Quantum Gravity and the Standard Model (Sundance + PI)

[f-h]

I see, you are saying we should think of the process as more symmetric between geometry and particles, we are also mixing up several different questions here, I don't think your objection relates to marcus original point that particle/spinfoam interactions should naturally replace/explain the Higgs mechanism, does it?

I still don't understand how this avoids the scale argument. If spacetime should emerge the same way as particles as configurations of this pregeometry (the loop spaces), then you need to explain why particles emerge and drastically different Energy scale then geometry.

On the other hand wrt your point, our best idea about how geometry emerges from Spinfoams is LQG in particular the Area/Volume operators, that particles emerge at all is surprising but it seems for the moment that there is nothing on the table that could explain why they should emerge at such different energy scales either, right?

 

[Careful]

** At the other end, one may expect that the emergent spacetime is the collection of events that are the interactions of the excitations of an underlying pre-spacetime quantum theory, with matter being also emergent as these same excitations.[/i] **


I am glad you underline this point yourself; since I remember you or Hossi being sceptic when I told that it is meaningless to study vacuum ``quantum gravity´´. Option (a) is simply bogus.

**
They then refer to Lloyd, the well-known Quantum Computation person. **

I see you still did not detect the mistakes in his ``fundamental´´ paper; probably you did not read that one either.


Careful

 

[rtharbaugh1]

Consider an infinitesimal dust evenly distributed in an open, infinite universe...(Mach space?) Is there any gravity? In other words, can we think of gravity except in

the local, perturbative sense?

Consider the scale in such a universe at which perturbations just begin. That scale must expand as the effect of the perturbations reaches longer and longer lengths.

So the scale at which perturbations just begin must be an expanding scale. Below this scale, the infinitesimal dust loses its continuity and begins to become

discrete...that is, local effects begin to predominate. Particulate matter then would be a sort of clumping or curdling of the infinitesimal dust.

Now as the perturbations expand and as the dust clumps, we observe that there are some consistant parameters. The clumps tend to settle into certain classes of sizes.

There are preferred scales. They may be expanding scales, but they have a consistant particulate horizon as they expand. The neutrino, the electron, and the proton

are examples of these horizons. The preferred scales also occur at much lower energies, as planets, stars, galaxies, but at some scale the number of possible states

becomes essentially infinite, and the appearence of continuity is restored. So we have all sorts of masses and sizes of rocks. At a slightly higher energy the clumps

have much more limited possible states, and we have the table of elements.

We have to ask ourselves then why some states are preferred, and why the number of preferred states seems to vary with scale. Is there some fundamental geometric

relationship in the infinitesimal dust which results in preferred states? Might the infinitesimal dust mites have some definite shape that causes them to prefer to pack

in certain groupings, such that certain numbers of them tend to be stable, while a few more or less tend to be unstable?

How can we make sense of the standard model, at which scale there seems to be a perfect uniformity, with every clump of matter as far as the eye can see existing in

one or another of only a few possible states?

Just wondering.

R.

 

[Kea]

...since I remember you or Hossi being sceptic when I told that it is meaningless to study vacuum ``quantum gravity´´.

Careful, I have always thought it was meaningless to study vacuum QG. I'm sorry if I somehow gave you another impression at some point.

 

[Kea]

...then you need to explain why particles emerge and drastically different Energy scale then geometry.

f-h

No one is claiming to have solved the (reverse) heirarchy problem. Also, remember that energy scales themselves are part of the geometric question.

 

[Kea]

I see you still did not detect the mistakes in his ``fundamental´´ paper; probably you did not read that one either.

I'm quite happy to admit that I have no interest whatsoever in reading Lloyd's papers. This thread is about what Sundance+PI are thinking.

 

[f-h]

f-h

No one is claiming to have solved the (reverse) heirarchy problem.

Ok, so I guess we are in agreement after all and I just misunderstood what you were saying.

My original statement was that:
"We can not expect Planck scale physics to replace the Higgs mechanism." where I meant gravitational spin networks with "Planck scale physics" and I erronously read your statements as a response to that.

Back on topic, do you have references to ideas how to get geometry from this kind of ribbon graph quantum theories in a non LQG way? I would be highly interested in that...

 

[Kea]

Back on topic, do you have references to ideas how to get geometry from this kind of ribbon graph quantum theories in a non LQG way? I would be highly interested in that...

