Braid Advance by Yidun Wan (Feynman diagrams for interacting braids)

[marcus]

This is a major advance in the inclusion of matter in four-valent spin networks of Loop quantum gravity.

things analogous to matter appear as topological excitations (braids) in the network

these things can propagate thru the network, and interact among themselves, in the normal course of the local reconnection moves by which the networks evolve. these include socalled Pachner moves from knot theory.

http://arxiv.org/abs/0809.4464
Effective Theory of Braid Excitations of Quantum Geometry in terms of Feynman Diagrams
Yidun Wan
24 pages, 7 figures
(Submitted on 25 Sep 2008)

"We study interactions amongst topologically conserved excitations of quantum theories of gravity, in particular the braid excitations of four-valent spin networks. These have been shown previously to propagate and interact under evolution rules of spin foam models. We show that the dynamics of these braid excitations can be described by an effective theory based on Feynman diagrams. In this language, braids which are actively interacting are analogous to bosons, in that the topological conservation laws permit them to be singly created and destroyed. Exchanges of these excitations give rise to interactions between braids which are charged under the topological conservation rules."

so this is real basic stuff. the spin network represents a quantum state of the geometry of the universe
and the idea is that matter particles are nothing but little accidental complications in the network, little knots or twisty-tangles or braids----call them knot-icles, or braid-icles.
And it is possible to classify them and find analogs of charge and chirality and P,C,T symmetries. He did this earlier.

Now in this paper Yidun Wan finds analogs of Feynman diagrams describing the interactions of these things.

And he also does something very helpful in the conclusions section: he looks forward to three possible ways this could develop further.

One idea of a future development would free the model from the need to talk about an embedding. I'll discuss the suggested future lines of development later in another post. Or others can if they want.

my feeling is well, Yidun Wan is a grad student working for Lee Smolin, and he is just working on one particular braid concept---a certain four-valent version of braids. If he can get so much mileage out of this one concept of spin network braids (so far not even using the spin labels), then it might be worth other people's while to explore the potential of some other braid concepts. Who knows unltimately what the right approach is, if there is one, a lot of gambits should be explored.

 

[marcus]

the impression I get from reading more in this paper is that there's new mathematics in it.
I suppose from time to time new mathematics is created that will find its way into physics and
this has that feel.

my own intuition does not extend far enough to allow a guess
as to whether or not this, or something like it, is a correct representation of matter
(as a local tangle in the quantum state of spatial geometry)

but it is remarkable and original. guided by mature insight.
impressive work

 

[MTd2]

I had been thinking and thinking about this... Why I was so drawn to this...

The first thought that came up to me, on Friday night, was that Wan invented a kind of cellular automata, because the way he he moved and draw the braids resembled me Conway's Game of Life, that is, you have a cell that changes the states according to how the state of the neighbor cells change. But this is not a deterministic model, but quantum... nevermind then.

Then, yesterday, I wondered that this could a kind of new kind of making mathematics using Cvitanovic's birdtracks - this is a kind of feynmann diagrams, except that it has also the advantage of writing ALL kinds of equations. http://www.nbi.dk/GroupTheory/ (free download) .He also wrote a book in 83 about feynman diagrams using birdtracks ( free download - http://www.cns.gatech.edu/FieldTheory/ ), and although it was way more primitive form than today, he could never, even with today's stat of the art birdtrack, define any kind of ghost particles. Check the last pages of his field theory book: because of that, he could never give a pure birdtrack treatment to QCD.

So, the least mathematical feat Wan accomplished here it is that he could at least define some kind ghost particles using birdtracks. This is a great thing, because birdtracks are usualy used to calculate perturbative expasions of spimfoam models, since 2 lines of calculus in this representation are worth even 10 or 20 pages of algebraic formulas, no kidding.

 

[MTd2]

If you read on his conclusions sugestions for further work, he sugests reference 13, string net condensation. In this case, the trivalent would resamble Cral Brannen 3-preon on a checkerboard model, including his analogy with elastic media.

 

[marcus]

thanks for thinking about this. different things stand out for you, and you draw different connections, so it helps me
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BTW the essential thing about GFT is that the underlying manifold is a direct product of copies of a group G x G x G x...x G
picture a quantum wave function on that. That wavefunction on the group manifold could describe a quantum state of geometry. How? Well think about a certain spin network, labeled with group elements. Amongst networks of that pattern, geometry is determined only by the labels. So a state of geometry is given by an n-tuple of group elements (g1, g2, g3, ..., g17) This is a shoddy explanation, forget I said it. But the general idea of GFT is to have a field defined on a direct product of copies of a group.
It has alwys seemed a bit too abstract to me. Now apparently some people recognize it has potential, and I have to learn more about it
if I want to keep up with the field. Maybe some others do as well.