====== Observable ======

<tabbox Why is it interesting?> 

<tabbox Layman> 

<note tip>
Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.
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<tabbox Student> 

<note tip>
In this section things should be explained by analogy and with pictures and, if necessary, some formulas.
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<tabbox Researcher> 

In the path integral approach to gauge theory, observables are gauge invariant functions on the space $\mathcal A$ of a $G$-connections on $E$, where $G$ denotes the structure group and $E$ the fiber bundle. Therefore, an observable $f$ is a function on the space $\mathcal A / \mathcal G$, of connections modulo gauge transformations.

As a result, vacuum expectation values are no longer defined as integrals with Lebesgue measure $ \mathcal A$, but instead with a Lebesgue measure on $ \mathcal A/ \mathcal G$. We obtain this measure by pushing forward the Lebesgue measure on $ \mathcal A$ by the map  $ \mathcal A \to \mathcal A/ \mathcal G$ that sends each connection to its gauge equivalence class, and then $ A$ denotes a gauge equivalence class of connections in the integral.

The simplest example of an observable in gauge theory are Wilson loops. 

Take note that this procedure of modding out $\mathcal G$ from $\mathcal A$ is what leads to [[advanced_notions:quantum_field_theory:ghosts|Ghosts]]. To do this properly requires to make use of the [[advanced_tools:gauge_symmetry:brst|BRST]] formalism. 
(Source: Baez, Munian; Gauge Fields, Knots and Gravity, page 342)



  
<tabbox Examples> 

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<tabbox FAQ> 
  
<tabbox History> 

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