There are four (apart from rearrangements and minor changes) nearly identical papers claiming in different degrees to have solved the 7th Clay Millennium Problem.

http://arxiv.org/abs/1308.6571

Mass gap in quantum energy-mass spectrum of relativistic Yang-Mills fields

Alexander Dynin

http://arxiv.org/abs/1205.3187

Quantum energy-mass spectra of relativistic Yang-Mills fields in a functional paradigm

Alexander Dynin

http://arxiv.org/abs/1110.4682

Quantum energy-mass spectrum of Yang-Mills bosons

Alexander Dynin

http://arxiv.org/abs/0903.4727

Energy-mass spectrum of Yang-Mills bosons is infinite and discrete

Alexander Dynin

The most recent paper (version v3 of the above listed first) claims at

the beginning of Section 1:

``A mathematically rigorous solution is given for both parts of the 7th Millennium problem of Clay Mathematics Institute''

**This claim is wrong. Neither are the explicit requirements of the problem definition satisfied (not even a discussion of Poincare invariance and causality), nor is the paper mathematically rigorous in a crucial part of the construction (it is not proved that there is an operator with the anti-normal symbol specified in the construction).**

In the following, I concentrate on the last (oldest) of these papers (more specifically its version arXiv:0903.4727v4) since I read this in most detail. However, it is easily seen that everything also applies to the more recent versions.

The paper starts with citing the short version of the formulation of the 7th Clay Millennium problem:

``Prove that for any compact (semi-)simple global gauge group,

a nontrivial quantum Yang-Mills theory exists on $R^{1+3}$ and has

a positive mass gap. Existence includes establishing axiomatic

properties at least as strong as the Wightman axioms of the axiomatic

quantum field theory. (Slightly edited)''

and goes on dismissing the second sentence of the requirement as impossible to solve, by reference to a 1993 book by Strocchi. The content of that book is widely known in the algebraic QFT community and was not overlooked when the authors of the millennium problem formulated it in 2000, as witnessed by their specific requirements. There seems to be be a misunderstanding on Dynin's part. Strocchi only argues that **gauge fields** cannot satisfy the Wightman axioms and would require an indefinite metric. However, Strocchi's arguments** do not apply** to local **observable** fields inside a quantum YM theory, namely to gauge invariant fields formed from gauge field strength (such as those mentioned in the citation below), are expected to satisfy the unaltered Wightman axioms in any valid construction of the vacuum representation.

The official statement in http://www.claymath.org/sites/default/files/yangmills.pdf says on p.5 explicitly:

A quantum field, or local quantum field operator, is an operator-valued generalized function on spacetime obeying certain axioms. The quantum fields act in a Hilbert space H that furnishes a positive energy representation of the Poincare group [...]

At any rate, for purposes of the CMI Millenium Problem, an existence proof for a quantum field theory must establish axioms at least as strong as those cited in [36, 29]. [...]

To establish existence of four-dimensional quantum gauge theory with gauge group $G$, one should deﬁne (in the sense of the last paragraph) a quantum ﬁeld theory with local quantum ﬁeld operators in correspondence with the gauge-invariant local polynomials in the curvature F and its covariant derivatives, such as $Tr F_{ij}F_{kl}(x)$. [...]

Correlation functions of the quantum field operators should agree at short distances with the predictions of asymptotic freedom and perturbative renormalization theory, as described in textbooks. Those predictions include among other things the existence of a stress tensor and an operator product expansion, having perscribed local singularities predicted by asymptotic freedom.

This leaves as minimal solution requirements (apart from the mass gap):

- a positive energy representation of the Poincare group,
- local quantum ﬁeld operators in correspondence with $Tr F_{ij}F_{kl}(x)$,
- correlation functions that agree at short distances with the predictions of perturbative renormalization theory,
- the existence of a stress tensor and an operator product expansion, with the local singularities predicted by asymptotic freedom.

Dynin's papers do nothing towards a verification of any of these properties.

Now let us look at the content of paper arXiv:0903.4727v4. First two important claims made but nowhere substantiated by even a rudimentary discussion:

``The spectrum is both Poincare and gauge invariant.''

(last sentence of abstract)

This statement is nowhere discussed. One would need to quantize all generators of the Poincare group and the gauge group, not only the Hamiltonian, and show that the commutation rules survive intact.

The same unresolved difficulties as for the Hamiltonian quantization (discussed below) can be expected.

