Abstract :
We propose a new, discretized model for the study of (3 + 1)-dimensional canonical quantum gravity, based on the classical SL(2, C)-connection formulation. The discretization takes place on a topological N3 lattice with periodic boundary conditions. All operators and wave functions are constructed from one-dimensional link variables, which are regarded as the fundamental building blocks of the theory. The kinematical Hilbert space is spanned by polynomials of certain Wilson loops on the lattice and is manifestly gauge- and diffeomorphism-invariant. The discretized quantum Hamiltonian H̆ maps this space into itself. We find a large sector of solutions to the discretized Wheeler-DeWitt equation H̆ψ = 0, which are labelled by single and multiple Polyakov loops. These states have a finite norm with respect to a natural scalar product on the space of holomorphic SL(2, C) -Wilson loops. We also investigate the existence of further solutions for the case of the 13 lattice. Our results provide for the first time a rigorous, regularized framework for studying non-perturbative canonical quantum gravity.
