# Observable

## Layman

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.

## Student

In this section things should be explained by analogy and with pictures and, if necessary, some formulas.

## 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 Ghosts. To do this properly requires to make use of the BRST formalism. (Source: Baez, Munian; Gauge Fields, Knots and Gravity, page 342)

Example1
Example2: