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

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Referee this paper: arXiv:0903.4727 by Alexander Dynin

Please use comments to point to previous work in this direction, and reviews to referee the accuracy of the paper. Feel free to edit this submission to summarise the paper (just click on edit, your summary will then appear under the horizontal line)

paper authored Mar 27, 2009

I polished my earlier review to take account of the discussions and offline correspondence with the author (who still thinks he solved the problem).

## 2 Reviews

+ 7 like - 0 dislike

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

reviewed Aug 9, 2014 by (15,468 points)
edited Aug 17, 2014

This review is completely wrong. There is no "gap" or "missing lemma" or "nonrigorous ambiguity" in this paper, it is simply garbage.

You are WRONG about the formalism of white noise calculus and Hida distributions! First, it is familiar to physicists, in the cases where they are used in this paper, they are equivalent to creation and annihilation operators, i.e. "second quantization of free field theory". They are NOT equivalent to second quantization in general, just to creation and annihilation operators of free field theory. They don't serve to define the path integral, as they do no renormalization.

A proper formalism to do this is provided by Harer's "regularity structures", which actually takes the limit of white noise stochastic stuff to the continuum doing an appropriate renormalization along the way.

Further, in this paper, the white noise calculus is used to define free-field style creation and annihilation operators on objects which are classical field theory solutions! It has absolutely no relation to second quantizing Yang-Mills theory.

The paper does not have small flaws, it has an ENORMOUS flaw that the formalism it uses does not bear any relation to the quantum theory which it is attempting to describe.

While you are right that there is an implicit path integral in the Einstein summation convention in the paper, you are wrong that this renders the paper non-rigorous. The implicit path integral in this case is interpreted in the stochastic formalism, and so the results that are obtained are not problematic, they are results which are true for quadratic path integrals. It's just that the author lifted these results by defining creation and annihilation operators associated to classical solutions, and declaring this to be a quantum field theory of Yang-Mills type!

It's really what I said in my answer, and really, there is nothing non-rigorous in the stuff he is doing--- he is using standard theorems from stochastic mathematics, just applying them to a new model which is essentially free particles indexed by solutions to classical equations, rather than anything to do with quantum field theory.

Further, the program of symplectic quantization is deeply flawed in the case of field theory, and there is no theorem, or even heuristic, that suggests that any field theory with a nontrivial running can ever be described by symplectic quantization. The reason is simply that the classical equations at short distances bear no relations to the long-distance properties of the typical configurations in the path integral, which in the case of QCD randomize everything.

If you would like to show that symplectic quantization works, I would suggest to first do the same thing in 2+1 dimension $\phi^4$ theory, where we have a rigorous Glimm-Jaffe description, recently reproduced by Harer. This arena allows you to see whether the quantum $\phi^4$ theory (much simpler than QCD) can be described in any way from the solutions to classical $\phi^4$ theory. The answer is obviously "no", since the coupling runs in the quantum theory, and doesn't run classically, but I'll let you figure that out yourself.

The stuff in this paper is nothing remotely resembling any sensible description of QCD, it is a crazy paper.

In mathematical physics (and this is supposed to be a paper in this area), second quantization always means "second quantization of free field theory"; no matter what more general usage there may be elsewhere.

In nonrelativistic quantum field theory, where no renormalization is needed, it is clear that the Hamiltonian dynamics happens in the Fock space obtained by second quantization of a system of infinitely many harmonic oscillators. The dynamics has nothing to do with this Fock space - all field theories with the same free field content have the same Fock space as Hilbert space. The dynamics depends solely on the Hamiltonian defined on it.

All (relativistic or nonrelativistic) Hamiltonian approaches to quantum field theory start with such a Fock space, though often on a lattice befoire taking the continuum limit.

Since Dynin is taking a Hamiltonian approach, it is natural that he starts with a Fock space, too. His Fock space (27) is the Fock space whose configuration space is the space of all (nice) functions on the classical space (23) of constrained Cauchy data. This space has no intrinsic dynamics, as the Cauchy data are given at $t=0$ only.

Thus your claim that he employs creation and annihilation operators on solutions is as unfounded as claiming that the Fock space of a single harmonic oscillator mode - in which an anharmonic oscillator is studied -  acts on solutions of the classical anharmonic oscillator.

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.

The lack of rigor is thus like that of integrating $e^x$ over the real line, based on the background knowledge that the Lebesgue integral has a meaningful definition. It has for integrable functions, but not for this particular case.

