The Effective Action and 1PI Generating Functional

The generating functional $Z[J]$ produces all correlators. Two further Legendre-related generating functionals distill this down to the physically essential pieces: $W[J]$ generates only the connected correlators, and its Legendre transform $\Gamma [\varphi ]$ — the effective action — generates the one-particle-irreducible (1PI) vertices that are the true building blocks of renormalization. The effective action also packages all quantum corrections into a single object whose minimization gives the true vacuum, via the effective potential.

Conventions: $\hslash =c=1$, ${\eta }_{\mu \nu }=\mathrm{diag}(+1,-1,-1,-1)$.

Connected correlators: $W[J]$

Define $W[J]$ by

$$\boxed{Z[J]=e^{iW[J]},W[J]=-i\ln Z[J].}$$

Whereas derivatives of $Z[J]$ generate all diagrams, derivatives of $W[J]$ generate only the connected ones — the logarithm removes disconnected pieces, the functional analogue of $\ln$ turning a product of exponentials into a sum. In particular the connected two-point function is

$$\frac{{\delta }^{2}W}{\delta J(x)\delta J(y)}{|}_{J=0}=i⟨0|T\varphi (x)\varphi (y)|0{⟩}_{\text{conn}}=i{D}_F(x-y).$$

$W[J]$ is the field-theory version of the free energy in statistical mechanics (recall the Euclidean analogy).

The classical field and the Legendre transform

Define the classical field as the source-dependent expectation value

$${\varphi }_{\text{cl}}(x)=\frac{\delta W[J]}{\delta J(x)}=⟨0|\varphi (x)|0{⟩}_J,$$

the vacuum expectation of $\varphi$ in the presence of $J$. The effective action $\Gamma [{\varphi }_{\text{cl}}]$ is the Legendre transform of $W$ that trades the source $J$ for the field ${\varphi }_{\text{cl}}$:

$$\boxed{\Gamma [{\varphi }_{\text{cl}}]=W[J]-\int d^{4}xJ(x){\varphi }_{\text{cl}}(x),\frac{\delta \Gamma }{\delta {\varphi }_{\text{cl}}(x)}=-J(x).}$$

The last relation is the quantum equation of motion: in the absence of a source, the physical field configurations extremize the effective action, $\delta \Gamma /\delta {\varphi }_{\text{cl}}=0$. $\Gamma$ is thus the full quantum-corrected version of the classical action $S$ — it reduces to $S$ at tree level and receives loop corrections order by order in $\hslash$.

1PI vertices

The decisive property: the functional derivatives of $\Gamma$ are exactly the one-particle-irreducible (1PI) correlation functions — those diagrams that cannot be split by cutting one internal line:

$$\frac{{\delta }^n\Gamma }{\delta {\varphi }_{\text{cl}}(x_1)\cdots \delta {\varphi }_{\text{cl}}(x_n)}{|}_{{\varphi }_{\text{cl}}=0}=i{\Gamma }^{(n)}(x_1,\dots ,x_n)=(\text{sum of 1PI diagrams}).$$

Two cases carry standard names:

• ${\Gamma }^{(2)}$ is the inverse full propagator: ${\Gamma }^{(2)}(p)=p^2-m^2-\Sigma (p)$, where $\Sigma$ is the self-energy (sum of 1PI two-point diagrams). The physical mass is the zero of ${\Gamma }^{(2)}$.
• ${\Gamma }^{(n)}$ for $n\ge 3$ are the proper vertices — the amputated, irreducible interaction vertices that renormalization renormalizes.

Because any diagram can be assembled by joining 1PI blobs with full propagators, the pair $({\Gamma }^{(2)},{\Gamma }^{(n\ge 3)})$ contains all the information of the theory with maximal economy — this is why renormalization is organized around 1PI functions.

The effective potential

For a constant classical field ${\varphi }_{\text{cl}}={\varphi }_0$ (translation-invariant vacuum), the effective action reduces to spacetime volume times the effective potential:

$$\Gamma [{\varphi }_0]=-(\text{vol})V_{\text{eff}}({\varphi }_0).$$

The true vacuum minimizes $V_{\text{eff}}$, and $⟨\varphi ⟩$ is its location. At tree level $V_{\text{eff}}=V({\varphi }_0)$, the classical potential; loops add corrections. The one-loop correction is a functional determinant (a boson Gaussian integral, or a fermion determinant) around the background ${\varphi }_0$:

$$V_{\text{eff}}({\varphi }_0)=V({\varphi }_0)+\frac{i}{2}\mathrm{Tr}\ln [{\partial }^2+V^{\prime \prime }({\varphi }_0)]+\cdots ,$$

the Coleman–Weinberg potential. Its central use is spontaneous symmetry breaking: even when the classical $V$ has a symmetric minimum, radiative corrections can shift the minimum of $V_{\text{eff}}$ to ${\varphi }_0\ne 0$, breaking the symmetry dynamically — the subject of the effective potential page.

The loop expansion is the $\hslash$ expansion

Restoring $\hslash$, the weight is $e^{iS/\hslash }$ and a stationary-phase (saddle-point) evaluation organizes $\Gamma$ in powers of $\hslash$:

$$\Gamma [\varphi ]=S[\varphi ]+\hslash {\Gamma }_{1\text{-loop}}[\varphi ]+{\hslash }^2{\Gamma }_{2\text{-loop}}[\varphi ]+\cdots .$$

The number of loops counts powers of $\hslash$: the classical action is the tree-level (${\hslash }^0$) term, and $\Gamma$ is literally the quantum-corrected action. This is the precise sense in which the loop expansion is the semiclassical expansion.

Summary

Where this leads

• Radiative symmetry breaking: the effective potential (Coleman–Weinberg).
• Renormalization of 1PI functions: renormalization and counterterms.
• Symmetry constraints on $\Gamma$: Ward–Takahashi identities.

References

• Peskin & Schroeder, An Introduction to Quantum Field Theory, Ch. 11.3–11.4.
• Weinberg, The Quantum Theory of Fields, Vol. 2, Ch. 16.
• Coleman & Weinberg, Phys. Rev. D 7, 1888 (1973).
• Srednicki, Quantum Field Theory, Ch. 21.