Schwinger-Dyson equation

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

Related Topics:
Julian Schwinger - Freeman Dyson - Quantum field theory - Polynomially bounded - Functional - Field configuration - State vector

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:=-i

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where S is the action functional and mathcal{T} is the time ordering operation.

Related Topics:
Action - Time ordering

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Equivalently, in the density state formulation, for any (valid) density state ρ, we have

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: ho(mathcal{T}{rac{delta}{deltaphi}F})=-i ho(mathcal{T}{Frac{delta}{deltaphi}S})

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This infinite set of equations can be used to solve for the correlation functions nonperturbatively.

Related Topics:
Correlation functions - Nonperturbative

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To make the connection to diagramatical techniques (like feynman diagrams) clearer, it's often convenient to split the action S as S=1/2 D-1ij φi φj+Sint 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

Related Topics:
Feynman diagram - DeWitt notation - Bare propagator

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:=

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If F is a functional of φ, then for an operator K, F is defined to be the operator which substitutes K for φ. For example, if

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:F=rac{partial^{k_1}}{partial x_1^{k_1}}phi(x_1)cdots rac{partial^{k_n}}{partial x_n^{k_n}}phi(x_n)

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and G is a functional of J, then

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:FG=(-i)^n rac{partial^{k_1}}{partial x_1^{k_1}}rac{delta}{delta J(x_1)} cdots rac{partial^{k_n}}{partial x_n^{k_n}}rac{delta}{delta J(x_n)} G.

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If we have an "analytic" (whatever that means for functionals) functional Z (called the generating functional) of J (called the source field) satisfying

Related Topics:
Analytic - Functional - Generating functional - Source field

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:rac{delta^n Z}{delta J(x_1) cdots delta J(x_n)}=i^n Z ,

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then, the Schwinger-Dyson equation for the generating functional is

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:rac{delta S}{delta phi(x)}Z+J(x)Z=0

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If we expand this equation as a Taylor series about J = 0, we get the entire set of Schwinger-Dyson equations.

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