Key ideas related to the twistor lift of TGD

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.

1. It applies only if the particles are massless. In TGD particles are massless in 8-D sense but the projection of 8-momentum to given M⁴ is in general massive in 4-D sense. This solves the problem. Note that the 4-D M⁴ momenta can be light-like for a suitable choice of M⁴⊂ H. There exist even a choice of M² for which this is the case. For given M² the choices of quaternionic M⁴ are parametrized by CP₂.
   
   
2. The twistor approach has second problem: it works nicely in signature (2,2) rather (1,3) for Minkowski space. For instance, twistor Fourier transform cannot be defined as an ordinary integral. The very nice results by Nima Arkani-Hamed et al about positive Grassmannian follow only in the signature (2,2).
   

   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.

1. They must lie in some quaternionic plane containing fixed M², which corresponds to the plane spanned by real octonion unit and some imaginary unit. . This condition is analogous to the condition that the space-time surfaces as preferred extremals in M⁸ have quaternionic tangent planes.
   
   
2. In particular, the wave functions can be expressed as products of plane waves in M², wave functions in the plane of transverse momenta in E²⊂ M⁴, where M⁴ is quaternionic plane containing M² and wave function in the space for the choices of M⁴, which is CP₂. One obtains exactly the same result in M⁴× CP₂ if delocalization in transversal E² momenta taking place of quarks inside hadrons takes place.
   Transversal wave function can also concentrate on single momentum value.
   

   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.
   
   

3. What is amusing that the geometric twistor space M⁴× S² of M⁴ having bundle projections to M⁴ and ordinary twistor spaces is nothing but the space of choices of causal diamonds with preferred M² and fixed rest frame (time axis connecting the tips). M⁴ point fixes the tip of causal diamond (CD) and S² the spatial direction fixing M² plane. In case of CP₂ the point of twistor space fixes point of CP₂ as analog for tip of CD: the complex CP₂ coordinates have origin at this point. The point of twistor sphere of SU(3)/U(1)× U(1) codes for the selection of quantization axis for hypercharge Y and isospin I₃. The corresponding subgroup U(1)× U(1) affects only the phases of the preferred complex coordinates transforming linearly under SU(2)× U(1).
   

   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.

1. There are topological "lines" corresponding to light-like orbits of partonic 2-surfaces playing the role of lines of Feynman diagrams. The topological diagram formed by these lines gives boundary conditions for 4-surface: at these light-like partonic orbits Euclidian space-time region changes to Minkowskian one. Vertices correspond to 2-surfaces at which these 3-D lines meet just like line in the case of Feynman diagrams.
   
   
2. There are also fermion lines assignable to fundamental fermions serving as building bricks of elementary particles. They correspond to the boundaries of string world sheets at the orbits of partonic 2-surfaces. Fundamental fermion-fermion scattering takes place via classical interactions at partonic 2-surfaces: there is no 4-vertex in the usual sense (this would lead to non-renormalizable theory).
   

   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.

1. Clearly, one must assume something. If all momenta along at vertices along fermion line are in same M² then they parallel as light-like M²-momenta. Kinematical conditions allow the gliding of two vertices of this kind past each other as is easy to show. The scattering would mean only redistribution of parallel light-like momenta in this particular M².
   

   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.
   
   

2. The condition would be that all four-fermion vertices along given fermion line correspond to the same preferred M². M²:s can differ only for fermionic sub-diagrams which do not have common vertices.
   

   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.

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