Twistors and TGD: a summary

The encounter between twistors and TGD turned out to be extremely fruitful. I spent some time with the idea about replacing loop momenta in Feynman diagrams with light-like ones in order to achieve twistorialization (or rather spinorialization) of Feynman graphs but - as it sometimes happens - a silly idea stimulated the right question, and after 31 years of hard work I have a proposal for precise rules of Feynman diagrammatics producing UV finite and unitary S-matrix. I glue below the introduction to the new chapter Twistors, N=4 Super-Conformal Symmetry, and Quantum TGD of "Towards M-matrix" in the hope that it give an overall view about the situation.

Twistors - a notion discovered by Penrose - have provided a fresh approach to the construction of perturbative scattering amplitudes in Yang-Mills theories and in N=4 supersymmetric Yang-Mills theory. This approach was pioneered by Witten. The latest step in the progress was the proposal by Nima Arkani-Hamed and collaborators that super Yang Mills and super gravity amplitudes might be formulated in 8-D twistor space possessing real metric signature (4,4). The questions considered below are following.

1. Could twistor space could provide a natural realization of N=4 super-conformal theory requiring critical dimension D=8 and signature metric (4,4)? Could string like objects in TGD sense be understood as strings in twistor space? More concretely, could one in some sense lift quantum TGD from M⁴×CP₂ to 8-D twistor space T so that one would have three equivalent descriptions of quantum TGD.

2. Could one construct the preferred extremals of Kähler action in terms of twistors -may be by mimicking the construction of hyper-quaternionic resp. co-hyper-quaternionic surfaces in M⁸ as surfaces having hyper-quaternionic tangent space resp. normal space at each point with the additional property that one can assign to each point x a plane M²(x) subset M⁴ as sub-space or as sub-space defined by light-like tangent vector in M⁴. Could one mimic this construction by assigning to each point of X⁴ regarded as a 4-surface in T a 4-D plane of twistor space satisfying some conditions making possible the interpretation as a tangent plane and guaranteing the existence of a map of X⁴ to a surface in M⁴×CP₂. Could twistor formalism help to resolve the integrability conditions involved?

3. Could one modify the notion of Feynman diagram by allowing only massless loop momenta so that twistor formalism could be used in elegant manner to calculate loop integrals and whether the resulting amplitudes are finite in TGD framework where only fermions are elementary particles? Could one modify Feynman diagrams to twistor diagrams by replacing momentum eigenstates with light ray momentum eigenstates completely localized in transversal degrees of freedom?

The arguments of this chapter suggest some these questions might have affirmative answers.

Twistors at space-time level

Consider first the twistorialization at the classical space-time level.

1. One can assign twistors to only 4-D Minkowski space (also to other than Lorentzian signature). One of the challenges of the twistor program is how to define twistors in the case of a general curved space-time. In TGD framework the structure of the imbedding space allows to circumvent this problem.

2. The lifting of classical TGD to twistor space level is a natural idea. Consider space-time surfaces representable as graphs of maps M⁴→ CP₂. At classical level the Hamilton-Jacobi structure required by the number theoretic compactification means dual slicings of the M⁴ projection of the space-time surface X⁴ by stringy word sheets and partonic two-surfaces. Stringy slicing allows to assign to each point of the projection of X⁴ two light-like tangent vectors U and V parallel to light-like Hamilton-Jacobi coordinate curves. These vectors define components tildeμ and λ of a projective twistor, and twistor equation assigns to this pair a point m of M⁴. The conjecture is that for preferred extremals of Kähler action this point corresponds to the M⁴ projection of the point in the natural M⁴ coordinates associated with the upper or lower tip of causal diamond CD. If this conjecture is correct one can lift the M⁴ projection of the space-time surface in CD×CP₂ subset M⁴×CP₂ to a surface in PT×CP₂, where CP₃ is projective twistor space PT=CP₃. Also induced spinor fields and induced gauge fields can be lifted to twistor space.

