General ideas about octonions, quaternions, and twistors

Octonions, quaternions, quaternionic space-time surfaces, octonionic spinors and twistors and twistor spaces are highly relevant for quantum TGD. In the following some general observations distilled during years are summarized.

There is a beautiful pattern present suggesting that H=M⁴× CP₂ is completely unique on number theoretical grounds. Consider only the following facts. M⁴ and CP₂ are the unique 4-D spaces allowing twistor space with Kähler structure. M⁸-H duality allows to deduce M⁴× CP₂ via number theoretical compactification. Octonionic projective space OP₂ appears as octonionic twistor space (there are no higher-dimensional octonionic projective spaces). Octotwistors generalise the twistorial construction from M⁴ to M⁸ and octonionic gamma matrices make sense also for H with quaternionicity condition reducing OP_{2 to to the twistor space of H.}

A further fascinating structure related to octo-twistors is the non-associated analog of Lie group defined by automorphisms by octonionic imaginary units: this group is topologically six-sphere. Also the analogy of quaternionicity of preferred extremals in TGD with the Majorana condition central in super string models is very thought provoking. All this suggests that associativity indeed could define basic dynamical principle of TGD.

See the new chapter Unified Number Theoretical Vision or the article.