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advanced_notions:observable

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.

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

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:

advanced_notions/observable.txt · Last modified: 2018/01/02 12:13 (external edit)

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