Archive for October, 2006

Mt Aspiring

Carl Brannen is currently enjoying Hawaii where he will be speaking about his derivation of the lepton masses. It is a large conference with a String theory session, which has some interesting sounding talks such as Sakurai on the geometric Langlands program. Amongst bloggers, at least Gordon Watts appears to be there. I guess we’re heading into a busy conference season!

The Euler characteristic for a Coxeter complex based on A_n goes like

chi = (-1)^n.2n.(n – 2)!!(n – 2)!!

which grows quickly but is only -6 for the case of A_3. Goodness me, that number does have a tendency to crop up, doesn’t it now?

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Fairy Fields

Weinberg described a theory of electroweak forces in 1967. He shared the Nobel prize for this unification with Glashow and Salam in 1979. Another gauge theory, quantum chromodynamics, took much longer to be accepted as experimental verification slowly came in. Gluons were only discovered at PETRA II at DESY in 1979.

The electroweak theory required a Higgs boson to explain the aquisition of mass of particles. It is a shame that these events occurred in the order that they did, although of course it had to be. For a long time many physicists took the Higgs mechanism seriously and failed to investigate clues from QCD. QCD is, after all, a theory for quarks which participate in the weak interactions.

Replacing the Higgs mechanism within the framework of rigorous QFT has proven to be a daunting task. It was, however, quite clearly never an explanation for mass quantum numbers, which by definition must arise in a quantum gravitational theory.

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Upcoming Events

On Monday January 22 2007 the NSF Distinguished Lecture will be given by the respected cosmologist Sean Carroll. The lecture has the title: Dark Energy, or Worse: Was Einstein Wrong?

From October 29 to November 3 2006 the Joint Meeting of the Pacific region Particle Physics communities will be held in Honolulu. Make sure you hear Carl Brannen’s talk if you are lucky enough to be in Hawaii.

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Euler’s Enchiladas

Leonhard Euler lived from 1707 til 1783. He published such an astonishing amount of mathematics that the St. Petersburg Academy continued publishing his work for more than 30 years after his death. Eventually he went blind, but continued doing enormous calculations in his head. He could recite the entire Aeneid of Virgil. One thing he did was study the multiple zeta values. He proved the two argument (depth 2) version of the result that the value of the Riemann zeta function at the 1-ordinal n was the sum over (depth k , weight n) MZVs such that the first argument was greater than 1. The depth 3 case was proved in 1996.

Euler’s MZVs were largely forgotten until recent times, but since their appearance in QFT structures they have arisen in many contexts. Multiple polylogarithms are a natural generalisation. Now we know that the MZVs are algebraic integrals for the cohomology of moduli of punctured spheres.

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The details of the new Standard Model of Connes, Marcolli and Chamseddine is now out. Recall that John Barrett also has a recent paper out on a Lorentzian version of the Connes model. These ideas bring neutrino mass generation into the SM in a natural way, but the number of generations is really put in by hand. Should we be focusing on the NCG language in order to interpret this new SM?

The physical problems of QG and Yang-Mills and precise mass values are closely related to the Riemann hypothesis, which is naturally what Connes and Marcolli are trying to solve, as is well known. One important ingredient in this program is the notion of Grothendieck-Teichmuller group, as discussed in this lecture by Schneps. Note the pretty tree diagram on page 11. Many people, such as Kontsevich and Cartier, have thought about this structure from different angles. Kontsevich said we should think of the GT group as the quotient of the motivic Galois group by its action on the spectrum of an algebra generated by (2 pi i), its formal inverse and all the MZVs. Apparently there are some problems with this idea. But the question is, do we really need to define this GT group? Perhaps we can get physical parameters much more directly.

In AQFT one prefers to think about Tomita-Takesaki modular theory. I went to an interesting NCG seminar on this yesterday by Paolo Bertozzini. He has been trying to understand the basic NCG geometry/algebra correspondence from a more categorical point of view. In fact, he made it clear that the categorical duality is far from understood. The correspondence usually works only on the level of objects: take a spin manifold and get a spectral triple, or take a spectral triple and find its spectrum. But to describe the correspondence properly the adjunction natural transformations need to be fully described. The categories need morphisms. Now one can do this, but it leads to the question that Paolo is thinking about: the spectral triples appear to be approximations to something higher categorical. They are recovered as endofunctors of some kind in a richer structure. What is this structure? Paolo was thinking along the lines of a quantum topos theory. Funny thing was that just after I was introduced to Paolo we realised that we had met on the internet, in a discussion on the fqxi funds on Woit’s blog. Paolo was one of those rejected for his over enthusiastic category theoretic proposal.

It was a nice day yesterday. There was a Feynman film night run by the Physics group. I was talking to a quantum optics guy and he told me they are getting six new staff in Quantum Information next year. Given the small size of the department at present, this is just flabbergasting. I stood there like a stunned mullet and he pointed out that there was an awful lot of money in this game. None of these guys seemed to have the least interest in what the Category Theory group are doing, even though they haunt the same corridors. One of the new guys will be Terno who has been at Perimeter, so I’m looking forward to meeting him in a few weeks when he arrives.

The pizza was yummy, too.

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The Future is Here

I continue to be alarmed at the disrespect that many of my colleagues pay to their climate science colleagues, even now, as the once green pastures of New South Wales turn to dust and atmospheric CO2 levels rise above anything they would believe possible.

