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3.11 Euclidean path integrals and quantum cosmology

Lorentzian path integrals allow us to compute interesting topology-changing amplitudes, in which the universe tunnels from one spatial topology to another. They do not, however, directly address a principle issue of quantum cosmology, the problem of describing the birth of a universe from “nothing”. Here, most of the literature has focused on the Hawking’s Euclidean path integral  [154] and the Hartle-Hawking “no boundary” proposal  [152], which describes the universe in terms of a path integral over Riemannian metrics on manifolds with a single, connected boundary S . As in 3+1 dimensions, most of the work in 2+1 dimensions has concentrated on the saddle point approximation. So far, the main benefit of the lower-dimensional model has been the possibility of treating topology more systematically, revealing interesting effects that are only now being explored in 3+1 dimensions.

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 M with a single spatial boundary S, as in Figure  4View Image .

View Image

Figure 4: A manifold M with a single boundary S describes the birth of a universe in the Hartle-Hawking approach to quantum cosmology.
In 2+1 dimensions, the selection rules of  [6] imply that such a process can be described in Lorentzian signature only if x(S) = 0, that is, only for S a torus. Moreover, the known examples of such metrics always yield a degenerate metric on S . If one allows Riemannian signature, on the other hand, such a path integral makes sense for any spatial topology, and if one further requires that S be totally geodesic - that is, that the extrinsic curvature of S vanish - one can smoothly join on a Lorentzian metric at S   [141] . Hartle and Hawking therefore propose a “ground state” wave function
integral sum Y[h, f|S;S] = [dg][df] exp (- IE[g, f]), (75) M:@M=S
where the value of the path integral is determined by a specified induced metric h and matter configuration f|S on the boundary. The summation represents a sum over topologies of M ; in the absence of any basis for picking out a preferred topology, all manifolds with a given boundary S are assumed to contribute. The wave function Y is to be interpreted as an amplitude for finding a universe, with a prescribed spatial topology S, characterized by an “initial” geometry h and a matter configuration f | @M . This approach finesses the question of initial conditions for the universe by simply omitting an initial boundary, and it postpones the question of the nature of time in quantum gravity: Information about time is hidden in the boundary geometry h, but the path integral can be formulated without making a choice of time explicit.

The path integral (75View Equation) 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 2f gmn --> e gmn contributes to IE 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  [100Jump To The Next Citation Point99] 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 (75View Equation) is a sum of terms of the form

( ) ( volg(M ) ) exp - IE = DM exp sign(/\) --------1/2 , (76) 4pGh |/\ |
where g is an Einstein metric on M, volg(M ) is the volume of M with the metric rescaled to constant curvature ± 1, and the prefactor D M is related, as in Section  3.10, to the Ray-Singer torsion of M .

For /\ > 0, 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 volg(M ) comes from the three-sphere itself, and one might expect it to dominate the sum over topologies. As shown in  [71Jump To The Next Citation Point], though, the number of topologically distinct lens spaces with volumes less than 3 volg(S ) 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 /\ < 0, three-manifolds that admit Einstein metrics are hyperbolic, and the single largest contribution to Equation (76View Equation) 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 DM), 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  [79Jump To The Next Citation Point80Jump To The Next Citation Point22919] .

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  [132Jump To The Next Citation Point] . 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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