In gauge theories, one can often use duality to relate a strongly coupled theory, where perturbation theory is bad, to a weakly coupled one, in which it is good. The situation seems to be similar in gravity, with the relation between ultra-violet and infra-red cut o s, in the AdS-CFT correspondence.
I shall therefore not worry about the higher loop divergences, and use eleven dimensional supergravity as the local description of the universe. This also goes under the name of M theory, for those that rubbished supergravity in the 80s and don’t want to admit it was basically correct. In fact, as I shall show, it seems the origin of the universe is in a regime in which rst order perturbation theory is a good approximation.
The second pillar of quantum cosmology is boundary conditions for the local theory. There are three candidates, the pre big bang scenario, the tunnelling hypothesis, and the no boundary proposal.
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Boundary conditions for Quantum Cosmology
1. Pre big bang scenario
2. Tunnelling hypothesis
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3. No boundary proposal
The pre big bang scenario claims that the boundary condition is some vacuum state in the innite past. But, if this vacuum state develops into the universe we have now it must be unstable. And if it is unstable, it wouldn’t be a vacuum state, and it wouldn’t have lasted an innite time before becoming unstable.
The quantum tunneling hypothesis is not actually a boundary condition on the spacetime elds, but on the Wheeler-Dewitt equation. However, the Wheeler-Dewitt equation acts on the innite dimensional space of all elds on a hyper-surface and is not well dened. Also, the 3 + 1, or 10 + 1, split is putting apart that which God, or Einstein, has joined together. In my opinion, therefore, neither the pre bang scenario, nor quantum tunneling hypothesis, are viable.
To determine what happens in the universe, we need to specify the boundary conditions, on the eld congurations, that are summed over in the path integral. One natural choice would be metrics that are asymptotically Euclidean, or asymptotically Anti de Sitter. These would be the relevant boundary conditions for scattering calculations, where one sends particles in from innity and measures what comes back out.
However, they are not the appropriate boundary conditions for cosmology. We have no reason to believe the universe is asymptotically Euclidean or Anti de Sitter. Even if it were, we are not concerned about measurements at innity, but in a nite region in the interior. For such measurements, there will be a contribution from metrics that are compact, without boundary. The action of a compact metric is given by integrating the Lagrangian.
Thus, its contribution to the path integral is well dened. By contrast, the action of a non compact, or singular, metric involves a surface term at innity, or at the singularity. One can add an arbitrary quantity to this surface term. It therefore seems more natural to adopt what Jim Hartle and I called, the ‘no boundary proposal’. The quantum state of the universe is dened by a Euclidean path integral over compact metrics. In other words, the boundary condition of the universe, is that it has no boundary.
No Boundary Proposal
The boundary condition of the universe is
that it has no boundary
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There are compact Reechi at metrics of any dimension, many with high dimensional moduli spaces. Thus eleven dimensional supergravity, or M theory, admits a very large number of solutions and compactications. There may be some principle, that we haven’t yet thought of, that restricts the possible models to a small sub class, but it seems unlikely. Thus I believe that we have to invoke the Anthropic Principle. Many physicists dislike the Anthropic Principle. They feel it is messy and vague, that it can be used to explain almost anything, and that it has little predictive power. I sympathize with these feelings, but the Anthropic Principle seems essential in quantum cosmology. Otherwise, why should we live in a four dimensional world and not eleven, or some other number of dimensions.
The anthropic answer is that two spatial dimensions are not enough for complicated structures, like intelligent beings.
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On the other hand, four, or more, spatial dimensions would mean that gravitational and electric forces would fall o faster than the inverse square law. In this situation, planets would not have stable orbits around their star, nor would electrons have stable orbits around the nucleus of an atom. Thus intelligent life, at least as we know it, could exist only in four dimensions. I very much doubt we will nd a non anthropic explanation.