To Commute or not to Commute...

Sorry for the lack of posting this summer, but I have been trying to write up my thesis. In fact I still am trying, and for no sensible reason I am now doing this at the Noncommutative Geometry Workshop at The Isaac Newton Institute for Mathematical Sciences in Cambridge. The institute is a wonderful place, although I haven't looked around much I have already heard about the lawn on the roof (where you can sometimes see someone mowing, which must look very peculiar from the road) and seen the bust of Paul Dirac in the foyer. But I have been most impressed by the blackboard which is mounted in the lavatory, should you have a maths dispute in the bathroom - it is by far the geekiest thing I have ever seen. It is wonderful and then horrifying and finally wonderful again.

It is also very nice to be visting Cambridge again, and King's College looked especially pretty today in the sunshine. I refer you to my picture below (see there was sunshine today!)

The programme is available online and today was the first day of five days of talks. There's also a public debate on thursday at 8pm in Queen's Lecture Theatre at Emmanuel College entitled "The Nature of Space and Time: An Evening of Speculation" invoving a panel of Alain Connes, Roger Penrose, Shahn Mahjid, Michael Heller and John Polkinghorne which should be interesting. If you are in Cambridge and want to come along you might benefit from registering at the above link. Or maybe you won't benefit - it is not clear.


The first talk this morning was "The Quest for Non Commutative Field Theory" by Vincent Rivasseau and we heard about noncommutative field theory in review. The talk began with a reminder about why noncommutative geometry is an interesting approach to quantum gravity, it went like this:

Quamtum Mechanics (Non commutativity) + General Relativity (Geometry) = Non commutative geometry.

Noncommutative field theory is the generalisation of well-known quantum field theories such as the phi^4 theory to noncommutative spacetimes. The approach is to upgrade the normal scalar product to the simplest non-commutative product which is known as the Moyal product and denoted by an asterisk. One can read all about this in a review paper from 2001 by Michael Douglas and Nikita Nekrasov called, you guessed it, "Noncommutative Field Theory". Rivasseau described the problems of renormalisations of such a naive upgrade to noncommutative geometry, while the planar Feynman diagrams and their ultra-violet divergence remain renormaliazable the non-planar ones pick up an infra-red divergence. This goes by the name of UV/IR mixing and some more complicated terms are needed before the noncommutative version of the theory can be made renormalizable. See Rivasseau's paper with Gurau, Magnen and Vignes-Tourneret for the detail on the renormalizability of noncommuting phi^4 field theory. We were also introduced to the modifications of the Feynman diagrams resulting from the noncommutative promotion. In the commuting field theory one uses the heat kernel as the propagator, while in noncommutative geometry the Mehler Kernel (which is far more complicated than the heat kernel) is the starting point. Interactions, which we are used to describing by one spacetime point, become dependent upon four and a vertex is promoted to a box, the four points specifying the corners. Rivasseau et al also have a paper entitled "Propagators for Noncommutative Field Theories". The end of the talk was dedicated to the parametric space which is a new approach to noncommuative field theory described by Gurau and Rivasseau in their paper. Since I am trying to get a small understanding of the tools used in noncommutative geometry and the motivations I would like to mention a couple of recurrent topics, whose importance I was unable to understand during the talk. The first is that the quantum hall effect seems to be a very important physical example cited by the noncommutative geometers. The second tool that was apparently of great practical value is the so-called Langmann-Szabo duality, which I think was introduced in their paper "Duality in Scalar Field Theory on Noncommutative Phase Spaces".

At 11.35pm Albert Schwarz began talking to us under the title "Space and Time from Translation Symmetry". The talk followed very closely his paper of the same title. He did not talk about noncommutativity much but gave us an axiomatic description of quantum mechanics as a unital, associative algebra of observables, A, over the complex space. He described translations as acting as automorphisms of the algebra A, and soon generalized the idea of a tranlsation generator to a commutative subalgebra. He said he was not trying to give solutions but rather to formulate problems. Alain Connes was interacting with Schwarz from the front row and at one point Connes asked repeatedly about the observables of string theory, culminating with "...but what are the observables?" To which Schwarz replied "There is no question: 'what is observables?'". It was rather like a Jedi mind trick. Schwarz expressed a strong interest in the notion that all physical numbers should be rational, while anything else is just used for felicity. He advocated using p-adic numbers instead of real numbers and the functioning of this proposal can be read about in his recent papers with Kontsevich, Vologodsky and Shapiro [1 and 2].

