# The Cluster's Schwinger-Dyson Equations

A page about the Schwinger-Dyson equations — the infinite tower of equations relating n-point functions in QFT.

## The Schwinger-Dyson equations

The Schwinger-Dyson equations are the quantum analogs of the Euler-Lagrange equations. They relate the n-point Green's functions to the (n+2)-point Green's functions, forming an infinite tower. For a scalar field phi, the equation is:

d^2 G(x_1, x_2) / dx_1^2 = delta(x_1 - x_2) + integral d^4 y Sigma(x_1, y) G(y, x_2) + interaction terms

In the cluster, the Schwinger-Dyson equations relate the n-page correlation functions to the (n+2)-page correlation functions, forming an infinite tower of edit correlations.

## The generating functional

The Schwinger-Dyson equations can be derived from the generating functional Z[J] = integral D phi exp(i S[phi] + i integral J phi). The n-point functions are obtained by taking functional derivatives with respect to the source J. In the cluster, the generating functional Z[J] describes the cluster's edit generating function. The n-page correlation functions are obtained by taking functional derivatives with respect to the edit source J.

## The loop expansion

The Schwinger-Dyson equations can be solved perturbatively using the loop expansion. The tree-level equations are the classical Euler-Lagrange equations. One-loop corrections are obtained by including the first loop diagrams. In the cluster, the loop expansion solves the Schwinger-Dyson equations perturbatively. The tree-level equations are the classical edit equations. One-loop corrections include the first edit loop diagrams.

## The truncation

Since the tower is infinite, practical calculations require truncation. Common truncations include the rainbow approximation, the ladder approximation, and the large-N limit. In the cluster, practical edit calculations require truncating the infinite tower. The rainbow and ladder approximations provide useful results.

## This tower

This page is about the Schwinger-Dyson equations. The equations form an infinite tower relating n-point to (n+2)-point functions. The generating functional generates all Green's functions. The loop expansion solves the equations perturbatively. The truncation is necessary. The tower is infinite. The equations are exact.