- Schwinger-Dyson equation
The Schwinger-Dyson equation, named after
Julian Schwinger andFreeman Dyson , is an equation ofquantum field theory (QFT). Given apolynomially bounded functional "F" over thefield configuration s, then, for anystate vector (which is a solution of the QFT), |ψ>, we have:leftlanglepsileft|mathcal{T}left{frac{delta}{deltaphi}F [phi] ight} ight|psi ight angle = -ileftlanglepsileft|mathcal{T}left{F [phi] frac{delta}{deltaphi}S [phi] ight} ight|psi ight angle
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:holeft(mathcal{T}left{frac{delta}{deltaphi}F [phi] ight} ight) = -i holeft(mathcal{T}left{F [phi] frac{delta}{deltaphi}S [phi] ight} ight)
This infinite set of equations can be used to solve for the correlation functions
nonperturbative ly.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 thedeWitt notation whose inverse, D is called thebare propagator and Sint is the "interaction action". Then, we can rewrite the SD equations as:langlepsi|mathcal{T}{F phi^j}|psi angle=langlepsi|mathcal{T}{iF_{,i}D^{ij}-FS_{int,i}D^{ij}}|psi angle
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_1phi(x_1)cdots frac{partial^{k_n{partial x_n^{k_nphi(x_n)
and "G" is a functional of "J", then
:Fleft [-ifrac{delta}{delta J} ight] G [J] =(-i)^n frac{partial^{k_1{partial x_1^{k_1frac{delta}{delta J(x_1)} cdots frac{partial^{k_n{partial x_n^{k_nfrac{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 thesource 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) angle,
then, the Schwinger-Dyson equation for the generating functional is
:frac{delta S}{delta phi(x)}left [-i frac{delta}{delta J} ight] 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)^4 ight )
for a real field φ.
Then,
: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
:"x"1, "x"2 and "x"3,
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)}|psi angle=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)}|psi angle
:langlepsi|mathcal{T}{phi(x_0)phi(x_1)phi(x_2)phi(x_3)}|psi angle = iD(x_0,x_1)langlepsi|mathcal{T}{phi(x_2)phi(x_3)}|psi angle + iD(x_0,x_2)langlepsi|mathcal{T}{phi(x_1)phi(x_3)}|psi angle + iD(x_0,x_3)langlepsi|mathcal{T}{phi(x_1)phi(x_2)}|psi angle ::::: 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)}|psi angle
etc.
(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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