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{{Quantum field theory|cTopic=Tools}}
{{no footnotes|date=May 2014}}
{{short description|Quantum version of the classical action}}
In ], the '''effective action''' is a modified expression for the ], which takes into account ] corrections, in the following sense:


In ], the ] can be derived from the ] by the ]. This is not the case in ], where the amplitudes of all possible motions are added up in a ]. However, if the action is replaced by the effective action, the ] for the ]s of the ]s can be derived from the requirement that the effective action be stationary. For example, a field <math>\phi</math> with a ] <math>V(\phi)</math>, at a low temperature, will not settle in a ] of <math>V(\phi)</math>, but in a local minimum of the '''effective potential''' which can be read off from the effective action. In ], the '''quantum effective action''' is a modified expression for the ] ] taking into account quantum corrections while ensuring that the ] applies, meaning that extremizing the effective action yields the ] for the ] of the quantum fields. The effective action also acts as a ] for one-particle irreducible ]. 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 ].


It was first defined ] by ] and ] in 1962,<ref>{{cite journal|last1=Weinberg|first1=S.|authorlink1=Steven Weinberg|last2=Goldstone|first2=J.|authorlink2=Jeffrey Goldstone|date=August 1962|title=Broken Symmetries|url=https://link.aps.org/doi/10.1103/PhysRev.127.965|journal=Phys. Rev.|volume=127|issue=3|pages=965–970|doi=10.1103/PhysRev.127.965|bibcode=1962PhRv..127..965G |access-date=2021-09-06}}</ref> while the non-perturbative definition was introduced by ] in 1963<ref>{{cite book|last1=DeWitt|first1=B.|author-link1=Bryce DeWitt|last2=DeWitt|first2=C.|date=1987|title=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|location=|publisher=Gordon and Breach|isbn=0677100809}}</ref> and independently by ] in 1964.<ref>{{cite journal|last1=Jona-Lasinio|first1=G.|authorlink1=Giovanni Jona-Lasinio|date=31 August 1964|title=Relativistic Field Theories with Symmetry-Breaking Solutions|url=https://doi.org/10.1007/BF02750573|journal=Il Nuovo Cimento|volume=34|issue=6|pages=1790–1795|doi=10.1007/BF02750573|bibcode=1964NCim...34.1790J |s2cid=121276897 |access-date=2021-09-06}}</ref>
Furthermore, the effective action can be used instead of the action in the calculation of ]s, and then only ] should be taken into account.


The article describes the effective action for a single ], however, similar results exist for multiple scalar or ] fields.
== Mathematical details ==


== Generating functionals ==
{{Confusing|date=December 2006}}


''Everything in the following article also applies to ]. However, the signs and factors of i are different in that case.'' ''These generating functionals also have applications in ] and ], with slightly different factors of <math>i</math> and sign conventions.''


A quantum field theory with action <math>S</math> can be fully described in the ] formalism using the ]
Given the ] ''Z'' in terms of the ] ''J'', the energy functional is its logarithm.


:<math>E = i\ln Z</math> :<math>
Z = \int \mathcal D \phi e^{iS + 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 ]. It is also the generating functional for correlation functions
Some physicists use ''W'' instead where ''W''&nbsp;=&nbsp;&minus;''E''. See ]s


:<math>
]s with two-] ]s,
\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},
the above ]s arise at first order in the ] of both ''Z'' and ''E''. The perturbation expansion for ''Z'' consists of all diagrams which are closed, while the perturbation expansion for ''E'' consists of all diagrams which are both closed and connected.]]
</math>


where the scalar field operators are denoted by <math>\hat \phi(x)</math>. One can define another useful generating functional <math>W = -i\ln Z</math> responsible for generating connected correlation functions
In multiple areas of mathematics and ], including statistical mechanics, one writes the ] as


:<math>E=-\ln Z \,</math> :<math>
\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},
</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 ], 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.
Just as ''Z'' is interpreted as the ] (aka ](al)/](al) of the ](al) ''e''<sup>&minus;''S''</sup>/''Z'') of the ] ]s/] (aka ]) (see ]), ''E'' (a.k.a. the ](al)/](al)) is the generator of "connected" ] ]s/connected Schwinger functions (i.e. the ]s) where connected here is interpreted in the sense of the ] which means that these functions approach zero at large spacelike separations, or in approximations using ]s, ] of the graph.


