A Millenium Problem issue


As my readers know, a recurring question in this blog is the solution to the Millenium Problem on Yang-Mills theory. So far, we have heard no fuzz about this matter and the page at the Clay Institute is no more updated since 2004. But in these years, activity on this problem has been significant and my aim here is to take this to your attention. In these days, a revision to a paper by Alexander Dynin is appeared (see here). The main conclusion in this paper is theorem 3.1 on page 17 that makes a clear statement about the spectrum of Yang-Mills theory: This must go like the one of a harmonic oscillator at least. This agrees perfectly well with the conclusions in my paper published in Physics Letters B (see here). As I prove that the theory is trivial in the infrared limit, a result that could be inferred but it is not stated in the Dynin’s paper, this limit gives a theory that exists, being it free. Anyhow, Dynin claims a complete proof of existence and this must be true all the way down the infrared limit starting from the ultraviolet one. Having the theory two trivial fixed points, at high and low energies, in these limits we are certain that the theory must exist. This is so because the theory becomes free. This should make easier a proof of existence of the theory for all the energy range. In any case, it would appear rather strange if the theory would exist just in its limit cases and not otherwise. So, if Dynin’s proof is correct, this should be checked promptly as all this would represent a significant breakthrough in our current understanding of quantum field theory. Dynin’s paper does not provide neither an explicit mass spectrum nor a techniques to do computations in quantum field theory. These are given in my papers and, all in all, we have here a complete new mathematical setup to manage also strongly coupled quantum field theories. It is interesting to show here a clear evidence of this situation for Yang-Mills theory through  the following picture obtained from lattice computations for the running coupling (see here and Physics Letters B)

This picture gives a blatant evidence of the scenario I was able to obtain mathematically and that is consistent with the mathematical proof given in Dynin’s paper.

On the other side, the spectrum is the one of a harmonic oscillator at lower energies. This means that, at lower energies. a Yukawa propagator should fit the bill rather well and a plateau must be observed from lattice computations. This is indeed the real situation (see here for a recent review).  To support further this scenario, the ghost propagator is the one of a free particle in the same limit. This shows again that the theory is infrared free. So, there is a situation, both from a mathematical side and a physical one clearly showing an explicit solution to the Yang-Mills question and that should be addressed rapidly.

Of course, behind all this, there is a lot of work of very good people that moved our knowledge to the present point and that is cited in the papers I presented here. It is my view that, whatever would be any other contribution to this research area, the acquired scenario is the one I described above as strongly emerged from lattice computations. My hope is that this will become part of our knowledge in a reasonable time.

Alexander Dynin (2009). Energy-mass spectrum of Yang-Mills bosons is infinite and discrete arXiv arXiv: 0903.4727v4

Marco Frasca (2007). Infrared Gluon and Ghost Propagators Phys.Lett.B670:73-77,2008 arXiv: 0709.2042v6

I. L. Bogolubsky, E. -M. Ilgenfritz, M. Müller-Preussker, & A. Sternbeck (2009). Lattice gluodynamics computation of Landau-gauge Green’s functions in the deep infrared Phys.Lett.B676:69-73,2009 arXiv: 0901.0736v3

Ph. Boucaud, J. P. Leroy, A. Le Yaouanc, J. Micheli, O. Péne, & J. Rodríguez-Quintero (2011). The Infrared Behaviour of the Pure Yang-Mills Green Functions arXiv arXiv: 1109.1936v1

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