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 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 tree diagrams should be taken into account. Everything in the following article also applies to statistical mechanics. However, the signs and factors of i are different in that case. Given the partition function Z in terms of the source field J, the energy functional is its logarithm. Some physicists use W instead, W = −E. See sign conventions In multiple areas of mathematics and information theory, including statistical mechanics, one writes the partition function as Just as Z is interpreted as the generating functional (aka characteristic function(al)/moment-generating function(al) of the probability distribution function(al) e−S/Z) of the time ordered VEVs/Schwinger function (aka moments) (see path integral formulation), E (a.k.a. the second characteristic function(al)/cumulant-generating function(al)) is the generator of 'connected' time ordered VEVs/connected Schwinger functions (i.e. the cumulants) where connected here is interpreted in the sense of the cluster decomposition theorem which means that these functions approach zero at large spacelike separations, or in approximations using Feynman diagrams, connected components of the graph.