Directly from the Peskin & Shroeder, page 63:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

S_F(x-y) = \int\frac{d^4p}{(2\pi)^4}\frac{i(\displaystyle{\not}p + m)}{p^2-m^2+i\epsilon} e^{-ip\cdot(x-y)}

[/tex]

I'm slightly confused with the notation with the contractions. Things like [itex]\overline{\psi}(x)\psi(y)[/itex] and [itex]\psi(x)\overline{\psi}(y)[/itex] get written carelessly although it doesn't really make sense to put psi bar on the right. On the other hand the propagator itself is a 4x4 matrix, so when I try to make sense out of this, this is the only conclusion I've succeeded to come up with: The contraction should be carried out always with fixed indexes of the fermion operators, and then we choose the corresponding element from the matrix in the propagator. That means, that when [itex]a,b\in\{1,2,3,4\}[/itex], then

[tex]

\textrm{contraction}(\psi_a(x),\overline{\psi}_b(y)) = (S_F(x-y))_{ab} = \int\frac{d^4p}{(2\pi)^4}\frac{i(\displaystyle{\not}p +m)_{ab}}{p^2-m^2+i\epsilon} e^{-ip\cdot(x-y)}

[/tex]

Is this correct?

If it was correct, what happens when the psi bar is on the left? What is

[tex]

\textrm{contraction}(\overline{\psi}_a(x),\psi_b(y)) ?

[/tex]

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# Fermion contraction

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