### Octonions at Institute of Advanced Study

Atiyah should have given last week a talk entitled

*Quantum Gravity and the Riemann Hypothesis*in Instutute of Advance Study. This talk was canceled and Kea tells that the title of the talk was

*Octonions and the four forces of Physics*. In the conference program the title of his talk held at Friday November fifth was Exploring the Geometry Behind the Quantum Universe. So what Atyiah did really talk about? Kea also informs that Atiyah concluded his talk with

*You can regard what I say as nonsense, or you can claim that you know it already, but you cannot make these two claims together*.

This does not give any clue.

Why I am worrying about what Atiyah really did at Friday November fifth is very TGD-centered. As you should already know, TGD is more or less what I am - or more precisely, what has been left of me under the squeeze of cruel academic forces. If Atiyah indeed talked about octonions, I have the courage (in my pitifully crackpottish manner of course) to wonder whether my humble and shy octonionic message from the bottomest bottom of the hierarchy could have reached the Olympian heights so that even Atiyah gets interested;-). In my heart I of course know that I must understand that I am just a poor classless pariah as compared to these Brahmins of Science and it is incredibly blasphemous to even imagine that they might be interested on something that I have said!

The basic problem with octonions and quaternions has been how to bring them to physics in a manner consistent with what we already know.

- One should build a connection with standard model quantum numbers. Here the solution comes from the observation that SU(3) subgroup of octonion automorphisms can take the role of color group. This observation leads in TGD to what I call M
^{8}-M^{4}× CP_{2}duality giving M^{4}× CP_{2}and therefore standard model symmetries a unique number theoretic status. - The signature of the imbedding space metric and space-time remains the problem and here complexified octonions and hyper-octonionic sub-spaces with Minkowskian signature is here a way out and means the replacement of number field with algebra. Same about hyper-quaternions.
- Non-associativity is the third tough problem: the ideas about octonionic and quaternionic quantum do not work. Here however a simple solution suggests itself: associativity as the number theoretical realization of the fundamental variational principle selecting the 4-D space-time surface as quaternionic and thus associative sub-manifolds of an 8-D space-time with octonionic structure. Everything would reduce to number theory: space-time dimension, standard model symmetries, dynamics, and much more.
- A further tough problem is what octonionic and quaternionic structure really means. The attempts to make sense of the notions of quaternion/ octonion analyticity (restricted to real analyticity) lead to conflict with what we know about wave equations. The notion of (hyper-) quaternionic and (hyper-)octonionic representations of gamma matrices turned out to be the optimal solution to the question what one means with quaternionic and octonionic structures. The first guess is that induced gamma matrices span quaternionic subspace at each point of space-time surface.
This works if volume defines the action behind space-time dynamics but not for Kähler action. One must replace the induced gammas with modified gammas. Modified gamma matrices would span a hyper-quaternionic sub-space of octonionic gammas at each point of space-time surface. Do these surfaces define preferred extremals of Kähler action (does the same holds for any general coordinate invariance action principle)? This is the question.

**Addition**. From Gil Kalai's blog I learned that Atyiah indeed talked about classical number fields and physics and proposed that the four number fields could correspond to fundamental interactions with gravitation assigned with octonions. This idea looks to me more like a numerology. Notice however that other interactions than gravitation could be described in field theory framework using 4-D Minkowski space which can be interpreted in terms of hyper-quaternionic flat space. When gravitation comes into play one must have more general hyper-quaternionic sub-manifolds of hyper-octonionic space with hyper-quaternionicity defined in terms of modified gamma matrices and their octonionic representation.

## 14 Comments:

Stop talk like that. What has happened?

Nothing. I like a little bit irony.

Was that irony? I did not understood that. In my ears it sounded more like - projections (I know you hate it). Some old wounds that made you talk like that. I have myself done that kind of talk too, so ... stop it. Pls.

If you don't appreciate your work yourself, how can then other do it? What would a man like Atiyah think if he read it?

http://uduality.blogspot.com/2010/11/atiyah-on-magic-square.html

And a TGD proof of neutrino stars forming a big nuclei. "This measurement tells us that if any quarks are present in a neutron star core, they cannot be free, but rather must be strongly interacting with each other as they do in normal atomic nuclei,"

http://www.astronomy.com/News-Observing/News/2010/10/Astronomers%20discover%20most%20massive%20neutron%20star%20yet%20known.aspx?utm_source=SilverpopMailing&utm_medium=email&utm_campaign=ASY_NEWS_NonSub_101029_final&utm_content=

I know, my nose...

