Student Friendly Introduction to QFT · Problem 8.7

The Vacuum Revisited

Physics — QFT · compton-scattering-amplitude

Problem

Prove that

s'=s

in (8-76). Recall that

iS_F(p-k) = \dfrac{(p_\alpha - k_\alpha)\gamma^\alpha + m}{(p-k)^2 - m^2 + i\varepsilon}

, and from Chap. 4 Appendix A (pg. 122), that

\left[\gamma^\alpha, \gamma^\nu\right]_+ = 2g^{\alpha\nu}

(\not{p} - m)u_r(\mathbf{p}) = 0

g_{\mu\nu}\gamma^\mu\gamma^\nu = \gamma_\nu\gamma^\nu = 4

, and

\bar{u}_r(\mathbf{p})\gamma^\mu u_s(\mathbf{p}) = \dfrac{p^\mu}{m}\delta_{rs}

. (Hint: Start with the integrand of (8-76) and use the gamma matrices anti-commutation relations to switch the order of

\gamma^\alpha

and

\gamma^\nu

. Then use the relation after that above to get certain terms to drop out.)