Linearised regge calculus

Here is a paper that I wrote a draft of back in 1987 but never made it to publication: Linearised Regge Calculus.

A long long time ago, I worked on a discrete version of general relativity called Regge Calculus. This is a rather elegant theory invented by the late great Tullio Regge using simplicial manifolds with Riemannian (or Lorentzian) metrics that are flat on each simplex (“piecewise flat”). I was particularly interested in how this could mimic general relativity on large scales, i.e., I wanted to show that if you worked to a finite resolution so that the (smaller) simplicial structure wouldn’t be apparent, then the metrics approximated nice smooth ones obeying Einstein’s equations.

This is a hard problem, but can be simplified by looking a weak gravitational fields, those that are close to flat space. By linearising the equations, one gets something that can be analysed. I got my result in a paper I was rather proud of, A convergence result for linearised Regge calculus. From the abstract: “…solutions of the linearised Regge equations converge to analytic solutions of the linearised Einstein equations.”

However it needed a technical result that is sort of “obvious” but actually highly technical, that the calculation of the curvature from the metric has the same properties in the linearised Regge case as it does for linearised Einstein. I worked it all out, with heavy use of the simplicial approach to algebraic topology which I had learnt by giving supervisions (tutorials to students) on C.R.F Maunder’s course in Cambridge, following his book Algebraic Topology. The results are summarised very briefly in the paper The fundamental theorem of Regge calculus. I called it that because it seemed pretty fundamental, and probably at that age I didn’t care whether anyone else agreed or not (but obviously the editor of the journal didn’t object).

The actual proof is in the preprint posted above. I don’t know where I sent it, but it seems it didn’t make the CERN or the KEK preprint libraries (and the arXiv didn’t exist then, of course). I always meant to revise the presentation a little before publishing. But I never got around to it; there must have been something more important to do. And now I no longer have the tex file I can’t upload it to arXiv (can I?) People keep asking me for it, so here it is!

Dichromatic state sum models

Manuel Bärenz and I have just finished a paper on 4-dimensional topological state sum models. It is posted here (since 12 Jan 2016), and on arXiv. The idea of the paper is to squeeze more out of the Crane-Yetter state sum model and perhaps indicate how to get a viable quantum gravity model (or condensed matter model) from it.

The original CY was defined just using the quantum group version of SU(2), and turns out to be rather too simple be an interesting physics model. The reason is that the group SU(2) is used to “colour” both one- and two-dimensional dual edges. This has the effect that the quantum theory can’t “see” the difference between the one- and two-dimensional stuff.

All this is best seen using a handle decomposition rather than a triangulation. There’s an operation that changes 1-handles into 2-handles (thus changing the topology of the manifold) and the problem with the original CY is that it is invariant under this operation – which is a property that isn’t wanted. As a consequence, CY is the same on lots of different manfolds, which is why it is “too simple”.

One of the things that Manuel and I have is an efficient translation between the triangulation picture and the handle picture. In the handle picture, Jerome Petit had the idea that the 1- and 2-handles can be coloured differently, to give a new set of models that he called “dichromatic”. We have understood that in the triangulation picture, this corresponds to the CY invariant being “nonmodular” (the original CY is “modular”). Interestingly, there are also models in the handle picture that don’t have a CY description at all. We calculated a few simple examples and found that one of them has configurations that are a plausible analogue of “teleparallel gravity” in the formulation given by Baez and Wise. This doesn’t yet mean we have a new quantum gravity model because, firstly, we only used finite groups instead of Lie groups (to keep things simple) and secondly, it isn’t clear that the action will be the gravity action. Still, it is an interesting direction.

What it needs next is to do a lot more examples. Probably any really interesting examples will involve representations of a group or quantum group that are non-unitary. New territory indeed!

Como lectures

I’m giving some lectures at the SIGRAV school in Como on “Non-commutative geometry and quantum gravity”. I’ll put the slides up here as they happen.

Lecture 1: The Planck scale. This lecture reviews the experimental evidence for a fundamental Planck scale. Various theoretical approaches to modelling the Planck scale are mentioned very briefly, mainly concentrating on the predictions of the spectral action suggested by non-commutative geometry.

Lecture 2: Non-commutative geometry: axioms, examples. The axioms of non-commutative geometry are given in this lecture alongside an explanation of how they are satisfied by an ordinary (commutative) manifold. The example given by the standard model internal space is also summarised.

Lecture 3: Matrix geometries and fuzzy spheres. The lecture explains my project to replace the usual space-time manifold with a non-commutative geometry based on finite-dimensional matrices. Examples with spherical symmetry are considered.

Lecture 4: State sum models, using 2d examples. A different topic is introduced in this lecture: 2d topological state sum models defined using a diagrammatic calculus. Some recent progress in defining models that are sensitive to the spin structure of the surface (and are thus fermionic) is explained.

Lecture 5: Non-commutative geometry in 2d state sum models. The final lecture connects the topic of state sum models and non-commutative geometry. It shows how defects in a 2d state sum model can carry a non-commutative geometry.

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Giving the lectures turned out to be enormously useful. I’m writing a paper called “matrix geometries and fuzzy spaces as spectral triples“, which I hope will appear soon is out now. The paper will be much better for having had to explain the ideas in the lectures.