Had Dynin shown that the path integral exists in the case he needs it, the paper would have been rigorous, but he hasn't shown it. And I believe he cannot show it.

''the classical equations at short distances bear no relations to the long-distance properties of the typical configurations in the path integral, which in the case of QCD randomize everything.''

This is irrelevant for Dynin's approach as he makes no use at all of classical solutions.  If one removes the discussion in Section 2 about classical solutions, his remaining arguments remain completely unaffected.

Thanks Arnold, This review seems to fit the bill properly. I would like to hear from the author. I wrote to him with no avail. I think that could be satisfactory also as an answer for MO.

The formalism he is using does not define anything like a gauge theory, this is not an arguable point. The raising and lowering operators he is using are defining "states" indexed by integers which are nothing like the states of gauge theory anything.

These integer indexed states are associated to solutions of the classical theory, but they have no relation to actual quanta of anything! This is the way he proves the "mass gap", by bounding the "energy" by "N" the excitation number.

The (nonsensical) procedure is exactly as I said, and upon further reading you will realize this is so.

By the same argument it would follow that, for any field theory, the states in canonical QFT via second quantization in Fock space defines integer-indexed states that have no relation to actual quanta of anything. In canonical quantization they are the bare building blocks of the renormalized theory.

Dynin's renormaization recipe is inadequate, but his state space is as good as that of any of the 4D state spaces considered in the literature.

That's right! The free-field quanta have no real relation to the actual states of the theory, this is the essence of that theorem in the 1960s that the free-field Hilbert space and the interacting field Hilbert space are not related unitarily.

The description starting from fock-space is only roughly ok for getting the basic picture in weakly interacting field theory. In this case, even if he were to do the quantization right, he would be getting gluon fock states, with an N=0 vacuum consisting of no gluons, around the solution gauge field=0.

This would not have any relation to the actual ground state, which is a randomly twisting gauge field every which way, and consists of N=infinity gluons even in the vacuum. The only place a fock-description of gluons is roughly appropriate is in a short-distance bag, at long distances there are lots of 'wee gluons', infinitely many particles.

The N=0 vacuum is nonsense in this case. But the paper makes worse nonsense. In this paper, the quantized states are not really fock states at all, they are free-field raising lowering operator fock states with no relation to the actual theory one is trying to quantize, they are just a tower of states associated to classical solutions. They are total nonsense, they don't even define the wrong ass perturbative vaccuum for gauge theory, they are nothing at all.

Here is the answer by Alexander Dynin that I post on his behalf:

The Schroedinger paradigm in QFT is a quantization of functionals on the initial data. Under certain conditions the latter parametrize the solutions of classical YM equations but not the latter are quantized. Functionals may be non-linear but the Shroedinger operator is linear of course. No disentanglement is required.

Certainly, my procedure  is not relativistic but the energy-mass component of the classical relativistic  energy-momentum vector is not either. That is very important. In particular,  Poincare generators have no role here even when an  action functional is relativistic. The  results are the same qualitatively in all  Lorentz coordinates.

The energy-mass functional is a tame polynomial in infinite dimensions. As such it is a symbol of a tame operator in the  Gelfand-Kree triple and defines an unbounded self-adjoint operator in the corresponding Fock space. As such  the quantum Yang-Mills operator does exist.  My latest main theorem about the structure of its spectrum achieves  much more  than a solution of the YM mass gap problem.

Alexander

Hi @Polarkernel, do you have an idea why neither I nor JonLester who posted it can edit this comment? The formatting should be improved ...

Here is the answer by Alexander Dynin that I post on his behalf:

The Schroedinger paradigm in QFT is a quantization of functionals on the initial data. Under certain conditions the latter parametrize the solutions of classical YM equations but not the latter are quantized. Functionals may be non-linear but the Shroedinger operator is linear of course. No disentanglement is required.

Certainly, my procedure  is not relativistic but the energy-mass component of the classical relativistic  energy-momentum vector is not either. That is very important. In particular,  Poincare generators have no role here even when an  action functional is relativistic. The  results are the same qualitatively in all  Lorentz coordinates.

The energy-mass functional is a tame polynomial in infinite dimensions. As such it is a symbol of a tame operator in the  Gelfand-Kree triple and defines an unbounded self-adjoint operator in the corresponding Fock space. As such  the quantum Yang-Mills operator does exist.  My latest main theorem about the structure of its spectrum achieves  much more  than a solution of the YM mass gap problem.

