Well, a couple of weeks ago I was up in Waterloo at the Perimeter Institute with Dan Christensen and his grad student Wade Cherrington for a couple of days for the “Young Loops and Foams” conference. It actually ran all week, but we only took the time out to go for the first couple of days. The talks that we were there for dealt mainly with the loop-quantum-gravity and spin-foam approaches to quantum gravity.

These are not really what I’m working on, though I certainly have thought about these approaches, and Dan and his grad students have done significant work on them. Wade Cherrington has been applying spin-foam methods to lattice gauge theory, and Igor Khavkine has been working on the “new” spin foam models. Both of these guys are in the Applied Mathematics department here at UWO (though Igor is graduating this year), and a lot of their work has been about getting efficient algorithms for doing computations with these models. This seems like great stuff to me – certainly it’s a step in the direction of getting predictions and comparing them to experiments (i.e. “real physics”, though as a “mathematician” who’s only motivated by physics, I clearly don’t say this to be snobby)

Many of the talks were a bit over my head – for one thing, a lot of the significant new stuff involves fairly substantial calculation, which is by nature rather technical. There were some more introductory talks about Group Field Theory – Etera Livine and Daniele Oriti gave talks about Group Field Theory which described the main concepts of this subject. Livine’s talk was fairly introductory – explaining how GFT describes a field theory on a background which consists of a product of a few copies of a Lie group, for instance on G^4. In that example, states of the theory

Oriti’s talk dealt more with issues about GFT, but also emphasized that it can be seen as a kind of “second quantization” of spin networks. That is, one can think of a spin network geometry in terms of a graph which is labelled with spins (in practice, half-integers). Given such a graph, there is a Hilbert space for such states on the graph, whereas in GFT, the graph itself emerges from the states. The total Hilbert space for the fields in GFT then includes many different graphs, with many different numbers of vertices. The analogy to second quantization, in which, for example, one takes the quantum mechanical theory of an oscillator with a given energy, and turns it

Oriti also made references to this paper, in which he proposes a way to get a continuum limit out of GFT (using methods, which I can hardly comment on, analogous to those used to describe condensates in solid-state physics). However, he didn’t have time to describe this in detail. I’ve only looked briefly at that paper, and it seems sort of impressionistic, but the impressions are interesting, anyway.

I managed to have a few conversations with Robert Oeckl about Extended TQFT’s on the one hand, and his general boundary formulation of QFT’s on the other (more here, and slides giving an overview here). These two points of view take the usual formalism of TQFT and run with it in two somewhat different directions. Since I’ve talked a lot here about Extended TQFT’s and categorification, I’ll just say a bit about what Oeckl calls the general boundary formulation. This doesn’t use categorical language, and it remains a theory at “codimension 1” (that is, it tells you about top-dimension “volumes” which connect codimension-1 “surfaces”, and that’s all). It does get outside what the functorial axiomatization of TQFT’s seems to ask, though. In particular, it doesn’t require you to be talking about a cobordism (“spacetime”) going from an input hypersurface (“space-slice”) to an output. Instead, it lets you talk about a general region with boundary, treating the whole boundary at once. Any part of it can be thought of as input or output.

One point of this way of describing a QFT is to help deal with the “problem of time”. His talk at the conference was a sort of “back to basics” discussion about the two basic approaches to quantum gravity – what he named the “covariant” (or perturbative) approach and the “canonical” (or “no-spacetime”) approach. One way to put the “problem” of time has to do with the apparently incompatible roles it plays in, respectively, general relativity and quantum mechanics, and these two approaches respect different portions of these roles.

The point is that in (non-quantum) relativity, a “state” is a whole world-history, part of which is the background geometry, which determines a causal order – a sort of minimal summary of time in that state. But in particular, it is part of the information contained in a state, which describes everything real. In QM, on the other hand, a “state” contains some information about the world in a maximal way (though IF you assume it represents all of reality, THEN you have to accept that reality isn’t local). But moreover, time plays a special role in QM outside any particular “world”.

In particular, the state vector in the Hilbert space \mathcal{H} encodes information about a system between measurements (chronologically!), an operator on \mathcal{H} changes a state \psi_1 into a new state \psi_2 (also chronologically), and composition of operators implies a temporal sequence (which gives the meaning of noncommuting operators – the result depends on the order in which you perform them). This all depends on a notion of temporal order which, in relativity, depends on the background metric, which is putatively depends on the state itself! So the two approaches to quantization try to either (a) keep the temporal order using a fixed background, and treat perturbations as the field (which can only be approximate), or (b) keep the idea that the metric is part of the state and hopefully recover the usual picture in some special cases (which is hard).

So as I understand it, the general boundary approach is meant to help get around this. It works by assigning data to both regions M, and their boundaries \Sigma = \partial M, subject to a few rules which are reminiscent of those which make a TQFT in the usual formulation into a monoidal functor. In particular, the theory assigns a Hilbert space \mathcal{H}_{\Sigma} to a boundary, and a linear functional \rho_M : \mathcal{H}_{\Sigma} \rightarrow \mathbb{C} to a region. This satisfies some rules such as that \mathcal{H}_{\Sigma_1 \cup \Sigma_2} = \mathcal{H}_{\Sigma_1} \otimes \mathcal{H}_{\Sigma_2}, that reversing the orientation of a boundary amounts to taking the dual of the Hilbert space, some gluing rules, and so on.

Then there is a way to recover a generalization of the probability interpretation for quantum mechanics. But it’s not a matter of first setting up a system in a state, and then making a measurement. Instead, it’s a way of asking a question, given some knowledge about the “system” at the boundary. Both knowledge and question take the form of subspaces (denoted \mathcal{A} and \mathcal{S}) of \mathcal{H}_{\Sigma}, and the formula for probability involves both \rho_M and the projection operators onto these subspaces. The “probablity of \mathcal{A} given \mathcal{S}” is:

P(\mathcal{A}|\mathcal{S}) = \frac{|\rho_M \circ P_\mathcal{S} \circ P_\mathcal{A}|^2}{|\rho_M \circ P_{S}|^2}

Then one of the rules defining how \rho_M behaves when M is deformed gives a sort of “conservation of probability” – the equivalent of unitarity of time evolution. If \Sigma decomposes as the union of an input and an output, and the subspaces \mathcal{A} and \mathcal{S} correspond to states on the input and the output surfaces, it gives exactly unitarity of time evolution.

Now, this seems like an interesting idea, assuming that it does indeed get over the shortcomings of both canonical and covariant approaches to quantum gravity. My main questions have to do with how to interpret it in category-theoretic terms, since it would be nice to see whether an extended TQFT – with 2-algebraic data for surfaces of codimension 2, and so on – could be described in the same way. The way Oeckl presents his TQFT’s is quite minimal, which is good for some purposes and avoids some complexity, but loses the organizing structure of TQFT-as-functor.

One thing that would be needed is a way of talking about some sort of n-category which has composition for morphisms with fairly arbitrary shapes – not just taking a source to a target. Instead of composition of arrows tip-to-tail, one has to glue randomly shaped regions together. Offhand, I don’t know the right way to do this.