Misplaced Pages

This is an old revision of this page, as edited by OpenScience709 (talk | contribs) at 22:01, 6 September 2021 (Rewrote the entire article since it is severely lacking. Expanded on the connection between the three generation functionals as well as introduced new sections on additional information concerning the effective action.). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

In quantum field theory, the quantum effective action is a modified expression for the classical action taking into account quantum corrections while ensuring that the principle of least action applies, meaning that extremizing the effective action yields the equations of motion for the vacuum expectation values of the quantum fields. The effective action also acts as a generating functional for one-particle irreducible correlation functions. The potential component of the effective action is called the effective potential, with the expectation value of the true vacuum being the minimum of this potential rather than the classical potential, making it important for studying spontaneous symmetry breaking.

It was first defined perturbatively by Jeffrey Goldstone 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.

Generating Functionals

These generation 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 ${\displaystyle S}$ can be fully described in the path integral formalism using the partition functional

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

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

${\displaystyle \langle {\hat {\phi }}(x_{1})\dots {\hat {\phi }}(x_{n})\rangle =(-i)^{n}{\frac {1}{Z}}{\frac {\delta ^{n}Z}{\delta J(x_{1})\dots \delta J(x_{n})}}{\bigg |}_{J=0}.}$

One can define another useful generating functional ${\displaystyle W=-i\ln Z}$ responsible for generating connected correlation functions

${\displaystyle \langle {\hat {\phi }}(x_{1})\cdots {\hat {\phi }}(x_{n})\rangle _{\text{con}}=(-i)^{n-1}{\frac {\delta ^{n}W}{\delta J(x_{1})\dots \delta J(x_{n})}}{\bigg |}_{J=0},}$

which is calculated perturbatively as the sum of all connected diagrams. Here connected is interpreted in the sense of the cluster decomposition theorem, 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 ${\displaystyle W}$

${\displaystyle \Gamma =W-\int d^{4}xJ_{\phi }(x)\phi (x),}$

where ${\displaystyle J_{\phi }}$ is the current for which the scalar field has the expectation value ${\displaystyle \phi (x)}$, hence is defined implicitly as the solution to

${\displaystyle \langle {\hat {\phi }}(x)\rangle _{J}={\frac {\delta W}{\delta J(x)}}=\phi (x).}$

The hat notation is used to distinguish between the scalar field operator ${\displaystyle {\hat {\phi }}(x)}$ and some particular expectation value of that operator ${\displaystyle \phi (x)}$, which depends on the state under consideration. Taking the functional derivative of the Legendre transformation with respect to ${\displaystyle \phi (x)}$ yields

${\displaystyle J_{\phi }(x)=-{\frac {\delta \Gamma }{\delta \phi (x)}}.}$

In the absence of an external current ${\displaystyle J_{\phi }(x)=0}$, 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 generation 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

${\displaystyle \langle {\hat {\phi }}(x_{1})\dots {\hat {\phi }}(x_{n})\rangle _{1PI}=-i{\frac {\delta ^{n}\Gamma }{\delta \phi (x_{1})\dots \delta \phi (x_{n})}}{\bigg |}_{J=0},}$

with ${\displaystyle \Gamma }$ being the sum of all 1PI Feynman diagrams. The close connection between ${\displaystyle W}$ and ${\displaystyle \Gamma }$ 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, is the inverse of the 1PI two-point correlation function

${\displaystyle \Delta (x,y)={\frac {\delta ^{2}W}{\delta J(x)\delta J(y)}}={\frac {\delta \phi (x)}{\delta J(y)}}={\bigg (}{\frac {\delta J(y)}{\delta \phi (x)}}{\bigg )}^{-1}=-{\bigg (}{\frac {\delta ^{2}\Gamma (\phi )}{\delta \phi (x)\delta \phi (y)}}{\bigg )}^{-1}=-\Pi ^{-1}(x,y).}$

Methods for calculating the effective action

A direct way to calculate the effective action ${\displaystyle \Gamma }$ 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 ${\displaystyle S}$. This works because any place where ${\displaystyle \phi _{0}}$ appears in any of the propagators or vertices is a place where an external ${\displaystyle \phi }$ line could be attached.

