Last night I was thinking about possible future project in TGD. The construction of scattering amplitudes has been the dream impossible that has driven me for decades. Maybe the understanding of fermionic M8-H duality provides the needed additional conceptual tools.
The challenge is to develop a concrete number theoretic hierarchy for scattering amplitudes: R→C→Q→O - actually their complexifications.
- M8 picture looks simple. Space-time surfaces in M8 can be constructed from real polynomials with real (rational) coefficients, actually knowledge of their roots is enough. Discrete data - roots of the polynomial!- determines space-time surface as associative or co-associative region! Besides this one must pose additional condition selecting 2-D string world sheets and 3-D light-like surfaces as orbits of partonic 2-surfaces. These would define strong form of holography (SH) allowing to map space-time surfaces in M8 to M4×CP2.
- Could SH generalize to the level of scattering amplitudes expressible in terms of n-point functions of CFT?! Could the n points correspond to the roots of the polynomial defining space-time region!
Algebraic continuation to quaternion valued scattering amplitudes analogous to that giving space-time sheets from the data coded SH should be the key idea. Their moduli squared are real - this led to the emergence of Minkowski metric for complexified octonions/quaternions) would give the real scattering rates: this is enough! This would mean a number theoretic generalization of quantum theory.
- One can start from complex numbers and string world sheets/partonic 2-surfaces. Conformal field theories (CFTs) in 2-D play fundamental role in the construction of scattering string theories and in modelling 2-D statistical systems. In TGD 2-D surfaces (2-D at least metrically) code for information about space-time surface by strong holography (SH) .
Are CFTs at partonic 2-surfaces and string world sheets the basic building bricks? Could 2-D conformal invariance dictate the data needed to construct the scattering amplitudes for given space-time region defined by causal diamond (CD) taking the role of sphere S2 in CFTs. Could the generalization for metrically 2-D light-like 3-surfaces be needed at the level of "world of classical worlds" (WCW) when states are superpositions of space-time surfaces, preferred extremals?
There is also the challenge to relate M8- and H-pictures at the level of WCW. The formulation of physics in terms of WCW geometry leads to the hypothesis that WCW Kähler geometry is determined by Kähler function identified as the 4-D action resulting by dimensional reduction of 6-D surfaces in the product of twistor spaces of M4 and CP2 to twistor bundles having S2 as fiber and space-time surface X4⊂ H as base. The 6-D Kähler action reduces to the sum of 4-D Kähler action and volume term having interpretation in terms of cosmological constant.
- In the case of fermions one can start from 1-D data at light-like boundaries LB of string world sheets at light-like orbits of partonic 2-surfaces. Fermionic propagators assignable to LB would be coded by 2-D Minkowskian QFT in manner analogous to that in twistor Grassmann approach. n-point vertices would be expressible in terms of Euclidian n-point functions for partonic 2-surfaces: the latter element would be new as compared to QFTs since point-like vertex is replaced with partonic 2-surface.
- The fusion (product?) of these Minkowskian and Euclidian CFT entities corresponding to different realization of complex numbers as sub-field of quaternions would give rise to 4-D quaternionic valued scattering amplitudes for given space-time sheet. Most importantly: there moduli squared are real! A generalization of quantum theory (CFT) from complex numbers to quaternions (quaternionic "CFT").
- What about several space-time sheets? Could one allow fusion of different quaternionic scattering amplitudes corresponding to different quaternionic sub-spaces of complexified octonions to get octonion-valued non-associative scattering amplitudes. Again scattering rates would be real. A further generalization of quantum theory?
The question is whether the Kähler function - an essentially geometric notion - can have a counterpart at the level of M8.
- SH suggests that the Kähler function identified in the proposed manner can be expressed by using 2-D data or at least metrically 2-D data (light-like partonic orbits and light-like boundaries of CD). Note that each WCW would correspond to a particular CD.
- Since 2-D conformal symmetry is involved, one expects also modular invariance meaning that WCW Kähler function is modular invariant, so that they have the same value for X4⊂ H for which partonic 2-surfaces have induced metric in the same conformal equivalence class.
- Also the analogs of Kac-Moody type symmetries would be realized as symmetries of Kähler function. The algebra of super-symplectic symmetries of the light-cone boundary can be regarded as an analog of Kac-Moody algebra. Light-cone boundary has topology S2× R+, where R+ corresponds to radial light-like ray parameterized by radial light-like coordinate r. Super symplectic transformations of S2× CP2 depend on the light-like radial coordinate r, which is analogous to the complex coordinate z for he Kac-Moody algebras.
The infinitesimal super-symplectic transformations form algebra SSA with generators proportional to powers rn . The Kac-Moody invariance for physical states generalizes to a hierarchy of similar invariances. There is infinite fractal hierarchy of sub-algebras SSAn⊂ SSA with conformal weights coming as n-multiples of those for SSA. For physical states SSAn and [SSAn,SSA] would act as gauge symmetries. They would leave invariant also Kähler function in the sector WCWn defined by n. This would define a hierarchy of sub- WCWs of the WCW assignable to given CD.
The sector WCWn could correspond to extensions of rationals with dimension n, and one would have inclusion hierarchies consisting of sequences of ni with ni dividing ni+1. These inclusion hierarchies would naturally correspond to those for hyper-finite factors of type II1.
See the chapter ZEO and matrices or the article Fermionic variant of M8-H duality.