Alexander

@Polarkernel I try reposting with identical results. The problem can be traced back to Thunderbird where there is no carriage return to end a line neither line feed. So, the line is never formatted. But it is not clear why there is no possibility of editing.

@JonLester: Please copy the message from Thunderbird into an ASCII editor and add there the line feeds before copying it into the comment window.

@JonLester Sorry, but I have to investigate what happens here. To do this, I have to download a backup of the database and load it into my local site. This will take a while. I hope you can solve it using the proposition of Arnold Neumaier.

@Polarkernel @Arnold Thanks, I'll do so. @ArnoldNeumaier meanwhile you can check MO where I posted as well and worked fine.

Here is the answer by Alexander Dynin that I post on his behalf:

The Schroedinger paradigm in QFT is a quantization of functionals on the initial data. Under certain conditions the latter parametrize the solutions of classical YM equations but not the latter are quantized. Functionals may be non-linear but the Shroedinger
operator is linear of course. No disentanglement is required.

Certainly, my procedure  is not relativistic but the energy-mass component of the classical relativistic  energy-momentum vector is not either. That is very important. In particular,  Poincare generators have no role here even when an  action functional
is relativistic. The  results are the same qualitatively in all  Lorentz coordinates.

The energy-mass functional is a tame polynomial in infinite dimensions. As such it is a symbol of a tame operator in the  Gelfand-Kree triple and defines an unbounded
self-adjoint operator in the corresponding Fock space.
As such  the quantum Yang-Mills operator does exist.
My latest main theorem about the structure of its spectrum achieves  much more  than a solution of the YM mass gap problem.

Arnold

While Dynin may think that Poincare invariance is irrelevant, it is one of the conditions of what it means to have solved the millennium problem. That he chooses not to address the issue means that he chooses his work not to be eligible as a solution of the millennium problem. Moreover, it is unacceptable that he claims under these circumstances in the abstract that the spectrum is Poincare-invariant without devoting later a word to it.

I also don't believe his statement that an operator with the given anti-normal symbol exists. At least the results he actually cites in his paper (the oldest or the newest) do not prove it (whereas some results that he cites are not even used). The least Dynin should do is to point precisely to the results from the literature needed to prove it, so that one can check his claim. He should also explicitly evaluate the corresponding normal symbol to prove me wrong, since I think it is infinite.

Dear @ArnoldNeumaier, I post these answers by Dynin for your consideration. Alexander knows you as a mathematician and so he addresses this to you:

I refere to my latest version 1009.3779v4 [math-ph] which is a  reedited my published paper to make exact reference to YM millennium problem.

1. The Clay official statement includes establishing axiomatic properties at least as strong as Wightman axioms of quantum field theory,  not  the Wightman axioms per se. In his own statement cited in the preprint, E. Wittendoes not mention them at all.  There is nothing about Poincare invariance in my abstract. Even in general relativity the non-relativistic initial values problem is a very important tool for existence of solutions of general Einstein equations. In my case the  dependence of the YM spectra does not depend on Lorentz coordinates.
2. I designate the energy-mass functional  of constrained initial data as the the anti-normal symbol of quantum Yang-Mills operator. On transverse  YM fields it is explicitly simplified  to the formula (69). The latter is a continuous polynomial functional on the transverse test space $\mathcal{H}^\infty$ so that the quantum YM operator with such anti-normal symbol is well defined as a continuous non-negative operator in  $\mathcal{K}^\infty$ (because it is a continual linear combination of the pointwise operators of creation and annihilation that have this property). As such the YM operator is essentially self-adjoint and I use its Friedrichs self-adjoint extension with a well defined spectrum.
3. My text is self-contained modulo well established mathematics. In the preprint I provide  precise references.

Regards,

Alxander

1. The 2009 paper (which is the one that I refereed) contains as last sentence of the abstract the claim about Poincare invariance.

''at least as strong as Wightman axioms'' means to a mathematician that the latter must be deducible. These include Poincare invariance and causality.

Even if the requirements were interpreted in a more generous sense I don't see any verification of axiomatic properties of any kind, with a discussion of why these should be regarded as being ''at least as strong as Wightman axioms''.

2. (63) in the 2009 paper defines a functional supposedly defining an operator that has this functional as its antinormal symbol. The reason why this operator exist cannot be something being ''a continual linear combination of the pointwise operators of creation and annihilation'' - using the properties of the operator to be constructed as proof for its existence is unsound circular reasoning.

