Archive for February, 2007

M Theory Lesson 15

Probably the first people to realise that the number of generations might have something to do with idempotents and Jordan algebras were Dray and Manogue in 1999, in The Exceptional Jordan Eigenvalue Problem, which was pointed out to me by kneemo. On pages 10 and 11 they discuss how the usual Dirac equation comes from the 9+1 dimensional one, which is written as a simple eigenvalue problem using a 2×2 octonion matrix, or again as a nilpotent equation using the Freudenthal product. The three generations fit into the Moufang plane, which are Jordan elements satisfying

$M \circ M = M$, tr$M = 1$

so the matrix components lie in a quaternion subalgebra of the octonions. These elements are primitive idempotents.

Naturally we should improve upon the reliance here on a higher dimensional Dirac equation, for which we see no real physical motivation. Brannen’s idempotents are a big step forward in this regard. But we can also reinterpret the higher dimensions in a categorical context, where they are not naively taken to mean spatial dimension.

Comments (2) »

Quote of the Week

Inspired by Tommaso’s regular column The Quote of The Week, I was amused by a comment on Carl Brannen’s not-so-busy blog:

lots of us 10 year olds really do want to learn physics… Al

It brings to mind the old adage about science progressing one death at a time. If you write more, Carl, perhaps more 10 year olds will join the club. Actually, I must confess, you do write a fair amount, such as this recent comment on Clifford’s post about Eureka moments:

My most recent such moment was when I realized that any non Hermitian projection operators in the Pauli algebra can be written in a unique way as a real multiple of a product of two Hermitian projection operators.

That’s nice.

Comments (4) »

M Theory Lesson 14

The higher categorical parity cube is a broken pentagon, such as the one that appeared in operad theory. It is simply the diagram
The top face really looks like
if we use trees to label choices of bracketings. Carl Brannen’s idempotents for lepton masses are also labelled by the parity cube, which describes three directions in space as well as three generations. It is nice to know that this agrees with the number of quark generations that we get by calculating an Euler characteristic for a gluon orbifold modelled by twistors.

Comments (4) »

No More Secrets II

Bernhard Riemann wrote only one 8 page paper in number theory [1]. It listed a number of conjectures, the most famous naturally being the Hypothesis. Let $\pi (N)$ be the number of primes less than or equal to $N$. Also recall the function

Li$(x) = \int_2^x \frac{dt}{\textrm{log} t}$

Another conjecture (later proved by von Mangoldt) that Riemann stated in his paper was that there should be a formula for $\pi (x)$ – Li$(x)$ valid for any $x > 1$. The only tricky term in the formula is a sum over the complex zeros $\rho$ of the $\zeta$ function,

$\sum_{\rho} \frac{x^{\rho}}{\rho}$

So the zeroes encode precise information about the distribution of the prime numbers. This imbues these points of the complex plane with a special significance, just as the ordinals along the real axis are also special because they represent ordinals, which we know are about counting elements of sets, or Euler characteristics for categories. Maybe the other numbers in the complex plane should be viewed this way as well. In other words, the complex plane is not given a priori as a boring set obeying the axioms of a field.

A topos theorist cannot escape such ideas, because number fields become heinously complicated. The ordinals make sense enough. Categories such as Set have an object of Natural Numbers. From this one can build the integers and the rationals, but then one has to worry about what one means by the real numbers because the set theoretic definitions don’t all necessarily agree in other toposes. And why should we care more about the reals, from the prime at infinity, than the p-adics? The p-adic number fields can fit into the complex plane via Chistyakov’s beautiful fractal patterns. From du Sautoy we also have some beautiful images of the $\zeta$ function:

[1] H. Davenport, Multiplicative Number Theory (LNM 74)

Comments (5) »

No More Secrets

Yesterday we were treated to a public lecture by the enigmatic Marcus du Sautoy on the music of the primes. It began with a discussion of the shirt numbers in European football teams, something that du Sautoy was particularly enthusiastic about. Fortunately, by the end of the hour we were well into the Riemann hypothesis.

He gave a wonderful explanation of what the zeroes mean. According to the hypothesis, the non-trivial zeros of the zeta function all lie on the critical line, the real part of $z$ equal to 1/2. Instead of plotting the axes of the complex plane, however, he showed the line on a plane marked by a vertical axis for frequency (of a Riemann harmonic component of the full prime counting function) and a horizontal axis for amplitude.

