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At the conclusion of a conference delivered by Alain Connes in 2000 (video in French at 1:19:25), an audience member posed a question. Below is a polished translated transcription:

Audience:

You have shown that with non-commutative geometry, we significantly simplify physics in a very elegant way, by giving up a property that seems quite simple. In the same approach, if we rely on algebras, complex numbers, quaternions, and octonions, there are researchers working, I believe, on exploring non-associative geometries.

Alain Connes:

Well, I can explain that to you right away. In fact, I think I've given the explanation before. Why? Because language is associative, words are associative. Associativity is essential to be able to represent something as an operator in a Hilbert space; it's crucial, right? So when I hear about non-associative geometry, I have exactly the same reaction that I think many mathematicians have regarding non-commutative geometry.

Question: What about nonassociative geometry?

Ten years on, Sir Michael Atiyah offered insights into Connes' theory (here at 58:03). Below is a polished transcription:

Connes' theory deals only with associative algebras, which means it can't address the octonions. That's why I believe it is not quite complete. His theory is indeed very beautiful, but it doesn't quite reach the core of gravity. I think the reason for this is that his theory is limited to associative algebras.

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    $\begingroup$ If you look on google you find some stuff, for example: arxiv.org/pdf/hep-th/9607086v2.pdf $\endgroup$
    – Qfwfq
    Commented Jun 8, 2013 at 15:17
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    $\begingroup$ Thank you for this reference. I also found 15 papers on arxiv containing nonassociative (or non-associative) geometry on their title, the most recent in 2013, but your reference is the first (1997). Just a note: on the 15 references found, 11 were tagged by "High Energy Physics", 2 by "Quantum algebra" and one by "Ring and algebra." It seems that the physicists are much more motivated by the nonassociative geometry than the mathematicians, why ? $\endgroup$
    – Sebastien Palcoux
    Commented Jun 8, 2013 at 16:18
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    $\begingroup$ Michael Atiyah speculates about the role for octonion-based geometries for quantum gravity around 30 minutes and 54 minutes into this talk: youtube.com/watch?v=I1ftUA17MZg. Also, after 58 minutes comes the thought that we need to go beyond cyclic homology for associative algebras to deal with nonassociativity. $\endgroup$
    – David Corfield
    Commented Jun 10, 2013 at 10:44
  • $\begingroup$ Thank you very much David, this video is very interesting. I will put, in the issue, a quotation of Michael Atiyah extract from this video. $\endgroup$
    – Sebastien Palcoux
    Commented Jun 10, 2013 at 14:20

1 Answer 1

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There has been progress in this area by mathematicians, as seen here: Jordan Operator Algebra.

(See also this Physics post: Non-associative Operators in Physics).

Note: Jordan operator algebras replace associativity with commutativity. Specifically, their product $\circ$, defined as $a_1 \circ a_2 = \frac{1}{2}(a_1*a_2 + a_2*a_1)$, is commutative but nonassociative, while $*$ is noncommutative and associative. This approach is advanced, yet not entirely satisfying. A truly comprehensive solution would involve algebras that are both nonassociative and noncommutative.

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