Effective action

Everything in the following article also applies to statistical mechanics. However, the signs and factors of i are different in that case.

Given the partition function Z[J] in terms of the source field J, the energy functional is its logarithm.

${\displaystyle E[J]=i\ln Z[J]}$

Some physicists use W instead, W = −E. See sign conventions

For systems with two-particle interactions, the above Feynman diagrams arise at first order in the perturbation expansion 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.

In multiple areas of mathematics and information theory, including statistical mechanics, one writes the partition function as

${\displaystyle E[J]=-\ln Z[J]\,}$

Just as Z is interpreted as the generating functional (aka characteristic function(al)/moment-generating function(al) of the probability distribution function(al) e^−S[φ]/Z) of the time ordered VEVs/Schwinger function (aka moments) (see path integral formulation), E (a.k.a. the second characteristic function(al)/cumulant-generating function(al)) is the generator of "connected" time ordered VEVs/connected Schwinger functions (i.e. the cumulants) where connected here is interpreted in the sense of the cluster decomposition theorem which means that these functions approach zero at large spacelike separations, or in approximations using Feynman diagrams, connected components of the graph.

${\displaystyle \langle \phi (x_{1})\cdots \phi (x_{n})\rangle _{\mathrm {con} }=(-i)^{n+1}\left.{\frac {\delta ^{n}E}{\delta J(x_{1})\cdots \delta J(x_{n})}}\right|_{J=0}}$

or

${\displaystyle \langle \phi ^{i_{1}}\cdots \phi ^{i_{n}}\rangle _{\text{con}}=(-i)^{n+1}E^{,i_{1}\dots i_{n}}|_{J=0}}$

in the deWitt notation

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,

${\displaystyle {\begin{aligned}&{}\quad \langle \phi (x_{1})\phi (x_{2})\phi (x_{3})\rangle \\&=\langle \phi (x_{1})\phi (x_{2})\phi (x_{3})\rangle _{\text{con}}+\langle \phi (x_{1})\phi (x_{2})\rangle _{\text{con}}\langle \phi (x_{3})\rangle _{\text{con}}+\langle \phi (x_{1})\phi (x_{3})\rangle _{\text{con}}\langle \phi (x_{2})\rangle _{\text{con}}\\&+\langle \phi (x_{1})\rangle _{\text{con}}\langle \phi (x_{2})\phi (x_{3})\rangle _{\text{con}}+\langle \phi (x_{1})\rangle _{\text{con}}\langle \phi (x_{2})\rangle _{\text{con}}\langle \phi (x_{3})\rangle _{\text{con}}\end{aligned}}}$

Assuming E is a convex functional (which is debatable), the Legendre transformation gives a one-to-one correspondence between the configuration space of all source fields and its dual vector space, the configuration space of all φ fields. If E is not convex, we take the Fenchel conjugate instead. φ here is a classical field and not a quantum field operator.

Slightly out of the usual sign conventions for Legendre transforms, the value

${\displaystyle \phi =-{\delta \over \delta J}E[J]}$

or

${\displaystyle \phi ^{i}=-E^{,i}\,}$

is associated to J. This agrees with the time ordered VEV <φ>_J. The Legendre transform of E is the effective action (this corresponds to the rate function, which is the Fenchel conjugate of the cumulant-generating function, a common construction in statistics; e.g. the Chernoff bound)

${\displaystyle \Gamma [\phi ]=-\langle J,\phi \rangle -E[J]\,}$

or

${\displaystyle \Gamma [\phi ]=-J_{i}\phi ^{i}-E[J]\,}$

where

${\displaystyle \phi =-{\delta \over \delta J}E[J]}$

and

${\displaystyle J=-{\delta \over \delta \phi }\Gamma [\phi ]}$

or

${\displaystyle J_{i}=-\Gamma _{,i}.\,}$

There are some caveats, though, the major one being we don't have a true one-to-one correspondence between the dual configuration spaces.

The Dyson equation relating the full propagator, bare propagator and the 1PI self-energy in the absence of tadpoles

Let us first consider the case without tadpoles, i.e. ${\displaystyle \langle \phi \rangle =0}$ for J=0. In that case, Γ[0] gives the zero-point energy, the first functional derivative of Γ at φ=0 is zero, the second functional derivative gives the inverse of the full propagator, and the n^th functional derivative for n ≥ 3 gives the one particle irreducible correlation functions or 1PI correlation functions. The Dyson equation 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 ≥ 3 1PI's as nodes and full propagators as edges.

But what if we have tadpoles? We can always adjust the source J so that there are no tadpoles, i.e. ${\displaystyle \langle \phi \rangle =0}$. 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 ${\displaystyle \langle \phi \rangle =0}$. Any graph without subtadpoles do not contain any couplings to the source. A Taylor expansion of the effective action about φ = 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 ${\displaystyle \langle \phi \rangle =0}$. 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 = 0. Then, we now perform a Taylor series expansion about this VEV after shifting the field φ to a new field redefinition ${\displaystyle \phi '=\phi -\langle \phi \rangle }$ (this is the background field method). Now we can compute the n-point correlations about the J = 0 vacuum.

The one-loop approximation to the effective (Euclidean) action is

${\displaystyle \Gamma [\varphi ]=S[\varphi ]+{\frac {1}{2}}\operatorname {Tr} \left[\ln {S^{(2)}[\varphi ]}\right]+\cdots .}$

where ${\displaystyle \varphi }$ is the VEV of the underlying quantum fields ${\displaystyle \phi }$, and ${\displaystyle S^{(2)}[\varphi ]}$ is the second functional-derivative of the classical action evaluated at the classical field configuration ${\displaystyle \phi =\varphi }$.

${\displaystyle S^{(2)}[\varphi ]:=\left.\left({\frac {\delta ^{2}\,S[\phi ]}{\delta \phi (x)\,\delta \phi (y)}}\right)\right|_{\phi =\varphi }}$

Note that the presence of spatial indices in the right-hand side of the above expression, but no spatial indices on the left-hand side, is no problem. Formally they must be present in the functional derivative, but they are ultimately summed over by the trace. This is why they are suppressed in the left-hand side.

• Goldstone, Jeffrey; Salam, Abdus; Weinberg, Steven (1 July 1962). "Broken Symmetries". Physical Review. American Physical Society (APS). 127 (3): 965–970. doi:10.1103/physrev.127.965. ISSN 0031-899X.
• Jona-Lasinio, G. (1964). "Relativistic field theories with symmetry-breaking solutions" (PDF). Il Nuovo Cimento. Springer Science and Business Media LLC. 34 (6): 1790–1795. doi:10.1007/bf02750573. ISSN 0029-6341.
• S.Weinberg: The Quantum Theory of Fields, Vol.II, Cambridge University Press 1996
• D.J.Toms: The Schwinger Action Principle and Effective Action, Cambridge University Press 2007