3. If the paper is self-contained: Where are the definition of tameness and its properties? Where is a sufficiently strong condition on a functional that guarantees that it defines an operator with this functional as antinormal symbol? (Does the tameness of a functional imply existence of this operator?) Where is the reference to a proof of this statement? And where is the verification that (63) satisfies this criterion? None of the three is present.

4. The 2010 paper 1005.3779v4 [the number in the Dynin's reply has a misprint] is irrelevant in the present context as it is not about YM quantization but about massless QCD. But its equation (69) is essentially the same as equation (63) in the 2009 paper commented on above. The existence of ${\mathbb Y}:=\widehat Y_\alpha$ at the bottom of page 19 is assumed without sufficient justification.

''The Clay official statement includes establishing axiomatic properties at least as strong as Wightman axioms of quantum field theory,  not  the Wightman axioms per se. In his own statement cited in the preprint, E. Wittendoes not mention them at all.''

The official statement in http://www.math.vt.edu/people/gao/physics/yan-mills/yang_mills.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 ... 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.

Thus without the verification of these properties Dynin might have constructed something, but has definitely not solved the challenge of the Millennium problem.

+ 3 like - 2 dislike

This paper gives a false construction, it proves theorems about a quantum system of no interest, whose kinematics is defined by a tower of harmonic-oscillator excitations labelled by classical solutions of Yang-Mills theory. These excitations have nonlinear interactions with themselves, but not with each other. Nothing about this construction has anything to do with quantum Yang-Mills theory, or with any field theory of any kind.

To explain what is going on, consider a free scalar field, with a mode-wise self-interaction between each mode and itself, as in the following quartic interaction:

$$H_0 = \int_k a^\dagger_k a_k dk$$

$$\phi_k = a^\dagger_k + a_k$$

$$H_i = \int_k \phi_k \phi_k \phi_k\phi_k dk$$

With appropriately relativistically normalized creation and annihilation operators, and appropriate relativistic measure. Notice that the spectrum of each mode is still discrete, you can prove this rigorously very easily, and also notice that each mode is still free.

For quantum fields, the interaction Hamiltonian links different modes:

$$H_i = \int \phi_{k_1} \phi_{k_2}\phi_{k_3}\phi_{k_4} \delta(k_1+k_2+k_3+k_4) dk_1 dk_2 dk_3 dk_4$$

With appropriately normalized delta function. This is what leads to scattering, and interactions.

The formalism Dynin uses is that of free-field creation and annihilation operators, and in his paper, these operators are indexed by the space of classical solutions, so they act to produce a free-field tower of excitations indexed by the occupation number integer N for each classical solution to the Yang-Mills field equations. Dynin then uses these towers to provide the kinematics of his purported construction of quantum Yang-Mills.

Such a construction, starting with a tower of occupation numbers on each classical solution, is a-priori impossible on general grounds. It is will known that the state space of an ordinary relativistic interacting theory has no unitary relation to the free Fock space of a free field theory, there are, if you like an intuitive picture, infinitely many virtual particle in the vacuum simply by translation invariance, so that even a finite density of virtual particles is an infinite number of particles overall, in infinite volume. This is the classical Haag theorem, which is very easy to prove.

The construction in this paper violates the Haag theorem, because it has a kinematic space which doesn't change depending on the coupling constant, the vacuum is the same in the interacting and free versions, and so it is manifestly incorrect.

But the construction is even more problematic than this, in that the creation and annihilation operators are indexed by the space of initial conditions, so that there is a separate operator for each initial condition. This is not even the correct kinematics for free field theory, where the creation and annihilation operators are indexed by modes of the initial conditions--- there is one photon creation/annihilation operator for each mode of a box, not for each initial condition. The photons corresponding to an initial condition are mixtures of the photons corresponding to the mode decomposition of this initial condition. The kinematic space described here is even more incorrect than just being identical in the free and interacting cases, it is introducing too many creation and annihilation operators.

To understand just how different the free and interacting Fock spaces are in Yang-Mills, is worthwhile to review the expected behavior of quantum Yang-Mills theory from the numerical simulations of lattice Yang-Mills, and the strong-coupling behavior discovered by Wilson in his 1974 paper introducing Lattice Gauge Theory. When you place a coarse lattice over the continuum Euclidean lattice theory and consider the holonomy along the links of the lattice as your variables (standard lattice gauge theory), assuming the scale of the lattice is much greater than the confinement radius, the gauge field becomes entirely random! The correlations of the link-matrices converge to zero. The convergence is exponentially fast at scales larger than the confinement radius $\Lambda^{-1}$.

