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

leftlanglepsileft|mathcal{T}left{frac{delta}{deltaphi}F[phi]right}right|psirightrangle = -ileftlanglepsileft|mathcal{T}left{F[phi]frac{delta}{deltaphi}S[phi]right}right|psirightrangle

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

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

rholeft(mathcal{T}left{frac{delta}{deltaphi}F[phi]right}right) = -irholeft(mathcal{T}left{F[phi]frac{delta}{deltaphi}S[phi]right}right)

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

langlepsi|mathcal{T}{F phi^j}|psirangle=langlepsi|mathcal{T}{iF_{,i}D^{ij}-FS_{int,i}D^{ij}}|psirangle

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

F[phi]=frac{partial^{k_1}}{partial x_1^{k_1}}phi(x_1)cdots frac{partial^{k_n}}{partial x_n^{k_n}}phi(x_n)

and G is a functional of J, then

Fleft[-ifrac{delta}{delta J}right]G[J]=(-i)^n frac{partial^{k_1}}{partial x_1^{k_1}}frac{delta}{delta J(x_1)} cdots frac{partial^{k_n}}{partial x_n^{k_n}}frac{delta}{delta J(x_n)} G[J].

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

frac{delta^n Z}{delta J(x_1) cdots delta J(x_n)}[0]=i^n Z[0] langlephi(x_1)cdots phi(x_n)rangle,

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

frac{delta S}{delta phi(x)}left[-i frac{delta}{delta J}right]Z[J]+J(x)Z[J]=0

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

S[phi]=int d^dx left (frac{1}{2} partial^mu phi(x) partial_mu phi(x) -frac{1}{2}m^2phi(x)^2 -frac{lambda}{4!}phi(x)^4right )

for a real field φ.


frac{delta S}{delta phi(x)}=-partial_mu partial^mu phi(x) -m^2 phi(x) - frac{lambda}{3!}phi(x)^3.

The Schwinger-Dyson equation for this particular example is:

ipartial_mu partial^mu frac{delta}{delta J(x)}Z[J]+im^2frac{delta}{delta J(x)}Z[J]-frac{ilambda}{3!}frac{delta^3}{delta J(x)^3}Z[J]+J(x)Z[J]=0

Note that since

frac{delta^3}{delta J(x)^3}

is not well-defined because

frac{delta^3}{delta J(x_1)delta J(x_2) delta J(x_3)}Z[J]

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 -partial^mu partial_mu-m^2 and so, the SD set of equation goes as

langlepsi|mathcal{T}{phi(x_0)phi(x_1)}|psirangle=iD(x_0,x_1)+frac{lambda}{3!}int d^dx_2 D(x_0,x_2)langlepsi|mathcal{T}{phi(x_1)phi(x_2)phi(x_2)phi(x_2)}|psirangle

langlepsi|mathcal{T}{phi(x_0)phi(x_1)phi(x_2)phi(x_3)}|psirangle = iD(x_0,x_1)langlepsi|mathcal{T}{phi(x_2)phi(x_3)}|psirangle + iD(x_0,x_2)langlepsi|mathcal{T}{phi(x_1)phi(x_3)}|psirangle + iD(x_0,x_3)langlepsi|mathcal{T}{phi(x_1)phi(x_2)}|psirangle
+ frac{lambda}{3!}int d^dx_4D(x_0,x_4)langlepsi|mathcal{T}{phi(x_1)phi(x_2)phi(x_3)phi(x_4)phi(x_4)phi(x_4)}|psirangle


(unless there is spontaneous symmetry breaking, the odd correlation functions vanish)

Examples outside the Physics:

(Deleted content that was cryptic and nonsensical.)

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