So there is a way to label the zeroes one by one, starting with the one lowest on the positive vertical axis at $y$ = 14.1347 (and remembering that there is a reflection symmetry across the horizontal axis). The label counts the number of nodes in a wave of some kind. All waves have the same amplitude. This fact must relate to the localisation we find with T-duality in twistor ribbon models. So a basic principle of quantum gravity forces the Riemann hypothesis to be true. What is this wave? The real world manifestation that du Sautoy mentioned was that of the energy levels for large atoms, such as uranium. But as we know, most analyses of the match between energy levels and Riemann zeroes involve the statistics of an ensemble. We would rather construct the zeroes one by one. The expectation is that the algorithm, if it exists at all, would become increasingly complex, so that the computation of all the primes would amount to a vast universal computation. But hang on a minute! Isn’t that what our rigorous QFT/QG program is all about?

We learn in kindergarten that the energy levels of atoms correspond to the allowed shells for electrons. Thus there is a correspondence between $E_n$ and particle number. Now recall, we saw that the tricategorical aspects of QCD suggest a correspondence between particle number and categorical dimension. This is good, since we expect categorical dimension to provide a measure of complexity for a problem naturally couched in that context, such as calculating the nth zero.

Another clear aspect of du Sautoy’s talk was his explanation of the logarithm function, as a method for converting multiplication into addition. The logarithm acts as the first mode of Riemann’s wave decomposition. We have met the higher dimensional analogues before in operad combinatorics.

Finally, there were a few fun film clips from movies about mathematics. As Robert Redford says when the detective slots the codebreaker box into the aircraft control computer, No More Secrets.

Comments (4) »

David’s No DE

As promised, David Wiltshire’s long awaited No DE cosmology paper is coming out. This arxiv link should work tomorrow. A poor person here was heard complaining that someone was hogging the printer to print a book. But don’t worry, it’s only a 72 page paper.

Comments (6) »

Valentine’s Day

I am the sort of person who has no awareness whatsoever of a day’s cause for celebration. And so, quite unaware of the day, I found myself boulder hopping down the Waimakariri. But at the road end I came across a lovely old couple sipping champagne and eating Belgian chocolates, an enjoyable activity that I soon found myself invited to partake in! So, yes, the absense here has been due to another little wander in the hills. The remainder of the day turned out excellent as well. By dinner time I had completely forgotten it was Valentine’s Day and couldn’t figure out why the restaurant, where I went to treat myself to a well-deserved steak, was so busy – that is until a sweet old Kiwi French ski instructor chatted me up and reminded me. Anyway, back to work, folks. All those guys who sent me email cards – I’m afraid they must have been chewed up by my vicious spam filter. Oh well!

Comments (6) »

M Theory Revision

The modern understanding of Feynman diagrams comes from a beautiful body of mathematical work, such as that of Kreimer et al on the Hopf algebra structure of renormalisation. In a category theory setting, we know that such structures rely on the concept of operad. Moreover, higher dimensional operads appear essential in moving beyond CFT and addressing the problem of describing mass quantum numbers.

The emphasis on renormalisation is a mistake. This picture still works in a Minkowski background QFT, the idea being that the Standard Model in its rigorous guise will not be much altered. But we have seen that a twistor correspondence must be implemented, for it is only in this setting that the physical logic has a chance of being written in a topos like language. The twistor point of view changes our use of operads. The need for higher dimensional structures becomes even more apparent as we match the 1-operad associahedra to mere real moduli.

Comments (7) »

Victoria Canterbury II

It turned out to be a NZ-Korea Gravity Workshop, because we had a number of visitors from Korea. Sung-Won Kim, who was here for the Kerrfest, was visiting again, this time with greatly improved weather. He took many photographs which he has promised to send us. Yesterday we also heard excellent talks by Jong-Hyuk Yoon, Silke Weinfurtner, Alex Nielsen and Petarpa Boonserm.

I learnt that Yongmin Cho’s Restricted Gravity decomposition has a very interesting application in lattice gauge theory. Yongmin Cho and Jong-Hyuk Yoon are off to Wanaka today to start the Rabbit Pass tramp, which they learnt about on the internet. I highly recommend it.

Leave a comment »

Victoria Canterbury

The joint UVW and UC Gravity Workshop kicks off today with a talk by Yongmin Cho from Korea entitled Restricted Gravity. David Wiltshire will then talk about his no DE cosmology, although this isn’t clear from the title. Will report later.

Comments (3) »