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.

1. 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 M⁸-M⁴× CP₂ duality giving M⁴× CP₂ and therefore standard model symmetries a unique number theoretic status.

2. 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.

3. 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.

4. 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.