PS12-solutions.pdf - Phys 253a 1 Problem Set 12 Solutions December 4 2010 1(a The kinetic term for \u03c6 is(D\u00b5 \u03c6)\u2217 D\u00b5 \u03c6 =(\u2202\u00b5 \u2212 ieA\u00b5)\u03c6\u2217(\u2202

PS12-solutions.pdf - Phys 253a 1 Problem Set 12 Solutions...

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Phys 253a 1 Problem Set 12 Solutions December 4, 2010 1.(a) The kinetic term forφis(Dμφ)*Dμφ=(μ-ieAμ)φ*(μ+ieAμ)φIf we switchφandφ*, then clearly we should takeAμ→ -Aμto leave theLagrangian invariant.(b) We have an expression for the correlation function in terms of a path integralh0|T{Aμ1(q1). . . Aμn(qn)}|0i=1ZZDADφDφ*nYj=1Aμj(qj)eiS=1ZZDADφDφ*nYj=1Zd4xjAμj(xj)e-ixj·qjeiSwhereZ=RDADφDφ*eiS(and we could take the fourier transforms out ofthe path integral if we like). To prove Furry’s theorem, consider the change ofvariablesφ0=φ*, φ*0=φ, andA0=-A.The action is invariantS0=Sbyconstruction. The measure is also invariantD[-A]Dφ*Dφ=DADφDφ*.1Thus, we getZDADφDφ*nYj=1Aμj(qj)eiS=ZD[-A]Dφ*DφnYj=1-Aμj(qj)eiS0=ZDADφDφ*nYj=1-Aμj(qj)eiS=(-1)nZDADφDφ*nYj=1Aμj(qj)eiS1 This study resource was shared via CourseHero.com
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