Saturday, 8 November 2008

Gribov copies.

There's an interesting discussion on BRST going on over at Not Even Wrong. Someone mentioned Gribov copies, in which I have an interest, and I put my two pence worth in.

I think I inadvertently caused the discussion to split into two groups -- BRST and copies, and it ended with people apologising on the comments board. Rolls eys. Physicists apologizing for their views on physics -- if only the string theorists would go for that idea in a big way.

Anyway, I'll summarise my thoughts here.

Gribov copies have physical effects -- they do have something to do with confinement. This is really very easy to show -- see work by Bagan, McMullan and Lavelle. What they have shown is that although you can write down the wavefunction for a physical quark (so not simply the fermion in the Lagrangian, as that isn't anything physical) in perturbation theory, you cannot do so non-perturbatively. If you could, there would exist a globally well defined gauge fixing for Yang Mills, and there isn't, because we know there are copies.

Seeing this from a slightly different angle, your supposedly gauge invariant quark picks up a non--perturbative gauge dependence (and hence becomes unphysical) precisely through the action of the Gribov copies. In other words, you can have physical quarks in perturbation theory (weak coupling, which makes perfect sense since we have asymptotic freedom), but not non--perturbatively (strong coupling, when we expect the flux tube to form). Non-perturbatively, you can't have single quarks, but the above argument doesn't go through for multiquark states, so you can have mesons and hadrons -- well great, this is exactly what we see!

However, this doesn't give you the confinement scale or anything like it. For that you need dynamical, quantum, arguments, and probably non--perturbative arguments at that. So, just like every other approach to confinement, it doesn't go all the way, but it's a very simple, very physical argument.

The way most people are introduced to Gribov copies is, probably, as an overcounting in the path integral. That's not the fundamental origin of the copies, it's just a symptom of their existence. If there is some other, operator, picture in which the copies don't appear --as mentioned in the discussion linked above-- then either there is something amiss with that picture (it isn't capturing all the physics, in particular non-pert. physics) or there is something wrong with our understanding of that picture (which is probably easier to accept).

But then, I'm going up against Nakanishi -- and he invented the Nakanishi Lautrup field, compare: what have I done?, so you should probably take what I say with a pinch of salt.


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