advanced_notions:observable

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+ | ====== Observable ====== | ||

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+ | <tabbox Why is it interesting?> | ||

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+ | <tabbox Layman> | ||

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+ | <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. | ||

+ | </note> | ||

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+ | <tabbox Student> | ||

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+ | <note tip> | ||

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

+ | </note> | ||

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+ | <tabbox Researcher> | ||

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+ | 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. | ||

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+ | 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. | ||

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+ | 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) | ||

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+ | <tabbox Examples> | ||

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+ | --> Example1# | ||

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+ | <-- | ||

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+ | --> Example2:# | ||

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+ | <-- | ||

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+ | <tabbox FAQ> | ||

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+ | <tabbox History> | ||

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+ | </tabbox> | ||

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