A guide to quantum field theory by Peeters K.

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By Peeters K.

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P −1 δp δp · · · δJ (y3 ) p δJ (y1 ) p−1 δJ (y2 ) p d4 x1 J ( x1 ) GF ( x1 − y1 ) exp (. 35) If we take a further derivative, we can either act again on the exponential, or act on the J that was brought down earlier. The latter action corresponds to d4 y1 d4 y2 · · · d4 y3 ↓ d4 y1 d4 y 2 · · · d4 y3 δ p −1 δ p −1 δp · · · δJ (y3 ) p δJ (y1 ) p−1 δJ (y2 ) p−1 GF (y2 − y1 ) exp (. ) . 36) Playing a bit with these procedures quickly leads to the following set of graphical rules. We represent every integration over an yi variable by a dot (these integration variables are dummies, so we do not have to label the dots).

39). For interacting theories the story is of course more complicated, and generically it is not possible to do the path integral explicitly. We hence resort, as in the previous chapter, to a perturbative analysis valid for small values of the coupling constant which sets the interaction strength. 30) . 3 Path integrals in field theory where S0 [φ] is the part of the action quadratic in the fields, and S I [φ] is the rest. We have assumed that a small coupling constant λ can be extracted from this latter part.

24) to write correlation functions: simply take a functional derivative with respect to the source. So we can also write Z[ J ] = 1+ + iλ h¯ 1 2 d4 y1 L I iλ h¯ 2 δ δJ (y1 ) δ δJ (y1 ) d4 y1 L I × D φ exp d4 y 2 L I δ +... δJ (y2 ) i S0 [ φ ] + h¯ d4 x J ( x )φ( x ) . 32) The only φ-dependence now sits in the last factor, and we can thus do the path integral over φ just as in the computation of Z0 [ J ]. We obtain Z[ J ] = 1+ + iλ h¯ 1 2 d4 y1 L I iλ h¯ 2 δ δJ (y1 ) d4 y1 L I × exp δ δJ (y1 ) 1 2 d4 y 2 L I δ +...

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