# qft_ex4 - Quantum Field Theory Example Sheet 4 Prof N.S...

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Copyright © 2010 University of Cambridge. Not to be quoted or reproduced without permission.Quantum Field Theory: Example Sheet 4Prof N.S. Manton, Michaelmas Term 20101.A real scalar field withφ4interaction has the Lagrangian densityL=12μφ∂μφ12m2φ2λ4!φ4.(1)Use Dyson’s formula (the expression forSas a time-ordered exponential of the integralofiHI, and its perturbative expansion) and Wick’s theorem to show that the leadingorder contribution to 3-particle3-particle scattering includes the amplitudep32pp1p12pp3///= ()2i(p1+p2+p3)2m2.(2)Check that this result is consistent with the Feynman rules for the theory. What otherdiagrams also contribute to this process?2.Examine(0|S|0)to orderλ2inφ4theory. Identify the different diagrams aris-ing from an application of Wick’s theorem. Confirm that to orderλ2, the combinatoricfactors work out so that the vacuum to vacuum amplitude is given by the exponentialof the sum of distinct vacuum bubble types,(0|S|0)= exp(+++...).(3)3.Consider the Lagrangian density for three scalar fieldsφi,i= 1,2,3, given byL=3summationdisplayi=112(μφi)(μφi)12m2(3summationdisplayi=1φ2i)λ8(3summationdisplayi=1φ2i)2.(4)Show that the Feynman propagator for the free field theory (i.e.λ= 0) is of the form(0|i(x)φj(y)|0)=δijDF(xy)(5)whereDF(xy) is the usual scalar propagator. Write down the Feynman rules of thetheory. Compute the amplitude for the scatteringφiφjφ