Atiyah has some work on how CP2 covers S4. This is known only to Atiyah, I asked a lot of people, even to Baez and nobody knew! It is probably a key result because S4 fibered with S3 is the octonion ball while CP2 fibered with S3 is Witten's manifold for non chiral standard model.

Matti,

I just found this posting thru another blog. Now, I posted today more along crude thoughts on complex numbers and logic. This octonian and quaternion relation I have long considered. Today I asked a rather childlike question. I can imagine the rotations in three space and actually hold a model in my hand- is there some sort of model for a four space orthogon that applies to octonians?

This probably A4 but the "quasic" plane for me was always n-dimensional.

ThePeSla pesla.blogspot com

To Alejandro:

Maybe mathematicians are finally beginning to realize;-)! In TGD approach CP_2 parametrizes quaternionic planes of octonion space containing preferred complex plane. This plane corresponds physically to non-physical polarizations. Mathematically it corresponds to a maximal commutative subspace. Physics seems to require the ladder non-associative-associative-commutative.

This generalizes trivially to subspace of complexified octonions assignable to M^8=M^4xE^4. I can map four-surface of M^8 to that of H= M^4xCP_2 by mapping point of M^4 to M^4 and the point of CP_2 parametrizing the quaternionic tangent space of the surface to CP_2.

One can ask whether partonic two-surfaces are co-commutative in the sense that the generalized gamma matrices associated their tangent spaces multiplied with a fixed octonionion imaginary unit are hyper-complex sub-spaces of octonions. Also the stringy 2-surfaces appearing in the proposed slicing of 4-surfaces with Minkowski signature could be hyper-complex in the proposed sense.

S^4 has SO(4) as group of isometries as also E^4 in M^4xE^4. In M^8-H duality I interpert SO(4) assigned with M^8 as symmetries of low energy hadron physics and SU(3) of CP_2 as color group acting as symmetries of highr energy hadron physics. These descriptions of hadron physics would be dual. Just a proposal;-).

To Pesla; I think that 8-dimensionality makes it hard to imagine any concrete model of octonions.

Probably is a typo, but the group of isometries of S4 is SO(5), not SO(4).

Of course. E^4 has rotation group of S^3 as isometries besides translations. Atyiah's picture seems to be different from mine. It would be nice to see article about his ideas.

Matti,

Sorry for the delay in responding to your comment to me.

I am not seeing a deeper octonian space surrounded by a quaternion surface- but the two relate as if the same space- more of a consequence of notation as information theory rather than just core number theory considered alone.

BTW one day, as soon as my own retirement becomes stable- I would like to buy your books if you will notify me at LoversOfWisdom@yahoo.com what is there. I have not been able yet to send money on line.

Dear ThePesla,

only one book is available and is out of date by several years;-)! There are fifteen online books at my homepage and viXra archives. They can be printed but you get a horrible pile of paper and regret that you ever decided to learn TGD!

Updating of these books have become for me just a manner to avoid ending up with chaos when new ideas emerge. They also serve as a testimony about my life work since academic science community still refuses to admit that TGD and also me really do exist.

There are quite recent articles in JCER and Prespacetime Journal about TGD as it is now. It is easy to find from web these journals founded by Huping Hu. They might provide the quickest way to TGD.

One voice from 'out there':

"It would be interesting to read your account of Matti, who he is and how he got to the present situation, also why his work is important. People on the web thought he was an unknown self taught amateur, one of the 40,000 less qualified people."

On discussions of a book about TGD and biology, my prospect. I can tell you that if you really try you can do it, but it is difficult. Matti writes no easy texts. But you will be rewarded by a better insight and less chaos :)

Everyone should learn TGD.

From Kea on Atiyah: http://scgp.stonybrook.edu/presentations/20101103_Atiyah_-_From_Algebraic_Geometry_to_Physics.pdf

And from the talk: http://arxiv.org/abs/1009.3176

And what he says: So it is left to the older generation like me to speculate. The same friend who likened string theory to poetry encouraged me to have wild ideas, saying ”you have nothing to lose!” That is true, I have my PhD. I do not need employment and all I can lose is a bit of my reputation. But then allowances are made for old-age, as in the case of Einstein when he persistently refused to concede defeat in his battle with Niels Bohr.

How many of those old guys think as him? If they only would talk.

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