# 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 (15,488 points)
edited Aug 17, 2014

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.

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.

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

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

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.

------------------------------------------------------------------------------

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?

Dear Ron, I appreciated your effort to kill Dynin's work even if it does not seem from your review that you ever understand the formalism he uses. Please, check my question here and be so fine to explain it to me and people at MO where I asked a similar question.  Dynin 's claim does not refer at all at any classical solution of any kind. Rather he says his proof of a mass gap is purely quantum.

@JiaYiyang: Please post this (and anyone, please post other independent questions) as a separate question in the Q/A part, in order to disentangle the discussion. I have a lot to say about this.

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.

@ArnoldNeumaier Please, let me know how do you like the disentanglement as I think the question of the classical solutions in QFT is a really fundamental one.

It just means posting the question ''Do classical solutions of field equation have any bearing on the quantization?'' under Q/A, together with a link to the present page.

@ArnoldNeumaier, I'll do that, give me a few minutes.

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.

Arnold, Please could you mirror your conclusions on MO as this could be helpful also to the community of mathematicians. Thanks a lot.

Hi @JonLester, I think you could also summarize what Arnold said as a self-answer for the MO question...

That should wait as I haven't said anything serious, which I'll do in my own review. Here I just corrected some misunderstandings about the paper.

Hi Dilaton, Surely I will be happy to post on MO a summary of Arnold's conclusions as soon as these will become available. Thanks a lot to both.

@ArnoldNeumaier: Your interpretation is due to having a more charitable eye, because you can see some sensible thing to do starting from classical solutions of the quantum effective action. The quantum effective action for Yang-Mills theory is only locally similar to the equations for classical Yang-Mills theory, the quantum effective is defined as a Legendre transform with respect to gauge field sources (in appropriate gauge) of the logarithm of the full partition function, and the equations one gets from this do produce the true correlation functions. One can ask in the case of Yang-Mills theory what this effective action looks like, and what type of classical solutions could describe quantum Yang-Mills.

But if you look at this paper, it is not doing this! It is not considering the quantum effective action defined by the path integral. It first defines the classical global solution space for classical Yang Mills, and then introduces an obfuscated method of defining independent creation and annihilation operators on the different classical solutions in section 3.3.

The bound he gives on the energy is impossible, as the vaccuum in QCD, in a coherent state formalism, will have $N=\infty$ for any local coherent state description. So will any excited state. The states of QCD are infinitely full of "particles" as defined by any type of near-classical description, the vacuum is a sea of twisting gauge field, it is not a small fluctuation around the N=0 vacuum except locally, if you zoom in at any one point.

The number of "particles" as defined by excitations on top of classical equations is infinite at long distances, there's a density of them, the vaccuum is full of condensates which randomize the gauge field. But because he uses a free-field creation and annihilation formalism, he can define the "vacuum" as having zero occupation numbers, and states of independent excitation number, which are precisely states of noninteracting bosons indexed by classical field mode solutions.

His paper has a wrong kinematics in exactly the way I say. The sections on classical theory are entirely fine, and the results he cites for quantization are correct in their context, but their context is free-theory creation and annihilation operators, as described by Langevin noise in stochastic mathematics. The coherent states he introduces are coherent number states of occupation in different classical modes, with no relation to actual coherent semi-classical states in a real non-free field theory.

This paper is not correct, really.

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.

Initial condition == classical solution, each initial condition produces one classical solution, each classical solution has one initial condition. They are in one-to-one correspondence (he spends a lot of time discussing this).

This is not the procedure in wavefunction quantum mechanics, which involves a wavefunction on configuration space, not initial condition space. The classical initial data on a Cauchy surface involves both position and momentum, not just one. But this doesn't bother me, because this is the usual business in phase-space quantization nonsense.

What bothers me is that he doesn't quantize gauge theory, he uses a stochastic formalism which is equivalent to defining creation and annihilation operators. These things define a tower of states sitting on top of anything you make stochastic.

He uses these creation and annihilation operators to place a tower of states on top of each classical solution/classical initial condition. This defines a theory of artificial particles sitting on top of classical data. You can do this, but it has no relation to the gauge theory, which has vacuum bubbles.

He then goes on to prove a trivial property about this, namely that the energy of any N state with N nonzero is greater than the energy of the N=0 state (because the classical energy is positive).

If the anharmonic oscillator is treated in the coherent state picture, wave functions psi are analytic functions in the phase space of the harmonic oscillator. hence given by the initial conditions of the free oscillator solutions.

