On the Coupling of Relativistic Particle to Gravity and Wheeler-DeWitt Quantization

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 $t\equiv x^0$ and serves as the Hamiltonian which gives the correct equations of motion. Besides that, the system satisfies the mass shell constraint, $p^\mu p_\mu - m^2 = 0$, which is the generator of the worldsheet reparametrizations, where the momenta $p_\mu$, $\mu=0,1,2,3$, generate infinitesimal changes of the particle's position $X^\mu$ in spacetime. Consequently, the Hamiltonian contains $p_0$, which upon quantization becomes the operator $- i \partial/\partial T$, occurring on the r.h.s. of the Wheeler-DeWitt euqtion. Here the role of time has the particle coordinate $X^0 \equiv T$, which is a distinct concept than the spacetime coordinate $x^0 \equiv t$. 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.