Dear me, f-h, this is just what category theoretic M-theory is all about. Yes, I mean String theory. I'm afraid a lot of people would consider that very off-topic.

 

[Careful]

Careful, I have always thought it was meaningless to study vacuum QG. I'm sorry if I somehow gave you another impression at some point.

Ah, then it was die schone Hossi. But don't misunderstand me, I find some points of the Sundance paper(s) interesting, but they do not serve for LQG purposes...

Cheers,

Careful

 

[Kea]

But don't misunderstand me, I find some points of the Sundance paper(s) interesting, but they do not serve for LQG purposes...

I completely agree.

 

[Careful]

I'm quite happy to admit that I have no interest whatsoever in reading Lloyd's papers. This thread is about what Sundance+PI are thinking.

Good good, the latter are definitely better

 

[Kea]

...references to ideas how to get geometry from this kind of ribbon graph quantum theories...

Where to begin? The edifice is mighty high. I guess one could do worse than look at:

TFT construction of RCFT correlators I: Partition Functions
J. Fuchs, I. Runkel, C. Schweigert
http://arxiv.org/PS_cache/hep-th/pdf/0204/0204148.pdf

 

[marcus]

Baez comment on "QG and SM" paper

comment #32 on christine dantas "QG and SM" thread
http://christinedantas.blogspot.com/2006/03/quantum-gravity-and-standard-model.html

right near the end so scroll down nearly all the way thru the comments

 

[selfAdjoint]

comment #32 on christine dantas "QG and SM" thread
http://christinedantas.blogspot.com/2006/03/quantum-gravity-and-standard-model.html

right near the end so scroll down nearly all the way thru the comments

Very interesting! The modular group, the Pythagorean spinors, and the connection to the Lorentz group! As he says, there has got to be something neat in there somehow.

Much more productive than empty "it can't be so because of all this QFT knowledge we have!" talk. On the wonderful QFT knowledge see the Schroer papers Woit links to, or read Cao's book.

 

[Mike2]

Dear me, f-h, this is just what category theoretic M-theory is all about. Yes, I mean String theory. I'm afraid a lot of people would consider that very off-topic.

Would you recommend I start off with :

Elementary Categories, Elementary Toposes (Oxford Logic Guides) (Paperback)
by Colin McLarty

at:
https://www.amazon.com/gp/product/0198514735/?tag=pfamazon01-20

Or do you know an easier introduction? Thanks.

 

[Kea]

Elementary Categories, Elementary Toposes (Oxford Logic Guides) (Paperback)
by Colin McLarty

Excellent choice. More a logician's viewpoint than a category theorist's, but probably the best book out there for working through.

 

[Kea]

Very interesting! The modular group, the Pythagorean spinors, and the connection to the Lorentz group! As he says, there has got to be something neat in there somehow.

John also mentioned the trefoil knot. This has a nice simple Jones polynomial, namely

J(t) = t + t^{3} - t^{4}

Now it so happens that a very nice HEP guy here mentioned today the logarithmic relation between m_{e}, m_{\mu}, m_{\tau} which I will write in the form

\sqrt{m_{e}} = r e^{\pi a} \hspace{1cm} \sqrt{m_{\mu}} = r<br /> e^{2 \pi a} \hspace{1cm} \sqrt{m_{\tau}} = r e^{3 \pi a}

This means we can plug it into the Koide formula

m_{e} + m_{\mu} + m_{\tau} = 4(\sqrt{m_{e}} \sqrt{m_{\mu}} +<br /> \sqrt{m_{e}} \sqrt{m_{\tau}} + \sqrt{m_{\mu}}<br /> \sqrt{m_{\tau}})

to get

1 + e^{2 \pi a} + e^{4 \pi a} = 4(e^{2 \pi a} + e^{3 \pi a} +<br /> e^{\pi a})

or rather,

1 = 4e^{\pi a} + 3e^{2 \pi a} + 4e^{3 \pi a} - e^{4 \pi a}<br />

which for t = e^{\pi a} reads as the very simple

r(1 - J(t)) = 3(\sqrt{m_{e}} + \sqrt{m_{\mu}} + \sqrt{m_{\tau}})

Hmmm...

 

[Mike2]

Does category theory and topos theory try to define sets independent of any underlying point set topology? Is this why it is useful in background independent efforts? Is topos theory the underlying mathematics of Algebraic QFT which define an algebra of operators (and not states) in order to get away from the background dependence of states?

