## Triviality

One of the questions that is not that easy to answer is: When is a quantum field theory exactly solved? Of course, we have the example of a free theory. When one is able to put the generating functional into a Gaussian form, the spectrum of the theory is that of a harmonic oscillator and when the coupling is zero, one is left with a possibly solved theory. But this case is trivial and does not say anything about the case of an exactly solved but interacting quantum field theory. An immediate answer to this question is: When one is able to get all the n-point functions. This implies that, if you are able to solve all the hierarchy of Dyson-Schwinger equations, you are done. Solving this set of equations is practically impossible in almost all the interesting case. But there is an exception and a notable one. So, consider the case of a massless quartic scalar field theory. Lattice computations in d=3+1 strongly hint toward triviality in the low-energy limit. Better, for d>3+1 there is a beautiful proof by Michael Aizenman that went published here. In this case the hierarchy is exactly solved (see here) and this is true also for d=3+1. So far, triviality and exact solution indeed are the same thing. But why does an interacting theory become trivial? The reason is in the behavior of the running coupling as the energy varies. We have learned from quantum field theory that couplings have not always the same value. Rather, their value is varying depending on the energy scale they are measured. In a trivial theory, couplings happen to go to zero in the given limit and an interacting theory becomes free!

For the scalar theory in the low-energy limit (infrared) in d=3+1, evidence is becoming wider that the beta function, the function that determines the behavior of the running coupling, goes like

$\beta(\lambda)=d\lambda$

being $\lambda$ the coupling and $d$ space-time dimension. I have proved this firstly here for d=3+1 but other authors arrived to an identical conclusion by different means (see here and here). But there is a surprise here: Some authors, a few years ago, proved an identical result for Yang-Mills theory (see here) with lattice computations. So, this is again a striking proof of the correctness of my mapping theorem but an indirect one. Then, we can conclude this post by stating a shocking result: Yang-Mills theory is trivial in the infrared even if QCD is not. But this result is enough to make QCD manageable at very low-energies.