Naive question about WIlson loops in electromagnetism
I'm a mathematician trying to get some very basic physical intuition on gauge theories, so I apologize if what follows is really naive. My first super elementary question is:
Am I right to think that in magneto-static, the <PERSON> loop for a simple curve is nothing but the exponential of the magnetic flux through any surface bounded by that curve ?
This seems obvious but I've never seen it explicitly written so I fear I'm missing something obvious. Now of course this is special to electromagnetism because the gauge group is abelian (hence holonomy of the one form $A$ describing the magnetic potential is exp of the integral of $A$ along the curve, which by <PERSON> theorem is the integral of $dA$ over any surface bounded by the curve), and as far as I understand classical non-abelian <PERSON>-Mills theories do not quite describe any "real" physics, still:
Do classical <PERSON> loops have some sort of physical meaning?
<PERSON> loops are still well-defined for self-intersecting loops. But if I understand correctly, when going to the quantum picture using the <PERSON> integral formulation there is some "regularization" issue, and one way to deal with it is to equip the curve at hand with a framing, which in particular forces to consider only simple loops. This is beautifully explained in <PERSON>'s <PERSON> polynomial paper, and in a way is the very reason why <PERSON>-Simons knot invariants are non-trivial (which as you may have guessed is the reason why I'm asking those questions).
Is there some intuitive/layman physical explanation as to why quantum <PERSON> loops are only well defined on simple curves? I would already be interested in an answer in the abelian case, perhaps something related to the fact that in that case it's only for simple curves that the classical ones are related to the magnetic flux?
Edit I realize my question was unclear, mostly because I don't really know how to use the jargon accurately. 346
By "quantum <PERSON> loop" I really meant its vacuum expectation value. Is there a simple explanation of what this quantity means ? Now this has, I believe, a divergence when you take the limit as the size of the loop goes to 0. For the classical <PERSON> loop for a given $A$, this basically gives the field strength at the point which is the limit of the loop. As far as I understand, the reason why the VEV is divergent in that situation, is that it doesn't quite make sense physically to measure the field at one point. Rather, all we can do is measure the flux going through a very small loop near that point, and somehow quantum physics is sensitive to that. Does that make sense ? Is there a similar explanation as to why the VEV of a <PERSON> loop diverge for self-intersecting loops ? 346