Correlators, Sources, and Effective Actions

Correlation functions carry different information depending on whether disconnected products have been removed and whether source variables have been replaced by mean fields. This chapter connects the generating functional, cumulants, Wick factorization, the effective action, and exact field-equation identities.

Enter this chapter

Begin with The Generating Functional after the finite Gaussian source calculation. It supplies the common definition from which the other pages branch.

Topics

1. The Generating Functional — Generates ordered field insertions with their normalization and source factors.
2. Connected Correlators and Cumulants — Uses the logarithm to isolate connected correlation functions.
3. Wick’s Theorem and Free Gaussian Factorization — Derives free-field pairing rules and their bosonic or fermionic signs.
4. The 1PI Effective Action and Mean-Field Equations — Trades sources for mean fields where the Legendre map is invertible.
5. Schwinger–Dyson Identities — Derives contact-term identities by regulated integration by parts.

How the ideas fit together

For the normalized Lorentzian in–out functional, $W[J]=-i\log Z[J]$ generates connected quantities and $\bar\phi=\delta W/\delta J$ is the mean field. The local Legendre transform $\Gamma[\bar\phi]=W[J]-\int J\bar\phi$ gives $\delta\Gamma/\delta\bar\phi=-J$. These operations reorganize information; Wick factorization additionally requires Gaussianity, while Schwinger–Dyson identities require the regulated change-of-variables assumptions. See Schwartz 2014, §§ 14.3–14.5 and 16.1.

Inspect the solid path in the following figure, then the separate dashed relations. A zero mode can obstruct the Legendre inversion, and an exact Schwinger–Dyson hierarchy need not be closed or solved.

Schematic information map for the normalized Lorentzian in–out convention. Solid arrows show $Z\to W\to\Gamma$. Dashed return arrows mark centered free-bosonic Wick factorization and regulated Schwinger–Dyson identities as separate relations, not further transforms; the diagram is not to scale.

Review the chapter

If $Z$ is Gaussian, why can a four-point function be nonzero when its connected four-point function vanishes?

Solution

The logarithm of a centered Gaussian is quadratic, so its fourth derivative vanishes. Differentiating $Z$ itself gives the three pairings of two-point functions. Connectedness removes those disconnected products; it does not require every higher moment to vanish.

Where to go next

Continue to Lorentzian, Euclidean, and In-In Formulations before using an effective action for causal expectation-value evolution.

References

• Matthew D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014. DOI.