Quantum Electrodynamics Mediated by a Photon with Generalized (Continuous) Spin

We present rules for computing scattering amplitudes of charged scalar matter and photons, where the photon has non-zero spin Casimir $\rho$, and is therefore a continuous spin particle (CSP). The amplitudes reduce to familiar scalar QED when $\rho \to 0$. As a concrete example, we compute the pair annihilation and Compton scattering amplitudes in this theory and comment on their physical properties, including unitarity and scaling behavior at small and large $\rho$.