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We can now do the Gaussian path integral over φ, which is nothing else but Z0 [ J = 0]. We thus finally get Z0 [ J ] = Z0 [ J = 0] × exp 1 2 d4 x d4 y J ( x ) GF ( x − y) J (y) . 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.

J (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 +... δJ (y2 ) d4 x1 d4 x2 J ( x1 ) GF ( x1 − x2 ) J ( x2 ) . 33) The expression above is quite complicated, but fortunately we can find a convenient graphical notation for the various manipulations which are contained in it.

13). 2 of M. Peskin and D. Schroeder, “An introduction to quantum field theory”, Perseus, 1995. 2. Correlation functions and Wick’s theorem Now that we have expressed the time evolution of the field φˆ (t, x ) entirely in terms of the field φˆ 0 (t, x ), we can go and compute correlation functions in the interacting theory at λ = 0. 39), but now with interactions. 15) and interpret this as the Feynman propagator in the presence of interactions. We have to be a bit careful with what we mean with |0 .

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