Editing Effective action (section)

==Symmetries==

[[Symmetries in quantum mechanics|Symmetries]] of the classical action <math>S[\phi]</math> are not automatically symmetries of the quantum effective action <math>\Gamma[\phi]</math>. If  the classical action has a [[continuous symmetry]] depending on some functional <math>F[x,\phi]</math>

:<math>
\phi(x) \rightarrow \phi(x) + \epsilon F[x,\phi],
</math>

then this directly imposes the constraint

:<math>
0 = \int d^4 x \langle F[x,\phi]\rangle_{J_\phi}\frac{\delta \Gamma[\phi]}{\delta \phi(x)}.
</math>

This identity is an example of a [[Slavnov–Taylor identities|Slavnov–Taylor identity]]. It is identical to the requirement that the effective action is invariant under the symmetry transformation

:<math>
\phi(x) \rightarrow \phi(x) + \epsilon \langle F[x,\phi]\rangle_{J_\phi}.
</math>

This symmetry is identical to the original symmetry for the important class of [[Linear form|linear]] symmetries 

:<math>F[x,\phi] = a(x)+\int d^4 y \ b(x,y)\phi(y).</math>

For non-linear functionals the two symmetries generally differ because the average of a non-linear functional is not equivalent to the functional of an average.