How do you derive the Bethe-Salpeter equation / Dyson-Schwinger equations?
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I am wondering how to derive the (homogenous) Bethe-Salpeter equation
Γ(P,p)=∫d4k(2π)4K(P,p,k)S(k−P2)Γ(P,k)S(k+P2)Γ(P,p)=∫d4k(2π)4K(P,p,k)S(k−P2)Γ(P,k)S(k+P2)
and from what I understand, it's possible to do this via the Dyson-Schwinger equations. But there lies the next problem: while trying to derive the Dyson-Schwinger equation (say for the electron propagator in QED), I get so far as
[δS[φ]δφk∣∣∣φ→δ/(iδJ)+Jk]Z[J]=0,[δS[φ]δφk|φ→δ/(iδJ)+Jk]Z[J]=0,
but I simply can't derive the "typical" form
S(p)=S0(p)+S0(p)(e2∫d4k(2π)4γμDμν(p−k)S(k)Γν(p,k))S(p),S(p)=S0(p)+S0(p)(e2∫d4k(2π)4γμDμν(p−k)S(k)Γν(p,k))S(p),
I would be really grateful if someone could link me a PDF/document/webpage where I can follow the steps.
Γ(P,p)=∫d4k(2π)4K(P,p,k)S(k−P2)Γ(P,k)S(k+P2)Γ(P,p)=∫d4k(2π)4K(P,p,k)S(k−P2)Γ(P,k)S(k+P2)
and from what I understand, it's possible to do this via the Dyson-Schwinger equations. But there lies the next problem: while trying to derive the Dyson-Schwinger equation (say for the electron propagator in QED), I get so far as
[δS[φ]δφk∣∣∣φ→δ/(iδJ)+Jk]Z[J]=0,[δS[φ]δφk|φ→δ/(iδJ)+Jk]Z[J]=0,
but I simply can't derive the "typical" form
S(p)=S0(p)+S0(p)(e2∫d4k(2π)4γμDμν(p−k)S(k)Γν(p,k))S(p),S(p)=S0(p)+S0(p)(e2∫d4k(2π)4γμDμν(p−k)S(k)Γν(p,k))S(p),
I would be really grateful if someone could link me a PDF/document/webpage where I can follow the steps.
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The quark Schwinger-Dyson equation for the Faddeev–Popov La-
grangian. Filled dots indicate full propagators and vertices. Springs
show gluons and unbroken lines are for quarks. .
grangian. Filled dots indicate full propagators and vertices. Springs
show gluons and unbroken lines are for quarks. .
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