On the coupling of relativistic particle to gravity and Wheeler–DeWitt quantization

Abstract

A system consisting of a point particle coupled to gravity is investigated. The set of constraints is derived. It was found that a suitable superposition of those constraints is the generator of the infinitesimal transformations of the time coordinate [Formula: see text] and serves as the Hamiltonian which gives the correct equations of motion. Besides that, the system satisfies the mass shell constraint, [Formula: see text], which is the generator of the worldline reparametrizations, where the momenta [Formula: see text], [Formula: see text], generate infinitesimal changes of the particle’s position [Formula: see text] in spacetime. Consequently, the Hamiltonian contains [Formula: see text], which upon quantization becomes the operator [Formula: see text], occurring on the right-hand side of the Wheeler–DeWitt equation. Here, the role of time has the particle coordinate [Formula: see text], which is a distinct concept than the spacetime coordinate [Formula: see text]. It is also shown how the ordering ambiguities can be avoided if a quadratic form of the momenta is cast into the form that instead of the metric contains the basis vectors.