Thanks for the response, I now understand what you are misunderstanding. I am not "using" the strong coupling expansion, rather I am describing what the results of this are. The strong coupling expansion is not used because there is nothing more to learn from it! It's served its purpose. The purpose was to describe what strong coupling lattice gauge theory looks like, it is now understood, and there is no more controversy about this.

The issues are about taking a continuum limit, where the coupling is weak. It is probably just as hopeless to try to approach the continuum limit from the strong coupling expansion as it is to approach the long-distance strong coupling limit from the continuum.

But the conclusions I am saying don't require any expansion at all, they are simply facts about simulations of lattice gauge theory at weak (or strong) coupling! The simulations randomize at long distances, and this means something precise--- it means that if you calculate the correlation function of <G(x) G(y)> where G is the lattice link gauge field, then it approaches zero as x and y become separated as exp(-|x-y|/L), where L is the inverse mass gap. This is the very statement of what mass-gap means.

This are set in stone, because it is simply a numerical fact. One part of the millenium problem is to prove that this is what happens at long distances, but that it is what happens at long distances is not controversial.

The rate of decay of the correlations is described well by the strong coupling expansion, but you don't need to use it to conclude that the correlations decay. When the correlations decay completely, at distances much larger than the confinement length, the configurations of the gauge fields are entirely uncorrelated.

If you superpose a large grid on top of the continuum theory, and define the lattice field as the parallel transport along the link, then the configuration of the lattice gauge theory so defined is a random sampling from an action which approaches the strong-coupling (zero action) fixed point. What that means is that the parallel transport along parallel lines is completely statistically independent when the lines are far apart (this follows from the mass gap).

Nothing I am saying can be challenged, because it is simply a numerical fact, I can show you this on a computer if you would like, but it is easier if you simulate it, or ask a lattice QCD person. The randomness of the long-wavelength gauge field is not proved by the strong coupling expansion, it is described by the strong coupling expansion. It is "proved" because it is numerically what happens in simualtions, and the goal of the Millenium problem is to get a mathematical proof, rather than a numerical demonstration through simulation.

Aug 11, 2014
by
Ron Maimon
I should say "approaches zero action according to Wilson's strong coupling expansion law" rather than "has zero action". The result is obvious--- the gauge fields are random, and become more random when you multiply them to define coarser lattice link variables.

Aug 10, 2014
by
Ron Maimon
Why are you asking about continuum limit? I am totally confused. The statement that the long-wavelength limit of pure lattice gauge theory is described by the action "0" is not controversial, it is the statement of mass-gap--- at distances larger than the inverse mass gap, all correlation functions in Euclidean space die off exponentially, i.e., the statistical fields are completey uncorrelated.

Taking the continuum limit makes the lattice small and the coupling logarithmically small simultaneously. If you multiply the gauge fields to find the transport matrix along a line, you can define the lattice link-variable for a coarse matrix sitting on top of the fine lattice. The coarse lattice, when it is coarser than the confinement scale, is a configuration of a strong coupling gauge theory, it has zero action.

These observations are trivial, I am completely confused why you keep giving irrelevant papers and trying to get me to read them. They are irrelevant, because I observed these things for myself, I don't need to read a thing.

Aug 10, 2014
by
Ron Maimon
Oh, no, I didn't. I see now. Thanks! I was asking because the name "Jon Lester" sounded like a real name. I won't bother you again.

Aug 9, 2014
by
Ron Maimon
Why is the picture you use here identical with the picture Marco Frasca uses on his website? Are you the same person? It's ok if you are, but please say so.

Aug 9, 2014
by
Ron Maimon