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.
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)).
- There are several notions of twistor. Twistor space for M4 is T(M4) =M4× S2 (see this) having projections to both M4 and to the standard twistor space T1(M4) often identified as CP3. T(M4)=M4× S2 is necessary for the twistor lift of space-time dynamics. CP2 gives the factor T(CP2)= SU(3)/U(1)× U(1) to the classical twistor space T(H). The quantal twistor space T(M8)= T1(M4)× T(CP2) assignable to momenta. The possible way out is M8-H duality relating the momentum space M8 (isomorphic to the tangent space H) and H by mapping space-time associative and co-associative surfaces in M8 to the surfaces which correspond to the base spaces of in H: they construction would reduce to holomorphy in complete analogy with the original idea of Penrose in the case of massless fields.
- The standard twistor approach has problems. Twistor Fourier transform reduces to ordinary Fourier transform only in signature (2,2) for Minkowski space: in this case twistor space is real RP3 but can be complexified to CP3. Otherwise the transform requires residue integral to define the transform (in fact, p-adically multiple residue calculus could provide a nice manner to define integrals and could make sense even at space-time level making possible to define action).
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!
- A further problem of the standard twistor approach is that in M4 twistor approach does not work for massive particles. In TGD all particles are massless in 8-D sense. In M8 M4-mass squared corresponds to transversal momentum squared coming from E4⊂ M4× E4 (from CP2 in H). In particular, Dirac action cannot contain anyo mass term since it would break chiral invariance.
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.
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 λ tilde;i.
- At WCW level there is a perturbative functional integral over small deformations of the 3-surface to which space-time surface is associated. The strongest assumption is that this 3-surface corresponds to maximum for the real part of action and to a stationary phase for its imaginary part: minimal surface extremal of Kähler action would be in question. A more general but number theoretically problematic option is that an extremal for the sum of Kähler action and volume term is in question.
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.
- Second piece corresponds to the construction of twistor amplitude from fundamental 4-fermion amplitudes. The diagrams consists of networks of light-like orbits of partonic two surfaces, whose union with the 3-surfaces at the ends of CD is connected and defines a boundary condition for preferred extremals and at the same time the topological scattering diagram.
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.
- Number theoretic universality suggests that scattering amplitudes have interpretation as representations for computations. All space-time surfaces giving rise to the same computation wold be equivalent and tree diagrams corresponds to the simplest computation. If the action exponentials do not appear in the amplitudes as weights this could make sense but would require huge symmetry based on two moves. One could glide the 4-vertex at the end of internal fermion line along the fermion line so that one would eventually get the analog of self energy loop, which should allow snipping away. An argument is developed stating that this symmetry is possible if the preferred M20 for which 8-D momentum reduces to light-like M2-momentum having unique direction is same along entire fermion line, which can wander along the topological graph.
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.
See the new chapter Some Questions Related to the Twistor Lift of TGD of "Towards M-matrix".