We have discovered some rather unexpected features, such as the difficulties caused by spatial diffeomorphism invariance and the consequent nonlocality in Wheeler-DeWitt quantization, and the necessity of understanding the representations of the group of large diffeomorphisms in almost all approaches. For particular quantization programs, (2+1)-dimensional models have also offered more specific guidance: Special properties of the loop operators (55), methods for treating noncompact groups in spin foam models, and properties of the sums over topologies described in Section 3.11 have all been generalized to 3+1 dimensions.
The idea that “frozen time” quantum gravity is a Heisenberg picture corresponding to a fixed-time-slicing Schrödinger picture is a central insight of (2+1)-dimensional gravity. In practice, though, we have also seen that the transformation between these pictures relies on our having a detailed description of the space of classical solutions of the field equations. We cannot expect such a fortunate circumstance to carry over to full (3+1)-dimensional quantum gravity; it is an open question, currently under investigation, whether one can use a perturbative analysis of classical solutions to find suitable approximate observables [50] .
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