Wheeler-DeWitt quantization for point-particles

We present the Hamiltonian formulation of a relativistic point-particle coupled to Einstein gravity and its canonical quantization \`a la Wheeler-DeWitt. In the resulting quantum theory, the wave functional is a function of the particle coordinates and the 3-metric. It satisfies a particular Hamiltonian and diffeomorphism constraint, together with a Klein-Gordon-type equation. As usual in the Wheeler-DeWitt theory, the wave function is time-independent. This is also reflected in the Klein-Gordon-type equation, where the time derivative is absent. Before considering gravity, we consider the coupling of a particle with electromagnetism, which is treated similarly, but simpler.