Dyson-Schwinger Equations

The Dyson-Schwinger equations, also called the Schwinger-Dyson equations, are an infinite hierarchy of coupled functional differential equations and integral equations satisfied by the Green's functions of a quantum field theory (Dyson 1949, Schwinger 1951ab). They are the quantum equations of motion and follow formally from invariance of a functional integral under the change of variables theorem. This step is called functional integration by parts: it is the functional-integral analogue of ordinary integration by parts, with a field as the integration variable and a functional derivative in place of an ordinary derivative. In both cases, the boundary term is assumed to vanish.

In Euclidean conventions for bosonic fields, let be the classical action, i.e., the functional whose stationary points give the classical equations of motion, and let be an external source coupled linearly to the field. Square brackets in expressions such as indicate dependence on the entire field configuration, rather than on the value of an ordinary function at one point. The generating functional , mean field , and quantum effective action are defined by

			(1)
			(2)
			(3)

The expression for is a functional integral. The notation is its formal functional integration measure over field configurations and is conventionally written directly after the integral sign. Here is the source-dependent expectation value of the field and is the Legendre transformation of . Repeated field labels include the appropriate summation or integration. Functional integration by parts then gives the master relation

(4)

Repeated functional derivatives of this identity generate equations relating Green's functions of different orders. The resulting hierarchy is exact but generally does not close at any finite order, so applications usually replace it by a truncated system.

For example, the zero-dimensional scalar model with

(5)

has generating integral

(6)

Integration by parts gives the Dyson-Schwinger equation

(7)

which illustrates how an interaction of degree four couples derivatives of different orders.