In general, in the presence of interactions, the equations of motions cannot be solved exactly, and one has to resort to a perturbative expansion in a small parameter. We discuss the scalar case first, as it is as always the simplest. We add to the Lagrangian density L a so-called source term, which couples linearly to the fieldϕ (compare the driving force term for a harmonic oscillator)
L = 12 (∂µϕ)2 − V(ϕ) − J (x)ϕ(x). (4.1) For sake of explicitness, we will take the following expression for the potential
V(ϕ) = 12 m2ϕ2(x) + g 3!