Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 4 / 10C (Dynamics and Relativity)

With signature $(+,-,-,-)$, the four-momenta areIn the rest frame of a future timelike $P_1$, one has $P_1=(Mc,0)$ and $P_2^0>0$, so $P_1\cdot P_2=Mc{P}_2^0>0$; Lorentz invariance proves the assertion in every frame.

The impossibility of photon decay into two massive particles follows because a photon has squared four-momentum zero. If it decayed into an electron and positron, conservation would givea contradiction.

Momentum conservation makes the sum of the two photon momenta parallel to $u$, so all three vectors are coplanar. Put the photons on opposite sides of the incident direction. Transverse and longitudinal momentum and energy conservation giveEliminating $E_1,E_2$ yieldsUsing the half-angle identities on the left gives exactly