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
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.
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Discrete Approaches to Quantum Gravity in Four Dimensions
Renate Loll http://www.livingreviews.org/lrr-1998-13 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |