Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 301 / 3

For metric ${\eta }^{\mu \nu }=\mathrm{diag}(1,-1,-1,-1)$, the defining Clifford algebra relation is

A positive-frequency on-shell Dirac spinor obeys

With $D_{\mu }={\partial }_{\mu }+ie{A}_{\mu }$ and $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$, the quantum electrodynamics Lagrangian is

The required QED Feynman rules are the electron-photon vertex $-ie{\gamma }^{\mu }$, the internal electron propagatorthe external spinors $u(p,s)$ and $\bar{u}(p^{\prime },s^{\prime })$, and photon factors ${ϵ}_{\mathrm{in}}^{\nu }$ and ${ϵ}_{\mathrm{out}}^{\ast \mu }$. Momentum conservation is imposed at both vertices.

There are two tree-level Feynman diagrams for Compton scattering: an $s$-channel ordering in which the electron first absorbs the incoming photon, with internal momentum $p+q$, and a crossed $u$-channel ordering in which it first emits the outgoing photon, with internal momentum $p-q^{\prime }$.

Multiplying the QED Feynman rules in fermion-line order gives, up to the common convention for $i\mathcal{M}$,The two terms are both required by the Ward identity; replacing either external polarization by its photon momentum makes their sum vanish after using momentum conservation and the external Dirac equations.

Average over the two initial electron spins and two initial photon polarizations, and sum over the final ones. The supplied fermion spin sum and photon polarization sum turn the result into gamma-matrix traces. The Clifford algebra impliesby anticommuting the outside matrix through the product. Together withthe two channel squares reduce, at $m=0$, to $-2e^{4}u/s$ and $-2e^{4}s/u$; the remaining cross terms cancel. Thus, in terms of the Mandelstam variables,Physical Compton kinematics has $s>0$ and $u<0$, so the displayed expression is nonnegative.