The approach of the long-range lattice field to a random field means that any naive description of the vacuum in a picture starting from the kinematics of a free field theory necessarily fails in a way much worse than in traditional QED. A random-gauge vacuum contains infinitely many soft gluons, the notion of a single gluon no longer exists. The Haag theorem is not particular to gauge theory, but in the case of gauge theory, the nontrivial vacuum is especially striking, because the long-range statistically random Euclidean gauge field has no relation to the classical Euclidean vacuum of everywhere zero gauge field, up to gauge transformations.

Putting aside general considerations, one can examine the specific arguments in the paper. The sections dealing with classical Yang-Mills are uncontroversial and unoriginal, they are summarizing previous results about the classical theory. The quantization formalism of this paper introduced in section 3.2 is using a mathematically rigorous version of the traditional creation and annihilation operators, where the coherent states are used as the non-orthonormal eigenstates of the annihilation operator. This is reviewed in physics sources in several places under the keyword coherent state path-integral.

The formalism on page 13 defines normal ordered products of field operators. The normal ordering is a bit obscured by the unfamiliar notation, but you can see the effect of the normal ordering multiplication on the bottom of page 13, and in particular, in proposition 3.2, which claims that the operators preserve the vacuum.

The claim that the N=0 state is still the vacuum is a mistake. When you normal order the interactions, you can make the N=0 state preserved by the interaction, but you are doing something extremely non-covariant, something which has no relation to relativistic quantum field theory. You are choosing a reference set of free-states relative to which you define your normal ordering, and defining the vacuum and dynamics relative to these reference states.

The proof of mass gap is then by rearranging an intermediate-step number-type operator defined to bound the energy from below from anti-normal ordered, to normal ordered, introducing a 1/2, just as in the Harmonic oscillator ground state energy, while the rest is positive definite. This is incomprehensible, because the rearrangement is done on a single mode only, using the fact that each mode is only interacting with itself, much as in the scalar example I started with.

This is a complete rewrite of an earlier review, in which I made a mistake, caused by working backward from equation 73. First, I saw that the vacuum is preserved by the form of the interaction, and the interacting theory would be unitarily related to the free one if the construction is correct. Since this is impossible in an interacting relativistic field theory, I assumed that the theory is free. Since equation 73 is simply reordering the Hamiltonian to introduce a 1/2, like the ground state energy of a Harmonic oscillator, I falsely deduced that Dynin's unjustified assumption is that the energy is the number operator times the classical energy. This is not the case. Dynin's mode-energy contains interactions, the actual unjustified assumption is that each mode only interacts with itself, so that he can do the rearrangements mode by mode to get a discrete spectrum for each mode.

The Hamiltonian in section 3.2 is manifestly not free, but it also manifestly only connects each mode number-operator tower with itself, it does not lead to interactions between different modes. This is a serious error, and there are no potential applications of this analysis to any field theory.

reviewed Aug 9, 2014 by (7,720 points)
edited Apr 11, 2015

+1. Judging from the propositions he posed(I'm not yet able to figure out the detailed mathematics, although they largely look like a formalization of what I already know), I completely agree with you. Just let me try to repeat the points in my own words to see if I misunderstand anything:

Firstly the author put Cauchy data of YM equation as the "wavefunction(al)s" to construct the Hilbert space, but then the apparent paradox is that according to YM equation, these wavefunctions evolve nonlinearly in time, but to be qualified as Hilbert space vectors, they have to evolve linearly in time, I don't see the author addressing the obvious paradox anywhere in the main article, except in the signpost section 4.3, a very brief discussion is given, where I can't extract too much useful information. To try my best, by equation (77) it seems the author forced the wavefunctions to evolve linearly, ignoring the nonlinear character of YM equation, the result probably is a free field dynamics(not sure if this is right after second thought. Actually, @RonMaimon, are you sure the resulting theory is a non-interacting one? It seems as long as the Hamiltonian constructed is not trivial, it has a chance to contain non-trivial scattering).

Secondly, claiming proposition 3.3 implies a mass gap is astonishingly absurd, to me it at best implies the spectrum is bounded below.

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Update on 15/April/2015: After reading Arnold I'm convinced I didn't understand Dynin's content at the first read, due to the unfamiliar notations Dynin used. Hence my comment above the dashed lines is most probably a result of confusions.

