What the properties of octonionic product can tell about fundamental physics?

In developing the view about M⁸-H duality reducing physics to algebraic geometry for complexified octonions at the level of M⁸, I became aware of trivial looking but amazingly profound observation about basic arithmetics of of complex, quaternion, and octonion number fields.

1. Imaginary part for the product z₁z₂ of complex numbers is
   

   Im(z₁z₂)= Im(z₁)Re(z₂)+Re(z₁)Im(z₂)
   

   and linear in Im(z₁) and Im(z₂).
   
   

2. Real part
   
   Re(z₁z₂)= Re(z₁)Re(z₂)-Im(z₁)Im(z₂).
   

   is not linear in real parts:
   
   

This generalizes to the product of octonions with Re and Im replaced by RE and IM in the decomposition to two quaternions: o= RE(o)+J IM(o), J is octonion imaginary unit not belonging to quaternionic subspace.

This extremely simple observation turns out to contain amazingly deep physics.

1. Space-time surfaces can be identified as IM(P)= loci or RE(P)=0 loci. When one takes product of two polynomials P₁P₂ the IM(P₁P₂)=0 locus as space-time surface is just the union of IM(p₁)=0 locus and IM(P₂) locus. No interaction: free particles as space-time surfaces! This picture generalizes also to rational functions R=P₁/P₂ and an their zero and infty loci.
   
   
2. For RE(P₁P₂)=0 the situation changes. One does not obtain union of RE(P₁)=0 and RE(P₂) space-time surfaces. There is interaction and most naturally this interaction generates wormhole contacts connecting the space-time surfaces (sheet) carrying fermions at the throats of the wormhole contact!
   
   

The entire elementary particle physics emerges from these two simple number theoretic properties for the product of numbers!

For details see the articleDo Riemann-Roch theorem and Atyiah-Singer index theorem have applications in TGD?.

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

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