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Sojourner01

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The chapter begins with applying Wick's Theorem to produce six S-Matrix components with a variety of no-equal-time contractions. It then details the contribution of one:

[tex]S^{(2)}_{B}=-\frac{e^{2}}{2!}\int\frac{d^{4}}{dx_{1}}\frac{d^{4}}{dx_{2}}N(\overline{\psi}A\psi)_{x_{1}}(\overline{\psi}A\psi)_{x_{2}}+...][/tex]

LaTeX can't adequately represent contraction marks as far as I know, so imagine there's a cntraction between the first psi and the second psi-bar, and a repetition of same with a contraction between the first psi-bar and the second psi.

The derivation goes on to break this down for the case of Compton scattering by electrons; I am struggling to understand how the two complementary components for the two photon fields produce the final two equations; the contractions in each case are reduced to [tex]iS_{F\alpha\beta}(x_{1}-x_{2})[/tex] with, presumably, the A-slash 1 and 2 on each side. Somehow though the definitions of the slash operator migrate:

[tex]\gamma^{\alpha}iS_{F}(x_{1}-x_{2})\gamma^{\beta}A_{\alpha}^{-}(x_{1})A_{\beta}^{+}(x_{2})[/tex]

and

[tex]\gamma^{\alpha}iS_{F}(x_{1}-x_{2})\gamma^{\beta}A_{\beta}^{-}(x_{2})A_{\alpha}^{+}(x_{1})[/tex]

I am afraid I can't see how the normal ordering resolves to this sequence. Anyone help me revise my Einstein notation and normal-ordering?