Arcadian Functor

Recent posts by bloggers Bergman, Mottle, Carroll and Woit suggest a mood of weariness in the physics blogosphere. The latter cannot even be bothered to write anymore and has resorted to phrases like, “blah, blah, blah, this pseudo-science is on hep-th because of blah, blah, blah.” But the demise of the arxiv is nothing new and the game is just beginning! Heck, I’ve barely warmed up.

The blogosphere is wonderful! In a comment at the Everything Seminar the remarkable Terence Tao points to a short article on the mathematics of Gian-Carlo Rota. On page 9 there is a brief comment on profinite combinatorics, a subject that Rota dreamed of developing. Unfortunately there is only a single obscure reference to Rota’s own writing on the subject.

Given a finite field of order $q$, a continuous geometry in the sense of von Neumann is a profinite limit of (lattices of subspaces of) projective geometries

$P(1,q) \rightarrow P(2,q) \rightarrow P(4,q) \rightarrow \cdots \rightarrow P(2^{n}, q) \cdots$

and this limit contains subspaces of any dimension $d \in [0,1]$. This led Rota to consider the Riemann zeta function, and for those with library access the (two and a half page) paper to look at is:

K. S. Alexander, K. Baclawski, G-C. Rota
A stochastic interpretation of the Riemann zeta function
Proc. Nat. Acad. Sci. USA 90 (1993) 697-699

Wait, I found a reprint! A stochastic process $Z_s$ indexed by $s \in \mathbb{N}^{+}$ is found such that a probability distribution for $Z_{s} = n$ is given by $n^{-s} \zeta (s)^{-1}$. Section 3 discusses in general Mobius inversion for an infinite lattice (recall that inversion was crucial to Leinster’s concept of Euler characteristic for a finite category). By specialising to the sequence of cyclic groups $\mathbb{Z}_n$ and the profinite integers (mentioned in the last lesson) one obtains the Riemann zeta correspondence for $Z_s$ a suitable random variable on the $s$-th power of the profinite integers.

Tegmark’s recent arxiv paper on the mathematical universe contains some interesting insights, but I was most struck by the discussion beginning on page 3 of a comparison between the bird perspective and the frog perspective. This immediately brings to mind the figure of Aristophanes, the ancient dramatist, well known for his comedies. These plays often contained political parody. Two well known plays by Aristophanes are The Frogs (405 BC) and The Birds (414 BC). The former begins with a line from the character Xanthias, accompanying his master Dionysus to Hades:

Say, master mine, would you that I should crack one of those standing jokes upon the stage, which always make the tickled audience laugh.

Surprisingly, I have not seen this coincidence remarked upon in the physics blogosphere.

The lack of suppression of spectral lines blueward of Lyman $\alpha$ for high redshift quasars indicates that most of the intergalactic hydrogen is ionised. That’s a lot of protons to worry about. As Carl Brannen pointed out, protons may be thought of as mixtures of quark triples: uud or udu or duu. Similarly, a neutron is a mixture of triples such as the one drawn.
The numbers at the bottom indicate the number of twists on an (appropriately coloured) strand, each representing a charge of e/3. Thus the neutron has a total zero charge. In the full triple of diagrams there are three -1 charges to make up an electron, three 0 charges to make up an antineutrino and three +1 charges to make up a proton. On the other hand, for the proton each such diagram is labelled by charges 0,1 or 2. This cannot be split up into a positron and neutrino set because that would leave three strands with a +2 charge, which certainly could not make up a neutron.

In a 2002 paper, Kennison proves the following result. First, let Stone be the category of Stone spaces and Bool be the category of Boolean algebras. There is a categorical equivalence $P$ from Bool to Stone which takes the space of all points in an object $B$. An arrow $t: X \rightarrow X$ in Stone is called Boolean cyclic if the corresponding arrow in Bool is cyclic, which means that the supremum over all equalisers of $t^{n}$ and $1_{X}$ is just $X$. Intuitively, these represent dynamical processes that eventually cycle. Now let $\mathbb{N}$ be the ordinals. An action is a map $\mathbb{N} \times X \rightarrow X$ given by $(n,x) \mapsto (t^{n}(x))$. Kennison showed that the property of being Boolean cyclic was equivalent to actually having an action by Z, the profinite integers, namely the product over ordinal primes $\prod \mathbb{Z}_{p}$ of the p-adic integers.

