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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