Exact solution to a classical spontaneously broken scalar theory


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.


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