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.