The Schwinger–Dyson equations (SDEs) or Dyson–Schwinger equations, named after
Julian Schwinger and
Freeman Dyson, are general relations between
correlation functions in
quantum field theories (QFTs). They are also referred to as the
Euler–Lagrange equations of quantum field theories, since they are the
equations of motion corresponding to the Green's function. They form a set of infinitely many functional differential equations, all coupled to each other, sometimes referred to as the infinite tower of SDEs.
In his paper "The S-Matrix in Quantum electrodynamics",[1] Dyson derived relations between different
S-matrix elements, or more specific "one-particle Green's functions", in
quantum electrodynamics, by summing up infinitely many
Feynman diagrams, thus working in a perturbative approach. Starting from his
variational principle, Schwinger derived a set of equations for Green's functions non-perturbatively,[2] which generalize Dyson's equations to the Schwinger–Dyson equations for the Green functions of
quantum field theories. Today they provide a non-perturbative approach to quantum field theories and applications can be found in many fields of theoretical physics, such as
solid-state physics and
elementary particle physics.
Schwinger also derived an equation for the two-particle irreducible Green functions,[2] which is nowadays referred to as the inhomogeneous
Bethe–Salpeter equation.
Derivation
Given a polynomially bounded
functional over the field configurations, then, for any
state vector (which is a solution of the QFT), , we have
Equivalently, in the
density state formulation, for any (valid) density state , we have
This infinite set of equations can be used to solve for the correlation functions
nonperturbatively.
To make the connection to diagrammatic techniques (like
Feynman diagrams) clearer, it is often convenient to split the action as
where the first term is the quadratic part and is an invertible symmetric (antisymmetric for fermions) covariant tensor of rank two in the
deWitt notation whose inverse, is called the bare propagator and is the "interaction action". Then, we can rewrite the SD equations as
If is a functional of , then for an
operator, is defined to be the operator which substitutes for . For example, if
R.J. Rivers (1990). Path Integral Methods in Quantum Field Theories. Cambridge University Press.
V.P. Nair (2005). Quantum Field Theory A Modern Perspective. Springer.
There are some review article about applications of the Schwinger–Dyson equations with applications to special field of physics.
For applications to
Quantum Chromodynamics there are