Editing Effective action (section)

== Generating functionals ==

''These generating functionals also have applications in [[statistical mechanics]] and [[information theory]], with slightly different factors of <math>i</math> and sign conventions.''

A quantum field theory with action <math>S[\phi]</math> can be fully described in the [[path integral formulation|path integral]] formalism using the [[partition function (quantum field theory)|partition functional]]

:<math>
Z[J] = \int \mathcal D \phi e^{iS[\phi] + i \int d^4 x \phi(x)J(x)}.
</math>

Since it corresponds to vacuum-to-vacuum transitions in the presence of a classical external current <math>J(x)</math>, it can be evaluated perturbatively as the sum of all connected and disconnected [[Feynman diagrams]]. It is also the generating functional for correlation functions

:<math>
\langle \hat \phi(x_1) \dots \hat \phi(x_n)\rangle = (-i)^n \frac{1}{Z[J]} \frac{\delta^n Z[J]}{\delta J(x_1) \dots \delta J(x_n)}\bigg|_{J=0},
</math>

where the scalar field operators are denoted by <math>\hat \phi(x)</math>. One can define another useful generating functional <math>W[J] = -i\ln Z[J]</math> responsible for generating connected correlation functions 

:<math>
\langle \hat \phi(x_1) \cdots \hat \phi(x_n)\rangle_{\text{con}} = (-i)^{n-1}\frac{\delta^n W[J]}{\delta J(x_1) \dots \delta J(x_n)}\bigg|_{J=0},
</math>

which is calculated perturbatively as the sum of all connected diagrams.<ref>{{cite book|last=Zinn-Justin|first=J.|author-link=Jean Zinn-Justin|date=1996|title=Quantum Field Theory and Critical Phenomena|url=|doi=|location=Oxford|publisher=Oxford University Press|chapter=6|pages=119–122|isbn=978-0198509233}}</ref> 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 <math>W[J]</math>

{{Equation box 1
|title=
|indent=:
|equation = <math>\Gamma[\phi] = W[J_\phi] - \int d^4 x J_\phi(x) \phi(x),</math>
|border
|border colour =#50C878
|background colour = #ECFCF4
}}

where <math>J_\phi</math> is the [[Source field|source current]] for which the scalar field has the expectation value <math>\phi(x)</math>, often called the classical field, defined implicitly as the solution to
{{multiple image
 | align = right
 | direction = vertical
 | width = 200
 | image1 = Not 1PI Feynman graph example.svg
 | alt1 = An example of a Feynman diagram that can be cut into two separate diagrams by cutting one propagator. 
 | caption1 = Example of a diagram that is not one-particle irreducible.
 | image2 = 1PI Feynman graph example.svg
 | alt2 = An example of a Feynman diagram that can not be cut into two separate diagrams by cutting one propagator. 
 | caption2 = Example of a diagram that is one-particle irreducible.
}}

:<math>
\phi(x) = \langle \hat \phi(x)\rangle_J = \frac{\delta W[J]}{\delta J(x)}.
</math>

As an expectation value, the classical field can be thought of as the weighted average over quantum fluctuations in the presence of a current <math>J(x)</math> that sources the scalar field. Taking the [[functional derivative]] of the Legendre transformation with respect to <math>\phi(x)</math> yields

:<math>
J_\phi(x) = -\frac{\delta \Gamma[\phi]}{\delta \phi(x)}.
</math>

In the absence of an source <math>J_\phi(x) = 0</math>, 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

:<math>
\langle \hat \phi(x_1) \dots \hat \phi(x_n)\rangle_{\mathrm{1PI}} = i \frac{\delta^n \Gamma[\phi]}{\delta \phi(x_1) \dots \delta \phi(x_n)}\bigg|_{J=0},
</math>

with <math>\Gamma[\phi]</math> being the sum of all 1PI Feynman diagrams. The close connection between <math>W[J]</math> and <math>\Gamma[\phi]</math> 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]] <math>\Delta(x,y)</math>, is the inverse of the 1PI two-point correlation function

:<math>
\Delta(x,y) = \frac{\delta^2 W[J]}{\delta J(x)\delta J(y)} = \frac{\delta \phi(x)}{\delta J(y)} = \bigg(\frac{\delta J(y)}{\delta \phi(x)}\bigg)^{-1} = -\bigg(\frac{\delta^2 \Gamma[\phi]}{\delta \phi(x)\delta \phi(y)}\bigg)^{-1} = -\Pi^{-1}(x,y).
</math>