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.
- 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.
- 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.
- 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.
- 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.
- 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.
To develop this idea one must understand what scattering diagrams are. The scattering diagrams involve two kinds of lines.
- 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.
- 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.
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.
- 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.
- 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.
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.