General Ideas about Octonions, Quaternions and Twistors

Abstract

An updated view about M⁸-H duality is discussed. M⁸-H duality allows to deduce M⁴ x CP₂ via number theoretical compactification. One important correction is that octonionic spinor structure makes sense only for M⁸ whereas for M⁴ x CP₂ complexified quaternions characterized the spinor structure.

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 x 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 Kahler structure. Octonionic projective space OP₂ appears as octonionic twistor space (there are no higher-dimensional octonionic projective spaces). Octotwistors generalize the twistorial construction from M⁴ to M⁸ and octonionic gamma matrices make sense also for H with quaternionicity condition reducing OP₂ to 12-D G₂/U(1) x U(1) having same dimension as the twistor space CP₃ x SU(3)/U(1) x U(1) of H assignable to complexified quaternionic representation of gamma matrices.

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.

Number theoretical vision about quantum TGD involves both p-adic number fields and classical number fields and the challenge is to unify these approaches. The challenge is non-trivial since the p-adic variants of quaternions and octonions are not number fields without additional conditions.  The key idea is that TGD reduces to the representations of Galois group of algebraic numbers realized in the spaces of octonionic and quaternionic adeles generalizing the ordinary adeles as Cartesian products of all number fields: this picture relates closely to Langlands program. Associativity would force sub-algebras of the octonionic adeles defining 4-D surfaces in the space of octonionic adeles so that 4-D space-time would emerge naturally. M⁸-H correspondence in turn would map the space-time surface in M⁸ to M⁴ x CP₂.