Anomalies [The Physics Travel Guide]

--- advanced_notions:quantum_field_theory:anomalies [2019/01/31 11:49]
jakobadmin
+++ advanced_notions:quantum_field_theory:anomalies [2019/07/01 09:37] (current)
jakobadmin [Concrete]
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 <tabbox Concrete>
+A classical theory possesses a symmetry if the action $S(\phi)$ is unchanged by a transformation $\phi \to \delta \phi$. In a quantum theory, however, we have a symmetry if the path integral $\int D \phi e^{iS(\phi)}$ is invariant under a given transformation $\phi \to \delta \phi$. The key observation is now that invariance of the action $S(\phi)$ does not necessarily imply invariance of the path integral since the measure $D \phi$ can be non-invariant too. In more technical terms, the reason for this is that whenever we change the integration variables, we need to remember that the Jacobian can be non-trivial.
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 An anomaly is an obstruction to the construction of a quantum theory that has the same symmetry group as its action.