Topologically Massive Gravity in dS3 (arxiv:1105.4733): detailed analysis

[mitchell porter]

Sorry for the proliferation of threads, but I wanted to have a thread devoted specifically to analysis of how this paper works, and not for https://www.physicsforums.com/showthread.php?t=503605" to higher dimensions. What I want to do is to bring out exactly how its conclusion is obtained, and discover exactly what that conclusion is good for.

The central result of the paper is a deduction: we define a certain 3d gravity theory, make various deductions (and introduce a few other assumptions along the way...), and come to the conclusion that the partition function is finite. So, I want to see how that deduction works; make it as plain as an exercise in high school algebra. For professional researchers, the paper itself should already do that, so this is partly an exercise in learning how to decode the literature. But there is some reason to think that the paper is noteworthy in itself, and not just as a case study.

What the conclusion is good for: I can't say in advance what this might be, with the same precision. But as an initial orientation, let me say that I hope it will tell us something about how dS/CFT works. The major problem in dS/CFT is the lack of a concrete example, comparable to the examples of AdS/CFT supplied by Maldacena's original paper. Here we have the partition function of a theory of quantum gravity on dS3; how far can we get in trying to interpret it as the partition function of a CFT2? But that discussion should be deferred until the paper's central argument is properly understood.

 

[mitchell porter]

First observation: to understand the paper, one has to understand the conceptual framework in which this partition function acquires its meaning - and this will require a detour to examine the precursor paper arxiv:1103.4620, and possibly others.

It seems that quantum gravity here is really approached via quantum cosmology according to Hartle and Hawking. There, you have a "wavefunction of the universe" in which an amplitude is attached to every possible spacelike configuration of metric and fields, and this amplitude is computed by a path integral over all Euclidean manifolds which have that configuration as a boundary. I am not really a fan of this concept - I think it's clever, but somewhat disconnected from observable reality. Where is time in this picture? However, mathematical results in physics sometimes have a significance outside the particular philosophical framework employed by their discoverers, so let us persist with the Hartle-Hawking approach for now, so as to understand what Maloney et al are up to. (1103.4620, pages 2-3, has a remark about the computation of correlation functions, which is exactly the sort of concrete application which might survive a reinterpretation of the formalism.)

Now at this point I will confess an elementary puzzle that I haven't solved. The Hartle-Hawking "quantum ground state of the universe" is computed by the path integral of manifolds-with-boundary that I just described. But the partition function is a path integral over manifolds without boundary. We are told that this helps us to define the norm of the universal wavefunction - but I don't see how. Do we consider manifolds-without-boundary in which the relevant "boundary condition" occurs on a slice somewhere inside them? Maybe, but then how, exactly? And I'd prefer not to guess. So the next thing to do is to find, somewhere in the literature, an exposition of exactly how this partition function helps us interpret the wavefunction.

 

[mitchell porter]

I went all the way back to http://prd.aps.org/abstract/PRD/v28/i12/p2960_1" , the paper which first introduced the wavefunction of the universe. In section II D, I find the reverse of what I want: that is, they construct the equivalent of a partition function over manifolds without boundary, by forming a scalar product of wavefunctions-of-the-universe (see also the paragraph following their equation 4.3). The geometry-amplitudes in one wavefunction come from a sum over manifolds with a compact past which terminate in the present with that geometry, the other from a sum over manifolds with a compact future which originate in the present with that geometry... my phrasing is a little awkward, but the concept does make sense.

All that worries me now is the process of working backwards from the partition function to the wavefunctions. It seems like we might have to take functional derivatives in superspace (in the Wheeler-DeWitt sense of the word - a configuration space of spacelike geometries - not in the supersymmetry sense of the word). Here's what I mean. The partition function will assign amplitudes to three-geometries (since we are concerned with dS3 here), the wavefunction will assign amplitudes to two-geometries. So Wheeler-DeWitt superspace will be the space of two-geometries, and a particular three-geometry corresponds to a surface in superspace - the set of all two-geometries which occur as slices through the given three-geometry. The partition function will have a superspace representation as an assignment of amplitudes to these surfaces. How do we use that to define a norm on wavefunctions (assignments of amplitudes to two-geometries, i.e. points in superspace)? It seems like it ought to be possible; I still don't quite see it; but the answer may be elsewhere in Hartle-Hawking.