How do Holonomies in Gauge Theory Compare to Ordinary Loops?

[rodsika]

How do Holonomies or ideas of closed-loops in Gauge Theory compare to the ordinary? What is its advantage and disadvantage? And how does it scale in the plausibility rating?

 

[tom.stoer]

In (non-abelian) gauge theories closed Wilson loops have been introduced especially in lattice gauge theories. The advantage ist that these closed loops are gauge invariant by construction. They can be used as "canonical variables" defining the theory, but unfortunately they are uncountable and do not allow for separable Hilbert spaces. This can be fixed in gravity due to the diffeomorphsims invariance of the theory (but not in gauge theory, so Wilson loops are not used as fundamental objects).

 

[rodsika]

In (non-abelian) gauge theories closed Wilson loops have been introduced especially in lattice gauge theories. The advantage ist that these closed loops are gauge invariant by construction. They can be used as "canonical variables" defining the theory, but unfortunately they are uncountable and do not allow for separable Hilbert spaces. This can be fixed in gravity due to the diffeomorphsims invariance of the theory (but not in gauge theory, so Wilson loops are not used as fundamental objects).

What is the relationship of "holonomy" to "wilson loop"?

 

[tom.stoer]

Up to mathematical subtleties they are the same

$$h_C[A] = \mathcal{P}\,\text{exp}\left[i \oint_C dx_\mu A^\mu(x)\right]$$