# Solving phi-fourth theory using Fourier analysis

1. Dec 30, 2015

### spaghetti3451

The equation of motion of $\phi^4$ theory is $(\partial^{2}+m^{2})\phi = -\frac{\lambda}{3!}\phi^{3}$.

Why can't this equation be solved using Fourier analysis? Can't we simply write the equation in Fourier space and take it from there?

2. Dec 30, 2015

### stevendaryl

Staff Emeritus
Well, the right-hand side of the equation is nonlinear, so performing a Fourier transform will produce a mess---a triple integral. You end up replacing a nonlinear differential equation by a nonlinear integral equation. That might be a good first step, but it doesn't solve the equation. You would have to resort to perturbation theory in any case.

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