``As shown in the present paper the constrained initial data form an

infinite dimensional Kähler manifold'' (4th paragraph of Section 4.3)

To show this, the Kähler potential would have to be constructed, or a compatible symplectic and metric structure. No discussion of this appears in the paper. Since this only concerns the classical part, it can be fixed; all information needed is already there, just not spelled out.

I mention this as it gives valuable insight into Dynin's approach. Essentially, Dynin attempts to perform the geometric quantization of the infinite dimensional Kähler manifold mentioned.

Geometric quantization is the program of quantizing a classical theory given on a Kähler manifold. This works fully automatically and with full rigor for finite-dimensional manifolds, giving 1+0 dimensional quantum field theory (aka quantum mechanics). There it can be done neatly in a coherent state formalism called Berezin quantization. But its extension to infinite dimensions is at present more an art than a science. Even for field theories in 1+1 dimensions, it can at present be done rigorously only in nice, completely integrable cases.

The relevant manifold for the quantization of a classical field theoryis the manifold of solutions of the classical field equations, which is a symplectic space with the so-called Peierls bracket. For a classical theory with a well-posed initial-value problem, the space of solutions is parameterized by the space of initial conditions. In the case of a classical theory with constraints (gauge theories and general relativity), the initial conditions must satisfy a consistency condition preserved by the dynamics. Thus working with the manifold of solutions is equivalent to working with the nonlinear manifold of consistent initial conditions. This is the setting actually used by Dynin, and this explains why he discusses the unique solvablility of the Yang-Mills initial-value problem. (In fact, he makes nowhere use of these results; the quantum part is completely independent of the classical existence results.)

To proceed in analogy with the finite-dimensional case one needs to equip the symplectic manifold with a Kähler structure and a Liouville measure, construct the corresponding coherent states, and express the operators to be quantized (in this case the generators of the Poincare group and of the gauge group), by defining consistently how they operate on the coherent states. Then one must check their commutation relations and resolve any anomalies that might be encountered in this process.

Dynin skips the Kähler structure and goes directly to the coherent states (from which the former can probably be obtained). The coherent states are constructed in the terminology of white noise calculus and Hida distributions, unfamiliar to physicists but equivalent to the standard second quantization formalism. (Comparing the action of creation and annihilation operators on Glauber coherent states with the formulas (40)-(44), one sees that $\hat\xi$ is an annihilation and $\hat\xi^\dagger$ a creation operator; the normal (antinormal) symbol is in a physical coherent state context the lower (upper) symbol or Q-(P-) representation. Making mentally the corresponding changes makes the paper much more readable.)

Dynin's notation is further obscured by adhering to a rarely used form of the Einstein summation convention, applying it to drop from the notation every functional integration over fields appearing twice. This makes the definition of the quantization operations in (45) look harmless, whereas in fact they involve a functional integration over the field $\zeta$.

This is the source of a gap in the later construction of the quantization of the Hamiltonian. Dynin quotes theorems stating the existence of the various symbols for every continuous operator from the bottom to the top of the Gelfand triple defining the Fock space of interest. However, this direction (from the operator to the symbol) is dequantization, while later he utilizes the opposite direction (from the symbol to the operator) for quantization. This direction is used in (64) unsupported by quoted theorems and in fact unsupported by any discussion.

As a consequence, it is unproved that an operator exists whose antinormal symbol is (64), only that if it exists it is unique. Since the operator in question is the putative quantized Hamiltonian, its existence is unproved.

**This is the essential gap in the construction.**

Note that the lack of rigor doesn't come from using a path integral, which is indeed a well-defined concept in stochastic calculus, but that he doesn't check whether the path integral he needs is actually convergent.

In fact, I think the gap cannot be cured without importing further ideas.

For by the theorems quoted, each good operator has all three symbols. Thus it should be possible to reorder the Hamiltonian with antinormal symbol (64) to normal form. But it seems to me that this produces infinite coefficients, which would show that the construction is faulty, i.e., the Hamiltonian would not exist. This would ruin the quantization program; and indeed this is the point where all rigorous techniques stumbled so far. Additional nonperturbative renormalization techniques would be needed to fix the gap.

However, if the Hamiltonian could be proved to exist, the remaining discussion appears to be valid. In particular, the discussion surrounding (75)-(76) would be consistent with the existence of a single massive particle and a dynamics that preserves the number of such particles and leads to scattering only within each sector.

Finally, note that there is also a published paper on massless QCD, which shares the virtues and flaws of the papers discussed here:

Alexander Dynin,

Quantum Yang-Mills-Weyl Dynamics in Schroedinger paradigm,

Russian Journal of Mathematical Physics 21 (2014),No.2,169-188.

http://arxiv.org/abs/1005.3779