Schwinger-Dyson equation
Encyclopedia
The Schwinger–Dyson equation (SDE's), also Dyson-Schwinger equations, named after Julian Schwinger
and Freeman Dyson
, are general relations between Green function
s in quantum field theories
(QFT's). They are also referred to as the Euler-Lagrange equation
s of quantum field theories, since they are the Equation of motion
s of the corresponding 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 SDE's.
In his famous 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 diagram
s, 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 function
s 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
.
F over the field configurations, then, for any state vector
(which is a solution of the QFT), |ψ>, we have
where S is the action
functional and
is the time ordering operation.
Equivalently, in the density state formulation, for any (valid) density state ρ, we have
This infinite set of equations can be used to solve for the correlation functions
nonperturbatively.
To make the connection to diagramatical techniques (like Feynman diagram
s) clearer, it's often convenient to split the action S as S[φ]=1/2 D−1ij φi φj+Sint[φ] where the first term is the quadratic part and D−1 is an invertible symmetric (antisymmetric for fermions) covariant tensor of rank two in the deWitt notation whose inverse, D is called the bare propagator and Sint is the "interaction action". Then, we can rewrite the SD equations as
If F is a functional of φ, then for an operator
K, F[K] is defined to be the operator which substitutes K for φ. For example, if
and G is a functional of J, then
If we have an "analytic
" (whatever that means for functionals) functional
Z (called the generating functional) of J (called the source field) satisfying
,
then, from the properties of the functional integrals
,
the Schwinger–Dyson equation for the generating functional is
If we expand this equation as a Taylor series
about J = 0, we get the entire set of Schwinger–Dyson equations.
for a real field φ.
Then,
.
The Schwinger–Dyson equation for this particular example is:
Note that since
is not well-defined because
is a distribution
in
this equation needs to be regularized
!
In this example, the bare propagator, D is the Green's function
for
and so, the SD set of equation goes as
etc.
(unless there is spontaneous symmetry breaking
, the odd correlation functions vanish)
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
Julian Schwinger
Julian Seymour Schwinger was an American theoretical physicist. He is best known for his work on the theory of quantum electrodynamics, in particular for developing a relativistically invariant perturbation theory, and for renormalizing QED to one loop order.Schwinger is recognized as one of the...
and Freeman Dyson
Freeman Dyson
Freeman John Dyson FRS is a British-born American theoretical physicist and mathematician, famous for his work in quantum field theory, solid-state physics, astronomy and nuclear engineering. Dyson is a member of the Board of Sponsors of the Bulletin of the Atomic Scientists...
, are general relations between Green function
Green function
Green function might refer to:*Green's function of a differential operator.*Deligne–Lusztig theory in the representation theory of finite groups of Lie type.*Green's function in many-body theory....
s in quantum field theories
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
(QFT's). They are also referred to as the Euler-Lagrange equation
Euler-Lagrange equation
In calculus of variations, the Euler–Lagrange equation, Euler's equation, or Lagrange's equation, is a differential equation whose solutions are the functions for which a given functional is stationary...
s of quantum field theories, since they are the Equation of motion
Equation of motion
Equations of motion are equations that describe the behavior of a system in terms of its motion as a function of time...
s of the corresponding 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 SDE's.
In his famous paper "The S-Matrix in Quantum electrodynamics" Dyson
Freeman Dyson
Freeman John Dyson FRS is a British-born American theoretical physicist and mathematician, famous for his work in quantum field theory, solid-state physics, astronomy and nuclear engineering. Dyson is a member of the Board of Sponsors of the Bulletin of the Atomic Scientists...
derived relations between different S-Matrix-elements, or more specific "one-particle Green's functions", in Quantum electrodynamics
Quantum electrodynamics
Quantum electrodynamics 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 special relativity is achieved...
, by summing up infinitely many Feynman diagram
Feynman diagram
Feynman diagrams are a pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, first developed by the Nobel Prize-winning American physicist Richard Feynman, and first introduced in 1948...
s, thus working in a perturbative approach. Starting from his variational principle, Schwinger
Julian Schwinger
Julian Seymour Schwinger was an American theoretical physicist. He is best known for his work on the theory of quantum electrodynamics, in particular for developing a relativistically invariant perturbation theory, and for renormalizing QED to one loop order.Schwinger is recognized as one of the...
derived a set of equations for Green's functions non-perturbatively, which generalize Dyson's equations to the Schwinger–Dyson equations for the Green function
Green function
Green function might refer to:*Green's function of a differential operator.*Deligne–Lusztig theory in the representation theory of finite groups of Lie type.*Green's function in many-body theory....
s of quantum field theories
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
.
