R c h o

arivero

http://arxiv.org/abs/1002.1497 I just mentioned RCHO in other thread, and an article appears. Very predictive, I am :-DDD


I haven't read it. I will not, most probably, in a few days. But I feel it could be place for a thread on the topic of relationship between unification and normed, division, algebras.

You may know, or perhaps not, that it is actually a *mainstream* topic. Evans did a proof of the relationship between supersymmetry and this kind of algebras. Of course it is also a "lost cause". But perhaps it could be regained. Also the vector-"diagonal" generalisation of Evans argument builds the full BraneScan, which can be also told to be mainstream. I told of the brane scan here, http://www.physicsforums.com/showthread.php?t=181194

BTW, the guy is at Perimeter, with Sorkin.

arivero

OK I read it. Well, the references :DDDD Hey, I bet it is the first Perimeter preprint actually quoting F.D.T. Smith in the references, is it? No reference for my boss, althought :(

I am going to put some references and then I get the excuse to quote Boya too ;-)

Baez discussion, last year.
http://golem.ph.utexas.edu/category/...eryangmil.html
and Huerta homepage:
http://math.ucr.edu/~huerta/
Tony smith holistic webpage:
http://www.valdostamuseum.org/hamsmith/
Evans paper, you can download the preprint from HEP
http://www.slac.stanford.edu/spires/...=NUPHA,B298,92
Spires search by title
http://www.slac.stanford.edu/spires/...tecount%28d%29
Subtopic "octonions"
http://www.slac.stanford.edu/spires/...tecount%28d%29
RCHO as seen from Zaragoza
http://arxiv.org/pdf/hep-th/0301037v1

I should add some references to minor related topics, as for instance S7 and S13 and the Atiyah Arnold etc results on related fiberings there. But I refrain, hoping that some reader will also show interest on the topic...

humanino

You must be moving at quite some speed wrt to PF server, because a few days for you turned out to be 5 minutes here ! Are you running in circle ?

arivero

Not circle, but U(1) :D Really I have not read it yet, only look the references. I happen to be busy on Hopf fiberings nowadays, so it was not a good idea to leave it go.
Ok, I am going to send it to the printer, to read in the bus.

arivero

It is not strange to get the standard model out from the octonions. The space of unit octonions is the sphere S7. And this sphere is "branched covered" by Witten's manifolds in a peculiar way:

S3 ---> S7 -----> S4 is the (generalised) Hopf fiber bundle of the sphere
S3 ---> M ------> CP2 is the fiber bundle schema both of Witten Manifold and also of some Aloff-Wallach spaces. In the first case, both the fiber and the bundle provide symmetries: SU(2)xU(1) and SU(3), respectively, as isometries of each. Remember S4 is HP1.

You can also fiber CP2 with an extra U(1) to get S5, whose isometry group is SO(6)=SU(4). This is the "lepton as the fourth colour" approach, and probably is nearer of Furey, who looks for the gauge group in C \otimes O.

The real problem of the RCHO approaches is to get the Higgs. Alain Connes got near of it, by considering two actions by CxH and CxM3, the 3x3 matrix, instead of O. See the Red Book.

If you dont get the Higgs, another mechanism for symbreak should exist. For instance, deformations of the metric. It is interesting that Alof-Wallach spaces do not have SU(2) isometry.

The relationship between CP2 and S4, as well as other RCHO compositions, is explained in math/0206135 by Atiyah and Berndt.

MTd2

I saw that paper, but I didn't read it. Why was that put on general physics? At first site, it looks like a serious work.

arivero

Let me go back to the unit sphere of octonions, I mean [itex]S^7[/itex]. Note [itex]S^3, S^1, S^0[/itex] are groups, while [itex]S^7[/itex] is not. First clue that we are hitting the octonionic world.

A point that intrigues me is that we can see this sphere in four different spaces: [itex]R^8, C^4, H^2, O[/itex]. Just as the [itex]S^3[/itex] can be seen in [itex]R^4, C^2[/itex] or [itex]H[/itex].

For [itex]S^3[/itex], the Hopf fibration works by projecting [itex]C^2[/itex] in [itex]CP^1[/itex].
For [itex]S^7[/itex], the Hopf fibration works by projecting [itex]H^2[/itex] in [itex]HP^1[/itex]

But there are other projections. In [itex]S^3[/itex] I can also project in [itex]RP^3[/itex]. In [itex]S^7[/itex] I can project in [itex]RP^7[/itex] or also in [itex]CP^3[/itex].

