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", 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, 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, which is nowadays referred to as the inhomogeneous Bethe–Salpeter equation.
Derivation Given a polynomially bounded functional F {\displaystyle F} over the field configurations, then, for any state vector (which is a solution of the QFT), | ψ ⟩ {\displaystyle |\psi \rangle } , we have
⟨ ψ | T { δ δ φ F [ φ ] } | ψ ⟩ = − i ⟨ ψ | T { F [ φ ] δ δ φ S [ φ ] } | ψ ⟩ {\displaystyle \left\langle \psi \left|{\mathcal {T}}\left\{{\frac {\delta }{\delta \varphi }}F[\varphi ]\right\}\right|\psi \right\rangle =-i\left\langle \psi \left|{\mathcal {T}}\left\{F[\varphi ]{\frac {\delta }{\delta \varphi }}S[\varphi ]\right\}\right|\psi \right\rangle }
where δ δ φ {\textstyle {\frac {\delta }{\delta \varphi }}} is the functional derivative with respect to φ {\displaystyle \varphi } , S {\displaystyle S} is the action functional and T {\displaystyle {\mathcal {T}}} is the time ordering operation. Equivalently, in the density state formulation, for any (valid) density state ρ {\displaystyle \rho } , we have
ρ ( T { δ δ φ F [ φ ] } ) = − i ρ ( T { F [ φ ] δ δ φ S [ φ ] } ) . {\displaystyle \rho \left({\mathcal {T}}\left\{{\frac {\delta }{\delta \varphi }}F[\varphi ]\right\}\right)=-i\rho \left({\mathcal {T}}\left\{F[\varphi ]{\frac {\delta }{\delta \varphi }}S[\varphi ]\right\}\right).}
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 S {\displaystyle S} as
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