The Schwinger-Dyson equation
, named after Julian Schwinger
and Freeman Dyson
, is an equation of quantum field theory
(QFT). Given a polynomially bounded functional 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 diagrams) 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, 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.
An example: φ4
To give an example, suppose
for a real field φ.
The Schwinger-Dyson equation for this particular example is:
Note that since
is not well-defined because
is a distribution in
- x1, x2 and x3,
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
(unless there is spontaneous symmetry breaking, the odd correlation functions vanish)
Examples outside the Physics:
(Deleted content that was cryptic and nonsensical.)