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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 VEVs of the fields can be derived from the requirement that the effective action be stationary. For example, a field ${\displaystyle \phi }$ with a potential ${\displaystyle V(\phi )}$, at a low temperature, will not settle in a local minimum of ${\displaystyle V(\phi )}$, 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.

Mathematical details

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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 in terms of the source field J, Z, the energy functional is its logarithm.

${\displaystyle E=i\ln Z}$

Some physicists use W instead where W=-E. See sign conventions

In statistical mechanics,

${\displaystyle E=-\ln Z}$

Just as Z is interpreted as the generating functional (aka characteristic function(al)/moment-generating function(al) of the probability distribution function(al) e/Z) of the time ordered VEVs/Schwinger function (aka moments) (see path integral formulation), E (aka 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.

${\displaystyle <\phi (x_{1})\cdots \phi (x_{n})>_{con}=(-i)^{n+1}\left.{\frac {\delta ^{n}E}{\delta J(x_{1})\cdots \delta J(x_{n})}}\right|_{J=0}}$

or

${\displaystyle <\phi ^{i_{1}}\cdots \phi ^{i_{n}}>_{con}=(-i)^{n+1}E^{,i_{1}\dots i_{n}}|_{J=0}}$

in the deWitt notation

Then the n-point correlation function is the sum over all the possible partitions of the fields involved in the product into products of connected correlation functions. To clarify with an example,

${\displaystyle <\phi (x_{1})\phi (x_{2})\phi (x_{3})>=<\phi (x_{1})\phi (x_{2})\phi (x_{3})>_{con}+<\phi (x_{1})\phi (x_{2})>_{con}<\phi (x_{3})>_{con}+<\phi (x_{1})\phi (x_{3})>_{con}<\phi (x_{2})>_{con}+<\phi (x_{1})>_{con}<\phi (x_{2})\phi (x_{3})>_{con}+<\phi (x_{1})>_{con}<\phi (x_{2})>_{con}<\phi (x_{3})>_{con}}$

Assuming E is a convex functional (which is debatable), the Legendre transformation gives a one-to-one correspondence between the configuration space of all source fields and its dual vector space, the configuration space of all φ fields. If E isn't convex, we take the Fenchel conjugate instead. φ here is a classical field and not a quantum field operator.

Slightly out of the usual sign conventions for Legendre transforms, the value

${\displaystyle \phi =-{\delta \over \delta J}E}$

or

${\displaystyle \phi ^{i}=-E^{,i}}$

is associated to J. This agrees with the time ordered VEV <φ>_J. The Legendre transform of E is the effective action (this corresponds to the rate function, which is the Fenchel conjugate of the cumulant-generating function, a common construction in statistics; e.g. the Chernoff bound)

${\displaystyle \Gamma =-<J,\phi >-E}$

or

${\displaystyle \Gamma =-J_{i}\phi ^{i}-E}$

where

${\displaystyle \phi =-{\delta \over \delta J}E}$

and

${\displaystyle J=-{\delta \over \delta \phi }\Gamma }$

or

${\displaystyle J_{i}=-\Gamma _{,i}}$.

There are some caveats, though, the major one being we don't have a true one-to-one correspondence between the dual configuration spaces.

If we perform a Taylor series expansion of Γ about the VEV, φ=<φ>, the coefficients give us the amputated correlation functions, also called one particle irreducible correlation functions or 1PI correlation functions.

We could also perform a Taylor series expansion about φ=0 or some other value which is not the VEV. But in that case, we have to include tadpoles in all our analyses, complicating our calculations. This is done, for example, if we don't know what the true VEV is.

• Quantum field theory