Hi Mike

The answer to all your questions is yes.

I've noticed some similarities amoung various efforts that I'd like to consider. This effort by Sundance, et al, looks a lot like Torstens effort to connect geometry to the operator algebra of the SM. They state that it is common to use knot theory to develop an operator algebra used in the SM. I wonder if these efforts are connected? Are the ribbons in the work at hand just knot theory in disguise? Perhaps it is the same as knot theory with an added dimension. (Sorry, I've only read the abstract)

Also, the web of graphs in LQG also looks like piecewise linear knot theory. Perhaps just a subset of all the links and nodes can be interpreted as intertwining knots use to develope the operator algebra of the SM. Has anyone considered that?

And if we intertwine the loops in string theory, perhaps that also is knot theory in disguise.

And perhaps a subset of the lattice of CDT might be interpreted as piecewise linear knot theory and develop the operator algebra of the SM from that.

As you can see. I've not achieved a synthesis yet. I'd like your opinion as to how likely it is that this algebraic QFT developed with the use of knot theory underlies all these different efforts. Thanks.

 

[Kea]

...how likely it is that this algebraic QFT developed with the use of knot theory underlies all these different efforts.

Firstly, the name algebraic QFT is the subject of Schroer et al (talked about recently here and on NotEvenWrong). Although they do admire knots and CFT and such things, there is still a vast gulf between this way of thinking and the way of thinking of which I am thinking. It really is a matter of there being an awful lot of things that need sorting out before different approaches can be linked (excuse the pun).

But this is all OT. Sorry, Mike.

 

[Kea]

r(1 - J(t)) = 3(\sqrt{m_{e}} + \sqrt{m_{\mu}} + \sqrt{m_{\tau}})

The trefoil knot is the simplest example of a torus knot. These are created by winding a string m times about one axis and n times about the other. Let J(m,n) denote the Jones polynomial for a torus knot. The trefoil is the (2,3) knot. Now, considering (J(m,n) - 1), all torus knots are naturally normalised to the value of (J(2,3) - 1). So it seems that this normalisation is somehow associated with a choice of mass scale.

Hmmm...

 

[Kea]

Goodness, this is so distracting. I must go for a walk. It feels like being trapped inside an artist's impression of a Bohr atom, like one of those nauseating joy rides.

 

[arivero]

This means we can plug it into the Koide formula

Hmmm...

How happens a thread on Sundance gets Koide inserted along ?

 

[marcus]

How happens a thread on Sundance gets Koide inserted along ?

Is this off topic? You know far more than I do, so I ask the question sincerely. To my limited understanding, it looks actually ON topic in a sense-----because of braids -> knots (like the trefoil) -> Jones polynomial -> pretty algebra like Koide

the Kiwi wild parrot called a kea is a notorious mischief-making bird members of this species have been known to drop a hiker's boot off a cliff---when he left his boots outside his tent while he was taking a nap.
they are strong intelligent and destructive birds----problematical during the day much as raccoons are problems during the night, if you know raccoons

part of Kea's persona is this new zealand kea bird. and she in fact DOES sometimes drag thread irretreivably off topic and sometimes will even drop them off a cliff. As with the wild parrot, this happens inevitably and although it may cause dismay, is not a proper subject for complaint.

but IN THIS CASE it seemed to me that the chain of mental associations was exceptionally elegant and fascinating

or were you giving a back-handed compliment? Not seriously objecting?
Sometimes I can't tell.

 

[arivero]

I'd not say it is off topic, indeed I would prefer most of the rest of physics discussions online to be considered offtopic! I was amazed because I was looking Sundance restricted to one generation of particles and wondering about how the triplication was to be got.

I was also amazed about relating the formula to knots. Amateurs should notice that we physicists are educated against knotting during the introductory years, because of the bad legend of a previous attempt (XIXth century!) to explain the periodic system via knot classification. Still, Jones polinomial is always a subject of advanced seminars from time to time.

Ah, I missed the remark on logarithmic relationships between masses. It is something I have not classified yet?

 

[Kea]

As with the wild parrot, this happens inevitably and although it may cause dismay, is not a proper subject for complaint.

Hee, hee.

 

[Kea]

Of course, in the real world the mass ratios are not equal. But, arivero, you have discussed quantum-group like mass logarithms yourself over on the thread

https://www.physicsforums.com/showthread.php?t=46055&page=8

Remember CarlB's appearance with the mass matrix? To quote Carl:

I should mention what all this has to do with Higgs-free lepton masses.