The quantum effective action is defined using the ] of <math>W</math>
:<math>\langle\phi(x_1)\cdots\phi(x_n)\rangle_{con}=(-i)^{n+1}\left.\frac{\delta^n E}{\delta J(x_1)\cdots \delta J(x_n)}\right|_{J=0}</math>


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


where <math>J_\phi</math> is the ] for which the scalar field has the expectation value <math>\phi(x)</math>, often called the classical field, defined implicitly as the solution to
:<math>\langle\phi^{i_1}\cdots\phi^{i_n}\rangle_\text{con}=(-i)^{n+1}E^{,i_1\dots i_n}|_{J=0}</math>
{{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>
in the ]
\phi(x) = \langle \hat \phi(x)\rangle_J = \frac{\delta W}{\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 ] of the Legendre transformation with respect to <math>\phi(x)</math> yields
Then the ''n''-point correlation function is the sum over all the possible partitions of the fields involved in the product into products of connected correlation functions. To clarify with an example,


:<math>\begin{align} :<math>
& {} \quad \langle\phi(x_1)\phi(x_2)\phi(x_3)\rangle\\ J_\phi(x) = -\frac{\delta \Gamma}{\delta \phi(x)}.
</math>
&=\langle\phi(x_1)\phi(x_2)\phi(x_3)\rangle_\text{con}
+\langle\phi(x_1)\phi(x_2)\rangle_\text{con}\langle\phi(x_3)\rangle_\text{con}
+\langle\phi(x_1)\phi(x_3)\rangle_\text{con}\langle\phi(x_2)\rangle_\text{con} \\
&+\langle\phi(x_1)\rangle_\text{con}\langle\phi(x_2)\phi(x_3)\rangle_\text{con}
+\langle\phi(x_1)\rangle_{con}\langle\phi(x_2)\rangle_\text{con}\langle\phi(x_3)\rangle_\text{con}
\end{align}</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.
Assuming ''E'' is a ] ] (which is debatable), the ] gives a one-to-one correspondence between the ] of all source fields and its ], the configuration space of all φ fields. If ''E'' is not convex, we take the ] instead. φ here is a classical field and not a quantum field operator.


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
Slightly out of the usual ]s for Legendre transforms, the value


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


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


:<math>\phi^i=-E^{,i} \, </math> :<math>
\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}{\delta \phi(x)\delta \phi(y)}\bigg)^{-1} = -\Pi^{-1}(x,y).
</math>


==Methods for calculating the effective action==
is associated to ''J''. This agrees with the ] ] <φ><sub>J</sub>. The Legendre transform of ''E'' is the '''effective action''' (this corresponds to the ], which is the ] of the ], a common construction in ]; e.g. the ])


A direct way to calculate the effective action <math>\Gamma</math> 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 <math>S</math>. This works because any place where <math>\phi_0</math> appears in any of the propagators or vertices is a place where an external <math>\phi</math> line could be attached. This is very similar to the ] which can also be used to calculate the effective action.
:<math>\Gamma=-\langle J,\phi\rangle-E \, </math>


Alternatively, the ] approximation to the action can be found by considering the expansion of the partition function around the classical vacuum expectation value field configuration <math>\phi(x) = \phi_{\text{cl}}(x) +\delta \phi(x)</math>, yielding<ref>{{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}}</ref><ref>{{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}}</ref>
or


:<math>
:<math>\Gamma=-J_i \phi^i - E \, </math>
\Gamma = S+\frac{i}{2}\text{Tr}\bigg}{\delta \phi(x)\delta \phi(y)}\bigg|_{\phi = \phi_{\text{cl}}} \bigg]+\cdots.
</math>


==Symmetries==
where


] of the classical action <math>S</math> are not automatically symmetries of the quantum effective action <math>\Gamma</math>. If the classical action has a ] depending on some functional <math>F</math>
:<math>\phi=-{\delta\over\delta J}E</math>


:<math>
and
\phi(x) \rightarrow \phi(x) + \epsilon F,
</math>


then this directly imposes the constraint
:<math>J=-{\delta\over\delta \phi}\Gamma</math>


:<math>
or
0 = \int d^4 x \langle F\rangle_{J_\phi}\frac{\delta \Gamma}{\delta \phi(x)}.
</math>


