In the Hartle-Hawking approach to quantum
cosmology, the initial wave function of the universe is described
by a path integral for a compact manifold
with a single spatial boundary
, as in Figure
4
.
|
The path integral (75) cannot, in general, be evaluated exactly, even in 2+1
dimensions. Indeed, there are general reasons to expect the
expression to be ill-defined: A conformal excitation
contributes to
with the wrong sign, and the action is unbounded below
[142]
. In the (2+1)-dimensional Lorentzian dynamical triangulation
models of Section
3.7, however, it is known that these wrong sign contributions are
unimportant
[12]
; they are overwhelmed by the much larger number of well-behaved
geometries in the path integral. This has led to a
suggestion
[100
, 99]
that the conformal contribution is canceled by a Faddeev-Popov
determinant (see also
[198]), and some preliminary supporting computations have been made in
a proper time gauge
[100]
.
Assuming that the “conformal factor problem” is
solved, a saddle point evaluation of the path integral is
arguably a good approximation. For simplicity, let us ignore the
matter contribution to the wave function. Saddle points are then
Einstein manifolds, with actions proportional to the volume. An
easy computation shows that the leading contribution to
Equation (75) is a sum of terms of the form
For
, three-manifolds that admit Einstein metrics are all elliptic -
that is, they have constant positive curvature, and can be
described as quotients of the three-sphere by discrete groups of
isometries. The largest value of
comes from the three-sphere itself, and one might expect it to
dominate the sum over topologies. As shown in
[71
], though, the number of topologically distinct lens spaces with
volumes less than
grows fast enough that these spaces dominate, leading to a
divergent partition function for closed three-manifolds. The
implications for the Hartle-Hawking wave function have not been
examined explicitly, but it seems likely that a divergence will
appear there as well.
For
, three-manifolds that admit Einstein metrics are hyperbolic, and
the single largest contribution to Equation (76
) comes from the
smallest
such manifold. This contribution has been worked out in detail,
for a genus 2 boundary, in
[134]
. Here, too, however, manifolds with larger volumes - although
individually exponentially suppressed - are numerous enough to
lead to a divergence in the partition function
[71]
. In this case, the Hartle-Hawking wave function has been
examined as well, and it has been shown that the wave function
acquires infinite peaks at certain specific spatial geometries:
Again, topologically complicated manifolds whose individual
contributions are small occur in large enough numbers to dominate
the path integral, and “entropy” wins out over “action”
[69]
.
The benefit of restricting to 2+1 dimensions
here is a bit different from the advantages seen earlier. We are
now helped not so much by the simplicity of the geometry
(although this helps in the computation of the prefactors
), but by the fact that three-manifold topology is much better
understood than four-manifold topology. It is only quite recently
that similar results for sums over topologies have been found in
four dimensions
[79
, 80
, 229, 19]
.
As noted in Section
3.6, recent work on spin foams has also suggested a new
nonperturbative approach to evaluating the sum over topologies.
Building on work by Boulatov
[60], Freidel and Loupre have recently considered a variant of the
Ponzano-Regge model, and have shown that although the sum over
topologies diverges, it is Borel summable
[132]
. This result involves a clever representation of a spacetime
triangulation as a Feynman graph in a field theory on a group
manifold, allowing the sum over topologies to be reexpressed as a
sum of field theory Feynman diagrams. The model considered
in
[132]
is not exactly the Ponzano-Regge model, and it is not clear that
it is really “ordinary” quantum gravity. Moreover, study of the
physical meaning of the Borel resummed partition function has
barely begun. Nonetheless, these results suggest that a full
treatment of the sum over topologies in (2+1)-dimensional quantum
gravity may not be hopelessly out of reach.
There are also indications that string theory might have something to say about the sum over topologies [111] . In particular, the AdS/CFT correspondence may impose boundary conditions that limit the topologies allowed in the sum. Whether such results can be extended to spatially closed manifolds remains unclear.
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