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Inquantum field theory, thequantum effective action is a modified expression for theclassicalaction taking into account quantum corrections while ensuring that theprinciple of least action applies, meaning that extremizing the effective action yields theequations of motion for thevacuum expectation values of the quantum fields. The effective action also acts as agenerating functional for one-particle irreduciblecorrelation functions. The potential component of the effective action is called theeffective potential, with the expectation value of the true vacuum being the minimum of this potential rather than the classical potential, making it important for studyingspontaneous symmetry breaking.

It was first definedperturbatively byJeffrey Goldstone andSteven Weinberg in 1962,^[1] while the non-perturbative definition was introduced byBryce DeWitt in 1963^[2] and independently byGiovanni Jona-Lasinio in 1964.^[3]

The article describes the effective action for a singlescalar field, however, similar results exist for multiple scalar orfermionic fields.

These generating functionals also have applications instatistical mechanics andinformation 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 thepath integral formalism using thepartition 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 disconnectedFeynman 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.^[4] Here connected is interpreted in the sense of thecluster 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 theLegendre transformation of${\displaystyle W[J]}$

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

Example of a diagram that is not one-particle irreducible.

Example of a diagram that is one-particle irreducible.

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

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 thefunctional 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 forone-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 thepropagator${\displaystyle \mathrm{\Delta }(x,y)}$, is the inverse of the 1PI two-point correlation function

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

A direct way to calculate the effective action${\displaystyle \mathrm{\Gamma }[{\varphi }_0]}$ 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${\displaystyle S[\varphi +{\varphi }_0]}$. This works because any place where${\displaystyle {\varphi }_0}$ appears in any of the propagators or vertices is a place where an external${\displaystyle \varphi }$ line could be attached. This is very similar to thebackground field method which can also be used to calculate the effective action.

Alternatively, theone-loop approximation to the action can be found by considering the expansion of the partition function around the classical vacuum expectation value field configuration${\displaystyle \varphi (x)={\varphi }_{\text{cl}}(x)+\delta \varphi (x)}$, yielding^[5][6]

${\displaystyle \mathrm{\Gamma }[{\varphi }_{\text{cl}}]=S[{\varphi }_{\text{cl}}]+\frac{i}{2}\text{Tr}[\ln \frac{{\delta }^{2}S[\varphi ]}{\delta \varphi (x)\delta \varphi (y)}{|}_{\varphi ={\varphi }_{\text{cl}}}]+\cdots .}$

Symmetries of the classical action${\displaystyle S[\varphi ]}$ are not automatically symmetries of the quantum effective action${\displaystyle \mathrm{\Gamma }[\varphi ]}$. If the classical action has acontinuous symmetry depending on some functional${\displaystyle F[x,\varphi ]}$

${\displaystyle \varphi (x)\to \varphi (x)+ϵF[x,\varphi ],}$

then this directly imposes the constraint

${\displaystyle 0=\int d^{4}x⟨F[x,\varphi ]{⟩}_{J_{\varphi }}\frac{\delta \mathrm{\Gamma }[\varphi ]}{\delta \varphi (x)}.}$

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

${\displaystyle \varphi (x)\to \varphi (x)+ϵ⟨F[x,\varphi ]{⟩}_{J_{\varphi }}.}$

This symmetry is identical to the original symmetry for the important class oflinear symmetries

${\displaystyle F[x,\varphi ]=a(x)+\int d^{4}yb(x,y)\varphi (y).}$

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.

For a spacetime with volume${\displaystyle {\mathcal{V}}_4}$, the effective potential is defined as${\displaystyle V(\varphi )=-\mathrm{\Gamma }[\varphi ]/{\mathcal{V}}_4}$. With aHamiltonian${\displaystyle H}$, the effective potential${\displaystyle V(\varphi )}$ at${\displaystyle \varphi (x)}$ always gives the minimum of the expectation value of theenergy density${\displaystyle ⟨\mathrm{\Omega }|H|\mathrm{\Omega }⟩}$ for the set of states${\displaystyle |\mathrm{\Omega }⟩}$ satisfying${\displaystyle ⟨\mathrm{\Omega }|\hat{\varphi }|\mathrm{\Omega }⟩=\varphi (x)}$.^[7] 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 aconvex function${\displaystyle V^{″}(\varphi )\ge 0}$.^[8]

Calculating the effective potential perturbatively can sometimes yield a non-convex result, such as a potential that has twolocal 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${\displaystyle V_0(\varphi )}$ with two local minima whose expectation values${\displaystyle {\varphi }_1}$ and${\displaystyle {\varphi }_2}$ are the expectation values for the states${\displaystyle |{\mathrm{\Omega }}_1⟩}$ and${\displaystyle |{\mathrm{\Omega }}_2⟩}$, respectively. Then any${\displaystyle \varphi }$ in the non-convex region of${\displaystyle V_0(\varphi )}$ can also be acquired for some${\displaystyle \lambda \in [0,1]}$ using

${\displaystyle |\mathrm{\Omega }⟩\propto \sqrt{\lambda }|{\mathrm{\Omega }}_1⟩+\sqrt{1-\lambda }|{\mathrm{\Omega }}_2⟩.}$

However, the energy density of this state is meaning${\displaystyle V_0(\varphi )}$ cannot be the correct effective potential at${\displaystyle \varphi }$ since it did not minimize the energy density. Rather the true effective potential${\displaystyle V(\varphi )}$ is equal to or lower than this linear construction, which restores convexity.

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

• 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

• Quantum field theory

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