How I derive the Dyson-Schwinger equations for
the electron propagator $S(p)$,
${\displaystyle S(p)=S_{0}(p)+S_{0}(p)*\left(e^{2}\int {\frac {d^{4}k}{(2\pi )^{4}}}\gamma _{\mu }D^{\mu \nu }(p-k)S(k)\Gamma _{\nu }(p,k)\right)*S(p)} ,$
the photon propagator ${\displaystyle D^{\mu \nu }(p)}$
${\displaystyle D^{\mu \nu }(p)=D_{0}^{\mu \nu }(p)+D_{0}^{\mu \nu '}(p)*{\Bigl (}-e^{2}\operatorname {tr} {\Bigl [}\int {\frac {d^{4}k}{(2\pi )^{4}}}\gamma _{\nu '}S(k)\Gamma _{\mu '}(k,k+p)S(k+p){\Bigr ]}{\Bigr )}*D^{\mu '\nu }(p)} ,$ and the electron-photon-vertex' ${\displaystyle \Gamma _{\nu }(p',p)} $
${\displaystyle \Gamma _{\nu }(p',p)=\Gamma _{\nu \,0}(p',p)+\int {\frac {d^{4}k}{(2\pi )^{4}}}S(p'+k)\Gamma _{\nu }(p'+k,p+k)S(p+k)K(p+k,p'+k,k)} $.
???
I have no idea, because the definition of the Dyson-Schwinger equation
$(\frac{\partial S(-i \partial_J)}{\partial \phi} + J)Z(J) = 0$
seems to be completely different to the equations above. What I have to do to obtain these equations? Is there a way to derive these equations?
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