Arcadian Functor

occasional meanderings in physics' brave new world

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Location: New Zealand

Marni D. Sheppeard

Tuesday, October 31, 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?

Sunday, October 29, 2006

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.

Friday, October 27, 2006

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.

Thursday, October 26, 2006

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.


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.

Tuesday, October 24, 2006

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.

Monday, October 23, 2006

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.

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?

Sunday, October 22, 2006

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.

Friday, October 20, 2006

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!

Thursday, October 19, 2006

Yang-Mills at the LHC

The LHC schedule for 2007 means that we won't be looking at new physics until 2008. But there is a lot of work to do before then! The 450 Gev calibration run is now planned for November 2007. The LHC will be interesting for heavy ion physics. One important process is gg fusion, but it is said that final state effects such as energy loss will make quantification difficult unless we can better understand such processes.

Fortunately twistor string theory has made great advances with the MHV diagram technique, which creates Feynman trees from maximal helicity violating vertices. Maybe we have time to improve on this a little.

Wednesday, October 18, 2006

M-theory Revisited

This is really just another boring post about the speed of light rather than some comments on mathematical M theory. The string blogger MathPhys made an interesting comment on Woit's blog recently: you all missed c < 1. Silly me...I wasn't sure whether he was referring to the speed of light or central charge!

In rational CFT one considers a deformation parameter q which is a root of unity in the complex plane. For q = exp(2.pi.i/N), the basic case, this depends only on the positive integer N. The same N labels a triple of points (0,q,oo) on the Riemann sphere, which can be used to cover moduli, described by the q=1 case. And before one knows it there are modular tensor categories, Galois groups and all sorts of other goodies floating around, which might explain why Terence Tao has been interested in physical distance scales recently.

Brannen has looked at different scales in the Standard Model with such a varying c. If c was supposed to be the speed of light one might equally ask about the domain c > 1, which has of course been considered by Riofrio. So c could be very, very big, or it could be very, very small.

Monday, October 16, 2006

Colourful Coincidence

World climate change, environmental degradation, poverty, violence and more...and now we have to worry about problems at NASA. Where is that renewable resource of political Will that Al Gore spoke about? Apparently Lee Smolin's book The Trouble with Physics discusses such serious issues. I haven't seen a copy of it yet, but I'm looking forward to reading it. (Anyone feel like sending me a copy?)

When I was plotting the snow level data from the Snowy Mountain hydroelectric scheme twenty years ago, I observed that seasonal snow levels at 1800m had fallen 30% in 50 years. Everybody told me it was due to the formation of Jindabyne dam. They don't say that anymore.

To cheer us up, I thought I'd show a pretty and colourful picture, from Huterer, of the CMBR and its coincidence with the ecliptic

People tell me that this is just because of the way photons interact with stuff on their way here. Oh, really. If there were local future horizons defining an ecliptic, then by T-duality their signature might appear in the cosmic CMBR. Apparently this is a more radical interpretation.

Have a nice day!

Sunday, October 15, 2006

Motif of Motives II

Carl Brannen has reminded me of Cartier's classic paper, A Mad Day's Work. He discusses everything, from Grothendieck's biography to symmetry groups for a point. In particular, he points out that a sensible notion of symmetry group for a point comes from considering points as functors between toposes. Since there are natural transformations between functors, one might find a group of invertible natural transformations between a functor and itself.

The really cool thing about all this is that the group is not fundamental. Eat your heart out Gauge Theory!

Which reminds me that I meant to say something about Grothendieck's motives. As Cartier explains, motives are a part of Grothendieck's dream, a vision of unifying number theory and modern topology, and hence almost everything else as well. The theory of motives is still mysterious, although an impressive amount of progress in the related physics and mathematics has been made in the last 30 years. Consider for example the work of Kontsevich on motives and operads in deformation quantization. It's kind of funny that the mathematicians have chosen a word (motives) that starts with M. It's their version of M-theory!