Consider the Feynman diagrams (in the momentum representation) where each vertex has only two propagators, a massless electron propagator coming in, and a massless electron propagator coming out, and a vertex value of m_{e} is being generated. When you add up this set of diagrams, the result is just the usual propagator for the electron with mass. Feynman's comment on this, (a footnote in his book, "QED: The strange theory of matter and light"), is that "nobody knows what this means". Well the reason that no one knows what it means is because these vertices can't be derived from a Lorentz symmetric Lagrangian.

But what the above comment does show is that it is possible to remove the Higgs from the standard model (along with all those parameters that go with it), if you are willing to assume Feynman diagrams that don't come from energy conservation principles.

 

[Mike2]

Firstly, the name algebraic QFT is the subject of Schroer et al (talked about recently here and on NotEvenWrong). Although they do admire knots and CFT and such things, there is still a vast gulf between this way of thinking and the way of thinking of which I am thinking. It really is a matter of there being an awful lot of things that need sorting out before different approaches can be linked (excuse the pun).

But this is all OT. Sorry, Mike.

As I understand it, you think more in terms of category and topos theory. I wonder if there is a connection between this and knot theory in that these knots are defined on 3D support manifolds whose union and intersection relate to operator algebra with the use of knot theory.

 

[Kea]

As I understand it, you think more in terms of category and topos theory. I wonder if there is a connection between this and knot theory in that these knots are defined on 3D support manifolds whose union and intersection relate to operator algebra with the use of knot theory.

Yes, Mike. This is well known.

 

[Mike2]

As I understand it, you think more in terms of category and topos theory. I wonder if there is a connection between this and knot theory in that these knots are defined on 3D support manifolds whose union and intersection relate to operator algebra with the use of knot theory.

Yes, Mike. This is well known.

I'm guessing here, but it sounds like the intertwining of these "knots" is not a differential feature, but is a description of something global. So it sounds like they are defining the union and intersection of sets in terms of something global and not in terms of elements of the underlying sets. This sounds like background independent set theory, independent of the background of underlying elements. Does this sound correct? If this backgound independent set theory is the basis of category/topos theory, then how can you deny the relevance of knot theory to all these different efforts when you do acknowledge the use of category theory for the same goal? Thanks.

 

[arivero]

Of course, in the real world the mass ratios are not equal. But, arivero, you have discussed quantum-group like mass logarithms yourself over on the thread
https://www.physicsforums.com/showthread.php?t=46055&page=8

Yeah I have (and you ). It was only the "today" remark that drove me to think I was missing something. BTW, in https://www.physicsforums.com/showpost.php?p=908909&postcount=171 I remarked that the only published note I am aware about mass logarithms is

Andreas Blumhofer, Marcus Hutter Nucl.Phys. B484 (1997) 80-96 http://arxiv.org/abs/hep-ph/9605393

I got to locate to M Hutter but not Blumhofer. Both of them have left physics; it seems that the system generates a very high rejection (abandon) rate even within people following the standard study path, such a thing enerves me.

 

[arivero]

On the other offtopic of this thread, categories, let me tell that I was in love with them time ago as a physics pregraduate. I particularly was very interested on Topoi, as a alternative foundation both for mathematics and physics (well, differential geometry or plainly Mechanics). And I was very happy when some authors (eg Doplicher) started to use categorical language, even if only to relate intertwinning representations.

But just a random use of this language does not carry one, I think, more far than to use, say, string theoretical language. Another point it could be if we can use it to understand the origin and interplay of classical and quantum mechanics. This is the thing on-topic with sundance work, if the particle structure they are seeing in geometry is due to some deep fundational point about differential geometry itself.

 

[arivero]

A thing more. It is great that elementary matter has spin, ie with an intrinsic angular momentum which is a (half)multiple of Planck constant, because it makes a lot more puzzling the classical limit, either h->0 or N->infity: we must to ask what happens with this spin, how does it transmutes when coming to classical mechanics.

 

[garrett]

I was reading a bit about categories today, including this:

http://math.ucr.edu/home/baez/categories.html

in which I came across this sentence:

"It turns out that one can get a representation of the category of tangles from any finite-dimensional representation of a semisimple Lie group."

with papers referenced.

