Saturday, May 05, 2007

The First Edge of a Cube

I learned in Kea's blog about posting concerning something related to n-categories: the posting was The First Edge of the Cube. I did not understand much of it. I however tried to make something out of it and this boiled down to a posting to Kea's blog whose polished version with a couple of additional questions I attach below.
Dear Kea,

I wish I could connect these n-transports to something having a concrete physical meaning! For years ago I tried to understand n-parallel transport (or perhaps it was something related;-)) in terms of simple geometric mental images. I try to formulate my mis-understandings using the ancient terminology still used by physicists like me and these mental images. No arrows nor commuting diagrams which make me mad!

  • n=1: One starts with a parallel transport from point a to b along curve C1(a,b). 1-parallel transport defines a map between fibers.

  • n=2: 1-parallel transport along C1(a,b) is parallelly transported to a 1-parallel transport along curve C1(c,d). One can say that one parallelly transports curve instead of point. 2-connection would define this parallel transport of parallel transport. One obtains a kind of square like structure C2(a,b|c,d).

  • n>2: One can continue this and obtains at n:th level parallel transport of parallel transport of.....

Some comments.

  • The ordered exponential representation for parallel transport suggests that n=1 parallel transport could define n-parallel transport. Probably something trivial and un-interesting.

  • If the n-connection is non-flat, the n-parallel transport depends on how the curve evolves from the initial state to the final state.

  • A physically highly attractive possibility is generalized general coordinate invariance stating that the parallel transport depends only on the n-surface spanned by the curve. Is n-parallel transport induced by 1-parallel transport the only solution to this requirement?

  • One can wonder about the counterparts of geodesic lines. 1-parallel transport leaves the tangent vector field of geodesic line invariant. n-parallel transport should leave invariant the n-form defining tangent spaces of a geodesic n-surface? For n-parallel transport induced by 1-parallel transport geodesic sub-manifolds would probably result. What is the n-counterpart for the equations of geodesic line? Could one model the behaviour of extended objects in gravitational fields using these kind of equations? Could one model the effect of non-gravitational forces on the motion by using n-connection not induced from 1-parallel transport?

  • One could also generalize the notion of holonomy group. 2-holonomy group would be associated with cylinder-like surfaces C2(a,b|a,b) with topology D×S1. At higher levels you would have topology D×S1×S1 and so on. You could also consider closed curve at n=1 level and get hierarchy of n-holonomy groups associated with n-tori. Of course also other topologies can result if the parallel transport is such that the surface develops pinches. Could one generalize the notion so that one could assign say 2-parallel transport to a 2-torus. What to do when the curve for 1-parallel transport decomposes into two separate pieces? Just hop? Why not?

For years ago I assigned this kind of hierarchical structure of parallel transports to a hierarchical structure defined by infinite primes (see this). I believe that this kind of abstractions about abstractions about..., thoughts about thoughts about... , statements about statements about... , and repeated second quantization, represent fundamental new physics especially relevant for quantum consciousness theories.




CarlBrannen said...

I hope that this is what category theory is about because your post was the first one on the subject that (a) is not completely trivial and (b) I think I understand completely.

The Cambridge geometry group rewrote GR so as to put it on a flat space. In other words, one can use a single chart, and label the coordinates x,y,z,t.

There are some subtle theoretical reasons for preferring this version of GR. I think they are best discussed by Hestenes in his introduction to the Cambridge method. See especially section IX.

One way of explaining what they did was that they used parallel transport to move all the tangent vectors to the same point. Then they define local Cartesian coordinates from the local tangent space and extend these to the rest of the space.

This sounds impossible, but they did it. The effect is that for any GR situation, they can draw a Cartesian coordinate system on it. That is, there is still curvature in the equations of motion and all that, but there is no longer any curvature in the coordintes. And the coordinates are global, not just local.

The simplest GR problem is the isolated black hole. For this their method gives Painleve coordinates. Doran generalized this to rotating black holes.

One of the cool effects of using Painleve coordinates is that particles fall down black holes in finite coordinate time (as they should). To do this, note that the line element is not diagonal.


CarlBrannen said...

Ooops. I forgot to mention that the result of this sort of coordinate choice is that parallel transport becomes trivial (it is given by the underlying Cartesian coordinate system).

Kea said...

I hope that this is what category theory is about...

LOL, Carl. Category theory is about lots of things, but yes this is one of them. These 'higher gauge theory' guys are quite devoted to higher transports.

Matti Pitkänen said...


I think that these global 'Cambridge coordinates' must have coordinate singularities. For instance, in case of sphere complex coordinate having origin at north pole develops becomes infinite at south pole so that the circle at infinity corresponds to south pole.

Kea is certainly right that category theory involves a lot more and the commuting diagrams which one encounters in quantum group theory probably have no translation to this kind of simple geometric picture. It would be nice to have physics friendly non-trivial examples of category theoretical structures stimulating also ideas about physical applications.