Fermionic Variant of M8-H Duality

Abstract

The topics of this article is M⁸-H duality for fermions. The basic guideline is that also fermionic dynamics should be algebraic and number theoretical. Spinors should be octonionic. I have already earlier considered their possible physical interpretation. Dirac equation as linear partial differential equation should be replaced with a linear algebraic equation for octonionic spinors which are complexified octonions. The momentum space variant of the ordinary Dirac equation is an algebraic equation and the proposal is obvious: PΨ=0, where P is the octonionic continuation of the polynomial defining the space-time surface and multiplication is in octonionic sense. The conjugation in O_c is induced by the conjugation of the commuting imaginary unit i. The square of the Dirac operator is real if the space-time surface corresponds to the projection O_c -> M⁸ -> M⁴ with real time coordinate and imaginary spatial coordinates so that the metric defined by the octonionic norm is real and has Minkowskian signature. Hence the notion of Minkowski metric reduces to octonionic norm for O_c - a purely number theoretic notion. The masslessness condition restricts the solutions to light-like 3-surfaces m_klP^kP^l=0 in Minkowskian sector analogous to mass shells in momentum space - just as in the case of ordinary massless Dirac equation. P(o) rather than octonionic coordinate o would define momentum. These mass shells should be mapped to light-like partonic orbits in H. This picture leads to the earlier phenomenological picture about induced spinors in H. Twistor Grassmann approach suggests the localization of the induced spinor fields at light-like partonic orbits in H. If the induced spinor field allows a continuation from 3-D partonic orbits to the interior of X⁴, it would serve as a counterpart of virtual particle in accordance with quantum field theoretical picture.