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
![\left.+gf^{abc}\partial_\mu\bar c^a A^{b\mu}c^c\right]](https://s0.wp.com/latex.php?latex=%5Cleft.%2Bgf%5E%7Babc%7D%5Cpartial_%5Cmu%5Cbar+c%5Ea+A%5E%7Bb%5Cmu%7Dc%5Ec%5Cright%5D&bg=ffffff&fg=333333&s=0&c=20201002)
being 
being 
![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.](https://s0.wp.com/latex.php?latex=S%3D-6%5Cint+d%5E4x%5Cleft%5B%5Cfrac%7B1%7D%7B2%7D%5Cpartial_%5Cmu%5Cphi%5Cpartial%5E%5Cmu%5Cphi%2B%5Cpartial%5Cbar+c%5Cpartial+c%5Cright%5D%2B6%5Cint+d%5E4x%5Cfrac%7B2g%5E2%7D%7B4%7D%5Cphi%5E4.&bg=ffffff&fg=333333&s=0&c=20201002)
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
by
being sn Jacobi snoidal function, 
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
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.
If you want more information about their use in QCD see the following:
Infrared gluon and ghost propagators Phys. Lett. B
Yang-Mills propagators and QCD to appear in Nucl. Phys. B: Proc. Suppl.
Update: Together with Terry Tao, we agreed that these solutions hold in a perturbative sense, i.e.
being 
Thanks a lot to Terry for the very helpful criticism.
Posted by mfrasca 
As you can see one has again the propagator reaching a finite, non-null value in the infrared limit. About the ghost propagator they obtain again a result very near to the case of a free particle. In this case the agreement was perfect for SU(2). For SU(3) there is a tiny disagreement.![Z[j]=\int[d\phi]e^{i\int d^4x\frac{1}{2}[(\partial\phi)^2-m^2\phi^2]+i\int d^4xj\phi}.](https://s0.wp.com/latex.php?latex=Z%5Bj%5D%3D%5Cint%5Bd%5Cphi%5De%5E%7Bi%5Cint+d%5E4x%5Cfrac%7B1%7D%7B2%7D%5B%28%5Cpartial%5Cphi%29%5E2-m%5E2%5Cphi%5E2%5D%2Bi%5Cint+d%5E4xj%5Cphi%7D.&bg=ffffff&fg=333333&s=0&c=20201002)
![Z[j]=e^{\frac{i}{2}\int d^4xd^4yj(x)\Delta(x-y)j(y)}](https://s0.wp.com/latex.php?latex=Z%5Bj%5D%3De%5E%7B%5Cfrac%7Bi%7D%7B2%7D%5Cint+d%5E4xd%5E4yj%28x%29%5CDelta%28x-y%29j%28y%29%7D&bg=ffffff&fg=333333&s=0&c=20201002)
. This is well-knwon matter. But now we rewrite down the above integral introducing another spatial coordinate and write down![Z[j]=\int[d\phi]e^{i\int d\tau d^4x\frac{1}{2}[(\partial_\tau\phi)^2-(\partial\phi)^2-m^2\phi^2]+i\int d\tau d^4xj\phi}.](https://s0.wp.com/latex.php?latex=Z%5Bj%5D%3D%5Cint%5Bd%5Cphi%5De%5E%7Bi%5Cint+d%5Ctau+d%5E4x%5Cfrac%7B1%7D%7B2%7D%5B%28%5Cpartial_%5Ctau%5Cphi%29%5E2-%28%5Cpartial%5Cphi%29%5E2-m%5E2%5Cphi%5E2%5D%2Bi%5Cint+d%5Ctau+d%5E4xj%5Cphi%7D.&bg=ffffff&fg=333333&s=0&c=20201002)
of this propagator. From this we learn immeadiately two things:
we get the right spectrum of the theory: a pole at 
and Wick-rotates one of the four spatial coordinates we recover the right Feynman propagator.![Z[j]=\int[d\phi]e^{i\int d\tau d^4x\frac{1}{2}[(\partial_\tau\phi)^2-(\partial\phi)^2-m^2\phi^2-\frac{\lambda}{2}\phi^4]+i\int d\tau d^4xj\phi}](https://s0.wp.com/latex.php?latex=Z%5Bj%5D%3D%5Cint%5Bd%5Cphi%5De%5E%7Bi%5Cint+d%5Ctau+d%5E4x%5Cfrac%7B1%7D%7B2%7D%5B%28%5Cpartial_%5Ctau%5Cphi%29%5E2-%28%5Cpartial%5Cphi%29%5E2-m%5E2%5Cphi%5E2-%5Cfrac%7B%5Clambda%7D%7B2%7D%5Cphi%5E4%5D%2Bi%5Cint+d%5Ctau+d%5E4xj%5Cphi%7D&bg=ffffff&fg=333333&s=0&c=20201002)
![Z[j]=\int[d\phi]e^{i\int d\tau d^4x\frac{1}{2}[(\partial_\tau\phi)^2-\frac{\lambda}{2}\phi^4]-i\int d\tau d^4x\frac{1}{2}[(\partial\phi)^2+m^2\phi^2]+i\int d\tau d^4xj\phi}](https://s0.wp.com/latex.php?latex=Z%5Bj%5D%3D%5Cint%5Bd%5Cphi%5De%5E%7Bi%5Cint+d%5Ctau+d%5E4x%5Cfrac%7B1%7D%7B2%7D%5B%28%5Cpartial_%5Ctau%5Cphi%29%5E2-%5Cfrac%7B%5Clambda%7D%7B2%7D%5Cphi%5E4%5D-i%5Cint+d%5Ctau+d%5E4x%5Cfrac%7B1%7D%7B2%7D%5B%28%5Cpartial%5Cphi%29%5E2%2Bm%5E2%5Cphi%5E2%5D%2Bi%5Cint+d%5Ctau+d%5E4xj%5Cphi%7D&bg=ffffff&fg=333333&s=0&c=20201002)
![Z_0[j]=\int[d\phi]e^{i\int d\tau d^4x\frac{1}{2}[(\partial_\tau\phi)^2-\frac{\lambda}{2}\phi^4]+i\int d\tau d^4xj\phi}.](https://s0.wp.com/latex.php?latex=Z_0%5Bj%5D%3D%5Cint%5Bd%5Cphi%5De%5E%7Bi%5Cint+d%5Ctau+d%5E4x%5Cfrac%7B1%7D%7B2%7D%5B%28%5Cpartial_%5Ctau%5Cphi%29%5E2-%5Cfrac%7B%5Clambda%7D%7B2%7D%5Cphi%5E4%5D%2Bi%5Cint+d%5Ctau+d%5E4xj%5Cphi%7D.&bg=ffffff&fg=333333&s=0&c=20201002)
It has seen the greatest gamma-ray burst ever (see 
Well, I am not Columbus but I can say that here we have an entire continent to explore. As happened to Columbus, there are a lot of thinkers against such an idea but maybe it is worth a try taking into account the possible pay-off.
The Gauge Connection 