The Schwinger–Dyson equations (SDEs) or Dyson–Schwinger equations, named after
Julian Schwinger and
Freeman Dyson
Freeman John Dyson (15 December 1923 – 28 February 2020) was an English-American theoretical physicist and mathematician known for his works in quantum field theory, astrophysics, random matrices, mathematical formulation of quantum m ...
, are general relations between
correlation functions in
quantum field theories
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles ...
(QFTs). They are also referred to as the
Euler–Lagrange equation
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ...
s of quantum field theories, since they are the
equations of motion
In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (V ...
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
In physics, the ''S''-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT).
More forma ...
elements, or more specific "one-particle Green's functions", in
quantum electrodynamics
In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
, by summing up infinitely many
Feynman diagram
In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduc ...
s, thus working in a perturbative approach. Starting from his
variational principle
In science and especially in mathematical studies, a variational principle is one that enables a problem to be solved using calculus of variations, which concerns finding functions that optimize the values of quantities that depend on those funct ...
, 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
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles ...
. 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
Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how th ...
and
elementary particle physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) and b ...
.
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
Functional may refer to:
* Movements in architecture:
** Functionalism (architecture)
** Form follows function
* Functional group, combination of atoms within molecules
* Medical conditions without currently visible organic basis:
** Functional sy ...
over the field configurations, then, for any
state vector (which is a solution of the QFT),
, we have
:
where
is the
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
functional and
is the
time ordering operation.
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 diagram
In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduc ...
s) 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
Physics often deals with classical models where the dynamical variables are a collection of functions
''α'' over a d-dimensional space/spacetime manifold ''M'' where ''α'' is the " flavor" index. This involves functionals over the ''φs, functio ...
whose inverse,
is called the bare propagator and