Alternatively, the ${\displaystyle {\mathcal {O}}(\hbar )}$ approximation to the action can be found by considering the expansion of the partition function around the classical vacuum expectation value field configuration ${\displaystyle \phi (x)=\phi _{\text{cl}}(x)+\delta \phi (x)}$, yielding the one-loop approximation

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

Symmetries

Symmetries of the classical action ${\displaystyle S}$ are not automatically symmetries of the quantum effective action ${\displaystyle \Gamma }$. If the classical action has a continuous symmetry depending on some functional ${\displaystyle F}$

${\displaystyle \phi (x)\rightarrow \phi (x)+\epsilon F,}$

then this directly imposes the constraint

${\displaystyle 0=\int d^{4}x\langle F\rangle _{J_{\phi }}{\frac {\delta \Gamma }{\delta \phi (x)}}.}$

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

${\displaystyle \phi (x)\rightarrow \phi (x)+\epsilon \langle F\rangle _{J_{\phi }}.}$

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

${\displaystyle F=a(x)+\int d^{4}y\ b(x,y)\phi (y).}$

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.

Convexity

The effective potential ${\displaystyle V(\phi )}$ at ${\displaystyle \phi (x)}$ always gives the minimum of the expectation value of the energy density for the set of states ${\displaystyle |\Omega \rangle }$ satisfying ${\displaystyle \langle \Omega |{\hat {\phi }}|\Omega \rangle =\phi (x)}$. This definition over a large set of states is necessary because multiple different states may result in the same expectation value. It can further be shown that the effective potential is necessarily a convex function ${\displaystyle V''(\phi )\geq 0}$.

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 where the apparent effective potential fails to be convex. The contradiction occurs when one is dealing with a situation in which the vacuum is unstable, while perturbation theory necessarily assumes that the vacuum is stable. For example, consider an apparent effective potential ${\displaystyle V_{0}(\phi )}$ with two local minima whose expectation values ${\displaystyle \phi _{1}}$ and ${\displaystyle \phi _{2}}$ are the expectation values for the states ${\displaystyle |\Omega _{1}\rangle }$ and ${\displaystyle |\Omega _{2}\rangle }$, respectively. Then any ${\displaystyle \phi }$ in the non-convex region of ${\displaystyle V_{0}(\phi )}$ can also be acquired for some ${\displaystyle \lambda \in }$ using

${\displaystyle |\Omega \rangle \propto {\sqrt {\lambda }}|\Omega _{1}\rangle +{\sqrt {1-\lambda }}|\Omega _{2}\rangle .}$

However, the energy density of this state is ${\displaystyle \lambda V_{0}(\phi _{1})+(1-\lambda )V_{0}(\phi _{2})<V_{0}(\phi )}$ meaning ${\displaystyle V_{0}(\phi )}$ cannot be the correct effective potential at ${\displaystyle \phi }$ since it did not minimize the energy density. Rather the true effective potential ${\displaystyle V(\phi )}$ is equal to or lower than this linear construction, which restores convexity.

References

1. Weinberg, Steven; Goldstone, Jeffrey (August 1962). "Broken Symmetries". Phys. Rev. 127 (3): 965–970. doi:10.1103/PhysRev.127.965. Retrieved 2021-09-06.
2. DeWitt, B.; DeWitt, C. (1987). Relativité, groupes et topologie = Relativity, groups and topology : lectures delivered at Les Houches during the 1963 session of the Summer School of Theoretical Physics, University of Grenoble. Gordon and Breach. ISBN 0677100809.
3. Jona-Lasinio, Giovanni (31 August 1964). "Relativistic Field Theories with Symmetry-Breaking Solutions". Il Nuovo Cimento. 34 (6): 1790–1795. doi:10.1007/BF02750573. Retrieved 2021-09-06.
4. Kleinert, Hagen (2016). "22" (PDF). Particles and Quantum Fields. World Scientific Publishing. p. 1257. ISBN 9789814740920.
5. Weinberg, Steven (1995). "16". The Quantum Theory of Fields Volume 2. Vol. 2. Cambridge University Press. p. 72-74. ISBN 9780521670548.

Further Reading

• M.D. Schwartz: Quantum Field Theory and the Standard Model, Cambridge University Press 2014
• D.J. Toms: The Schwinger Action Principle and Effective Action, Cambridge University Press 2007
• S. Weinberg: The Quantum Theory of Fields, Vol.II, Cambridge University Press 1996

• Quantum field theory