Hawking Radiation 5

Quaternions are not commutative (AB is not equal to AB in general) and one can ask could one define commutative and co-commutative surfaces of quaternionic space-time surface and their variants with Minkowski signature. This is possible.

There is also a physical motivation. The generalization of twistors to 8-D twistors starts from generalization in the tangent space M8 of CP2. Ordinary twistors are defined in terms of sigma matrices identifiable as complexified quaternionic imaginary units. One should replaced the sigma matrices with 7 sigma matrices and the obvious guess is that they represent octonions. Massless irac operator and Dirac spinors should be replaced by their octonionic variant. A further condition is that this spinor structure is equivalent with the ordinary one. This requires that it is quaternionic so that one must restrict spinors to space-time surfaces.

This is however not enough - the associativity for spinor spinor dynamics forces them to 2-D string world sheets. The reason is that spinor connection consisting of sigma matrices replaced with octonion units brings in additional potential source of non-associativity. If induced gauge fields vanish, one has associativity but not quite: induce spinor connection is still non-associative. The stronger condition that induced spinor connection vanishes requires that the CP2 projection of string world sheet is not only 1-D but geodesic circle. String world sheets would be possible only in Minkowskian regions of space-time surface and their orbit would contain naturally a light-like geodesic of imbedding space representing point-like particle.