Summary

2.17 Summary

Some progress has been achieved in Hamiltonian lattice gravity, using discrete analogues of the Ashtekar variables. The quantization program is still open-ended, and no physically non-trivial solutions to the Wheeler–DeWitt equation are known. The fact that one can study and evaluate geometric operators provides useful characterization of quantum states.

An analysis of the spectrum of the volume operator is crucial for handling the non-polynomial terms in the quantum Hamiltonian constraint. It will be necessary to find a suitable truncation or approximation to simplify further the spectral analysis of the Wheeler–DeWitt operator. A suitable quantum analogue of the continuum limit $a → 0$ has not yet been established, and in this regard the Hamiltonian ansatz does not go beyond the results obtained in the Lagrangian formulations described earlier.

Why should one bother with a Hamiltonian quantization at all? Typical quantities one wants to study in a discrete path-integral approach to gravity are transition amplitudes between three-geometries on different spatial slices. This is not complete without a specification of the corresponding quantum states, which are in principle elements of Hilbert spaces of discrete three-geometries of the type described above.