Nevertheless one can quantize in this setting arbitrary anharmonic oscillators, including those (such as the Morse oscillator) that have a very different long-time behavor.

The reason is that the initial conditions only determine the kinematics (which is the same for all oscillators), not the dynamics (which depends on the Hamiltonian or action).

One can rewrite any quantum mechanical system with finitely many degrees of freedom in a similar way by making the $(p,q)$ to $(a,a^*)$ transformation for each coordinate separately, and then write the hilbert space as a space of analytic functions of a complex vector $z$, which takes values in phase space. This is just the Bargmann representation of conventional quantum theory.

In each case one begins with functions on the space of Cauchy data for the corresponding nonlinear classical system, which happens to agree with the space of Cauchy data for a linear classical system, too. Thus dynamical issues don't play the slightest role. It is just a matter of taste.

Berezin quantization does the same for classical dynamics on an arbitrary symmetric space. The coherent states are labelled by the points in the symmetric space (which is the phase space), but the dynamics can be given by an arbitrary element of the enveloping algebra of the Lie algebra of symmetries of that space. You can model arbitrary quantum mechanical systems (with finitely many degrees of freedom) in this setting, and with success. (There is a huge literature on dynamical symmetries proving that both in theory and in practice).

In infinite dimensions, there is some related successful work in the 1+1 dimensional case, but no general approach.

Dynin tackles the 4-dimensional case, which is harder (hence the millennium prize), though it should not be held against him that he failed. His Hamiltonian (if it exists) is a nonlinear Hamiltonian in a subspace of the standard Fock space of Hamiltonian field theory, but because of the constraints, which are in phase space rather than configuration space, the subspace cannot be described without the coherent state formalism.

That it can be done with coherent states is the positive result of his paper, and this is of value independent of the naiveness with which he ignores nontrivial renormalization issues.

Sorry, Arnold, but I believe you are misinterpreting what he is doing (although now that I look more closely, I see that he has four-creation and annihilation terms in his Hamiltonian). The quantization he is doing is based on classical solutions, and then placing a free-field tower on top of these using stochastic operators, and claiming these N states are somehow bounding the gauge theory particle states.

The result is a nonsense kinematic space, which has nothing to do with the kinematic space of real gauge theories, it's an entirely orthogonal fictitious Hilbert space. His "N=0" state corresponds to a free field vacuum constructed on top of classical solutions, and there is no way to relate this free field nonsense to the gauge field vacuum. He claims, however, in "proposition 3.2" that the vacuum is described by the N=0 state! You know this is nonsense, that this is a free-field construction, because the full gauge vacuum is defined by a path integral, and is approaching something completely random at long distances as described by the strong coupling expansion.

There is no point in saying "look how well phase-space techniques work in 0+1 dimensions", I know they work, the kinematics here are trivial. In this case, the kinematics are altered at long distances, the vacuum has infinitely many bare gluons in it, otherwise the gauge field couldn't get randomized they way it does, it would be similar to a QED vacuum with holonomies averaging to 0 at long distances, instead of to a random pick from the gauge group.

I still think you pay too little attention to what he writes, with resulting misjudgments.

Unlike the path integral view, which works covariantly in 1+3D, the Hamiltonian approach Dynin takes (in attempted extension of rigorous 1+1D work by Landsman, Hall, and others) works at fixed time, hence in a purely spatial 3D setting. Here the vacuum is described by the ground state of the Hamiltonian (as in the 1+0D case), not by a path integral.

The comment after Proposition 3.2 is about the bare vacuum, corresponding to a harmonic oscillator whose Hamiltonian is the number operator $N$. He didn't yet introduce the Hamiltonian, which contains the anharmonic terms (64), and therefore changes the vacuum to the ground state of this Hamiltonian $H$. For this ground state he only gets inequalities on the energy - certainly no equality! The inequality comes from the fact that $H-N$ is evidently positive semidefinite. After eliminating the continuous spectrum by considering (75), he ends up with a set of states labelled by $n$, corresponding to an $n$-particle sector of a theory with a single bound state. This part of the analysis would be sound if the Hamiltonian were well-defined.

Whether you call it "path integral" or "Hamiltonian" is irrelevant, the Hamiltonian is just a sliced path integral, and the configurations which contribute to the integral determine the vacuum correlation functions in the ground state of the Hamiltonian.