This identity is an example of a ]. It is identical to the requirement that the effective action is invariant under the symmetry transformation
:<math>J_i=-\Gamma_{,i}. \, </math>


:<math>
There are some caveats, though, the major one being we don't have a true one-to-one correspondence between the dual configuration spaces.
\phi(x) \rightarrow \phi(x) + \epsilon \langle F\rangle_{J_\phi}.
</math>


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


:<math>F = a(x)+\int d^4 y \ b(x,y)\phi(y).</math>
Let us first consider the case without ], i.e. <math>\langle \phi \rangle =0</math> for J=0. In that case, Γ gives the zero-point energy, the first ] of Γ at φ=0 is zero, the second functional derivative gives the inverse of the full propagator, and the ''n''<sup>th</sup> functional derivative for ''n''&nbsp;&ge;&nbsp;3 gives the '''one particle irreducible correlation functions''' or '''1PI correlation functions'''. The ] relates the full propagator, the bare propagator and the 1PI self-energy. The ''n''-point connected functions are given as the sum over all trees with ''n''&nbsp;&ge;&nbsp;3 1PI's as nodes and full propagators as edges.


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.
But what if we have tadpoles? We can always adjust the source J so that there are no tadpoles, i.e. <math>\langle \phi \rangle =0</math>. This corresponds to adding a Feynman rule corresponding to a coupling to the source. For any Feynman diagram, a subtadpole is a subgraph corresponding to a component not connected to any of the external legs which arises after cutting of an edge. Any Feynman diagram with a subtadpole can be evaluated as nonzero, but we can group these diagrams into equivalence classes (two connected diagrams are equivalent if they only vary in their subtadpoles). Therefore, we only need to consider the sum of all connected graphs without subtadpoles. The sum over all graphs in an equivalence class with subtadpoles is zero, since J is adjusted so that <math>\langle \phi \rangle =0</math>. Any graph without subtadpoles do not contain any couplings to the source. A Taylor expansion of the effective action about ''φ''&nbsp;=&nbsp;0 gives the 1PI's corresponding to these value of the source according to the rules of the previous paragraph. So, we compute the 1PI's to get the Taylor series about <math>\langle \phi \rangle =0</math>. Then, from the effective action that we get from the Taylor series, we find the value of φ which minimizes the effective action. This gives us the VEV of φ when ''J''&nbsp;=&nbsp;0. Then, we now perform a Taylor series expansion about this VEV after shifting the field φ to a new field redefinition <math>\phi'=\phi - \langle \phi \rangle</math> (this is the ]). Now we can compute the ''n''-point correlations about the ''J''&nbsp;=&nbsp;0 vacuum.


==Convexity==
==One loop approximation==
The one-loop approximation to the effective action is


]
:<math>\Gamma=S+\frac{1}{2}Tr\left}\right]+\cdots.</math>

For a spacetime with volume <math>\mathcal V_4</math>, the effective potential is defined as <math>V(\phi) = - \Gamma/\mathcal V_4</math>. With a ] <math>H</math>, the effective potential <math>V(\phi)</math> at <math>\phi(x)</math> always gives the minimum of the expectation value of the ] <math> \langle \Omega|H|\Omega\rangle</math> for the set of states <math>|\Omega\rangle</math> satisfying <math>\langle\Omega| \hat \phi| \Omega\rangle = \phi(x)</math>.<ref>{{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}}</ref> 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 ] <math>V''(\phi) \geq 0</math>.<ref>{{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}}</ref>

Calculating the effective potential perturbatively can sometimes yield a non-convex result, such as a potential that has two ]. 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 <math>V_0(\phi)</math> with two local minima whose expectation values <math>\phi_1</math> and <math>\phi_2</math> are the expectation values for the states <math>|\Omega_1\rangle</math> and <math>|\Omega_2\rangle</math>, respectively. Then any <math>\phi</math> in the non-convex region of <math>V_0(\phi)</math> can also be acquired for some <math>\lambda \in </math> using

:<math>
|\Omega\rangle \propto \sqrt \lambda |\Omega_1\rangle+\sqrt{1-\lambda}|\Omega_2\rangle.
</math>

However, the energy density of this state is <math>\lambda V_0(\phi_1)+ (1-\lambda)V_0(\phi_2)<V_0(\phi)</math> meaning <math>V_0(\phi)</math> cannot be the correct effective potential at <math>\phi</math> since it did not minimize the energy density. Rather the true effective potential <math>V(\phi)</math> is equal to or lower than this linear construction, which restores convexity.

