By Peeters K.
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This will be important when we compute correlation functions. Also note once more that all dots and crosses correspond to integrals, which is why the ‘external points’ are not labelled. 29). Interacting theories can be treated in perturbation theory. 32). The φ path integral then becomes a Gaussian integral. 4. Feynman diagrams again Having computed the generating functional Z [ J ] in perturbation theory, we now want to extend our toolbox and compute correlation functions. 24), and use the expressions for Z [ J ] worked out using the logic in the previous section.
E. 76) where GF ( x − y) is the single-field scalar propagator? e. 1 to the more complicated system under consideration here). (c) Compute, at order λ, the transition amplitude for the 2 → 2 scattering process in which one particle of type i and one of type j scatter to one particle of type k and one of type l. 53). 6. g. 66)), that a scattering cross section always contain integration measures for the outgoing momenta which take the form d3 k . g. 43). 77) is invariant under Lorentz transformations, despite the fact that it only involves a three-dimensional volume element in momentum space.
23) also for products of more than two fields. The generic expression is called Wick’s theorem and reads T φˆ 0 ( x1 ) · · · φˆ 0 ( xn ) = : φˆ 0 ( x1 ) · · · φˆ 0 ( xn ) + all possible contractions : . 27) In order to explain what this means, let us consider the case of four fields. 3 Feynman diagrams where the entire right-hand side is inside the normal ordering symbols. In this notation, it is still understood that any contraction that involves non-neighbouring fields yields a propagator, : φˆ 0 ( x1 )φˆ 0 ( x2 )φˆ 0 ( x3 )φˆ 0 ( x4 ) : = GF ( x1 − x3 ) : φˆ 0 ( x2 )φˆ 0 ( x4 ) : .
A guide to quantum field theory by Peeters K.