The transformation he writes down can't possibly be sound because the ground state of the interacting Hamiltonian has no relation to the free-field fock space of the noninteracting Hamiltonian, nor do any of the glueball states in the interacting theory have any relation to the Fock states of free gluons except locally, in a ball much smaller than the confinement length.

His "inequalities" are nonsense, because any "N" in the vacuum defined from classical solutions and quantizations around these is infinite, as is the "N" in any perturbations to the vacuum. His analysis explicitly states that the N=0 state is the vacuum in the theory, and that there is a well defined "N" operator counting excitations relative to this vacuum. This is false.

I don't understand why you are so charitable to this work, it is not even close to correct, it is a misunderstanding from start to finish. The relation to 1+1 work is irrelevant, since 1+1 gauge theory has no field theoretic degrees of freedom, and is just a quantum mechanics on topological variables.

Dynin never claims that the vacuum of the free theory (the ground state of his $N$) is the vacuum of the interacting theory (the ground state of his $H$) - you read this into his text. This would not even be the case for an anharmonic oscillator (where space is just a single point).

1+1 gauge theory shows that the infrared behavior of the quantum theory faithfully reflects the topological features of the classical theory. This is expected to remain valid in 4D, since theta angles are physically relevant in real QCD. This is the reason why I find your argument that the classical theory has nothing to say about the large-scale (infrared) behavior of QCD not convincing, no matter how persuasive it may sound to you.

Dynin's quantization has no theta angles, another indicator that his construction is faulty. But the millennium problem is a mathematical problem, so any criticism of claims to solutions must be on mathematical grounds, not on physical expectations. I only count errors in what he actually does and claims, compared to the actual requirements in the defining document by Jaffe and Witten.

I am not criticizing on physical grounds, I am criticizing on mathematical grounds. You say "Dynin never claims that the vacuum of the free theory (the ground state of his N) is the vacuum of the interacting theory", but he does exactly this. The Hilbert space that he starts with is a tower of particle states on top of classical solutions. This is how he claims to bound the energy from below by N, he claims that the vaccuum is unchanged from N=0. The vacuum in a real interacting theory at infinite volume is changed from N=0 to N=infinity instantly.

Regarding the relation between the quantum and classical theories, the topological behavior is never an issue, it's the fluctuating behavior that is the problem. There is no problem in quantizing topological quantum field theories by symplectic methods, and the classical and quantum theories are obviously as related as in ordinary 0+1 dimensional quantum mechanics, there are only finitely many degrees of freedom.

The relation between classical and quantum theory you see in topological gauge theory is not expected to remain valid in 3d or 4d, because The gauge fields at long distances in pure gauge theory are totally random. The path to the strong coupling limit is completely smooth.

The reason theta angles are relevant is because they associate a phase to any configuration which is topologically like an instanton. You don't have to think of these as classical, although that's the leading saddle point, they are perfectly well defined in a path integral, or even on a small enough lattice. The classical configurations are only for action-minimizing instantons, and these are not classical when their size is large. The completely random gauge field at long distances will also contains fluctuating instanton topological number.

The classical limit of quantum gauge theory is free gauge theory. The idea that classical gauge theory solutions can be used to produce quantized gauge theory is unjustified, and a few minutes with a lattice and strong coupling expansion will show this.

I cannot see why (if the renormalization issues that Dynin neglected were accounted for correctly) the missing things that you claim making the whole approach invalid couldn't  be a consequence of adding the quartic terms to the Hamiltonian. This suffices to make the vacuum structure completely nonclassical, because of the associated renormalization. But it seems that no new aspects come up, so I'd like to close this discussion soon.

@ArnoldNeumaier; I was totally insane, I don't know what I was thinking. Thanks for you patient correction. I stand by the comments and review as I edited them just now. The earlier versions were incomprehensibly inappropriate and even worse, also factually wrong.

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!

For "the gauge field becomes entirely random!", do you simply mean the VEV of Wilson loop decays as $\sim \exp{(-tr)}$ at large radius? Or is there more intended content summarized in this statement?

@JiaYiyang: The coarse lattice action converges to zero, so that if you look at the correlations of large distant Wilson loops, there are none. On a coarse lattice, each matrix link variable is independent and random, i.e., you choose a random link variable, then another, then another, independently, and fill the whole lattice, and that's a gauge field configuration for a coarse lattice! That's "zero action", i.e. no correlations between the sites, and completely random on any site, and this is the coarse lattice gapped limit of gauge theory which explains confinement as Wilson explained (although he used the Wilson loop to illustrate this behavior, he knew the full behavior).

