Editing Effective action (section)

==Methods for calculating the effective action==

A direct way to calculate the effective action <math>\Gamma[\phi_0]</math> perturbatively as a sum of 1PI diagrams is to sum over all 1PI vacuum diagrams acquired using the Feynman rules derived from the shifted action <math>S[\phi+\phi_0]</math>. This works because any place where <math>\phi_0</math> appears in any of the propagators or vertices is a place where an external <math>\phi</math> line could be attached. This is very similar to the [[background field method]] which can also be used to calculate the effective action.

Alternatively, the [[One-loop Feynman diagram|one-loop]] approximation to the action can be found by considering the expansion of the partition function around the classical vacuum expectation value field configuration <math>\phi(x) = \phi_{\text{cl}}(x) +\delta \phi(x)</math>, yielding<ref>{{cite book|first=H.|last=Kleinert|title=Particles and Quantum Fields|publisher=World Scientific Publishing|date=2016|chapter=22|chapter-url=http://users.physik.fu-berlin.de/~kleinert/b6/psfiles/Chapter-21-direffac.pdf|page=1257|isbn=9789814740920}}</ref><ref>{{cite book|last=Zee|first=A.|author-link=Anthony Zee|date=2010|title=Quantum Field Theory in a Nutshell|publisher=Princeton University Press|edition=2|pages=239–240|isbn=9780691140346}}</ref>

:<math>
\Gamma[\phi_{\text{cl}}] = S[\phi_{\text{cl}}]+\frac{i}{2}\text{Tr}\bigg[\ln \frac{\delta^2 S[\phi]}{\delta \phi(x)\delta \phi(y)}\bigg|_{\phi = \phi_{\text{cl}}} \bigg]+\cdots.
</math>