# What is the physical meaning of Wilson loops?

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 Wilson 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 Stokes theorem is the integral of $$dA$$ over any surface bounded by the curve), and as far as I understand classical non-abelian Yang-Mills theories do not quite describe any "real" physics, still:

Do classical Wilson loops have some sort of physical meaning?

Wilson loops are still well-defined for self-intersecting loops. But if I understand correctly, when going to the quantum picture using the Feynman 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 Witten's Jones polynomial paper, and in a way is the very reason why Chern-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 Wilson 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. By "quantum Wilson 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 Wilson 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 WIlson loop diverge for self-intersecting loops ?

## 2 Answers

I'm a physicist in training who doesn't know any math, so hopefully I'll have good naive answers for your naive questions.

Am I right to think that in magneto-static, the Wilson loop for a simple curve is nothing but the exponential of the magnetic flux through any surface bounded by that curve ?

Yes, for a purely spatial Wilson loop that's perfectly correct. For a temporal Wilson loop, such as one lying in the $$zt$$ plane, you instead get $$\int E(z, t) \, dz dt$$ which, roughly speaking, integrates the potential difference between the two temporal sides of the loop over time.

Do classical Wilson loops have some sort of physical meaning?

The quantum Wilson loop yields the phase a quantum particle picks up upon going around the loop, but the quantum phase is nothing but the classical on-shell action. So the spatial Wilson loop tells you the extra action cost for a spatial path when there's a magnetic field around. It's trickier to talk about a temporal Wilson loop since a single classical particle can't go back in time, but you can use it to compare the actions of two classical particles that start and end in the same place.

Upon quantization, the spatial case gives you the Aharanov-Bohm phase, while the temporal case simply gives you the dynamical $$e^{-iEt/\hbar}$$ phase where the energy $$E$$ includes the electric potential.

Is there some intuitive/layman physical explanation as to why quantum Wilson loops are only well defined on simple curves?

If you're asking whether you can understand this classically, I think the answer is no. Witten states that this restriction has to do with regularization of inherently quantum effects. I don't think the nonapplicability of Stokes' theorem really has anything to do with it, as you can still apply it if you split the curve into simple pieces.

• Thanks, this is useful ! As for the last point I precisely did not mean classically, sorry if this was unclear. On the contrary I was wondering what kind of purely quantum effect makes Wilson loops along self-intersecting curves ill defined. – Adrien Sep 16 '18 at 7:23
• The path-integral formulation is common in QFT due to QFT's focus on amplitude calculation, but is provably equivalent to the Schrödinger representation; how do Wilson loops look in the Schrödinger functional representation? – alexchandel Dec 11 '18 at 23:05
1. Yep, Stokes' theorem:

$$e^{i \int_{\partial D} A} = e^{i \int_D dA}.$$

1. Wilson loops measure the Aharonov-Bohm phase of an infinitely massive probe particle.

2. It's possible to consider networks of Wilson loops in 3d TQFT. See this paper. It's not obvious how to do it in Chern-Simons theory though.

• Sure, I'm aware of that: in TFT's one can, often has to, consider knotted graphs with vertices labeled by intertwiners, but this is not wuite what I'm wondering about. If you like my question could be: what is the physics intuition/explanation as to why quantizing CS theory magically turns an homotopy invariant into an isotopy one ? – Adrien Sep 16 '18 at 8:25
• Or, indeed, why is the change of AB phase ill-defined on self-intersecting trajectories ? Is there some quantum principle preventing a particle to go back somewhere it already has been ? Witten point out that this has to do with some regularization issue which is somehow a general fact of life about quantum Wilson loops but I'm wondering whether there is a "reason" for that. – Adrien Sep 16 '18 at 8:25
• It's hard to tell looking at the CS action what the theory is actually going to be like when you integrate over the gauge field. For one, the choice of measure is quite delicate. One needs at least to introduce a framing, which Witten explains resolves self-intersections. This is because of a divergence $\langle A(0) A(x) \rangle \sim 1/x$ in the 2-point function (cf. Gauss' linking number integral). It does not forbid branching Wilson lines though. If you want to understand gauge theory, I recommend beginning with electrodynamics, not Chern-Simons theory. – Ryan Thorngren Sep 16 '18 at 17:17
• That's kind of what my question is about :) I'm aware of those facts about CS, and the necessity of taking care of this divergence. I assumed something similar would happen in QED, and I'm asking about the "physical meaning" of that divergence, if that makes sense ? Something about some test charge interacting with itself, or emitting and absorbing photons that would'nt make sense on a self-intersecting trajectory. – Adrien Sep 17 '18 at 11:16