Octonians (new paper by Louis Kauffman and Jon Hackett)

[marcus]

http://arxiv.org/abs/1010.2979
Octonions
Jonathan Hackett, Louis Kauffman
11 pages, 11 figures
(Submitted on 14 Oct 2010)
"In this paper we review the topological model for the quaternions based upon the Dirac string trick. We then extend this model, to create a model for the octonions - the non-associative generalization of the quaternions."

This is primarily a math paper. But it might (like many other math innovations) have some application to physics.

Any comment?

As I recall John Baez has written about the Octonians---but I have not read what he had to say. Does anybody know the prior literature on this topic?

Can anyone speculate as to possible physics application?

 

[bcrowell]

I don't know much about physical applications of the octonions. Basically the quaternions are of interest physically because they relate to the algebra of rotations in three dimensions. The Pauli spin matrices are simply Pauli's reinvention of the quaternions, with an extra factor of i.

Some sources of information:

John H. Conway, Derek A. Smith, On Quaternions and Octonions. A.K. Peters, 2003.

John Baez, The Octonions, AMS 2001, http://math.ucr.edu/home/baez/octonions/

http://math.ucr.edu/home/baez/octonions/conway_smith/ -- is supposedly a review of Conway & Smith, but is actually a substantial discussion of the ideas; geometrical discussion in terms of lattices (e.g., Gaussian integers)

 

[arivero]

This particular paper is about a topological construct, the Dirac String. But if the general topic is to be octonions, or generically division algebras in physics, this thread should be moved to the BSM forum.

As Baez and Huerta have explained from time to time, the main role of division algebras is that they define the condition for supersymmetry (first noticed by Evans in the eighties? See also Cesar Gomez?) in extra dimensions, stressing the role of dimensions 3,4, 6 and 10, as well as some "diagonals" appearing in membrane theory "the brane scan".

In mathematics their dimensional, geometric, role is to allow for Hoft fibrations. In this way, we can fiber S0 on S1, S1 on S2 and S3 on S4. The last one in a seven dimensional space, coincidentally the maximum number of dimensions allowed in a theory of supergravity.

A puzzling detail is that the SM does not have to correspond to S3 on S4, but to S3 on CP2. There is a covering construction going from CP2 to S4, studied recently (ie in the XXIth century) by Atiyah. Octonions and quaternions should have a role explaining this kinde of constructions, but the physics side is still obscure. Chirality could have a word too.

 

[tiny-tim]

Octonions are non-associative, and I've never understood how a non-associative structure can represent any aspect of reality.

In reality, how can A followed by B-followed-by-C not be the same as A-followed-by-B followed by C? Don't the two phrases mean the same thing?

 

[MTd2]

This paper is related to a soon to be published TOE* by Seth Lloyd and Sundance, and this is not the information universe. Check reference [2] which is this paper:

http://arxiv.org/abs/1002.1497

Those octonion belt tricks will be used to find the Standard Model for the Sundance Bilson-Thompson preon model. Cohl Furey is also on this new line of research.

Check the acknowledgments of the Octonion's paper:

"We would like to thank both Sundance Bilson-Thompson and Lee Smolin for their helpful discussions regarding this work."

*I don't know if it is a TOE, but it claims SM + QG.

 

[arivero]

Octonions are non-associative, and I've never understood how a non-associative structure can represent any aspect of reality.

Because they are alternative The next step in the ladder, the sedenions, are not associative and not alternative.

More seriusly, the important property is to be a http://en.wikipedia.org/wiki/Normed_division_algebra"

On other hand, the fact of not being associative actually helps, because it truncates the ladder. So in some sense octonions are the biggest general structure for a lot of properties.

Still, it could be possible that nature only "almost builds" the octonions. You can notice that I have not mentioned the fiber of S7 over S8. It is there, but its role in a theory of extra dimensions is, to say the least, unclear. Nature seems to play S3 on S4, then does some discrete juggling, then adds a extra finite dimension (for M theory or SU(5) guts) and then perhaps an extra infinitesimal one (For F theory or for U(1)B-L).

 

[arivero]

This paper is related to a soon to be published TOE* by Seth Lloyd and Sundance, and this is not the information universe. Check reference [2] which is this paper:

http://arxiv.org/abs/1002.1497

EDIT: In some versions it is reference 3!

Those octonion belt tricks will be used to find the Standard Model for the Sundance Bilson-Thompson preon model. Cohl Furey is also on this new line of research.

Furey is courageus, in the sense that Smolin preachs (or, is it, "predicates"?). But I think it is not going be the same line of attack that Lloyd and Bilson-Thomson.

EDIT: actually, I have no clue what Lloyd and Bilson-Thompson are planning!

