# Four Dimensional Quantum Yang-Mills Theory and Mass Gap I: Quantization of the Solution of the Classical Equation

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Referee this paper: arXiv:1406.4177

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by Simone Farinelli

Abstract: A quantization procedure for the Yang-Mills equations for the Minkowski space $R^{1,3}$ is carried out in such a way that field maps satisfying Wightman's axioms of Constructive Quantum Field Theory can be obtained. Moreover, the spectrum of the corresponding QCD Hamilton operator is proven to be positive and bounded away from zero except for the case of the vacuum state which has vanishing energy level. The particles corresponding to all solution fields are bosons. As expected from QED, if the coupling constant converges to zero, then so does the mass gap.

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paper authored Jun 16, 2014
edited Mar 17, 2019

## 2 Reviews

+ 3 like - 0 dislike

Given the title, this paper should be about the mass gap problem in 4 dimensional quantum Yang-Mills theory. In fact, it is difficult to find a meaning to the "main result" which should be the Theorem 1.1. This statement is meaningless and simply shows a misundertsanding of what a quantum field theory is. The paper claims that there is a quantum Yang-Mills theory for each solution of the classical Yang-Mills equations, which is clearly a non-sense. The goal of the paper seems to prove that there is a mass gap for each of these "theories". Without reason, the paper considers what is claimed to be some solutions of classical Yang-Mills equations, the "Jormakka's solution" and tries to prove that there is a mass gap for the corresponding "theory". The proof of the Corollary 1.2, supposed to deduce the case of a general solution from the case of the Jormakka's solution, makes no sense, in particular does not use the definition of the "Jormakka's solution".

After some elementary reminders on classical Yang-Mills theory, the paper recalls what the "Jormakka's solution" is. It is supposed to be a solution of classical Yang-Mills equations. The Jormakka's paper claims to prove the absence of mass gap by constructing some "solutions" of the classical Yang-Mills equations of arbitrarly small positive energy. This "proof" has obviously nothing to do with quantum Yang-Mills theory and is just a (wrong, see below) proof of the trivial fact that classical Yang-Mills theory has no mass-gap. The paper under review disagrees with the Jormakka's conclusion but still used its classical "solutions". These solutions are highly suspect. Choosing very simple parameters, one can make the corresponding connection with values in U(1), i.e.  reduce to the case of Maxwell equations. The form of the proposed solution (something like a gaussian) has nothing to do with a possible solution of Maxwell equation (which is easy to check by a direct computation). The "Jormakka's solution" is so wrong , which is a rather bad starting point.

Then the paper devotes a large section to something apparently unrelated, the spectrum of Dirac operators. This is crazy because the theory under consideration is supposed to be pure Yang-Mills theory, in particular without fermions. Nevertheless, after elementary reminders, the paper proves a bound on spectrum of Dirac operators, in a way not unreasonable (a globally defined operator is controlled in terms of locally defined operators with boundary conditions) and probably true  (I did not check it completely).

The final section is about a combination of the two preceding one. Given a "Jormakka's solution", the paper wants to construct a quantum field theory satisfying the Wightman axioms. In particular, it wants a Hilbert space and it chooses, without explanation, the space of sections of a Dirac bundle. Here, some details are unclear on which Dirac bundle is chosen: as the theory is pure Yang-Mills, there is of course no matter representation and no associated vector bundle. Nevertheless, the paper considers "the representation of G" (maybe that the author thinks that a Lie group has a unique representation?). Anyway, to choose a space of sections of a Dirac bundle as Hilbert space and a (squared) Dirac operator as Hamiltonian has nothing to do with pure Yang-Mills theory.

The most meaningful interpretation of the constructed theory is that of a fermionic field in a classical gauge background, which is a free theory indeed controlled by the spectrum of the Dirac operator. Such construction are relevant if one wants to study theories with quarks and if one can prove bounds on the spectrum of the Dirac operator which are independent of the gauge field background, in order to have a result which survives a path integral over the gauge fields (this is the way that Vafa and Witten proved the absence of mass gap in quantum Yang-Mills theory with massless quarks).

But the interpretation of the paper is different: it really considers the Hilbert space as the space of gluons whereas there is no gluon in the theory because the gauge field is taken fixed to a classical solution. As the Hilbert space is a space of spinors, the paper logically concludes that gluons are fermions whereas there should be bosons. Is is argued that it just means that they are unphysical, as Faddeev-Popov ghosts. Manifestly, the author does not understand why the Faddeev-Popov ghosts can be considered as unphysical (because they decouple in a non-trivial way from the physical states) and has no problem with the fact that all the states of its theory are unphysical. The bound on spectrum of Dirac operator is used to "show" that this "theory" as a mass gap.

Conclusion: the section on Dirac operators has maybe a mathematical interest but all the physical considerations are just wrong. In particular, this paper has nothing to do with quantum Yang-Mills theory because the gauge field is kept fixed to some value (which is wrongly claimed to be a classical solution) whereas quantum theory is about integration over all the gauge field configurations. The paper is probably more about a free fermion theory in a classical gauge field background and about a totally wrong interpretation of this, rather trivial, theory.

reviewed Aug 9, 2014 by (5,120 points)
edited Aug 9, 2014 by 40227

To the last sentence. There is no classical gauge field background. It is just a free fermion theory with a scaled field operator.

+ 1 like - 0 dislike

The paper defines a field theory satisfying the Wightman axioms and claims wrongly that it is a proof of the conjecture that any quantum Yang-Mills theory has a mass gap.

The field construction is on p.52 under the heading ''Field maps''. It is obvious that this field lies in the Borchers class of the free spinor field and hence describes a collection of noninteracting free spin 1/2 particles. Thus Wightman's axioms are trivially satisfied.

The connection to Yang-Mills theory is spurious, simply consisting in using a particular classical solution of the Yang-Mills equations to scale the free spinor field.

In order to qualify as a quantization of the classical Yang-Mills field theory one would have to verify that in the limit $\hbar\to 0$, the field degenerates to a theory satisfying the classical Yang-Mills equation and their commutators (scaled by $\hbar^{-1}$) reduce to the corresponding Poisson bracket. This is not the case.

reviewed Aug 10, 2014 by (15,428 points)

@ ArnoldNeumaier

@40227

Dear all,

I am the author of this paper and I recognize all the criticism here above. What has been proved in this version is another result, but for sure not the Yang-Mills Mass Gap in 3+1 dimensions.

Meanwhile, after having realized this, I have worked on the true problem, and I believe I can provide a solution.

You can find it under arXiv:1406.4177

I would be very interested in your feedback.

Kind regards

Simone Farinelli

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