Effective action

It was first defined perturbatively by Jeffrey Goldstone, Abdus Salam and Steven Weinberg in 1962, while the non-perturbative definition was introduced by Bryce DeWitt in 1963 and independently by Giovanni Jona-Lasinio in 1964.

The article describes the effective action for a single scalar field, however, similar results exist for multiple scalar or fermionic fields.

Generating functionals

These generating functionals also have applications in statistical mechanics and information theory, with slightly different factors of ${\displaystyle i}$ and sign conventions.

A quantum field theory with action ${\displaystyle S[\varphi ]}$ can be fully described in the path integral formalism using the partition functional

${\displaystyle Z[J]=\int \mathcal{D}\varphi e^{iS[\varphi ]+i\int d^{4}x\varphi (x)J(x)}.}$

Since it corresponds to vacuum-to-vacuum transitions in the presence of a classical external current ${\displaystyle J(x)}$, it can be evaluated perturbatively as the sum of all connected and disconnected Feynman diagrams. It is also the generating functional for correlation functions

${\displaystyle ⟨\hat{\varphi }(x_1)\dots \hat{\varphi }(x_n)⟩=(-i{)}^n\frac{1}{Z[J]}\frac{{\delta }^{n}Z[J]}{\delta J(x_1)\dots \delta J(x_n)}{|}_{J=0},}$

where the scalar field operators are denoted by ${\displaystyle \hat{\varphi }(x)}$. One can define another useful generating functional ${\displaystyle W[J]=-i\ln Z[J]}$ responsible for generating connected correlation functions

${\displaystyle ⟨\hat{\varphi }(x_1)\cdots \hat{\varphi }(x_n){⟩}_{\text{con}}=(-i{)}^{n-1}\frac{{\delta }^{n}W[J]}{\delta J(x_1)\dots \delta J(x_n)}{|}_{J=0},}$

which is calculated perturbatively as the sum of all connected diagrams. Here connected is interpreted in the sense of the cluster decomposition, meaning that the correlation functions approach zero at large spacelike separations. General correlation functions can always be written as a sum of products of connected correlation functions.

The quantum effective action is defined using the Legendre transformation of ${\displaystyle W[J]}$

${\displaystyle \mathrm{\Gamma }[\varphi ]=W[J_{\varphi }]-\int d^{4}x{J}_{\varphi }(x)\varphi (x),}$

where ${\displaystyle J_{\varphi }}$ is the source current for which the scalar field has the expectation value ${\displaystyle \varphi (x)}$, often called the classical field, defined implicitly as the solution to

${\displaystyle \varphi (x)=⟨\hat{\varphi }(x){⟩}_J=\frac{\delta W[J]}{\delta J(x)}.}$

As an expectation value, the classical field can be thought of as the weighted average over quantum fluctuations in the presence of a current ${\displaystyle J(x)}$ that sources the scalar field. Taking the functional derivative of the Legendre transformation with respect to ${\displaystyle \varphi (x)}$ yields

${\displaystyle J_{\varphi }(x)=-\frac{\delta \mathrm{\Gamma }[\varphi ]}{\delta \varphi (x)}.}$

In the absence of an source ${\displaystyle J_{\varphi }(x)=0}$, the above shows that the vacuum expectation value of the fields extremize the quantum effective action rather than the classical action. This is nothing more than the principle of least action in the full quantum field theory. The reason for why the quantum theory requires this modification comes from the path integral perspective since all possible field configurations contribute to the path integral, while in classical field theory only the classical configurations contribute.

The effective action is also the generating functional for one-particle irreducible (1PI) correlation functions. 1PI diagrams are connected graphs that cannot be disconnected into two pieces by cutting a single internal line. Therefore, we have

${\displaystyle ⟨\hat{\varphi }(x_1)\dots \hat{\varphi }(x_n){⟩}_{1\mathrm{P}\mathrm{I}}=i\frac{{\delta }^n\mathrm{\Gamma }[\varphi ]}{\delta \varphi (x_1)\dots \delta \varphi (x_n)}{|}_{J=0},}$

with ${\displaystyle \mathrm{\Gamma }[\varphi ]}$ being the sum of all 1PI Feynman diagrams. The close connection between ${\displaystyle W[J]}$ and ${\displaystyle \mathrm{\Gamma }[\varphi ]}$ means that there are a number of very useful relations between their correlation functions. For example, the two-point correlation function, which is nothing less than the propagator ${\displaystyle \mathrm{\Delta }(x,y)}$, is the inverse of the 1PI two-point correlation function

${\displaystyle \mathrm{\Delta }(x,y)=\frac{{\delta }^{2}W[J]}{\delta J(x)\delta J(y)}=\frac{\delta \varphi (x)}{\delta J(y)}=(\frac{\delta J(y)}{\delta \varphi (x)}{)}^{-1}=-(\frac{{\delta }^2\mathrm{\Gamma }[\varphi ]}{\delta \varphi (x)\delta \varphi (y)}{)}^{-1}=-{\mathrm{\Pi }}^{-1}(x,y).}$