1. Calculate directly the cross section for $\gamma\gamma\rightarrow e^+ e^-$ in the non polarised case in the ultrarelativistic limit $m_e \ll \sqrt{s}$.
2. Calculate the cross section for $e^+ e^-\rightarrow\gamma\gamma$ in the non polarised case in the nonrelativistic limit $2 m_e\approx \sqrt{s}$.
3. Gauge invariance tells us that $A_\mu$ and $A_\mu+\partial_\mu\Lambda$ give equal results. In $k$ space, this means that the polarisation vector of a photon of momentum $k_\mu$, $\epsilon_\mu$, and $\epsilon_\mu+c k_\mu$ give identical results. Show that this is the case for the Compton scattering amplitude.
4. Imagine that the photon is a massless scalar $\phi$ that interacts with electrons via ${\mathcal L}_{int}=g\bar\psi\psi\phi$. How would that affect the Compton cross section?