An important intuition behind motives is that of projective geometry. Motives obey powerful relations, an example of which is the equation

M(projective plane) = M(plane) + M(line) + M(point)

which expresses the usual grading of a projective plane (over any field) into an affine space with a line and point at infinity. This feature of a grading in dimension is typical of motives, as it is for categorical dimension.

Saturday, October 14, 2006

Motif of Motives

It is said that Grothendieck, one of the greatest mathematicians of the 20th century, is now mad. A piece of evidence often cited in support of this hypothesis is his fixation with the speed of light, a mental exercise that might be recommended to many of the critics.

The arbitrary local numerical value of this quantity depends on the arbitrary old definition of the metre from Napolean's time. After some international political wrangling, some French guys measured the meridian from Dunkerque to Barcelona in the years 1792 to 1798. If they had chosen a different geographical location the platinum metre bar would no doubt have come out slightly differently and maybe, with a little stretch of the imagination, we would not be plagued with awkward values for c today. As Einstein said in a lecture in 1921:

In order to complete the definition of time we may employ the principle of the constancy of the velocity of light in a vacuum.

With emphasis on the word may. The constancy of c was not to be taken as a fundamental consideration, but as a convenient means of defining clocks for observers in uniform motion. To assume that the constancy of c should suffice for quantum gravitational clocks is rather stupid. Fortunately people have considered alternatives. Louise Riofrio has some very pretty pictures and graphs which use a varying c to explain away the magical Dark Energy.

Tuesday, October 10, 2006

Some Chitchat

The biology blogger Dcase complimented me recently on my knowledge of biomathematics. Now, whether talking about the biology or the fancy String mathematics, either way my knowledge is actually very poor. But the point is that we both recognise a direction here, which I allude to in many of my posts. The application of trees, networks or categories to genetics, linguistics, computer science, physics, physiology or whatever else is not merely a coincidental appearance of a new type of calculus. Certainly this is one way to see things, because this combinatorics does open vast new vistas, mathematically speaking. But the biologists are not just talking about modelling systems. They are talking about a unified theory for understanding systems; something they have never had before.

Physicists are quite used to the idea of unifying laws of nature. Ever since the ancient Greeks they have worked with the unreasonable effectiveness of mathematics (to quote Weyl). Most physicists are therefore convinced that a theory of Quantum Gravity (a loose term for something that unifies QFT and GR) exists. Moreover, this theory must be predictive. The idea of a Landscape is outrageous and, since we already have better ideas anyway, one wonders why people persist with such investigations.

A theory of Quantum Gravity will say some radical things. Many physicists are now happy with the idea that spacetime disappears and is in some sense generated by the matter degrees of freedom. Einstein could not, in the end, incorporate a Machian inertia into GR, but we expect Quantum Gravity to be able to achieve this. After all, it only fell over with Einstein's commitment to a classical differential geometry. However, to believe that any old background independent description of quantum covariance which yields roughly the standard cosmology would be radical enough is, perhaps, to underestimate the meaning of the word radical.

For starters, some of us are now fully convinced that Quantum Gravity will do for biology what QM did for chemistry. Of course, this is an arrogant physicist's point of view. A biologist might say that the unified theory of biology happens to provide Quantum Gravity as well. Whatever. It's the same theory.

A little while back we were talking about the number of generations in the Standard Model. It was pointed out that this follows from the orbifold Euler characteristic of the moduli of the six punctured sphere. Actually, the higher n-operads of Batanin highlight the fact that something special happens when one considers the statistics on six objects. Tamarkin showed that for n > 1 the polytopes that are usually considered cannot stabilise moduli. But Batanin's can! The simplest Tamarkin example is for six points as branches of a two level nine edged tree. Since in this setting 2-operads are used to study points in the real plane, this enters into the correct combinatorics for the six punctured sphere. Look out for a paper on this soon!

