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 $-i\mathrm{\partial }\mathrm{/}\mathrm{\partial }T$, occurring on the r.h.s. of the Wheeler-DeWitt euqtion. Here the role of time has the particle coordinate $X^0 \equiv T$,whichisadistinctconceptthanthespacetimecoordinate$x^0 \equiv t$. 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.