Effective action

In quantum field theory, the effective action is a modified expression for the action, which takes into account quantum-mechanical corrections, in the following sense:
In classical mechanics, the equations of motion can be derived from the action by the principle of stationary action. This is not the case in quantum mechanics, where the amplitudes of all possible motions are added up in a path integral. However, if the action is replaced by the effective action, the equations of motion for the vacuum expectation values of the fields can be derived from the requirement that the effective action be stationary. For example, a field



ϕ


{\displaystyle \phi }
with a potential



V
(
ϕ
)


{\displaystyle V(\phi )}
, at a low temperature, will not settle in a local minimum of



V
(
ϕ
)


{\displaystyle V(\phi )}
, but in a local minimum of the effective potential which can be read off from the effective action.
Furthermore, the effective action can be used instead of the action in the calculation of correlation functions, and then only one-particle-irreducible correlation functions should be taken into account.

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