advanced_notions:observable
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| - | In the path integral approach to gauge theory, observables are gauge invariant functions on the space $A$ 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 $A/G$, of connections modulo gauge transformations. | + | 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 $A$, but instead with a Lebesgue measure on $A/G$. We obtain this measure by pushing forward the Lebesgue measure on $A$ by the map $A \to A/G$ that sends each connection to its gauge equivalence class, and then A$ denotes a gauge equivalence class of connections in the integral. | + | 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. | The simplest example of an observable in gauge theory are Wilson loops. | ||
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| + | 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) | (Source: Baez, Munian; Gauge Fields, Knots and Gravity, page 342) | ||
advanced_notions/observable.1511070136.txt.gz · Last modified: 2017/12/04 08:01 (external edit)
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