## A set of exact classical solutions of Yang-Mills equations

28/02/2009

Following the discussion with Rafael Frigori (see here) and returning to sane questions, I discuss here a class of exact classical solutions that must be considered in the class of Maximal Abelian Gauge.  As usual, we consider the following Yang-Mills action

$S=\int d^4x\left[\frac{1}{2}\partial_\mu A^a_\nu\partial^\mu A^{a\nu}+\partial^\mu\bar c^a\partial_\mu c^a\right.$

$-gf^{abc}\partial_\mu A_\nu^aA^{b\mu}A^{c\nu}+\frac{g^2}{4}f^{abc}f^{ars}A^b_\mu A^c_\nu A^{r\mu}A^{s\nu}$

$\left.+gf^{abc}\partial_\mu\bar c^a A^{b\mu}c^c\right]$

being $c,\ \bar c$ the ghost field, $g$ the coupling constant and, for the moment we omit the gauge fixing term. Let us fix the gauge group being SU(2). We choose the following (Smilga’s choice, see the book):

$A_1^1=A_2^2=A_3^3=\phi$

being $\phi$ a scalar field. The other components are taken to be zero. It easy to see that the action becomes

$S=-6\int d^4x\left[\frac{1}{2}\partial_\mu\phi\partial^\mu\phi+\partial\bar c\partial c\right]+6\int d^4x\frac{2g^2}{4}\phi^4.$

This is a very nice result as if we have a solution of the scalar field theory we get immediately a classical solution of Yang-Mills equations while the ghost field decouples and behaves as that of a free particle. But such solutions do exist. We can solve exactly the  equation

$\partial_t^2\phi-\partial_x^2\phi+2g^2\phi^3=0$

by

$\phi = \mu\left(\frac{2}{2g^2}\right)^\frac{1}{4}{\textit sn}(p\cdot x+\theta,i)$

being sn Jacobi snoidal function, $\mu,\ \theta$ two arbitrary constants, if holds

$p^2=\mu^2\left(\frac{2g^2}{2}\right)^\frac{1}{2}.$

We see that the field acquired a mass notwithstanding it was massless and the same happens to the Yang-Mills field. These are known as non-linear waves. These solutions do not represent a new theoretical view. A new theoretical view is given when they are used to build a quantum field theory. This is the core of the question.

What happens when we keep the gauge fixing term as

$\frac{1}{\xi}(\partial\cdot A)^2?$

If you substitute Smilga’s choice in this term you will find a correction to the kinematic term implying a rescaling of space variables. This is harmless for the obtained solutions resulting in the end into a multiplicative factor for the action of the scalar field.

The set of Smilga’s choice is very large and increases with the choice of the gauge group. But such solutions always exist.

$A_\mu^a(x)=\eta_\mu^a\phi(x)+O(1/g)$
being $\eta_\mu^a$ a constant and $g$ the coupling taken to be very large. These become exact solutions when just time dependence is retained. So, the theorem contained in the above papers is correct for this latter case and approximate in the general case. As the case of interest is that of a large coupling, these results permit to say that all the conclusions drawn in the above papers are correct. This completed proof will appear shortly in Modern Physics Letters A (see http://arxiv.org/abs/0903.2357 ).