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 the basic arithmetics of complex, quaternion, and octonion number fields.
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.
- Imaginary part for the product z1z2 of complex numbers is
and linear in Im(z1) and Im(z2).
- Real part
is not linear in real parts:
This extremely simple observation turns out to contain amazingly deep physics.
- 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.
- 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!
See the chapter Does M8-H duality reduce classical TGD to octonionic algebraic geometry? or the article Do Riemann-Roch theorem and Atyiah-Singer index theorem have applications in TGD?.