To my humble opinion twistor approach to the scattering amplitudes is plagued by some mathematical problems. Whether this is only my personal problem is not clear.
In number theoretic vision something rather unexpected emerges and I will propose that this unexpected might allow to solve the above problems and even more, to understand color and even color confinement number theoretically. First of all, a new view about color degrees of freedom emerges at the level of M8.
- As Witten shows, the twistor transform is problematic in signature (1,3) for Minkowski space since the the bi-spinor μ playing the role of momentum is complex. Instead of defining the twistor transform as ordinary Fourier integral, one must define it as a residue integral. In signature (2,2) for space-time the problem disappears since the spinors μ can be taken to be real.
- The twistor Grassmannian approach works also nicely for (2,2) signature, and one ends up with the notion of positive Grassmannians, which are real Grassmannian manifolds. Could it be that something is wrong with the ordinary view about twistorialization rather than only my understanding of it?
- For M4 the twistor space should be non-compact SU(2,2)/SU(2,1)× U(1) rather than CP3= SU(4)/SU(3)× U(1), which is taken to be. I do not know whether this is only about short-hand notation or a signal about a deeper problem.
- Twistorilizations does not force SUSY but strongly suggests it. The super-space formalism allows to treat all helicities at the same time and this is very elegant. This however forces Majorana spinors in M4 and breaks fermion number conservation in D=4. LHC does not support N=1 SUSY. Could the interpretation of SUSY be somehow wrong? TGD seems to allow broken SUSY but with separate conservation of baryon and lepton numbers.
M20 momentum deserves a separate discussion.
- One can always find a decomposition M8=M20× E6 so that the complex light-like quaternionic 8-momentum restricts to M20. The preferred octonionic imaginary unit represent the direction of imaginary part of quaternionic 8-momentum. The action of G2 to this momentum is trivial. Number theoretic color disappears with this choice. For instance, this could take place for hadron but not for partons which have transversal momenta.
- One can consider also the situation in which one has localized the 8-momenta only to M4 =M20× E2. The distribution for the choices of E2 ⊂ M20× E2=M4 is a wave function in CP2. Octonionic SU(3) partial waves in the space CP2 for the choices for M20× E2 would correspond ot color partial waves in H. The same interpretation is also behind M8-H correspondence.
- The transversal quaternionic light-like momenta in E2⊂ M20× E2 give rise to a wave function in transversal momenta. Intriguingly, the partons in the quark model of hadrons have only precisely defined longitudinal momenta and only the size scale of transversal momenta can be specified.
The introduction of twistor sphere of T(CP2) allows to describe electroweak charges and brings in CP2 helicity identifiable as em charge giving to the mass squared a contribution proportional to Qem2 so that one could understand electromagnetic mass splitting geometrically.
The physically motivated assumption is that string world sheets at which the data determining the modes of induced spinor fields carry vanishing W fields and also vanishing generalized Kähler form J(M4) +J(CP2). Em charge is the only remaining electroweak degree of freedom. The identification as the helicity assignable to T(CP2) twistor sphere is natural.
- In general case the M2 component of momentum would be massive and mass would be equal to the mass assignable to the E6 degrees of freedom. One can however always find M20× E6 decomposition in which M2 momentum is light-like. The naive expectation is that the twistorialization in terms of M2 works only if M2 momentum is light-like, possibly in complex sense. This however allows only forward scattering: this is true for complex M2 momenta and even in M4 case.
The twistorial 4-fermion scattering amplitude is however holomorphic in the helicity spinors λi and has no dependence on λtilde;i. Therefore carries no information about M2 mass! Could M2 momenta be allowed to be massive? If so, twistorialization might make sense for massive fermions!
What happens to the twistorialization at the level of M8 if one accepts that only M20 momentum is sharply defined?
- A sharp localization of 8-momentum to M20 means vanishing E2 momentum so that the action of U(2) would becomes trivial: electroweak degree of freedom would simply disappear, which is not the same thing as having vanishing em charge (wave function in T(CP2) twistorial sphere S2 would be constant). Neither M20 localization nor localization to single M4 (localization in CP2) looks plausible physically - consider only the size scale of CP2. For the generic CP2 spinors this is impossible but covariantly constant right-handed neutrino spinor mode has no electro-weak quantum numbers: this would most naturally mean constant wave function in CP2 twistorial sphere.