==See also==
*]
*]
*]
*]
*]


==References== ==References==
{{reflist|50em}}
* J.Goldstone, A.Salam, and S.Weinberg, Phys.Rev.127, 965 (1962)

* G.Jona-Lasinio, Nuovo Cimento 34, 1790 (1964)
==Further reading==
* S.Weinberg: ''The Quantum Theory of Fields'', Vol.II, Cambridge University Press 1996
* Das, A. : ''Field Theory: A Path Integral Approach'', World Scientific Publishing 2006
* D.J.Toms: ''The Schwinger Action Principle and Effective Action'', Cambridge University Press 2007
* 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


] ]

Latest revision as of 13:33, 1 May 2024

Quantum field theory
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Quantum version of the classical action

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.

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 i {\displaystyle i} and sign conventions.

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

Z [ J ] = D ϕ e i S [ ϕ ] + i d 4 x ϕ ( x ) J ( x ) . {\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 J ( x ) {\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

ϕ ^ ( x 1 ) ϕ ^ ( x n ) = ( i ) n 1 Z [ J ] δ n Z [ J ] δ J ( x 1 ) δ J ( x n ) | J = 0 , {\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},}

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

ϕ ^ ( x 1 ) ϕ ^ ( x n ) con = ( i ) n 1 δ n W [ J ] δ J ( x 1 ) δ J ( x n ) | J = 0 , {\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, 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 ] {\displaystyle W}

Γ [ ϕ ] = W [ J ϕ ] d 4 x J ϕ ( x ) ϕ ( x ) , {\displaystyle \Gamma =W-\int d^{4}xJ_{\phi }(x)\phi (x),}

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

An example of a Feynman diagram that can be cut into two separate diagrams by cutting one propagator.Example of a diagram that is not one-particle irreducible.An example of a Feynman diagram that can not be cut into two separate diagrams by cutting one propagator.Example of a diagram that is one-particle irreducible.
ϕ ( x ) = ϕ ^ ( x ) J = δ W [ J ] δ J ( x ) . {\displaystyle \phi (x)=\langle {\hat {\phi }}(x)\rangle _{J}={\frac {\delta W}{\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 ) {\displaystyle J(x)} that sources the scalar field. Taking the functional derivative of the Legendre transformation with respect to ϕ ( x ) {\displaystyle \phi (x)} yields

J ϕ ( x ) = δ Γ [ ϕ ] δ ϕ ( x ) . {\displaystyle J_{\phi }(x)=-{\frac {\delta \Gamma }{\delta \phi (x)}}.}

In the absence of an source J ϕ ( x ) = 0 {\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 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

ϕ ^ ( x 1 ) ϕ ^ ( x n ) 1 P I = i δ n Γ [ ϕ ] δ ϕ ( x 1 ) δ ϕ ( x n ) | J = 0 , {\displaystyle \langle {\hat {\phi }}(x_{1})\dots {\hat {\phi }}(x_{n})\rangle _{\mathrm {1PI} }=i{\frac {\delta ^{n}\Gamma }{\delta \phi (x_{1})\dots \delta \phi (x_{n})}}{\bigg |}_{J=0},}

with Γ [ ϕ ] {\displaystyle \Gamma } being the sum of all 1PI Feynman diagrams. The close connection between W [ J ] {\displaystyle W} and Γ [ ϕ ] {\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 Δ ( x , y ) {\displaystyle \Delta (x,y)} , is the inverse of the 1PI two-point correlation function

Δ ( x , y ) = δ 2 W [ J ] δ J ( x ) δ J ( y ) = δ ϕ ( x ) δ J ( y ) = ( δ J ( y ) δ ϕ ( x ) ) 1 = ( δ 2 Γ [ ϕ ] δ ϕ ( x ) δ ϕ ( y ) ) 1 = Π 1 ( x , y ) . {\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 }{\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 Γ [ ϕ 0 ] {\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 S [ ϕ + ϕ 0 ] {\displaystyle S} . This works because any place where ϕ 0 {\displaystyle \phi _{0}} appears in any of the propagators or vertices is a place where an external ϕ {\displaystyle \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 ϕ ( x ) = ϕ cl ( x ) + δ ϕ ( x ) {\displaystyle \phi (x)=\phi _{\text{cl}}(x)+\delta \phi (x)} , yielding