Does this mean we could start with our favorite semisimple Lie group, SU(3)xSU(2)xU(1), in a chosen representation and translate it directly to something (exactly?) like Sundance's tangling-ribbon model?

 

[arivero]

the haiku

Now I think about... does our quantum haiku argues for or against Sundance? Because the Haiku says that if we ask anyone Planck area to take more than one Planck time to be orbited, then there are no gravity trajectories below Compton lenght.

Now this breaking of gravity could be argued to say than no theory of gravity can describe substructure below Compton length, and then argue against Sundance. But we could counter that it only implies that no classical gravity can go below Compton, and then it forces a new structure exactly at the energies/distances where we expect to need it, and this could be Sundance+PI.

 

[Kea]

Christine Dantas has posted on the Sundance+PI paper:

http://christinedantas.blogspot.com/

 

[Kea]

Does this mean we could start with our favorite semisimple Lie group, SU(3)xSU(2)xU(1), in a chosen representation and translate it directly to something (exactly?) like Sundance's tangling-ribbon model?

Why do you want to start with a Lie group?

 

[CarlB]

Kea, if you like the Koide mass formula, and you like Sundance's braid theory, and are willing to stomach my comments on taking the Higgs out of masses, you're really going to like my freshly minted paper extending Koide's mass formula to the neutrinos:
http://brannenworks.com/MASSES.pdf

When I wrote the above (last week), Sundance's paper was still on my list of things to read. Vic Christianto pointed it out to me, and having read it, and read the comments on this thread, it is clear that the above mass paper solves the following problems with the Sundance version:

(a) It gives a reason for the generations.
(b) It puts the hierarchy inside each generation to a discrete symmetry as is suitable for Sundance.
(c) It puts the hierarchy between the charged and neutral leptons to a discrete symmetry that is particularly suitable for Sundance in that it is exactly a power of three.

And the Sundance preons give an explanation for all those powers of three that appear in the MASSES paper, as well as the intersting fact that the MNS matrix can be manipulated into a 24th root of unity by multiplying by a 3x3 matrix of eigenvectors of 3x3 circulant matrices.

Sundance's division of the elementary particles into preons appears also in my earlier paper here:
http://brannenworks.com/a_fer.pdf

By the way, if I'd known people were quoting me over here (on the subject of mass generation no less) I'd certainly have paid more attention!

Carl

 

[Kea]

...you're really going to like my freshly minted paper extending Koide's mass formula to the neutrinos:
http://brannenworks.com/MASSES.pdf

Hi Carl

Goodness, I had no idea that the known figures for neutrino oscillations were already that good. And you get precisely those values...as well as the Koide formula for charged leptons...and so simple. Well, I must say I am glad that you decided to visit us over here! By the way, I wouldn't exactly call the neutrino experiments 'primitive'...I suspect some of them are really quite sophisticated.

Here's to the next decimal place! Must fill my glass with some of that sweet Sri Lankan coconut brandy...

 

[marcus]

Baez TWF #233 is out, mentions the Sundance + PI paper

http://math.ucr.edu/home/baez/week233.html

===quote===
May 20, 2006
This Week's Finds in Mathematical Physics (Week 233)
John Baez

On Tuesday I'm supposed to talk with Lee Smolin about an idea he's
been working on with Fotini Markopoulou and Sundance Bilson-Thompson.
This idea relates the elementary particles in one generation of the
Standard Model to certain 3-strand framed braids:

1) Sundance O. Bilson-Thompson, A topological model of composite preons,
available as hep-ph/0503213.

2) Sundance O. Bilson-Thompson, Fotini Markopoulou, and Lee Smolin,
Quantum gravity and the Standard Model, hep-th/0603022.

It's a very speculative idea: they've found some interesting ...

===endquote===

 

[lqg]

I guess I should keep my mouth shut in this thread.

 

[marcus]

I guess I should keep my mouth shut in this thread.

here we have group-brain and group-mouth, so no one is responsible to say anything

anyone who wants (for any reason) to NOT say something, does not have to say it! I like this. the reason is that the group-mouth will take care of it----someone else will accidentally say what is necessary

so we live in a relaxed situation without those kind of requirements on us

Maybe I did not explain this clearly, but it doesn't matter: not an important thought

yes, well Baez wrote TWF on Saturday 20th, and on Tuesday 23rd he was planning to talk to the author(s) of the Sundance paper.

