Tuesday, September 12, 2006

A Penrose Universe

First, there have been a number of introductory posts on the Barrett-Connes standard model spectral triple over at the n-category cafe, in particular see posts I, II, III and IV. Second John Barrett has been talking about his approach to finding the appropriate spectral triple in Cambridge yesterday, his paper, "A Lorentzian version of the non-commutative geometry of the standard model of particle physics" appeared on the arxiv on the same day as Connes' and suggested identical alterations, and while I have been feverishly attacking my thesis, Alejandro Rivero attended the talk and has made some comments about it on physics forums. He also has uploaded his notes from the talk, but these are a little hard to read. Also the Newton Institute have audio of all the talks from last week's workshop for you to enjoy here.

On a different note, a while ago I attended the Winter School on the Attractor Mechanism in Frascati but was unable to write up much about it due to being very busy. Well Per Kraus, who gave a set of talks at the school has helped me out by publishing his lecture notes on the archive as "Lectures on black holes and the AdS3/ CFT2 correspondence". Thank-you Per!

You have probably heard of Penrose tilings, Penrose diagrams, Penrose limits, Penrose triangles and even a Penrose staircase, well last week, during Roger Penrose's talk at the Noncommutative Geometry Workshop, I heard a little about a Penrose Universe. (In my picture it looks like Roger Penrose is keeping the audience entertained with his shadow puppet routine)It is a peculiar thing and represents Penrose's approach to understanding the second law of thermodynamics, that entropy increases, on a cosmological scale. Penrose points out that there is a contradiction in the entropy increase picture of the big bang, which is that the background radiation matches a model that is in thermal equilibrium. Such a state is just about the highest entropy state you can imagine for a system. Almost any other distribution besides uniform would result in a smaller entropy. The Big Bang picture requires the restriction of phase space at early time and the consequence that entropy ought to be tiny at the Big Bang. Penrose wonders how to resolve this contradiction, and seeks to resolve it by separating the entropy of the universe into that arising from matter (the eneergy momentum tensor) and that encoded in gravitational degrees of freedom (the Weyl curvature, see below).

In a nutshell it is a universe without a big crunch, and if taken to the extreme limit without a big bang, but which gives the features of energy density fluctuations in the background radiation. Penrose suggested his Weyl curvature hypothesis in 1979 as a physical origin of the increasing entropy of the universe with time. The Weyl curvature tensor is the traceless part of the Riemann curvature, i.e. the parts which when contracted upon two indices give zero for the Ricci (two-form) tensor. To quote Penrose's description,
"In Einstein’s theory the Ricci curvature R_{ab} is directly determined by the gravitational sources, via the energy-momentum tensor of matter (analogue of the charge-current vector J_{a} in Maxwell’s electromagnetic theory) and the remaining part of the space-time Riemann curvature, namely the Weyl curvature C_{abcd}, describes gravitational degrees of freedom (analogue of the field tensor F_{ab} of Maxwell’s theory)."
The Weyl curvature hypothesis is that the Weyl curvature is zero at the big bang but rises gradually as the universe ages. Consequently the Weyl curvature will not be zero at black hole singularities and we may use the Weyl curvature in this picture to distinguish between cosmological singularities and other singularities. As time passes, the Weyl curvature increases and gravitational masses attract each other more strongly forming a less-homogeneous universe, with clumped masses and higher entropy encoded in the dense packing massive bodies. So that early uniform universe may be explained by there being zero Weyl curvature. Penrose talks about the Weyl curvature's growth as freeing up gravitational degrees of freedom that may then be excited. It is the excitation of these gravitaional degrees of freedom that is the real measure of entropy. It is a nice picture. But just what drives the Weyl curvature's variance is a mystery to me. It does allow us to describe gravitational entropy increase with a tensor field, and of course to associate the arrow of time with such a field. So, at least, algebraically it is appealing. It also offers an alternative to a fast period of inflation in the early universe, which some might find equally as arbitrary as a varying curvature field.

The latest idea is built upon the findings of Paul Tod in his paper "Isotropic cosmological singularities: other matter models", where it is shown that even though the Ricci curvature blows up at the cosmological singularity the Weyl curvature remains finite. Penrose takes this finding and argues that near the big bang gravity becomes a conformal theory, so that he may rescale the metric to infinity and blow up the big bang singularity. The justification for this is that near the big bang, when temperatures are extremely high, there is little difference between the dynamics of massive and massless particles, all particles are treated as massless, and respect conformal equations of motion. Once the description of physics is conformally invariant, Penrose says that a sense of time is lost, tying in neatly with the low entropy ideas. Having blown-up the cosmological singularity, and beleiving that the Weyl curvature remains finite, has lead Penrose to ponder the smooth continuation of the Weyl curvature at the boundary. Perhaps, he suggests, in what he refers to as his "outrageous" proposal, there is a "conformal cyclic cosmology", in which one may knit the conformal geometry at the big bang to another conformal geometry prior to the big crunch, and thereby create a series of universes with a long-lived/eternal conformal geometry.