After lunch, Samson Shatashvili talked under the title "Higgs bundles, gauge theories and quantum groups" who described his reasons for claiming that the so-called Yang-Mills-Higgs theories are dual to the nonlinear Schrodinger quantum system. The preprint (with A. Gevasinov) that the talk was based on is due to appear overnight at hep-th/0609024, but a fundamental paper in the literature, at almost ten years of age, is "Integrating Over Higgs Branches" by Greg Moore, Nikita Nekrasov and Shatashvili. At the end of his talk Shatashvili made the point that as far as he could tell his dual theories contained all the information required for geometric Langlands duality (although he also claimed to not know what geometric Langlands is) and both regimes of the duality are reasonably well understood. But I think we'll have to wait for the preprint...

Today's final talk was a big one. The speaker was the wonderful Alain Connes and he was talking about his recent short paper describing a theory of everything. Lubos Motl has commented extensively on this preprint which you can read by boosting to his Reference Frame. You can also read Alain Connes explanation of himself in the preprint, "Noncommutative Geometry and the Standard Model with Neutrino Mixing" but it will take a lot of work if you are of a more physical than mathematical constitution. Connes described his aim to encode the gravitational and the standard model Lagrangian in a purely geometric picture. The essence of the approach is not to use the metric to define the square of the line element, but rather to start with the line element, and not its square, by using the Dirac operator, D. In fact ds = 1/D. This approach was used to construct the standard model via the spectral action principle in work with Ali Chamseddine (see [1,2]. However the resulting theory was not able to match the standard model perfectly, it exhibited fermion doubling (as pointed out in the work of Lizzi, Mangano, Miele and Sparano) and the introduction of right-handed neutrinos caused Poincare duality to be violated. In his latest work Connes has fixed the problems and reproduced the standard model Lagrangian. This is no mean feat, at the beginning of his talk Connes bamboozled the audience by displaying the enormous Lagrangian of the standard model as written down by Veltman. It filled one page of A4 (single-spaced) and no-one in the audience could read it clearly. In Connes latest work one takes what is called the "finite space", F, of the standard model algebra which is 90-dimensional corresponding to 45 particles and 45 antiparticles. One then writes down the spectral action which has two terms, one for bosons and one for fermions, and one feeds in the spectral dimension....wait! What's the spectral dimension??? Well apparently this is the sequence of positive integers bounded above, and specified, by the metric dimension - and the metric dimension is our usual notion of dimension. There is also another type of dimension called the KO-dimension coming from K-theory, which I do not claim to understand, but Connes' fix of his theory involves allowing the metric dimension to take different values to the KO-dimension. In particular the conjugation properties of the relevant spinors and the necessity of removing his double fermions leads to picking the KO-dimension of the required space F, which Lubos has taken to calling the Connes manifold, to be 6mod8. From our experience of spacetime the metric dimension is 4 and in total the dimension of MxF becomes 10mod8 - which are dimensions that are exceedingly familiar from string theory. Connes strongly denied suggestions that his finite space F was anything like a Calabi-Yau manifold, but said that if someone showed that it was, then he would applaud. Having made these changes to the spectral dimension data that is fed into the spectral action formulation, Connes told us that he expanded out the explicit action and exactly reproduced the enormous Veltman Lagrangian. Due to the compactness of the notation this is an extremely elegent construction of the standard model, and while it may not answer the questions about why certain data are fed in, it is certainly a remarkable discovery. No doubt there is more to be uncovered along these lines. Connes told us that the preprint on the archive is a short version of a much more detailed paper to appear later on, again with Chamseddine. At the end of the talk Connes told the audience that the finiteness of the space F is really tantamount to there existing a basic unit of length, and it was revealed during the questions that it was really the Euclidean version of the standard model that had been constructed. Nevertheless the compact notation makes this approach worth some study.

It is clear to me after today that I wouldn't win the Krypton Factor challenge for observation: I have been surrounded by the words "noncommutative", "non commutative" and "non-commutative" and I still haven't worked out which is the officially endorsed spelling (see my non-renormalized spellings in the text). To hyphen or not to hyphen...that is really the question?