Difficult as it is, however, an understanding of matter couplings may be the key to many of the conceptual issues of quantum gravity. One can explore the properties of a singularity, for example, by investigating the reaction of nearby matter, and one can look for quantization of time by examining the behavior of physical clocks. Moreover, some of the deep questions of quantum gravity can be answered only in the presence of matter. For example, does gravity cut off ultraviolet divergences in quantum field theory? This idea is an old one [109, 165, 166], and it gets some support from the boundedness of the Hamiltonian in midi-superspace models [32], but it is only in the context of a full quantum field theory that a final answer can be given.
Second, the discovery that the sum over topologies can lead to a divergent partition function has been extended to 3+1 dimensions, at least for , and it has been argued that this behavior might signal a phase transition that could prohibit a conventional cosmology with a negative cosmological constant [79, 80] . The crucial case of a positive cosmological constant is not yet understood, however, and if a phase change does indeed occur, its nature is still highly obscure. It may be that the nonperturbative summation over topologies discussed at the end of Section 3.11 could cast light on this question.
One might also hope that a careful analysis of the coupling of matter in 2+1 dimensions could reveal useful details concerning the vacuum energy contribution to , perhaps in a setting that goes beyond the usual effective field theory approach. For example, there is evidence that the matter Hamiltonian is bounded above in (2+1)-dimensional gravity [27] ; perhaps this could cut off radiative contributions to the cosmological constant at an interesting scale.
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