Connection between quaternionicity and causality

The notion of quaternionicity is a central element of M⁸-H duality. At the level of momentum space it means that 8-momenta -, which by M⁸-H-duality correspond to 4-momenta at level of M⁴ and color quantum numbers at the level of CP₂ - are quaternionic. Quaternionicity means that the time component of 8-momentum, which is parallel to real octonion unit, is non-vanishing. The 8-momentum itself must be time-like, in fact light-like. In this case one can always regard the momentum as momentum in some quaternionic sub-space. Causality requires a fixed sign for the time component of the momentum.

It must be however noticed that 8-momentum can be complex: also the 4-momentum can be complex at the level of M× CP₂ already classically. A possible interpretation is in terms of decay width as part of momentum as it indeed is in phenomenological description of unstable particles.

Remark: At space-time level either the tangent space or normal space of space-time surface in M⁸ is quaternionic (equivalently associative) in the regions having interpretation as external particles arriving inside causal diamond (CD). Inside CD this assumption is not made. The two options correspond to space-time regions with Minkowskian and Euclidian signatures of the induced metric.

Could one require that the quaternionic momenta form a linear space with respect to octonionic sum? This is the case if the energy - that is the time-like part parallel to the real octonionic unit - has a fixed sign. The sum of the momenta is quaternionic in this case since the sum of light-like momenta is in general time-like and in special case light-like. If momenta with opposite signs of energy are allowed, the sum can become space-like and the sum of momenta is co-quaternionic.

This result is technically completely trivial as such but has a deep physical meaning. Quaternionicity at the level of 8-momenta implies standard view about causality: only time-like or at most light-like momenta and fixed sign of time-component of momentum.

Remark: The twistorial construction of S-matrix in TGD framework based on generalization of twistors leads to a proposal allowing to have unitary S-matrix with vanishing loop corrections and number theoretically determined discrete coupling constant evolution. Also the problems caused by non-planar diagrams disappear and one can have particles, which are massive in M⁴ sense.

The proposal boils down to the condition that the 8-momenta of many-particle states are light-like (in complex sense). One has however a superposition over states with different directions of the projection of light-like 8-momentum to E⁴ in M⁸= M⁴× E⁴). At the level of CP₂ one has massive state but in color representation for which color spin and hypercharge vanish but color Casimir operator can have value of the order of the mass squared for the state. This prediction sharply distinguishes TGD from QCD.

See the article The Recent View about Twistorialization in TGD Framework or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.