## Exact solution to a classical spontaneously broken scalar theory

02/08/2008

As promised in my preceding post I said that a classical spontaneously broken scalar theory can be exactly solved. This is true as I will show. Consider the equation

$\ddot\phi -\Delta\phi + \lambda\phi^3-m^2\phi=0.$

You can check by yourself that the exact solution is given by

$\phi(x)=v\cdot{\rm dn}(p\cdot x,i)$

being $v=\sqrt{2m^2/3\lambda}$ the v.e.v. of the field and ${\rm dn}$ an elliptical Jacobi function. As always the following dispersion relation must be true

$p^2=\frac{\lambda v^2}{2}$

giving a consistent classical solution. When one goes to see the spectrum of the theory, the Fourier series of the Jacobi dn function has a zero mass excitation, the Goldstone boson.

Update: A proper full solution is given by

$\phi(x)=v\cdot{\rm dn}(p\cdot x+\varphi,i)$

being $\varphi$ an integration constant.