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
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 the large-scale properties of solid materials result from...
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
Bethe-Salpeter equation
The Bethe–Salpeter equation, named after Hans Bethe and Edwin Salpeter, describes the bound states of a two-body quantum field theoretical system in a relativistically covariant formalism...
.
Derivation
Given a polynomially bounded functionalFunctional (mathematics)
In mathematics, and particularly in functional analysis, a functional is a map from a vector space into its underlying scalar field. In other words, it is a function that takes a vector as its input argument, and returns a scalar...
F over the field configurations, then, for any state vector
State vector
*A state vector in general control systems describes the observed states of an object in state space, e.g. in variables of the degrees of freedom for motion *A state vector in general control systems describes the observed states of an object in state space, e.g. in variables of the degrees of...
(which is a solution of the QFT), |ψ>, we have
where S is the action
Action (physics)
In physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy × time, and its unit is...
functional and
is the time ordering operation.Equivalently, in the density state formulation, for any (valid) density state ρ, we have
This infinite set of equations can be used to solve for the correlation functions
Correlation function (quantum field theory)
In quantum field theory, the matrix element computed by inserting a product of operators between two states, usually the vacuum states, is called a correlation function....
nonperturbatively.
To make the connection to diagramatical techniques (like Feynman diagram
Feynman diagram
Feynman diagrams are a pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, first developed by the Nobel Prize-winning American physicist Richard Feynman, and first introduced in 1948...
s) clearer, it's often convenient to split the action S as S[φ]=1/2 D−1ij φi φj+Sint[φ] where the first term is the quadratic part and D−1 is an invertible symmetric (antisymmetric for fermions) covariant tensor of rank two in the deWitt notation whose inverse, D is called the bare propagator and Sint is the "interaction action". Then, we can rewrite the SD equations as
If F is a functional of φ, then for an operator
Operator (physics)
In physics, an operator is a function acting on the space of physical states. As a resultof its application on a physical state, another physical state is obtained, very often along withsome extra relevant information....
K, F[K] is defined to be the operator which substitutes K for φ. For example, if
and G is a functional of J, then
If we have an "analytic
Analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...
" (whatever that means for functionals) functional
Functional (mathematics)
In mathematics, and particularly in functional analysis, a functional is a map from a vector space into its underlying scalar field. In other words, it is a function that takes a vector as its input argument, and returns a scalar...
Z (called the generating functional) of J (called the source field) satisfying
,then, from the properties of the functional integrals
,the Schwinger–Dyson equation for the generating functional is
If we expand this equation as a Taylor series
Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
about J = 0, we get the entire set of Schwinger–Dyson equations.
An example: φ4
To give an example, supposefor a real field φ.
Then,
.The Schwinger–Dyson equation for this particular example is:
Note that since
is not well-defined because
is a distribution
Distribution (mathematics)
In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...
in
- x1, x2 and x3,
this equation needs to be regularized
Regularization (physics)
-Introduction:In physics, especially quantum field theory, regularization is a method of dealing with infinite, divergent, and non-sensical expressions by introducing an auxiliary concept of a regulator...
!
In this example, the bare propagator, D is the Green's function
Green's function
In mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to specific initial conditions or boundary conditions...
for
and so, the SD set of equation goes asetc.
(unless there is spontaneous symmetry breaking
Spontaneous symmetry breaking
Spontaneous symmetry breaking is the process by which a system described in a theoretically symmetrical way ends up in an apparently asymmetric state....
, the odd correlation functions vanish)
Further reading
There are not many book that treat the Schwinger–Dyson equations. There are two standard references:There are some review article about applications of the Schwinger–Dyson equations with applications to special field of physics.
For applications to Quantum Chromodynamics
Quantum chromodynamics
In theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...
there are
See also
- Functional renormalization groupFunctional renormalization groupIn theoretical physics, functional renormalization group is an implementation of the renormalization group concept which is used in quantum and statistical field theory, especially when dealing with strongly interacting systems. The method combines functional methods of quantum field theory with...
- Dyson equation
- Path integral formulationPath integral formulationThe path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics...