How do Holonomies in Gauge Theory Compare to Ordinary Loops?

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Discussion Overview

The discussion explores the comparison between holonomies in gauge theory and ordinary closed loops, focusing on their advantages, disadvantages, and implications in theoretical frameworks such as lattice gauge theories and gravity.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants propose that closed Wilson loops in (non-abelian) gauge theories are gauge invariant by construction and can serve as "canonical variables" defining the theory.
  • Others note that while Wilson loops are advantageous, they are uncountable and do not permit separable Hilbert spaces, which presents a limitation in gauge theory.
  • A participant mentions that this issue can be addressed in gravity due to diffeomorphism invariance, but not in gauge theory, leading to the conclusion that Wilson loops are not utilized as fundamental objects in that context.
  • One participant questions the relationship between "holonomy" and "Wilson loop," suggesting they may be equivalent up to mathematical subtleties.
  • A later reply provides a mathematical expression relating holonomy to Wilson loops, indicating a potential equivalence in certain contexts.

Areas of Agreement / Disagreement

Participants express differing views on the implications and utility of holonomies and Wilson loops, with no consensus reached on their equivalence or fundamental status in gauge theory.

Contextual Notes

The discussion highlights limitations related to the mathematical properties of closed loops in gauge theories, including issues with separability and the implications of diffeomorphism invariance in gravity.

rodsika
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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?
 
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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).
 
tom.stoer said:
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"?
 
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]
 

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