Tuesday, September 21, 2010

Theory Update 3

Using cospan objects (edge wedges) in cubical path diagrams is a fun way to study arrow compositions. Recall the difference between horizontal and vertical composition for two arrows, where the final picture looks quite different in each case. After a few steps, further compositions lead to a wide range of diagrams.

Like in entropy triangulations, we can normalise the square areas to integral values $d$, and then the single edge lengths around a square go as $1$, $\sqrt{2}^{-1}$, $\sqrt{3}^{-1}$ and so on, just like the normalisation factors for the MUBs in dimension $d$. That's nice, because we want to think of the square areas as designating a dimension, namely that of the composed arrow. Observe that even for two $2$-arrow compositions one can have a dimension $3$ object. This is what happens for the Crans-Gray tensor product.

1. gotta love the cute example. :P

is it possible to define the composition more clearly with a set of rules, flow etc?

what is it used for?

2. Is this at all related to decimations in lattice gauge theory? This appears similar to a discrete verions of renormalization theory.

3. No, it is not intended to have anything to do with discretisations in the usual gauge theory setting. Renormalisation theory is written in the language of Hopf algebras a la Kreimer, Connes et al. When we go to twistor and information spaces, spacetime points are no longer the basic geometric elements, which is fortunate since they should be emergent. There will be a link between the Hopf algebra theory, which is about trees and loops, and the lower levels of this categorical combinatorics, a part of which is about trees and loops.

4. Then this is a decimation system, similar to Ising scaling rules.

5. I suppose that is a better analogy, yes.

6. The next question is whether it makes sense to do this sort of slicing and dicing on finer scales. If so then there might be a piling up of small tiles towards a boundary. This would be a case where this is wrapped into a disk, say a Poincare disk, or where this is mapped onto on a plane with the small bits approaching the 1/2 plane boundary.

7. Small bits at the ends is kind of the idea, but I am not really slicing up anything: I am composing arrows!

8. While you compose arrows on one side, why not do the opposite at the other. These might then be tesselation fragments which fill out a Poincare 1/2 plane by linear fractional transformations or modular functions. If this works it could lead to an interesting way of working certain results.

9. Yes, that's a good point. But to get the category theory working properly will be some work.

10. There are reasons for my thinking about this. Category theory, which I admit I am not that familiar with, would be more of a sort of tool. It would not be the intended goal. The reason for thinking about this is with regards to an unsolved problem in physics. In fact it is a multi-decades long unsolved physics problem.

The procedure outlined here could be continued in both directions it seems to me. You could have compositions occurring in the forwards direction. You could also do an inverse procedure as well with dividing up the unused squares. I presume this could be used with any polygonal or polyhedral geometry. Then the process will have features of the linear fractional transformation

x - -> (ax+b)/(cx+d).

For b = 0 under the adjoint representation this is a Heisenberg matrix group, and how sl(2,R) is the conformal quantum group. The diagonal elements of the matrix sl(2,R) matrix a and d are elements of a metaplectic (real valued as sympletic) group that define the basis of the quantum system.

11. Well, the backwards procedure would simply be the dual process (using spans rather cospans) so it is allowed. Then there is a diagonal line (of object spans and cospans) that acts as an anchor for the whole picture. Is this somehow connected to your matrix diagonal?

I'm afraid I don't quite see your whole $sl(2,R)$ story, but presumably the linear fractional transformations are acting on complex numbers, or two dimensional real vectors. I was in fact hoping these pictures would be useful in understanding complex geometry, so it would be great if you figured that out. One basic idea here was that the (QG inspired) category theory should 'automatically generate' complex spaces, which gradually get filled in by the $p$-adic geometries of the $p$-dits.

12. This is something I might get to in the next few weeks. The complex geometry comes in with the adjoint representation of the fractional linear transformations. For the lower left matrix in the 2x2 = 0 this ad-rep is a 3x3 Heisenberg matrix. This is also a form of a Borel set. It got a bit late so I will have to defer a more detailed discussion.

As for p-adic numbers, or prime extensions of fields and groups, some years ago I did this using prime numbers as "Godel" numbers for quantum numbers in twisted supersymmetry. There is also Matti's ideas along these lines, which frankly I have never completely understood. I guess I never quite figured out what the big idea is there.