Exact solutions on arxiv

24/07/2009

As promised in my preceding post (see here), I have posted yesterday a preprint on this matter on arxiv (see here). I have presented all the exact solutions I was able to obtain at a classical level and I have given a formulation of the quantum field theory for a massless quartic theory. The key point in this case is the solution of the equation for the propagator

$-\Box\Delta+3\lambda\phi_c^2(x)\Delta=\delta^D(x)$

being $\phi_c$ the given exact classical solution. As usual, I have used a gradient approximation and the solution of the equation

$\ddot\phi(t)+3\lambda\phi_c^2(t,0)\phi(t)=\delta(t)$

that I know when the phase in $\phi_c(t,0)$ is quantized as $(4n+1)K(i)$, being $n$ an integer and $K(i)$ an elliptic integral. This gives back a consistent result in the strong coupling limit, $\lambda\rightarrow\infty$, with my preceding paper on Physical Review D (see here).

The conclusion is rather interesting as quantum field theory, given from such subset of classical solutions, is trivial when the coupling becomes increasingly large as one has a Gaussian generating functional and the spectrum of a harmonic oscillator. This is in perfect agreement with common wisdom about this scalar theory. So, in some way, Jacobi elliptical functions that describe nonlinear waves behave as plane waves for a quantum field theory in a regime of a strong coupling.