Holonomies in Gauge Theory

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).

 

Up to mathematical subtleties they are the same

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