The generalization of twistor approach from M4 to H=M4× CP2 involves the replacement of twistor space of M4 with that of H. M8-H duality allows also an alternative approach in which one constructs twistor space of octonionic M8. Note that M4,E4, S4, and CP2 are the unique 4-D spaces allowing twistor space with Kähler structure. This makes TGD essentially unique.
Ordinary twistor approach has two problems.
One can always find M2⊂ M8 in which the 8-momentum lies and is therefore light-like in 2-D sense. Furthermore, the light-like 8-momenta and thus 2-momenta are prediced already at classical level to be complex. M2 as subspace of momentum space M8 effectively extends to its complex version with signature (2,2)!
At classical space-time level the presence of preferred M2 reflects itself in the properties of massless extremals with M4= M2× E2 decomposition such that light-like momentum is in M2 and polarization in E2.
4-D conformal invariance is restricted to its 2-D variant in M2. Twistor space of M4 reduces to that of M2. This is SO(2,2)/SO(2,1)=RP3. This is 3-D RP3, the real variant of twistor space CP3. Complexification of light-like momenta replaces RP3 with CP3.
Light-like M8-momenta are in question but they are not arbitrary.
It should be noticed that quaternionicity forces number theoretical spontaneous compactification. It would be very clumsy to realize the condition that allowed 8-momenta are qiuaternionic. Instead going to M4× CP2, “spontaneously compactifying”, description everyting becomes easy.
At the level of momentum space M4 twistor codes for the momentum and helicity of particle. For CP2 it codes for the selection of M4⊂ M8 and for em charge as analog of helicity. Now one has actually wave function for the selections of CP2 point labelled by the color numbers of the particle.
Number theoretical vision inspires the idea that scattering ampitudes define representations for algebraic computations leading from initial set of algebraic objects to to final set of objects. If so, the amplitudes should not depend on how the computation is done and there should exist a minimal computation possibly represented by a tree diagram. There would be no summation over the equivalent diagrams: one can choose any-one of them and the best choice is the simplest one.
To develop this idea one must understand what scattering diagrams are. The scattering diagrams involve two kinds of lines.
The conjecture is that he 4-vertex is described by twistor amplitude fixed apart from over all scaling factor. Fermion lines are along parton orbits. Boson lines correspond to pairs of fermion and antifermion at the same parton orbit.
As a matter fact, the situation is more complex for elementary particles since they correspond to pairs of wormhole contacts connected by monopole magnetic tubes and wormhole contacts has two wormhole throats – partonic 2-surfaces.
For the idea about diagrams as representations of computations to make sense, there should exist moves which allow to glide the 4-fermion vertex and associated flux tubes along the topological line of scattering diagrams in the vicinity of the second end of the loop. Second move should allow to snip away the loop. Is this possible? The possibility to find M2 for which momentum is light-like is central in the argument claiming that this is indeed possible.
The basic problem is that the kinematics for 4-fermion vertices need not be consistent with the gliding of vertex past another one so that this move is not possible.
This kind of scattering would be more general than the scattering in integrable quantum field theories in M2: in this case the scattering would not affect the momenta but would induce phase shifts: particles would spend some time in the vertex before continuing. What is crucial for having non-trivial scatterings, is that in the general frame M2⊂ M4 ⊂ M8 the momenta would be massive and also different.
Note however that tree diagrams for which lines can have different M2s can give rise to non-trivial scattering. One can take tree diagram and assign to the internal lines networks with same M2s as the internal line has. It is quite possible that for general graphs allowing different M2s in internal lines and loops, the reduction to tree graph is not possible.
At least this idea could define precisely what the equivalence of diagrams, if vertices in which M2:s can be different are allowed. One can of course argue, that there is not deep reason for not allowing more general loopy graphs in which the incoming lines can have arbitrary M2:s.
One implication is that the BCFW recursion formula allowing to generate loop diagrams from those with lower number of loops must be trivial in TGD – this of course only if one accepts that BCFW formula makes sense in TGD. This requires that the entangled removal appearing as second term in the right hand side of BCFW formula and adding loop gives zero. One can develop and argument for why this must be the case in TGD framework. Also the second term corresponding to removal of BCFW bridge should give zero so that allowed diagrams cannot have BCFW bridges.
In TGD Universe allowed diagrams would represent closed objects in what one might call BCFW homology. The operation appearing at the right hand side of BCFW recursion formula is indeed boundary operation, whose square by definition gives zero.
For background see the article Some questions related to the twistor lift of TGD.
For a summary of earlier postings see Latest progress in TGD.