassociatedwiththistransformationandverifyexplicitlythatitisconservedusingthefieldequationssatisfiedbyψ.14.VerifythattheLagrangiandensity1µ12L=∂µφa∂φa−mφaφa(7)22foratrip

5、letofrealfieldsφa(a=1,2,3)isinvariantundertheinfinitesimalSO(3)rotationbyθφa→φa+θǫabcnbφc(8)wherenisaunitvector.ComputetheNoethercurrentjµ.DeducethatthethreeaquantitiesZQ=d3xǫφ˙φ(9)aabcbcareallconservedandverifythisdirectlyusingthefieldequationssatisfiedbyφa.5.ALorentztransformat

6、ionxµ→x′µ=ΛµxνissuchthatitpreservestheMinkowskiνmetricη,meaningthatηxµxν=ηx′µx′νforallx.Showthatthisimpliesthatµνµνµνστηµν=ηστΛµΛν.(10)UsethisresulttoshowthataninfinitesimaltransformationoftheformµµµΛν=δν+ων(11)isaLorentztranformationwhenωµνisantisymmetric:i.e.ωµν=−ωνµ.Writedo

7、wnthematrixformforωµthatcorrespondstoarotationthroughanin-νfinitesimalangleθaboutthex3-axis.Dothesameforaboostalongthex1-axisbyaninfinitesimalvelocityv.6.ConsidertheinfinitesimalformoftheLorentztransformationderivedinthepreviousquestion:xµ→xµ+ωµxν.Showthatascalarfieldtransformsas

8、ν′µνφ(x)→φ(x)=φ(x)−ωνx∂µφ(x)(12)andhenceshowthatthevariationoftheLag