Further Study of Twistor Lift of TGD (Part I)

Abstract

In this article I study questions related to both classical and quantum aspects of twistorialization. The first group of questions relates to the twistor lift of classical TGD. What does the induction of the twistor structure really mean? Can the analog of Kahler form assignable to M⁴ suggested by the symmetry between M⁴ and CP₂ and by number theoretical vision appear in the theory? What would be the physical implications? How does gravitational coupling emerge at fundamental level?  Could one regard localization of spinor modes to string world sheets as localization to Lagrangian sub-manifolds of space-time surface with vanishing induced Kahler form? Lagrangian sub-manifolds would be commutative in the sense of Poisson bracket. How this relates to the idea that string world sheets correspond complex (commutative) surfaces of quaternionic space-time surface in octonionic imbedding space. During the re-processing of the details related to twistor lift, it became clear that the earlier variant for the twistor lift can be criticized and allows an alternative. This option led to a much simpler view about twistor lift, to the conclusion that minimal surface extremals of Kahler action represent only asymptotic situation (external particles in scattering), and also to a re-interpretation for the p-adic evolution of the cosmological constant: cosmological term would correspond to the entire 4-D action and the cancellation of Kahler action and cosmological term would lead to the small value of the effective cosmological constant. Second group of questions relates to the construction of scattering amplitudes. The idea is to generalize the usual construction for massless states. In TGD all single particle states are massless in 8-D sense and this gives excellent hopes about the applicability of 8-D twistor approach. M⁸-H duality turns out to be the key to the construction. Also the holomorphy of twistor amplitudes in helicity spinors λ_i and independence on tilde λ_i is crucial. The basic vertex corresponds to 4-fermion vertex for which the simplest expression can be written immediately. n>4-fermion scattering amplitudes can be also written immediately. If scattering diagrams correspond to computations as number theoretic vision suggests, the diagrams should be reducible to tree diagrams by moves generalizing the old-fashioned hadronic duality. This condition reduces to the vanishing of loops which in terms of BCFW recursion formula states that the twistor diagrams correspond to closed objects in what might be called WCFW homology.