Friday, October 06, 2006

Tree of Life

On the phylogenetic tree of life on earth, the oldest of the three main branches is that of the Archaea, a class of prokaryote including extremophiles, organisms that inhabit environments far outside the range which is comfortable for humans. For example, in the submarine volcanic environment of Loihi, which erupted violently in 1996, microbial mats have since been found (see picture). The evidence for life on earth dates back to the oldest rocks on earth, namely the Akilia island sediments of West Greenland. Although the evidence in carbon isotopes for life in these particular sediments has been open to question, there is plenty of evidence in other ancient rock sediments which typically date life back 3.55 billion years.

The theory that life originated in space, and was transported to the Earth's surface from space, is known as panspermia. But if the oldest life on Earth, perhaps as old as Earth, likes hot environments, is there perhaps a different explanation? Astrobiologists such as Lawrence Krauss think that extremophiles will radically alter our understanding of the origins of life.

If there was a Black Hole at the centre of the Earth, would it have anything to tell us about the evolution of life?

Thursday, October 05, 2006

A Celestial Sphere

Hearty congratulations to John C. Mather and George F. Smoot, who have just received the 2006 Nobel Prize for their leading roles in the COBE experiment measuring the CMBR.

Yesterday I was reading a new book by Joel R. Primack and Nancy Ellen Abrams, The View from the Center of the Universe. This book has a commendable grand vision: to look at how the current revolution in cosmology can benefit humanity as a whole by altering its conception of Nature itself. Primack was apparently one of the physicists who predicted the anisotropy of the CMBR, based on the existence of Dark Matter. Despite the book's relatively conservative, and hence quite erroneous, view of current cosmology, it offers brilliant physical insights in a very accessible way. I would like to quote a little:

We don't normally think of reality as funnelling from great galaxy clusters into us and spreading cell to cell, then soaring inward to the molecular level, the atomic, the quantum levels - and our humanness the fulcrum at the centre of the entire process. But we need to. We need to experience the universe from the inside. We have to imagine ourselves in our proper place, inside the symbols, part of the symbols, the point of the symbols.

There is also a reasonable discussion about the celestial sphere, the badly named surface of last scattering, and how this returns us in some sense to a cosmology with Earth at the centre, but in a way that the Greeks could never have imagined. In the standard modern cosmology (the one that is purported to be revolutionary) this celestial sphere is a fixed surface in a concrete reality that, despite Primack's promises, the mathematics has not escaped. We continuously receive light from this primordial sphere. Compare the COBE results to those of the more recent WMAP satellite. The blotches look roughly the same. Over great lengths of time on Earth it is supposed that the light reaching us will become more and more redshifted as the concrete universal spacetime itself expands.

There are other, even more profound, possibilities. If we accept that the temperature of the CMBR is an indicator of cosmic epoch, then its measurement is a kind of clock. The consequences of this simple observation are not considered in the standard cosmology.

The local standard for time is now the cesium clock, which is accurate to an incredible 2 nanoseconds per day or, equivalently, one second in 1400000 years. A second, by definition, is precisely the time it takes for 9192631770 cycles of microwave light (of a particular wavelength) from cesium133 atoms in their ground state to be absorbed or emitted. Observe that this definition requires only counting (of cycles) and an understanding of the measurement of wavelength.

In the overzealous use of the basic equations of GR people have forgotten that it was the mathematician Minkowski who packaged spacetime neatly into a box all tied up with string. It has always been physically clear that time and space, though necessarily related, are conceptually distinct.

Monday, October 02, 2006

Operads and a Broken Pentagon

In The Combinatorics of Iterated Loop Spaces, Batanin describes an operad based on the poset of faces of the nth Stasheff associahedron. The case of the pentagon looks like this:
It is a broken pentagon, but the top side is an identity if we want the sequence to form a 1-operad in the usual sense. Otherwise, the sequence of permutohedra form a kind of non-commutative operad. Moreover, there is a map of operads from these permutohedra to the diagrams with collapsing identities. This was outlined concretely by Loday.

This is an example of a low dimensional operadic map which we might otherwise have viewed as a broken parity cube in a tetracategorical context.