On the news yesterday I heard an interview with a local astronomer, who felt it necessary to defend funding for science at a time when food and commodity prices were rising. The unfortunate reality is that we cannot expect this problem to go away. You might think it unlikely. Look at the data yourself. Water shortages and forced migrations have always caused economic and political tension. They have never happened on the scale that they soon will. It’s a pretty simple story, really. It’s time to think about what you take for granted: the fresh water, long showers, luxury items, enough food to eat.

Yes, people like me are called alarmist. I’ve been hearing that for a long time. That’s why it’s all so depressing. We live at a time when pretty well everybody on earth needs to change their life. And they’re just not doing it.

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Trippy Trees

Tony Smith, who likes octonions and Clifford algebras a lot, has a nice page on the surreal numbers. The further one moves up the tree, the more rational numbers one gets!

We’ve also seen trees in phylogenetics and knot theory, but most importantly in Batanin’s operads. Recall that 1-level trees represented the Stasheff associahedra. These turn up everywhere, such as in tiling the real moduli M(0,n) of genus zero surfaces.

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Connes Kreimer Marcolli

There is an amazing series of papers by Connes, Marcolli and others on From Physics To Number Theory. See for example here or here or here. This goes back to work of Kreimer and Broadhurst, which is now very well known. Some of the older papers are here. I particularly recommend the paper: Broadhurst and Kreimer, Association of Multiple Zeta Values with Positive Knots via Feynman Diagrams up to 9 Loops, Phys. Lett. 393 B (1997) 403-412.

Its about turning knots into simple Feynman diagrams into Multiple Zeta Values. These MZVs satisfy all sorts of crazy relations, which the mathematicans have been studying like crazy. But really they’re quite simple. They act on a set of k ordinals (yes, that’s right, you should be thinking 1-ordinals) and are characterised by two numbers, namely the weight n, which is the sum of these, and k itself, the so called depth. Of course these naturally show up as special integrals of something called Mixed Tate Motives (don’t even ask), so we know that the weight n is the same n of M(0,n+3). Goodness, me. The Yang-Mills problem and the Riemann hypothesis seem to be related. Well, well.

The real question, however, is how to go beyond scalars to other entities in QFT. Any guesses?

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Yang-Mills Yumyums

The gallant kneemo gave me a link to some great slides by Zvi Bern who works on perturbative quantum gravity. Before twistor strings came along he was thinking about the KLT relations between gravity amplitudes and colour free diagrams, such as MHV tree level diagrams for n gluons.

These amplitudes are surprisingly simple, and apparently people don’t really understand why! For example, the MHV tree amplitude for 4 gluons in QCD looks like A4 = (k1 + k2)^2 / (k2 + k3)^2.

It does make one wonder about the modelling of Witten’s gluon spaces by projective twistor geometry in the context of M theory. Remember that there are three complex moduli of real dimension six, namely M(0,6), M(1,4) and M(2,0). We already know the first one is interesting because it has an orbifold Euler characteristic of minus six. What if we needed to draw little loops on representative Riemann surfaces? There is a nice mathematical way to think about this, but just imagine cutting up the two holed surface from end to end, straight through the two holes and at 90 deg to the correct way to cut a bagel. That cut marks six points on the two holed surface. It turns out that special loops on this surface map to ones on the M(0,6) by taking the six points to the six punctures. Zvi Bern would say that the graviton polarisation tensor is written as a square of gluon polarisation vectors.

Of course one can consider any number of punctures on the sphere to get tree amplitudes for n gluons. All one needs to know is that the compactified real moduli are all tiled by 1-operad Stasheff associahedra, as shown in the work of Devadoss. For example, the three dimensional case of six points is tiled by the 3D 14 vertex polytope.

From both a physical and mathematical point of view, we would like to better understand Yang-Mills theory in 4D. To quote Jaffe and Witten: we would like to prove that for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on R4 and has a mass gap d > 0. One of the ATLAS people recently said: our field must get some serious profit from LHC start-up and first data, and we better teach ourselves right now how to explain Higgs, SUSY and extra dimensions to the public and the media. Oops. This statement needs a little revision.

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Jordan and Pizza

The volume The Physicist’s Conception of Nature, edited by Mehra, is a collection of lectures given at the 70th birthday celebrations for Dirac in 1972. The list of contributors is impressive: Chandrasekhar, Dirac, Wheeler, Heisenberg, Wigner and Schwinger, to name a few.

Pascual Jordan’s contribution is entitled The Expanding Earth. He explains that, having been deeply impressed with Dirac’s 1937 idea of a varying G/c^2, he spent time investigating the possibility that the Earth had been expanding over time. The lecture includes some beautiful geological diagrams regarding mid-ocean drift, and he talks about the difficulty that Wegener had with geologists accepting the theory of continental drift. Now we understand that the value of G/c^2 is decreasing as we go back in time, and consequences of this should indeed be measurable on Earth. Jordan says:

There exists a great diversity between the mentalities of physicist’s and of geologists. Physicist’s are eager to learn about new facts and new ideas caused by new facts.

Pascual Jordan is one of the founders of Jordan algebras, which appear in M theory.

For anyone who happens to be around Sydney next week: come to the Feynman fest at the University of Macquarie at 5.30 pm on Wednesday Oct 25 for free pizza!

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