Exact solutions of quantum field theories are very rare and, normally, refer to toy models and pathological cases. Quite recently, I put on arxiv a pair of papers presenting exact solutions both of the Higgs sector of the Standard Model and the Yang-Mills theory made just of gluons. The former appeared a few month ago (see here) while the latter has been accepted for publication a few days ago (see here). I have updated the latter just today and the accepted version will appear on arxiv on 2 January next year.
What does it mean to solve exactly a quantum field theory? A quantum field theory is exactly solved when we know all its correlation functions. From them, thanks to LSZ reduction formula, we are able to compute whatever observable in principle being these cross sections or decay times. The shortest way to correlation functions are the Dyson-Schwinger equations. These equations form a set with the former equation depending on the higher order correlators and so, they are generally very difficult to solve. They were largely used in studies of Yang-Mills theory provided some truncation scheme is given or by numerical studies. Their exact solutions are generally not known and expected too difficult to find.
The problem can be faced when some solutions to the classical equations of motion of a theory are known. In this way there is a possibility to treat the Dyson-Schwinger set. Anyhow, before to enter into their treatment, it should be emphasized that in literature the Dyson-Schwinger equations where managed just in one way: 
Using their integral form and expressing all the correlation functions by momenta. It was an original view by Carl Bender that opened up the way (see here). The idea is to write the Dyson-Schwinger equations into their differential form in the coordinate space. So, when you have exact solutions of the classical theory, a possibility opens up to treat also the quantum case!
This shows unequivocally that a Yang-Mills theory can display a mass gap and an infinite spectrum of excitations. Of course, if nature would have chosen the particular ground state depicted by such classical solutions we would have made bingo. This is a possibility but the proof is strongly related to what is going on for the Higgs sector of the Standard Model that I solved exactly but without other matter interacting. If the decay rates of the Higgs particle should agree with our computations we will be on the right track also for Yang-Mills theory. Nature tends to repeat working mechanisms.
Marco Frasca (2015). A theorem on the Higgs sector of the Standard Model Eur. Phys. J. Plus (2016) 131: 199 arXiv: 1504.02299v3
Marco Frasca (2015). Quantum Yang-Mills field theory arXiv arXiv: 1509.05292v1
Carl M. Bender, Kimball A. Milton, & Van M. Savage (1999). Solution of Schwinger-Dyson Equations for ${\cal PT}$-Symmetric Quantum Field Theory Phys.Rev.D62:085001,2000 arXiv: hep-th/9907045v1
Posted by mfrasca 
theory and I solve exactly all the set of Dyson-Schwinger equations. I am able to do this as I know a set of exact solutions of the classical equation of the theory and I am able to solve an apparently difficult equation for the two point function. At the end of the day, one gets the exact propagator, the spectrum and the beta function. It is seen that this theory has only trivial fixed points. I was able to get these results on 
and let me explain why. When one does quantum field theory, the only real tool that she has to do any meaningful computation is small perturbation theory. The word “small” is never said but it should be said in any circumstance as this technique only works if you have a small parameter in your theory (a coupling) to use as a development parameter. Otherwise we are lost and all starts to become foggy and not so well-defined. Today, nobody knows how to manage a theory with a strong coupling. Parameter 
the given exact classical solution. As usual, I have used a gradient approximation and the solution of the equation
is quantized as 
, being 
an integer and 
an elliptic integral. This gives back a consistent result in the strong coupling limit, 
, with my preceding paper on Physical Review D (see 
and the parabola 
and this is enough to keep up the attention of the public for all the time.
.
making all the argument consistent. This asymptotic series should be modified as the limit ![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)
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.
. As usual, I rescale time variable as 
and I take a solution series
and 
. These are conjugate variables as you know. So, already at the leading order I have solved my equation. Indeed, I note that
being f and g arbitrary functions. In this case the gradient expansion gives immediately the exact result making its application trivial as should be. Indeed, I take 
in the perturbation series and put 
and I get
The Gauge Connection 