THE PHYSICAL HAMILTONIAN IN NONPERTURBATIVE QUANTUM-GRAVITY

A quantum Hamiltonian which evolves the gravitational field according to time as measured by constant surfaces or a scalar field is defined through a regularization procedure based on the loop representation, a nd is shown to be finite and diffeomorphism invariant. The problem of constructing this Hamiltonian is reduced to a combinatorial and algebr aic problem which involves the rearrangements of lines through the ver tices of arbitrary graphs. This procedure also provides a construction of the Hamiltonian constraint as a finite operator on the space of di ffeomorphism invariant states as well as a construction of the operato r corresponding to the spatial volume of the Universe.