### Site Tools

 advanced_notions:observable [2018/01/02 13:13] advanced_notions:observable [2018/01/02 14:13] (current)jakobadmin ↷ Links adapted because of a move operation Both sides previous revision Previous revision 2018/01/02 14:13 jakobadmin ↷ Links adapted because of a move operation2017/12/04 09:01 external edit2017/11/19 06:57 jakobadmin [Researcher] 2017/11/19 06:43 jakobadmin [Researcher] 2017/11/19 06:42 jakobadmin created 2018/01/02 14:13 jakobadmin ↷ Links adapted because of a move operation2017/12/04 09:01 external edit2017/11/19 06:57 jakobadmin [Researcher] 2017/11/19 06:43 jakobadmin [Researcher] 2017/11/19 06:42 jakobadmin created Line 1: Line 1: + ====== 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 [[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) + + + + ​ +  ​ + + --> Example1# + + + <-- + + --> Example2:# + + + <-- + +  ​ + ​ +  ​ + + ​ +