Wheeler-DeWitt quantization for point-particles

TL;DR

This paper develops a Hamiltonian and Wheeler-DeWitt quantization framework for a relativistic point-particle coupled to Einstein gravity, resulting in a time-independent wave functional satisfying specific constraints.

Contribution

It introduces a canonical quantization approach for a relativistic particle in gravity, extending Wheeler-DeWitt theory to include particle and electromagnetic couplings.

Findings

Wave functional depends on particle coordinates and 3-metric

Wave functional satisfies Hamiltonian and diffeomorphism constraints

Wave function is time-independent, following Wheeler-DeWitt formalism

Abstract

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.