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

An example: φ⁴

To give an example, suppose

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:S=int d^dx left (rac{1}{2} partial^mu phi(x) partial_mu phi(x) -rac{1}{2}m^2phi(x)^2 -rac{lambda}{4!}phi(x)^4 ight )

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

for a real field φ.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Then,

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The Schwinger-Dyson equation for this particular example is:

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Note that since

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:rac{delta^3}{delta J(x)^3}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

is not well-defined because

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:rac{delta^3}{delta J(x_1)delta J(x_2) delta J(x_3)}Z

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

=iD(x_0,x_1)+rac{lambda}{3!}int d^dx_2 D(x_0,x_2)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

=iD(x_0,x_1)+iD(x_0,x_2)+iD(x_0,x_3)+rac{lambda}{3!}int d^dx_4 D(x_0,x_4)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

etc.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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

~ ~ ~ ~ ~ ~ ~ ~ ~ ~