But I can not do "middle way" projections. I can not project [itex]S^7[/itex] in, say, [itex]CP^2[/itex] and get a meaningful fiber bundle. If I build U(1) fiber bundles over [itex]CP^3[/itex] I get the sphere again, but if I build U(1) fiber bundles over [itex]CP^2 \times CP^1[/itex] I get Witten's spaces, the ones with isometry group SU(3)xSU(2)xU(1).

It should be interesting to understand this game algebraically, down from the octonion sphere.

arivero

Some of this thread has been continued in this one
http://www.physicsforums.com/showthread.php?t=438585

MTd2

I told Tony Smith about the Octonion`s paper. He got really excited.

MTd2

Well, quoting from the other thread :)

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

arivero

Just a minor correction here: it is Kauffman, no Furey, who does the belt trick in the last paper.

The important point, at the end, is if they can bypass Salam's objection about the charges. I guess that the C in CxO has a role there, because Bailin and Love did the bypass by going up one dimension.
And of course, they should solve the issue of breaking a SU(2)xSU(2) into SU(2)xU(1), perhaps related to chirality, and the selection for colour of SU(3) instead of SO(5). I think that these details are minor, but people will consider them important.

MTd2

I was talking about both of them. Kauffman in 3d and Furey in 3+1.

page 2 here:

"The groups are unied with the vectors they transform, and further, those vectors: the scalars, spinors, 4-momenta and eld strength tensors, are all born from the same meagre algebra."

http://www.perimeterinstitute.ca/per...TI20100805.pdf

arivero

Thers is no belt trick there. It is not that it can not be performed, but Furey tells nothing about it, it just puts a generic reference to Hestenes due to the use of Lorentz group. You have been dreaming some extra pages in the article, it seems :-)

Moreover, that parragraph refers to C \otimes H. Even if there were a belt trick in the references, it would not be about octonions.

MTd2

I wasn't talking about belt trick on Furey's paper. 3d for belt trick and Kauffman's paper and 3d+1 for Furey's.

You asked me about having a line of research for a new kind of quantum gravity. The paper mentions that, but I forgot where. I found it now:)

page. 10

"Extending on the relationship of the quaternions with SU(2) is the question of whether this model could provide illumination to attempts to use the octonions to construct the standard model of particle physics - such as the attempt in [2]. Here again the resemblance of L to parity inversion is suggestive of something more profound. We will continue these considerations in a sequel to the present paper"

Anyway, if you are a beginner, and like you said before, a very courageous one, and you receive and invitation or sugestion from a Grand Master like Louis Kauffman, wouldn't you follow it?

arivero

Ah, but Furey's is not 3+1.

He takes R,C,H,O, then he mixes a bit the R and the C, then he uses CxH to generate the Lorentz Group (acting in 3+1) and _separately_ he uses CxO to generate the gauge group... and he does not tell where this gauge group is acting. But if you compare CxH and CxO, you should deduce that the group got from CxO is acting also in some space. You could expect to be an object of a dimension 7+1. Which is the right result, because 7 is the minimum for the standard model and 8 for the GUT groups, and probably it is something between, because he should use R and C instead of two times C, when putting all together.

Note that in the standard theory also the 11 dimensional space divides in a very natural way into 4+7. This is well known.

MTd2

Let's see how many dimensions this RCHO has.

On page 2:

"The generic element of CHO is [FORMULA]. Imaginary units of the diferent division
algebras always commute with each other; explicitly, the
complex i commutes with the quaternionic i; j; k, all four
of which commute with the octonionic feng."

That means 2x4x8=64 dimensions.

Everything there happens in a certain hypersurface with the property of being and ideal of the algebra.

arivero

I count the unit ball in C times the unit ball in H times the unit ball in O. That makes 11.

Consider the product of a line and a circle. It is a cylinder. When you multiply manifolds, the dimensions add. And the same happens when you multiply commutative algebras (eg, the algebra of complex functions over the circle times the algebra of complex functions over the line): the dimensions of their Gelfand Naimark dual add.

A different problem is when you have a non commutative algebra. In this case two approaches are known: Morita equivalence, which doest a sort of reduction of most finite algebras to be equivalent to the algebra of complex functions over a dot. Or group theory, where you look for symmetry groups, for instance isolating the unit ball: a circle, a S3 sphere, or a S7 sphere for the respective case C, H, O.