This is interesting because J. Borger has been looking at finite sets with Z actions in order to characterise $\Lambda$-rings in terms of finite sets with actions of $G \times \mathbb{N}^{+}$ for $G$ the absolute Galois group for the rationals. Recall that this group acts on Grothendieck’s ribbon graphs. A $\Lambda$-ring structure is a series of arrows $f_{p}: R \rightarrow R$ for a ring $R$. For example, one may take

$R = \frac{\mathbb{Z} [x]}{x^r – 1}$

along with the Frobenius maps $\psi_{p}: x \mapsto x^p$. Borger et al show that certain nice $\Lambda$-rings can only be a field if they are in fact the rationals. It turns out that all nice $\Lambda$-rings are subrings of products of the cyclotomic example above, for some $r$. This seems to be important somehow…

Smolin’s slides from Loops07 are now available. Skip the stuff about The Dark Force and look in particular at slide number 38. The important thing to note here is that (a) The Loopies have enlisted the help of none other than Louis Kauffman, an absolutely brilliant knot theorist, and (b) Kauffman has invented something called the Kauffman numbers for three stranded braids, which do this: turning elementary braids into codes of the kind that appear in Carl Brannen’s version of the Standard Model (eg. see Carl’s comment here). Thus it appears there is a growing consensus that the three generations arise not from more complicated knots, as originally proposed, but rather from the kind of combinatorics that appear in M Theory. Category Theory is not mentioned at all in this work, despite the increasing usage of both knot theory and quantum information language.

Update: Carl Brannen points out that his scheme for the generations is far more advanced than the one outlined in Smolin’s talk in later slides. I would have to agree.

Jason D. Padgett, a student at Tommaso Dorigo’s blog, who has to my knowledge previously been ignored, left an interesting comment, an edited version of which I shall post here.

Hello again, thanks for the comment. Some were not as kind … The equation I am referring to comes from the Planck constant and the pure geometry of space time. What I did initially was to try to draw … the structure of space time keeping in mind Planck’s constant. In other words … every time you move through space you must move exactly one or whole multiples of one Planck constant.

Don’t take this literally to mean spacetime, but think of it as a local model for spacetime, in which there is a proper notion of local Planck scale. Carrying on:

Planck constants also vibrate at the speed of light. What causes the Planck constants to vibrate is uncertainty … Anyway, once you have a grid drawn with Planck lengths and you start them vibrating (from uncertainty) you will see that at specific points the Plancks will collide with each other at the speed of light …

OK, so this is kind of fun, but is it telling us something interesting? Next we have:

When you draw this diagram the only shape that space time can take is a 2 dimensional hexagon or 3 dimensional cube …

Heisenberg’s hexagons from fractals! Now that’s cool. I’m hoping to take a closer look at his work.

It would be interesting to look at Zagier’s conjecture

$d_n = d_{n – 2} + d_{n – 3}$

using associahedra. After all, the relations between MZV values come from the decomposition of an associahedron face as a product of lower dimensional associahedra. For example, the square faces of the 14 vertex K4 polytope arise as products of the K2 intervals, which represent a basic associator. Thus 1-operad combinatorics gives a way to count MZV relations. The dimension of an MZV space of weight $n$ should be related to the difference between the total number of possible arguments (the ordered partitions of n) and the conditions imposed by the combinatorics.

Euler considered a generating function for the partition function, namely $P^{-1}$ for

$P(x) = \prod_{1}^{\infty} (1 – x^n)$

For $n = 3$ we have $P_3 = 3$ since 3 may be written as 1+1+1 or 1+2 or 3. Subtracting 2 kinds of relation for the K4 faces leads us to suspect that $d_3$ is in fact one. That was very rough, but it is nice to think about how generating functions for MZV spaces relate to generating functions from operads.

Louise Riofrio offers yet another reality check on The Dark Side, inspired by the reappearance of Lunsford in the blogosphere, and Carl Brannen continues with the ABC of the Standard Model. Meanwhile I have been a little lazy, having just discovered an amusing puzzle (diverting even despite its repulsive popularity).

I was hoping somebody could point me to papers on 2-categories and double shuffle relations for MZV algebras. These arise from the two shuffle products, on series or on integral forms, of zeta values. Both of these products may be thought to arise from associahedra combinatorics, as discussed in the work of Brown. A recent paper, by Zagier et al, extends these relations to a set that can characterise the full algebra of MZVs. It does so by introducing an infinite series

$A(x) = exp(\sum_{2}^{\infty} \frac{-1}{n} \zeta (n) x^n)$

with zeta value coefficients. New results include a proof that the weight 3 and 4 zeta values form one dimensional algebras. For example,

$4 \zeta (3,1) = \zeta (4) = \zeta (3,1) + \zeta (2,2) = \zeta (2,1,1)$

There is a conjecture due to Zagier which states that the dimensions of the $\mathbb{Q}$ vector spaces for weight $n$ obey the recurrence relation

$d_n = d_{n – 2} + d_{n – 3}$

with $d_{0} = 1$ and $d_{1} = 0$. The appendix discusses the weight $n$ and depth $d$ generalisation due to Broadhurst and Kreimer, which I believe appeared in the Feynman diagram paper listed as a reference.