JonLester

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Dear Marco Frasca:

After having been reading Ron Maimon's answers  for two years, I really believe he is an extremely honest researcher who only cares about science's sake. He may be wrong about this issue (I'm not competent enough regarding this discussion), but if so, he will admit it when he realizes that. When discussing, he uses a very particular style that you have already noticed. He has said several times that he does so on purpose to break political persuasion. He has also said he is aware he looks like a crackpot and that's the price to pay to avoid ad hominem arguments. This strategy is perhaps wrong.

But, please, there are many people (I'm one of them) who learn a lot from reading these discussions. I think you shouldn't care about his particular style and try to focus on the meat. Of course, you can do whatever you want. I'm just asking you to keep in mind that this discussion can benefit plenty of readers.

Thanks.

I have commented on the particular papers in question, and I will write a full review. You have to understand that these Landau gauge peculiarities (if they are real) are due to the Landau gauge fixing, which is rigid, and can artifactually introduce long-range correlations, just because it is a deterministic gauge fixing. It still shouldn't do that, but whether it does or doesn't, the gauge-unfixed simulations, which compute only gauge invariant quantities, are clear about the vanishing of all long-distance correlation, so that one can understand any correlations as emerging from the transition to Landau gauge (which in principle can be done after sampling, so that you first generate gauge-unfixed equilibrated configurations, then go to Landau gauge on each one numerically).

I always say what I think, and I don't care about "going far", just about understanding gauge theory.

Ok, I finally understand what you think is contradicting what I said--- the 2007 lattice conference where a bunch of groups saw that the gauge-gauge correlation function for Landau gauge is not going to zero at zero momentum in the large lattice simulations. That is completely nuts as they interpret it. They think it means that there is some residual correlations at large distances in their simulation. Then, further, in response to this, some moronic people actually postulated that the beta function for QCD turns around and makes the coupling small at large distances (as if it were possible for a one-dimensional parameter running to turn around).

This lattice behavior, if interpreted as physical correlations at long distance, is simply incompatible with the existence of a mass gap--- a mass gap requires the correlation functions in real space to fall off as exp(-M|x-y|) as x and y separate to longer distances. That is physically required, but it does not necessarily mean that the Landau gauge correlation function as defined in the simulation has to vanishes at p=0. The reason is that when you Fourier transform, there is still the purely local contribution to the g-g propagator when you integrate over all space, and they probably did the details of the gauge fixing so that there is still a residual constant in the g-g propagator. I'll look at the details to try to see exactly why it was the constant that they saw.

The vanishing of all long-distance Euclidean correlations in a theory with a mass-gap is certainly a rigorous theorem somewhere in the literature, because I know how to go about proving this. It's not controversial, and it is well known for gauge invariant observables in pure gauge theory simulations (this is what I was talking about, I didn't even bother to fix a gauge in anything I said)

The absence of the vanishing behavior in the propagator is just some stupid artifact in the way these lattice people go to Landau gauge. Landau gauge is difficult to implement numerically, because it is a very rigid condition. What I can say for sure is that they are NOT seeing any residual correlations of any physical kind at long distances.

I'll try to write a review of the paper as soon as I understand what is going on in the simulation. I have to think about simulating in Landau gauge, which is something so unnatural I never thought about it before.

Your comments have made me think about this, and there is an interesting puzzle in the case where you are simulating SO(3) gauge theory--- the lattice formulation contains monopole-like configurations which are not present in the actual theory, because the gauge group is not simply connected. In the continuum limit, the action of the configurations which twist around the gauge group when you make a sphere goes up and up without bound. Such configurations will make the lattice formulation less reliable, and they are analogous to the case of simulating Lattice QED.

There are some difficulties in cases where the gauge group is quotiented so that it is not simply connected. Other than this, the skepticism is unwarranted. In normal SU(3) gauge theory, any lattice configuration can be interpolated to a continuum solution in a natural way.

Also, I would appreciate you pointing me to what you THINK contradicts what I said in any of these papers. I read them all, and there is nothing in there that remotely begins to contradict anything I said.

Sorry, I meant "twice the confinement length" not "twice the lattice length".

I am saying it because the randomization is an easily observed fact, and it can't possibly get "updated" by anything. I read all the papers you put up for review, and none have any bearing on it at all. If you've done a simulation, good. Then do the following: take the product of successive gauge variables over a length equal to twice the lattice length, and see what the distribution is for this transport quantity. (it will cover the whole group space--- you can plot distance from the identity and compare to a sphere).

You can compute the gauge-gauge correlator at large distances (or anything else), and notice that it vanishes. There is nothing to dispute here, this is the statement (not the proof) of mass gap.

I would like to review any paper that claims that this randomization at long distances is wrong, so please post this paper to the reviews section. I have simulated pure gauge theory, and I know what it looks like (and this aspect cannot be controversial).

Pure gauge theory at long distances is "free" only in the trivial sense of being empty. It has no excitations below a critical energy, this is what "mass gap" means. The lack of excitations below a given gap is due to the randomization of the Euclidean theory, it is a synonymous with it. So if someone is saying the theory is not random at long distances, then they have disproved the Yang-Mills mass-gap conjecture.

It is really nonsense, you really are misinterpreting. Please, please, look at an actual configuration of equilibrated gauge field, you will see that at short distances it is correlated, and at long distances uncorrelated.

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