A new view about color, color confinement, and twistors

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.

1. 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.

2. 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?
3. 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.
4. 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.

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.

1. 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.

2. One can consider also the situation in which one has localized the 8-momenta only to M⁴=M²₀× E². In this case, the transversal quaternionic light-like momenta in E²⊂ M²₀× E² are labelled by the points of CP₂ labelling at the same time the choices of M⁴. Thus octonionic SU(3) partial waves could be interpreted as partial waves in the space of choices for M²₀× E² defining tangent space at given point of quaternionic space-time surface. The same interpretation is also behind M⁸-H correspondence.
3. Octonionic SU(3) partial waves would correspond to wave functions in the space of transverse momenta. M²₀ component of momentum would be in general massive: only when it is light-like, the orbit of transversal momenta is trivial. Could QCD color assigned with M⁴× CP₂ spinor harmonics and octonionic color assigned with 8-D quaternionic momentum space as subset of M⁸ consisting of G₂ orbit of fixed quaternionic M⁴ be regarded as dual? 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. This would of course be a profound and completely unexpected connection!

   At the level of M⁸ electroweak quantum numbres could be described in terms of helicities assignable to E⁴ degrees of freedom and describable in terms of CP₂ twistor space T(CP₂). One would have unified description of all standard model quantum numbers in terms of generalized twistors.

4. Could a sharp localization to M²₀ be possible only at the level space-time surfaces – both in M⁸ and H. Indeed, neither leptonic nor quark-like induced spinors carry color as a spin like quantum number. Color emerges only at the level of H and M⁸ as color partial waves. Could the interpretation be that the counterpart of 8-momentum as tangent space-vector is in M² at space-time level. Perhaps also the integrable local decompositions M⁴= M²(x)× E²(x) suggested by the general solution ansätze for field equations are possible.
5. What about color singlet particles in general at the level of M⁸? Could it be possible to choose M²₀ always so that the complex momentum is light-like but that this localization forces the state to be color singlet? Measurement would induce color confinement but not masslessness since the real momentum need not be light-like for M²₀ localization – also classically the momenta are complex. Even if the system is color singlet, its subsystems 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.

6. 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.

What happens to the twistorialization at the level of M⁸ if one accepts that only M²₀ momentum is sharply defined?

1. 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.

2. 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 P³_R – 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?

3. 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.

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.

For details see the chapter How the hierarchy of Planck constants might relate to the almost vacuum degeneracy for twistor lift of TGD? of “Towards M-matrix” or the article Some questions related to the twistor lift of TGD.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.