Γ [ ϕ cl ] = S [ ϕ cl ] + i 2 Tr [ ln δ 2 S [ ϕ ] δ ϕ ( x ) δ ϕ ( y ) | ϕ = ϕ cl ] + . {\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 S [ ϕ ] {\displaystyle S} are not automatically symmetries of the quantum effective action Γ [ ϕ ] {\displaystyle \Gamma } . If the classical action has a continuous symmetry depending on some functional F [ x , ϕ ] {\displaystyle F}

ϕ ( x ) ϕ ( x ) + ϵ F [ x , ϕ ] , {\displaystyle \phi (x)\rightarrow \phi (x)+\epsilon F,}

then this directly imposes the constraint

0 = d 4 x F [ x , ϕ ] J ϕ δ Γ [ ϕ ] δ ϕ ( x ) . {\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

ϕ ( x ) ϕ ( x ) + ϵ F [ x , ϕ ] J ϕ . {\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

F [ x , ϕ ] = a ( x ) + d 4 y   b ( x , y ) ϕ ( y ) . {\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

An example of a two local minima apparent effective potential and the corresponding correct effective potential which is linear in the non-convex region of the apparent potential.
The apparent effective potential V 0 ( ϕ ) {\displaystyle V_{0}(\phi )} acquired via perturbation theory must be corrected to the true effective potential V ( ϕ ) {\displaystyle V(\phi )} , shown via dashed lines in region where the two disagree.

For a spacetime with volume V 4 {\displaystyle {\mathcal {V}}_{4}} , the effective potential is defined as V ( ϕ ) = Γ [ ϕ ] / V 4 {\displaystyle V(\phi )=-\Gamma /{\mathcal {V}}_{4}} . With a Hamiltonian H {\displaystyle H} , the effective potential V ( ϕ ) {\displaystyle V(\phi )} at ϕ ( x ) {\displaystyle \phi (x)} always gives the minimum of the expectation value of the energy density Ω | H | Ω {\displaystyle \langle \Omega |H|\Omega \rangle } for the set of states | Ω {\displaystyle |\Omega \rangle } satisfying Ω | ϕ ^ | Ω = ϕ ( x ) {\displaystyle \langle \Omega |{\hat {\phi }}|\Omega \rangle =\phi (x)} . 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 ( ϕ ) 0 {\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 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 ( ϕ ) {\displaystyle V_{0}(\phi )} with two local minima whose expectation values ϕ 1 {\displaystyle \phi _{1}} and ϕ 2 {\displaystyle \phi _{2}} are the expectation values for the states | Ω 1 {\displaystyle |\Omega _{1}\rangle } and | Ω 2 {\displaystyle |\Omega _{2}\rangle } , respectively. Then any ϕ {\displaystyle \phi } in the non-convex region of V 0 ( ϕ ) {\displaystyle V_{0}(\phi )} can also be acquired for some λ [ 0 , 1 ] {\displaystyle \lambda \in } using

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

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

See also

References

  1. Weinberg, S.; Goldstone, J. (August 1962). "Broken Symmetries". Phys. Rev. 127 (3): 965–970. Bibcode:1962PhRv..127..965G. doi:10.1103/PhysRev.127.965. Retrieved 2021-09-06.
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  4. Zinn-Justin, J. (1996). "6". Quantum Field Theory and Critical Phenomena. Oxford: Oxford University Press. pp. 119–122. ISBN 978-0198509233.
  5. Kleinert, H. (2016). "22" (PDF). Particles and Quantum Fields. World Scientific Publishing. p. 1257. ISBN 9789814740920.
  6. Zee, A. (2010). Quantum Field Theory in a Nutshell (2 ed.). Princeton University Press. pp. 239–240. ISBN 9780691140346.
  7. Weinberg, S. (1995). "16". The Quantum Theory of Fields: Modern Applications. Vol. 2. Cambridge University Press. pp. 72–74. ISBN 9780521670548.
  8. Peskin, M.E.; Schroeder, D.V. (1995). An Introduction to Quantum Field Theory. Westview Press. pp. 368–369. ISBN 9780201503975.

Further reading

  • 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
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