I think that is time to make a point about the question of mass gap existence in the Yang-Mills theory. There are three lines of research in this area: Theoretical, numerical and experimental. I can suppose that the one that mostly interests my readers is the theoretical one. I would like to remember that, in order to get a Millenium Prize, one also needs to prove the existence of the theory. This makes the problem far from being trivial.

As for today, **the question of existence of the mass gap both for scalar field theories and Yang-Mills theory should be considered settled**. Currently there are two papers of mine, here and here both published in archival journals, proving the existence of the mass gap and give it in a closed analytical form. A proof has been also given by Alexander Dynin at Ohio State University here. Alexander does not give the mass gap in a closed form but gets a lower bound that permits him to conclude that Yang-Mills theory has a discrete spectrum with a mass gap. This is enough to declare this part of the problem solved. It is interesting to note that, differently from Poincaré conjecture, this solution does not require a mathematics that is too much complex. This can be understood from the fact that the corresponding classical equations of the theory already admit massive solutions of free particle. The quantum theory can be built on these solutions and all this boils down to a trivial fixed point in the infrared for the quantum theory. Such a trivial fixed point, that explains also the lower bound Alexander is able to find, is a good news: We have a set of asymptotic states at diminishing momenta that can be used to do perturbation theory and do computations for physics! The reason why these relevant mathematical results did not get the proper exposition so far escape me and enters into the realm of things that I do not know. It is true that in this area there is a lot of caution and this can be understood as this problem received a lot of attention after Witten and Jaffe proposed it for a big money prize.

But, as I have already said, this problem has two questions to be answered and while computing the mass gap is quite easy, the other question is rather involved. To prove the existence of a quantum field theory is not a trivial matter and, for sure, we know that the Wiener integral exists and the Feynman integral does not (so far and only for mathematicians). What I prove in my papers is that the Euclidean theory exists for the scalar field theory (thanks to Glimm and Jaffe that already proved this) and that this theory matches the Yang-Mills theory in the limit of the gauge coupling going to infinity. It should be an asymptotic existence… Alexander by his side proves existence in a different way but here unfortunately I cannot say too much but I would appreciate that Alexander would write down some lines here about his work.

Other theoretical attempts are based on some educated guess as a starting point as could be the vacuum functional, the beta function or other parts of the theory that, for a full proof, should be derived instead. These attempts give a strong support to my work and that of Alexander. In these papers you will see a discrete spectrum and this is the one of a harmonic oscillator or simply the very existence of the mass gap itself. But, for physicists, the spectrum is the relevant conclusion as from it we can get the masses of physical states to be seen in accelerator facilities. This is the reason why I do not worry too much for mathematicians fussing about my papers.

Finally, I would like to spend a few words about numerical and experimental results. Experiments show clearly always bound states of quarks and gluons that are never seen as free. This is the better proof so far Nature gave us of the existence of the mass gap. Numerically, people computed both Green functions and the spectrum of the theory. I am convinced that these lines should merge. The spectrum on the lattice, both quenched and unquenched, displays the mass gap. Green functions, when one considers just the decoupling solution, are Yukawa-like, both on the lattice and from Dyson-Schwinger equations, and this again is a proof of existence of the mass gap.

I hope I have not forgotten anyone. Please, let me know. If you need explicit references here and there I will be pleased to post here. A lot of people is involved in this kind of research and I am happy to acknowledge the good work.

Finally, I would like to remember that one cannot be skeptical about mathematics as mathematics can only be either right or wrong. No other way.

“Finally, I would like to remember that one cannot be skeptical about mathematics as mathematics can only be either right or wrong. No other way.”

Actually, mathematical truth is spacetime dependent (think about Euclid and Lobachevskii). Naturally, it depends on a mathematical context. The Clay Institute YM-problem partially addresses this issue. The famous Glimm-Jaffe constructive QFT methods have not succeeded beyond scalar quartic 2+1 models.

Happily for Quantum Mechanics, it appeared shortly after Hilbert work on infinite dimensional quadratic forms and followed by the definitive paper and book by von Neumann to providing QM with a mathematical context. In particular spectral theory of unbounded selfadjoint operators on Hilbert space.

Regretfully, Quantum Field Theory has been less fortunate. According to Dirac,

“The interactions that are physically important in quantum eld

theory are so violent that they will knock any Schrodinger state

vector out of Hilbert space in the shortest possible time interval.”

Thus von Neumann theory is not applicable, and one needs another definitition

for a QFT energy spectrum. My theory ventures from a Hilbert space (actually the first quantization) to a second quantization in nuclear Gelfand triples. In this I follow the seminal work of P.Kree in conjunction with White Noise Calculus and Berezin anti-normal quantization (all originated during 1970’s). My novelty is a variational spectrum of linear operators in Gelfand triples. Note that a second quantization produces linear operators from classical nonlinear Hamiltonian functionals.

This ignitible combination allowed me to provide a rigorous estimate from below to the variatonal YM spectrum by the von Neumann spectrum of the shifted number operator, the second quantization of the classical harmonic oscillator. Numerically the latter coincides with the spectrum of quantum harmonic oscillator but has infinite degeneracy of excited energy levels.

I still try to understand what kind yours YM energy spectrum is. Certainly,

it should be asymptotic (along with perturbation theory) but, due to self-interaction, not additive as is the case of harmonic oscillator.

Dear Alexander,

Thank you for your intervention. This was the one I expected for. Of course, if you would like to be more technical about your work and posting something more elaborated, you are welcome.

I agree with you that the spectrum is not additive. As far as I can tell, from the classical solutions I am not able yet to move from a single free particle to many particle case due exactly to the nonlinearity. For QFT this does not matter too much as I work in a different way. Besides, I was able to go to higher orders and find the NLO correction to the propagator (see here). I think this should help understanding.

Regards,

Marco

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