For last month or so I have been writing an article about reduction of quantum classical TGD to octonionic algebraic geometry. During writing process "octonionic" has been replaced with "super-octonionic" and it has turned out that the formalism of twistor Grassmannian approach could generalize to TGD. The twistorial scattering diagrams have interpretation as cognitive representations with vertices assignable to points of space-time surface in appropriate extension of rationals
In accordance with the vision that classical theory is exact part of quantum theory, the core part of the calculations of
scattering amplitudes would reduce to the determination of zero loci for real and imaginary parts of octonionic polynomials and finding of the points of space-time surface with M8 coordinates in extension of rationals. At fundamental level the physics would reduce to number theory.
A proposal for the description of interactions is discussed in the article. Here is a brief summary.
See the chapter Does M8-H duality reduce classical TGD to octonionic algebraic geometry? or the articles Does M8-H duality reduce classical TGD to octonionic algebraic geometry?: part I and Does M8-H duality reduce classical TGD to octonionic algebraic geometry?: part II.
- The surprise that RE(P)=0 and IM(P)=0 conditions have as singular solutions light-cone interior and its complement and 6-spheres S6(tn) with radii tn given by the roots of the real P(t), whose octonionic extension defines the space-time variety X4. The intersections X2= X4∩ S6(tn) are tentatively identified as partonic 2-varieties defining topological interaction vertices. S6 and therefore also X2 are doubly critical, S6 is also singular surface.
The idea about the reduction of zero energy states to discrete cognitive representations suggests that interaction vertices at partonic varieties X2 are associated with the discrete set of intersection points of the sparticle lines at light-like orbits of partonic 2-surfaces belonging to extension of rationals.
- CDs and therefore also ZEO emerge naturally. For CDs with different origins the products of polynomials fail to commute and associate unless the CDs have tips along real (time) axis. The first option is that all CDs under observation satisfy this condition. Second option allows general CDs.
The proposal is that the product ∏ Pi of polynomials associated with CDs with tips along real axis
the condition IM(∏ Pi)=0 reduces to IM(Pi)=0 and criticality conditions guaranteeing associativity and provides a description of the external particles. Inside these CDs RE(∏ Pi)=0 does not reduce to RE(∏ Pi)=0, which automatically gives rise to geometric interactions. For general CDs the situation is more complex.
- The possibility of super-octonionic geometry raises the hope that the twistorial construction of scattering amplitudes in N=4 SUSY generalizes to TGD in rather straightforward manner to a purely geometric construction. Functional integral over WCW would reduce to summations over polynomials with coefficients in extension of rationals and criticality conditions on the coefficients could make the summation well-defined by bringing in finite measurement resolution.
If scattering diagrams are associated with discrete cognitive representations, one obtains a generalization of twistor formalism involving polygons. Super-octonions as counterparts of super gauge potentials are well-defined if octonionic 8-momenta are quaternionic. Indeed, Grassmannians have quaternionic counterparts but not octonionic ones. There are good hopes that the twistor Grassmann approach to N=4 SUSY generalizes. The core part in the calculation of the scattering diagram would reduce to the construction of octonionic 4-varieties and identifying the points belonging to the appropriate extension of rationals.