In the course of time whatever is true in the Sundance paper will coagulate as pure metal and whatever is not true will burn up and go away as smoke. I think there ARE valuable ideas in that paper and I am very happy to wait until it is obvious what they are. I am not in a hurry, personally, to find out.

Maybe Smolin made an impossible leap in this paper (!) Creative people sometimes do this. they jump across the grand canyon and then discover a minute later that it was impossible to do that. We will find out sometime how it went.

The leap is this: CAN YOU HAVE PARTS OF A SPIN NETWORK WHICH HAVE SOME DISTINCTIVE TOPOLOGICAL CHARACTER WHICH MAINTAIN THEIR INTEGRITY? these have to be inclusions in the network which don't dissipate in the normal interaction with the rest----by ordinary "moves" performed on the network. these are like gems in a rock. or like fossil shells in clay. they have to propagate freely WITHOUT LOSING THEIR INTEGRITY by too much interaction.

this seems like a "tall order". this seems like something that is impossible to ask to happen in a spin network.

Anyway that is my understanding of it. I suppose this could happen by KNOTTING. When someone has long hair and gets small tangles in the hair that will not be combed out, it is called "snarls". Maybe a spin network can have "snarls"

Or maybe not. I feel happy about this, knowing that Baez and Smolin got together on Tuesday 23rd to talk about it. I think when you get creative people together who know how to talk business, that eventually WE WILL FIND OUT HOW TO GET MATTER TO ARISE IN SPIN NETWORK, or else in beef, and they will very likely have made some steps of progress!
================

if anyone actually knows anything about progress in that area, and wants to say anything about it, that would be delightful, but it is not necessary
what I post here are just my first guesses at the moment you mentioned it

 

[lqg]

You are right.

 

[lqg]

By the way, since Lee never stepped in this forum, I gave him the link today. I wish he would take a few minutes to come by and say something here, not just in this thread. He knows everything I have done on the project and surely much more than that.

 

[marcus]

By the way, since Lee never stepped in this forum, I gave him the link today. I wish he would take a few minutes to come by and say something here, not just in this thread. He knows everything I have done on the project and surely much more than that.

time too valuable IMHO
need him to pull the train and not take time talking to the passengers

you are actually better as a connection with that program, lqg.
it is a case of "impedance matching" (to make an electical engineer comparison)

from time to time Prof. Smolin posts at Woit's blog when there is a hard misunderstanding there and a real need for clarification in some argument between academic people. that is a good use of his time.

here it would be just like handing out free tickets to the circus.

but John Baez comes here sometimes, so why not?

 

[CarlB]

I saw Smolin's (crowded) lecture on the Sundance braid theory at the APS meeting in Dallas. The basic problem is that their theory fails to give three generations, and their method of expanding to multiple generations fails to explain why there are only three low mass neutrinos.

For a while I thought that I could make my preons into a braid theory. There is sort of a hint as to how to do this in the literature. I'm purely algebraic, no braids, but I'm using Clifford / Geometric algebra. There's a paper by the Cambridge geometric algebraists showing a relation between twistors and algebra here:
http://arxiv.org/abs/math-ph/0603037
Also see:
http://arxiv.org/abs/hep-th/0008072

Also, Koide has written a paper referencing mine on the neutrino masses. The papers are here:
http://arxiv.org/abs/hep-ph/0605074
http://brannenworks.com/MASSES2.pdf

also, Alexei Smirnov, an invited speaker at the LHP conference in Tehran, mentioned it in his lecture:
http://physics.ipm.ac.ir/conferences/lhp06/notes/smirnov3.pdf
The audio is apparently available here, though I haven't listened to it:
http://physics.ipm.ac.ir/conferences/lhp06/program.pdf

Carl

 

[Kea]

There's a paper by the Cambridge geometric algebraists showing a relation between twistors and algebra...

Hi CarlB

Alternatively, one might observe that circulants are naturally associated to the Hall polynomials that quantum group people like. Note also that 3x3 circulants A, in particular, are characterised by the relation

P_{3} A = A P_{3}

where P_{3} is the S_{3} permutation (circulant) with 0,1,0 in the first row.

Hmmm...

 

[Kea]

Update

Smolin's slides from Loops07 are available here:

http://www.matmor.unam.mx/eventos/loops07/talks/PL5/Smolin.pdf

They indicate that the old loopy idea on mass generation has been abandoned in favour of a much more sensible approach (see slide 38), although details are lacking.