I actually now know that this whole process of constructing gauge theory from the lattice approximation was actually carried out by Balaban in 1984-89. He probably solved the problem of existence already, and possibly mass gap, I need to read and review these papers pronto. It wouldn't be the first time that a solved problem was made out to look unsolved, but this might not be what happened, I haven't read the papers. But on the non-rigorous level, Wilson solved the whole thing.

Balaban's work is incomplete (with respect to solving the Millennium problem) as the estimates available are not strong enough to prove the existence of the infinite-volume limit. Nobody has found a way to overcome this problem; it is very hard analysis. The book by Glimm and Jaffe discusses the much simpler case of $\Phi^n$ theory in 2D, and already here the technical requirements are considerable.

For the state of the construction problem in 2000 when the Millennium problems were posed, see Section II.1 of Rivasseau, where Balaban's work is placed into context. Nothing significant in this direction happened since then, but two papers by Dimock, here and here, might be interesting.

@RonMaimon, I can only semi-demi-hemi-quasi understand what you are trying to say, could you formulate what you said in a more mathematically precise way? I suppose you are saying something like, the value of lattice action is $S(\text{link variable configuration}, a)$, where a is the lattice spacing, and then we have $\lim_{a\to\infty}S(\text{random link variable configuration}, a)=0$? Then my question becomes: what exactly does it mean by random? Does it simply by definition mean some correlation function that characterizes randomness is 0? What is this correlation function?  Is this correlation function calculated by weighting over the exponentiated continuum or lattice action?

@JiaYiyang: There is a lattice action for QCD, which is $1/g^2$ times the trace of all the plaquette holonomies. If you integrate out some appropriate links, and define matrices for links twice as long, you relate the lattice action to a lattice twice the size. This is a real-space RG transformation which is precisely defined on a lattice, it is like Migdal-Kadanoff RG, except without the 2-d specific rotation by 45 degrees.

As you integrate out the links and make the lattice bigger and bigger, the matrices associated with each link become more and more widely fluctuating, and the non-gauge-fixed action converges to 0, i.e. $g\rightarrow\infty$ meaning that the probability for any configuration is equal to the probability for any other configuration. So each link-matrix in the gauge-unfixed lattice formalism is an independent completely random choice from the gauge group by Haar measure. You can generate a gauge-unfixed gauge-field configuration for a lattice much coarser than $\Lambda^{-1}$ by simply picking a random independent SU(3) matrix for each link (zero-mass quarks wreck this property, since there are massless pions)

This is the zeroeth order approximation for the strong coupling expansion, that's the initial approximation of zero action. Wilson begins with this 0 action thing in the 1974 paper, and corrects the 0 action perturbatively in $1/g^2$ to include the effect of small correlations between neighboring sites, and shows that the zero coupling limit is attractive in the strong coupling expansion, so that when you get close to it, and you do RG to larger lattices, you get even closer.

This totally random large-lattice behavior is gapped and confining, since it is ultralocal, and the approach to this limit describes the gapping of all states, i.e. confinement of gluons. If you leave in a charged non-dynamical colored source somewhere, the Gauss law requires there to be a "string" of non-random plaquettes extending to infinity, to carry the flux of the colored source to infinity, and this produces a deformation on top of the initially totally random gauge vacuum, which makes the matrices along the flux-tube not totally random. That makes them infinitely improbable, because the line is infinitely long, so you get a line-tension, each length of line has an improbability cost per unit length--- this is the QCD string tension.

This is Wilson's 1974 explanation of confinement, and it works on lattice gauge theory in strong coupling expansion. You find the string at strong coupling and compute the tension. Wilson was the first to compute the string tension in pure SU(3) gauge theory in his numerical followup around 1980 (done on a 4x4x4x4 lattice at strong coupling). He found a linear quark potential, as expected from strong coupling expansion. This was the beginning of lattice gauge theory.

This approach to strong coupling expansion is only confirmed with larger lattices, and the transition from small lattices where the theory is nearly free to large lattices where it is strong coupling is the standard picture in lattice QCD.

This picture is starting from zero action, which only makes sense in a lattice formulation. It is a mathematically precise thing, but it has very little relation to other pictures of confinement, it is much simpler and direct. So this picture is not well advertized outside of the lattice gauge theory world, where it is the foundation of the field. The focus of lattice gauge theory is more accurate computation, so the insights do not always spread out to the larger HEP community.

that completely clarifies, thanks!

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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