Tuesday, July 07, 2020

M8-H duality for fermions

M8-H duality in bosonic sector is rather well understood but the situation is different in the fermionic sector. The basic guideline is that also fermionic dynamics should be algebraic and number theoretical.
  1. Spinors should be octonionic. I have already earlier considered their possible physical interpretation.
  2. 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 algebrac 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 masslessness condition restricts the solutions to light-like 3-surfaces mklPkPl=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.
  3. 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.

    Addition: A really pleasant surprise that came this morning-9.7.2020 - sounds melodramatic but I do not want to forget it - it could have come more than decade ago but did not. The octonionic inner product for complexified octonionic 8-momenta with conjugation with respect to commuting imaginary unit i gives 8-D Minkowski norm squared. Same about quaternonic norm for complexified quaternionic momenta. Minkowski space with signature of (1,-,-1,-1) for metric follows from number theory alone! This conforms with the very idea of M8-H duality that geometry and number theory are dual in physics. Already this single finding makes M8-duality a "must".

    See the article Fermionic variant of M8-H duality or the chapter Does M8 duality reduce classical TGD to octonionic algebraic geometry?: Part III.

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

    Articles and other material related to TGD.

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