Interactions and Feynman Diagrams

Abstract

We will start considering a theory with interactions with lagrangian density L = L 0 + L int. (10.1) where the first term L 0 refers to the free lagrangian and the sencond one L int. contains the interaction terms. For instance, for a real scalar theory we have L 0 = 1 2 ∂ µ φ∂ µ φ − 1 2 m 2 φ 2. (10.2) The generating functional in the presence of a linearly coupled source J(x) now is Z[J] = N Dφ e i d 4 x {L 0 +L int. +J(x)φ(x)} , (10.3) Defining the generating functional for the free theory as Z 0 [J] ≡ N Dφ e i d 4 x {L 0 +J(x)φ(x)} , (10.4) we can rewrite (10.3) as Z[J] = e i d 4 x L int. [−i δ δJ(x) ] Z 0 [J]. (10.5) In the expression above L int. [−iδ/(δJ(x))] means that the argument of the functional L int. [φ(x)] is obtained by functional derivative in the following way 1