Effective action

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

A quantum field theory with action $S[\backslash phi]$ can be fully described in the path integral formalism using the partition functional

:$$

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

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

:$$

\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},

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

:$$

\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},

which is calculated perturbatively as the sum of all connected diagrams.{{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}} 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 $W[J]$

{{Equation box 1

|title=

|indent=:

|equation = $\backslash Gamma[\backslash phi]\; =\; W[J\_\backslash phi]\; -\; \backslash int\; d^4\; x\; J\_\backslash phi(x)\; \backslash phi(x),$

|border

|border colour =#50C878

|background colour = #ECFCF4

}}

where $J\_\backslash phi$ is the source current for which the scalar field has the expectation value $\backslash phi(x)$, 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.

}}

:$$

\phi(x) = \langle \hat \phi(x)\rangle_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 $J(x)$ that sources the scalar field. Taking the functional derivative of the Legendre transformation with respect to $\backslash phi(x)$ yields

:$$

J_\phi(x) = -\frac{\delta \Gamma[\phi]}{\delta \phi(x)}.

In the absence of an source $J\_\backslash phi(x)\; =\; 0$, 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

:$$

\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},

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

:$$

\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).

A direct way to calculate the effective action $\backslash Gamma[\backslash phi\_0]$ perturbatively as a sum of 1PI diagrams is to sum over all 1PI vacuum diagrams acquired using the Feynman rules derived from the shifted action $S[\backslash phi+\backslash phi\_0]$. This works because any place where $\backslash phi\_0$ appears in any of the propagators or vertices is a place where an external $\backslash phi$ line could be attached. This is very similar to the background field method which can also be used to calculate the effective action.

Alternatively, the one-loop approximation to the action can be found by considering the expansion of the partition function around the classical vacuum expectation value field configuration $\backslash phi(x)\; =\; \backslash phi\_\{\backslash text\{cl\}\}(x)\; +\backslash delta\; \backslash phi(x)$, yielding{{cite book|first=H.|last=Kleinert|title=Particles and Quantum Fields|publisher=World Scientific Publishing|date=2016|chapter=22|chapter-url=http://users.physik.fu-berlin.de/~kleinert/b6/psfiles/Chapter-21-direffac.pdf|page=1257|isbn=9789814740920}}{{cite book|last=Zee|first=A.|author-link=Anthony Zee|date=2010|title=Quantum Field Theory in a Nutshell|publisher=Princeton University Press|edition=2|pages=239–240|isbn=9780691140346}}

:$$

\Gamma[\phi_{\text{cl}}] = S[\phi_{\text{cl}}]+\frac{i}{2}\text{Tr}\bigg[\ln \frac{\delta^2 S[\phi]}{\delta \phi(x)\delta \phi(y)}\bigg|_{\phi = \phi_{\text{cl}}} \bigg]+\cdots.

Symmetries of the classical action $S[\backslash phi]$ are not automatically symmetries of the quantum effective action $\backslash Gamma[\backslash phi]$. If the classical action has a continuous symmetry depending on some functional $F[x,\backslash phi]$

:$$

\phi(x) \rightarrow \phi(x) + \epsilon F[x,\phi],

then this directly imposes the constraint

:$$

0 = \int d^4 x \langle F[x,\phi]\rangle_{J_\phi}\frac{\delta \Gamma[\phi]}{\delta \phi(x)}.

This identity is an example of a Slavnov–Taylor identity. It is identical to the requirement that the effective action is invariant under the symmetry transformation

:$$

\phi(x) \rightarrow \phi(x) + \epsilon \langle F[x,\phi]\rangle_{J_\phi}.

This symmetry is identical to the original symmetry for the important class of linear symmetries

:$F[x,\backslash phi]\; =\; a(x)+\backslash int\; d^4\; y\; \backslash \; b(x,y)\backslash phi(y).$

For non-linear functionals the two symmetries generally differ because the average of a non-linear functional is not equivalent to the functional of an average.

File:effective_potential_SVG.svg

For a spacetime with volume $\backslash mathcal\; V\_4$, the effective potential is defined as $V(\backslash phi)\; =\; -\; \backslash Gamma[\backslash phi]/\backslash mathcal\; V\_4$. With a Hamiltonian $H$, the effective potential $V(\backslash phi)$ at $\backslash phi(x)$ always gives the minimum of the expectation value of the energy density $\backslash langle\; \backslash Omega|H|\backslash Omega\backslash rangle$ for the set of states $|\backslash Omega\backslash rangle$ satisfying $\backslash langle\backslash Omega|\; \backslash hat\; \backslash phi|\; \backslash Omega\backslash rangle\; =\; \backslash phi(x)$.{{cite book|first=S.|last=Weinberg|title=The Quantum Theory of Fields: Modern Applications|publisher=Cambridge University Press|date=1995|chapter=16|volume=2|pages=72–74|isbn=9780521670548}} This definition over multiple states is necessary because multiple different states, each of which corresponds to a particular source current, may result in the same expectation value. It can further be shown that the effective potential is necessarily a convex function $V\text{'}\text{'}(\backslash phi)\; \backslash geq\; 0$.{{cite book|last1=Peskin|first1=M.E.|author1-link=Michael Peskin|last2=Schroeder|first2=D.V.|date=1995|title=An Introduction to Quantum Field Theory|publisher=Westview Press|pages=368–369|isbn=9780201503975}}

Calculating the effective potential perturbatively can sometimes yield a non-convex result, such as a potential that has two local minima. However, the true effective potential is still convex, becoming approximately linear in the region where the apparent effective potential fails to be convex. The contradiction occurs in calculations around unstable vacua since perturbation theory necessarily assumes that the vacuum is stable. For example, consider an apparent effective potential $V\_0(\backslash phi)$ with two local minima whose expectation values $\backslash phi\_1$ and $\backslash phi\_2$ are the expectation values for the states $|\backslash Omega\_1\backslash rangle$ and $|\backslash Omega\_2\backslash rangle$, respectively. Then any $\backslash phi$ in the non-convex region of $V\_0(\backslash phi)$ can also be acquired for some $\backslash lambda\; \backslash in\; [0,1]$ using

:$$

|\Omega\rangle \propto \sqrt \lambda |\Omega_1\rangle+\sqrt{1-\lambda}|\Omega_2\rangle.

However, the energy density of this state is

• Background field method
• Correlation function
• Path integral formulation
• Renormalization group
• Spontaneous symmetry breaking

{{reflist|50em}}

• Das, A. : Field Theory: A Path Integral Approach, World Scientific Publishing 2006
• Schwartz, M.D.: Quantum Field Theory and the Standard Model, Cambridge University Press 2014
• Toms, D.J.: The Schwinger Action Principle and Effective Action, Cambridge University Press 2007
• Weinberg, S.: The Quantum Theory of Fields: Modern Applications, Vol.II, Cambridge University Press 1996

Category:Quantum field theory