The progress in the understanding of the classical aspects of twistor lift of TGD makes possible to consider in detail the quantum aspects of twistorialization of TGD and for the first time an explicit proposal for the part of scattering diagrams assignable to fundamental fermions emerges.
Also the positive Grassmannian requires (2,2) signature. In M8-H relies on the existence of the decomposition M2⊂ M2= M2× E2⊂ M8. M2 could even depend on position but M2(x) should define an integrable distribution. There always exists a preferred M2, call it M20, where 8-momentum reduces to light-like M2 momentum. Hence one can apply 2-D variant of twistor approach. Now the signature is (1,1) and spinor basis can be chosen to be real! Twistor space is RP3 allowing complexification to CP3 if light-like complex momenta are allowed as classical TGD suggests!
Furthermore, the ordinary twistor amplitudes are holomorphic functions of the helicity spinors λi and have no dependence on &lambda tile;i: no information about particle masses! Only the momentum conserving delta function gives the dependence on masses. These amplitudes would define as such the M4 parts of twistor amplitudes for particles massive in TGD sense. The simplest 4-fermion amplitude is unique.
Twistor approach gives excellent hopes about the construction of the scattering amplitudes in ZEO. The construction would split into two pieces corresponding to the orbital degrees of freedom in “world of classical worlds” (WCW) and to spin degrees of freedom in WCW: that is spinors, which correspond to second quantized induced spinor fields at space-time surface (actually string world sheets- either at fundamental level or for effective action implied by strong form of holography (SH)).
By Kähler geometry of WCW the functional integral reduces to a sum over contributions from preferred extremals with the fermionic scattering amplitude multiplied by the ration Xi/X, where X=∑i Xi is the sum of the action exponentials for the maxima. The ratios of exponents are however number theoretically problematic.
Number theoretical universality is satisfied if one assigns to each maximum independent zero energy states: with this assumption ∑ Xi reduces to single Xi and the dependence on action exponentials becomes trivial! ZEO allow this. The dependence on coupling parameters of the action essential for the discretized coupling constant evolution is only via boundary conditions at the ends of the space-time surface at the boundaries of CD.
Quantum criticality of TGD demands that the sum over loops associated with the functional integral over WCW vanishes and strong form of holography (SH) suggests that the integral over 4-surfaces reduces to that over string world sheets and partonic 2-surfaces corresponding to preferred extremals for which the WCW coordinates parametrizing them belong to the extension of rationals defining the adele. Also the intersections of the real and various p-adic space-time surfaces belong to this extension.
Fermionic lines correspond to boundaries of string world sheets. Fermion scattering at partonic 2-surfaces at which 3 partonic orbits meet are analogs of 3-vertices in the sense of Feynman and fermions scatter classically. There is no local 4-vertex. This scattering is assumed to be described by simplest 4-fermion twistor diagram. These can be fused to form more complex diagrams. Fermionic lines runs along the partonic orbits defining the topological diagram.
The vanishing of topological loops would correspond to the closedness of the diagrams in what might be called BCFW homology. Boundary operation involves removal of BCFW bridge and entangled removal of fermion pair. The latter operation forces loops. There would be no BCFW bridges and entangled removal should give zero. Indeed, applied to the proposed four fermion vertex entangled removal forces it to correspond to forward scattering for which the proposed twistor amplitude vanishes.
To sum up, the twistorial approach leads to a proposal for an explicit construction of scattering amplitudes for the fundamental fermions. Bosons and fermions as elementary particles are bound states of fundamental fermions assignable to pairs of wormhole contacts carrying fundamental fermions at the throats. Clearly, this description is analogous to a quark level description of hadron. Yangian symmetry with multilocal generators is expected to crucial for the construction of the many-fermion states giving rise to elementary particles. The problems of the standard twistor approach find a nice solution in terms of M8-H duality, 8-D masslessness, and holomorphy of twistor amplitudes in λi and their indepence on &lambda tilde;i.
See the new chapter Some Questions Related to the Twistor Lift of TGD of “Towards M-matrix”.
For a summary of earlier postings see Latest progress in TGD.