 

[MTd2]

There is an updated version on Furey`s website:

http://www.perimeterinstitute.ca/personal/cfurey/
http://www.perimeterinstitute.ca/personal/cfurey/UTI20100805.pdf

According to the acknowledgments:

"Thank you to: S. Lloyd at MIT
for suggesting the name of ideal representations during
my visit there, S. Lloyd and T. Konopka for steering me
away from some major stylistic disasters, P.L. Mana, B.
Hartmann and F. Caravelli for their comments, G. Dixon
and J. Koeplinger for telling me of reference [3]. A special
thanks to R. Sorkin for his careful insight during this
paper's nal stage."

 

[arivero]

In a third or four reading (I have already read v1, with its funny notation), it is a best approach that Dixon, in the sense that it uses a single algebra of octonions for the full standard model. It sound good because the space of unit octonions is S7, so back to 11 space time dimensions :-)

Dixon took a more logic approach, separating the chiral part into a CxH and leaving a vector part for colour in CxO. But then you get too much content. For instance, Connes used, time ago, C+H and C+M3(C) for the chiral and vector parts.

 

[arivero]

Hmm, is Furey around here in the forum? I could take a bit more time to write some comments, if he is going to hear them. In the meanwhile, here is a cc of a mail I sent last February to Lloyd, with a further copy to Furey at perimeter.

- Hide quoted text -
---------- Forwarded message ----------
From: Alejandro Rivero <al.rivero@ >
Date: Mon, Feb 15, 2010 at 12:20 PM
Subject: Re: Furey work
To: slloyd@


On Sun, Feb 14, 2010 at 9:09 PM, <slloyd@ > wrote:

> SL(2,C)). The `idealized' octonions then give a natural representation
> of GL(8), which contains SU(3) X SU(2) X U(1) as a subgroup.

I had not noticed the GL(8) representation, this point of view is very
interesting!. From the point of view of seven-dimensional manifolds, we
have on one side the sphere S which is a fibration of S3 over S4, with
isometry group SO(8), and on the other side the family M of U(1) fibrations over
CP1 x CP2, or alternatively, if I am not wrong, S3 fibrations over CP2.

The amusing is that the group of isometries of M contains
SU(3)xSU(2)xU(1), while the group of isometries of S does not
contain it. There is a misterious link between CP2 and S4, you
can get the sphere by quotient of CP2 by complex conjugation;
CP2 is a branched covering of S4.

I was concentrated in SO(8), so I didn't check about the
role of GL(8).

(There is a third, mysterious player, in seven dimensions, the
family of manifolds of the kind SU(3)/U(1). They do not have
SU(2) symmetry nor a clear formulation as fiber bundles.)

> In my opinion, this work represents a significant contribution
> to the existing work on the potential role of division algebras
> in physics.

Yes, it is very illuminating.

Thank you for your time,

Alejandro

Please check also the thread
https://www.physicsforums.com/showthread.php?t=376804
It was me who put MTd2 back on the track of this paper, who he had skipped I need more RAM in my brain!

 

[MTd2]

It sound good because the space of unit octonions is S7, so back to 11 space time dimensions :-)

But the belt trick is in 3 spatial dimensions and it doesn`t seem that Furey defines anything in other than 3+1d.

 

[arivero]

But the belt trick is in 3 spatial dimensions and it doesn`t seem that Furey defines anything in other than 3+1d.

Ah ok now I follow you. You are telling that Kauffman can/could do the trick without resourcing to extra dimensions, and then Furey octonions should appear from a 3d interpretation, and the seven extra dimensions (from the unit sphere acted by GL(8), say) would be a mathematical artifact. Well, it could be a valid interpretation.

Btw, how do you guess/know that Lloyd and Sundance are working in some joint paper on this kind of topics? Is there some draft or some lecture I have missed.

 

[MTd2]

Btw, how do you guess/know that Lloyd and Sundance are working in some joint paper on this kind of topics? Is there some draft or some lecture I have missed.

Sundance would not let this opportunity escape! This is something he was looking for. Seth Lloyd is always looking for a TOE. That`s why :)

 

[arivero]

Sundance would not let this opportunity escape! This is something he was looking for. Seth Lloyd is always looking for a TOE. That`s why :)

Hmm so it is your guess. You could put some pressure on them, if you have some acquittance. OK, let's go back to the original thread for further discussion. Actually, lacking more papers to read, I will try to sleep some hours. :zzz: It is 3:46 here.

 

[octoman]

For more literature on octonions and their application of physics, look here:
http://www.octospace.com/

 

[tiny-tim]

octoman !

octoman! octoman!
does whatever a spiderman can
is he strong? listen bud
he's got octane in his blood
vroom! here comes the octoman! ​

 

[arivero]

By the way, looking at my agenda, it seems that the 20th of December of 1996 I was in this talk.

16:30-17:00 Kauffman, LH (Illinois at Chicago) NCG
Non-Commutative worlds

But I was not impressed enough to remember. Probably it was because I was associating Kauffman both with a joint work with Noyes and with another approaches to NCG in the mid nineties. I think that also Crane was occasionally in this gang, was he?