Schwinger-Dyson equation Guide, Meaning , Facts, Information and Description

The Schwinger-Dyson equation 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

These sets of equations can be used to solve for the correlation functions nonperturbatively (although these equations happen to be highly nonlinear).

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.

Then,

.

The Schwinger-Dyson equation for this particular example is:

Note that since is not well-defined ( is a distribution in x₁, x₂ and x₃), this equation needs to be regularized!

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