Actually, I'm inspired to ask: Do classical solutions of YM equation have any bearing on the quantization? What is it?

Ron, the paper is not rubbish, although it falls short of its claims. Please read the paper more carefully. At least one of your claims is simply wrong: You write

Since the author defines the "quantum energy" as the occupation number times the classical energy, it concludes that any state with occupation number 1 always has C more energy than the "vacuum", with occupation number zero.

but this is not what he does. The number operator (60) is quadratic in a/c operators, while the Hamiltonian (64) is quartic. He uses the number operator only to get a lower bound on the Hamiltonian spectrum, resulting in Theorem 3.1, which says the spectrum is discrete and bounded below by a shifted occupation spectrum.

This is a misunderstanding. In (77), H must be an operator on functions of histories (as the next line suggests), otherwise the equation doesn't make sense at all. Thus the linear dynamics is in the space of all linear combinations of coherent states, as in all QFTs (even the free ones).  The flaws of the paper lie elsewhere, not in these superficial problems due to the fact that Dynin uses notation unfamiliar to physicists.

Dynin does not quantize the space of classical solutions; in fact, the construction in Sections 3 and 4 does not make use of any of the results about classical solutions.

Instead, he quantizes the space of initial conditions! This is the same procedure we follow in quantum mechanics and in nonrelativistic quantum field theory. The Hilbert space is the space of functions on zero time configuration space.

For everything else see my review.

But the construction is even more problematic than this, in that the creation and annihilation operators are indexed by the space of initial conditions, so that there is a separate operator for each initial condition. This is not even the correct kinematics for free field theory, where the creation and annihilation operators are indexed by modes of the initial conditions--- there is one photon creation/annihilation operator for each mode of a box, not for each initial condition. The photons corresponding to an initial condition are mixtures of the photons corresponding to the mode decomposition of this initial condition. The kinematic space described here is even more incorrect than just being identical in the free and interacting cases, it is introducing too many creation and annihilation operators.

Here it doesn't necessarily involve any inconsistency, because creation/annihilation operators can be understood as operator-valued functionals: if the amplitudes of the Fourier modes of the initial condition is $f(k)$, then $a^\dagger(f)=\int dk f(k) a^\dagger(k)$, where $a^\dagger(k)$ is the creation operator that creates a photon with definite momentum $k$. So "creation and annihilation operators are indexed by the space of initial conditions" could really be just a fancy way of saying you can superimpose $a^\dagger(k)$'s to get a new creation operator. This contains no more and no less information than what we physicists are already familiar with.

@JiaYiyang: Indeed, that could have been what he was doing, and then it would have been the correct free-field kinematics, but it's not. When he constructes the Hamiltonian with the "a breves", he replaces the values in the classical energy function with operators specific for the given initial condition, the replacement of a-breves by a's consitute the quantization procedure, and this replacement makes "interactions" (i.e. quartic terms) which contain a-a-a-a products for one given set of initial conditions only, i.e. each initial condition interacts only with itself. Since each initial condition is a separate a and a-dagger, a separate Harmonic oscillator, he can then do formal rearrangements on the quartic term and show that the spectrum is discrete and gapped solution by solution. But he never has products of operators linking together different modes, as when you have this, you immediately get infinite integrals over a mode-index when you try to rearrange the nonlinear terms, and you get all the normal behavior of quantum field theory, like nontrivial vacuum, and divergent expressions.

Each separate initial condition is a completely disconnected excitation tower whose interactions are not linked to any other tower, and the kinematics is much larger than the correct kinematics you describe in the statement above. This is disguised by the formalism--- it would be possible in the same formalism to do things right, but that's not what he does.

This is exactly what confused me in the first iteration of the review. I noticed equation 73 is rearranging creation and annihilation operators, as if he is dealing with a single decoupled Harmonic oscillator. So I assumed that the Hamiltonian is free because each creation/annihilation tower was disconnected from all the others, since he is rearranging the operators which in free field theory would normally correspond to one mode independently from every other mode. This is a property of mode-operators in free-field-theory, and not in interacting field theory. But his mode theory is not free, there are quartic terms, but these quartic terms after quantization only talk about the a's associated with a single solution, they don't multiply the a's of separate solutions, so the end result is still a completely disconnected set of anharmonic oscillators, one for each initial condition.

If he had made the identification that different initial conditions have a's which are not distinct, he wouldn't be able to rearrange the operators like in 0+1 d perturbations of the HO, as he would get integrals over k from the products of coinciding operators, as usual.

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