# Solution to nonlinear field equations

1. Mar 25, 2007

### wandering.the.cosmos

In the usual way to do QFT, we find the Green functions to the quadratic part of the lagrangian (usually with Feynman boundary conditions), and use this in the computation of n point (usually time ordered) correlation functions. Suppose one manages to solve instead the full nonlinear equation, is there (1) some sort of Green function for non-linear equation we can construct out of these solutions? and (2) is it possible, and if so, how do we use these solutions instead of the quadratic Green functions to do perturbative QFT?

An example of this is, suppose one knows the solution to the classical massless $\phi^4$ equations:

$$\partial_\mu \partial^\mu \phi = -\frac{g}{3!} \phi^3$$

$$D_F(x-y) = \int \frac{d^dk}{(2\pi)^4} \frac{ie^{-ik(x-y)}}{k^2+i\epsilon}$$
can we use the full solution to the $\phi^4$ equations to do calculations? If so, how? Say I wish to now calculate the $\phi + \phi \to \phi + \phi$ scattering amplitude to 1 loop -- what's the difference between computing it using the usual Feynman propagator $D_F$ and using the full solution? Or more generally, how do we go about computing the generating functional
$$\int \mathcal{D}\phi \exp\left[ i\int d^4x \left( \frac{1}{2} \partial_\mu\phi \partial^\mu \phi - \frac{g}{4!}\phi^4 + J \phi \right) \right]$$