For the preferred extremals of twistor lift of TGD either M4 or CP2 twistor sphere can effectively collapse to a point. This would mean disappearence of the degrees of freedom associated with M4 helicity or electroweak quantum numbers.
- The localization to M4⊃ M20 is possible for the tangent space of quaternionic space-time surface in M8. This could correlate with the fact that neither leptonic nor quark-like induced spinors carry color as a spin like quantum number. Color would emerge only at the level of H and M8 as color partial waves in WCW and would require de-localization in the CP2 cm coordinate for partonic 2-surface. Note that also the integrable local decompositions M4= M2(x)× E2(x) suggested by the general solution ansätze for field equations are possible.
- Could it be possible to perform a measurement localization the state precisely in fixed M20 always so that the complex momentum is light-like but color degrees of freedom disappear? This does not mean that the state corresponds to color singlet wave function! Can one say that the measurement eliminating color degrees of freedom corresponds to color confinement. Note that the subsystems of the system need not be color singlets since their momenta need not be complex massless momenta in M20. Classically this makes sense in many-sheeted space-time. Colored states would be always partons in color singlet state.
- At the level of H also leptons carry color partial waves neutralized by Kac-Moody generators, and I have proposed that the pion like bound states of color octet excitations of leptons explain so called lepto-hadrons. Only right-handed covariantly constant neutrino is an exception as the only color singlet fermionic state carrying vanishing 4-momentum and living in all possible M20:s, and might have a special role as a generator of supersymmetry acting on states in all quaternionic subs-spaces M4.
- Actually, already p-adic mass calculations performed for more than two decades ago forced to seriously consider the possibility that particle momenta correspond to their projections o M20⊂ M4. This choice does not break Poincare invariance if one introduces moduli space for the choices of M20⊂ M4 and the selection of M20 could define quantization axis of energy and spin. If the tips of CD are fixed, they define a preferred time direction assignable to preferred octonionic real unit and the moduli space is just S2. The analog of twistor space at space-time level could be understood as T(M4)=M4× S2 and this one must assume since otherwise the induction of metric does not make sense.
To sum up, these observation suggest a profound re-evalution of the beliefs related to color degrees of freedom, to color confinement, and to what twistors really are.
- What happens to the conformal group SO(4,2) and its covering SU(2,2) when M4 is replaced with M20⊂ M8? Translations and special conformational transformation span both 2 dimensions, boosts and scalings define 1-D groups SO(1,1) and R respectively. Clearly, the group is 6-D group SO(2,2) as one might have guessed. Is this the conformal group acting at the level of M8 so that conformal symmetry would be broken? One can of course ask whether the 2-D conformal symmetry extends to conformal symmetries characterized by hyper-complex Virasoro algebra.
- Sigma matrices are by 2-dimensionality real (σ0 and σ3 - essentially representations of real and imaginary octonionic units) so that spinors can be chosen to be real. Reality is also crucial in signature (2,2), where standard twistor approach works nicely and leads to 3-D real twistor space.
Now the twistor space is replaced with the real variant of SU(2,2)/SU(2,1)× U(1) equal to SO(2,2)/SO(2,1), which is 3-D projective space RP3 - the real variant of twistor space CP3, which leads to the notion of positive Grassmannian: whether the complex Grassmannian really allows the analog of positivity is not clear to me. For complex momenta predicted by TGD one can consider the complexification of this space to CP3 rather than SU(2,2)/SU(2,1)× U(1). For some reason the possible problems associated with the signature of SU(2,2)/SU(2,1)× U(1) are not discussed in literature and people talk always about CP3. Is there a real problem or is this indeed something totally trivial?
- SUSY is strongly suggested by the twistorial approach. The problem is that this requires Majorana spinors leading to a loss of fermion number conservation. If one has D=2 only effectively, the situation changes. Since spinors in M2 can be chosen to be real, one can have SUSY in this sense without loss of fermion number conservation! As proposed earlier, covariantly constant right-handed neutrino modes could generate the SUSY but it could be also possible to have SUSY generated by all fermionic helicity states. This SUSY would be however broken.
- The selection of M20 could correspond at space-time level to a localization of spinor modes to string world sheets. Could the condition that the modes of induced spinors at string world sheets are expressible using real spinor basis imply the localization? Whether this localization takes place at fundamental level or only for effective action being due to SH, is a question to be settled. The latter options looks more plausible.
For details see the new chapter Some Questions Related to the Twistor Lift of TGD or the article Some questions related to the twistor lift of TGD.