How can Penrose convince us that geometry may become conformal again at the end of the universe's lifetime? Well, he says, after most of the matter in the universe has been swallowed by black holes and has then been recycled back into the universe via massless Hawking radiation we are really only troubled by charged matter that escaped this process. Here we must presume that black holes can radiate away to pure radiation (which seems unlikely - no topology change, no unexcited microstates...) leaving a universe that may contain some unabsorbed charged matter (let's call all matter electrons) and photons. Now if we can come up with some way of doing away with the electrons, says Penrose, then we will be in business. For again without any massive particles left in the universe the scale of the metric has lost its meaning. This is the real weak point, since the mechanisms to get rid of electrons require either allowing their charge or their mass to dissipate over long time scales. But, of course, this may be possible. Once this position is arrived at one might imagine a conformal rescaling of the metric down to zero, so that a future infinite region is made finite and may be attached to the finite cosmological singularity of some other universe. Penrose argues that the appropriate conformally invariant verion of general relativity the spin-2 field picks up an inverse conformal factor when the conformal tranformation is applied to the metrc, while the Weyl curvature does not. Hence the matter density from the previous universe survives the conformal rescaling and passes over into the next universe. Penrose identifies this with the density fluctuations at the Big Bang - which is exceedingly appealing, and presumably testable.

The conformal rescaling marks the beginning of the "new" universe. In this picture there is also cosmological scale clock, whose ticks are the rescalings of the metric, so nothing to worry about on a local level. It is also imperitive that the conformal rescalings occur in the right way, i.e. to infinity at big bang singularity and to zero at late time. Effectively one must imagine that the previous universe occured at miniture scale comparatively, and the future universe will be built upon the swirling dust of ours at a gigantic scale. It doubles as a very nice picture for a science-fiction novel, as well as an exceedingly interesting proposal for the origin of the density fluctuations in the universe.

We have mentioned the assumptions, namely that black holes evaporate to pure radiation and that electron charge/mass dissipates. There are also questions about particle antiparticle pair creation, but which if we are able to argue in favour of some long term alteration of the properties of the electron, so that it eventually becomes pure radiation, this would not present a problem. Furthermore there seem to be mysterious forces driving the rescaling of the metric, for which it would seem some additional dilaton field may be necessary or some other argument presented.

You can hear Penrose talk on this in two places on the web, both of which took place at the Newton Institute. The first is from November 2005 at the Spitalfield's Day and the second occurred last
(you will have to wait until the end to hear about this cosmological model). Penrose also has written up his description of wha he refers to as "conformal cyclic cosmology" in the proceedings of the EPAC, 2006, conference, and one can read the pdf here.

On Thursday of last week we also suffered a panel discussion on the nature of space-time, being organised by the sponsors the notorious Templeton foundation, I was a little wary. I think on the whole the event worked very well, it was simply not to my personal taste, but I went along to enjoy the views of (Rev. Dr.)John Polkinghorne(Eclesiastical physicist), (Prof.) Shahn Majid, (Rev. Dr.) Michael Heller (of the Vatican observatory), (Sir) Roger Penrose and (Prof.) Alain Connes. As you might imagine there was a very strong representation of the religious apprecatiation of spacetime, and even Alain Connes couldn't resist talking of his interpretation of three pages of "ancient text" by which he meant the Veltman Lagrangian of the standard model. I do not think it was a night of much scientific progress. But there were some anecdotal highlights. The evening was organised so that each panelist took five minutes to mention their conception of spacetime, there was an overhead projector and it appeared that the speakers were well organised having prepared detailed slides. Throughout the first two talks Alain Connes looked a little preoccupied, occasionally staring at the desk and sometmes laying his head upon it. Suddenly after the second speaker Connes sprang to life, borrowed some OHP slides and multicoloured pens from the others, and began to prepare his own slides there and then. Since the chairman was supposedly inviting the presentations at random this seemed a wonderfully carefree approach. It made for some nice theatre. We also heard an anecdote from Roger Penrose, in response to the first question from the audience which was along the lines of 'which came first quantum mechanics or general relativity?'. Penrose replied by telling of a time he had listened to a wonderfully animated lecture by John Wheeler and at the end there came a similar question from the audience, which came first G.R. or the quantum principle? Penrose said that a small voice in the front of the audience piped up and asked 'what is the quantum principle?' The small voice belonged to Dirac. A final amusing interchange involved Shahn Majid, the chairman (Jeremy Butterfield) and a mischievous Alain Connes. Shahn Majid was summing up his disenchantment with the present understanding of spacetime with the Shakespearean line "there is something rotten in the state of Denmark" (Penrose said later he thought Majid was referring to the Copenhagen interpretation), and Alain Connes responded with another Shakespeare quotation "Throw physics to the dogs; I'll none of it." It was left to Jeremy Butterfield to point out that the actual line from Macbeth is about "physic" (referring to medecine) and not "physics". So there was some enjoyment to be had from the evening after all.

Monday, September 04, 2006

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?