3. If one can fix the scales of the tangent vectors U and V and fix the phase of spinor λ one can consider also the lifting to 8-D twistor space T rather than 6-D projective twistor space PT. Kind of symmetry breaking would be in question. The proposal for how to achieve this relies on the notion of finite measurement resolution. The scale of V at partonic 2-surface X² subset dCD×X³_l would naturally correlate with the energy of the massless particle assignable to the light-like curve beginning from that point and thus fix the scale of V coordinate. Symplectic triangulation in turn allows to assign a phase factor to each strand of the number theoretic braid as the Kähler magnetic flux associated with the triangle having the point at its center. This allows to lift the stringy world sheets associated with number theoretic braids to their twistor variants but not the entire space-time surface. String model in twistor space is obtained in accordance with the fact that N=4 super-conformal invariance is realized as a string model in a target space with (4,4) signature of metric. Note however that CP₂ defines additional degrees of freedom for the target space so that 12-D space is actually in question.

4. One can consider also a more general problem of identifying the counterparts for the preferred extremals of Kähler action with arbitrary dimensions of M⁴ and CP₂ projections in 10-D space PT×CP2. The key idea is the reduction of field equations to holomorphy as in Penrose's twistor representation of solutions of positive and negative frequency parts of free fields in M⁴. A very helpful observation is that CP₂ as a sub-manifold of PT corresponds to the 2-D space of null rays of the complexified Minkowski space M⁴_c. For the 5-D space N subset PT of null twistors this 2-D space contains 1-dimensional light ray in M⁴ so that N parametrizes the light-rays of M⁴. The idea is to consider holomorphic surfaces in PT_±×CP₂ (± correlates with positive and negative energy parts of zero energy state) having dimensions D=6,8, 10; restrict them to N×CP₂, select a sub-manifold of light-rays from N, and select from each light-ray subset of points which can be discrete or portion of the light-ray in order to get a 4-D space-time surface. If integrability conditions for the resulting distribution of light-like vectors U and V can be satisfied (in other words they are gradients), a good candidate for a preferred extremal of Kähler action is obtained. Note that this construction raises light-rays to a role of fundamental geometric object.

Twistors and Feynman diagrams

The recent successes of twistor concept in the understanding of 4-D gauge theories and N=4 SYM motivate the question of how twistorialization could help to understand construction of M-matrix in terms of Feynman diagrammatics or its generalization.

1. One of the basic problems of twistor program is how to treat massive particles. Massive four-momentum can be described in terms of two twistors but their choice is uniquely only modulo SO(3) rotation. This is ugly and one can consider several cures to the situation.
   1. Number theoretic compactification and hierarchy of Planck constants leading to a generalization of the notion of imbedding space assign to each sector of configuration space defined by a particular CD a unique plane M² subset M⁴ defining quantization axes. The line connecting the tips of the CD selects also unique rest frame (time axis). The representation of a light-like four-momentum as a sum of four-momentum in this plane and second light-like momentum is unique and same is true for the spinors λ apart from the phase factors (the spinor associated with M² corresponds to spin up or spin down eigen state).

   2. The tangent vectors of braid strands define light-like vectors in H and their M⁴ projection is time-like vector allowing a representation as a combination of U and V. Could also massive momenta be represented as unique combinations of U and V?

   3. One can consider also the possibility to represent massive particles as bound states of massless particles.

   It will be found that one can lift ordinary Feynman diagrams to spinor diagrams and integrations over loop momenta correspond to integrations over the spinors characterizing the momentum.

2. One assign to ordinary momentum eigen states spinor λ but it is not clear how to identify the spinor tildeμ needed for a twistor.
   1. Could one assign tildeμ to spin polarization or perhaps to the spinor defined by the light-like M² part of the massive momentum? Or could λ and tildeμ correspond to the vectors proportional to V and U needed to represent massive momentum?

   2. Or is something more profound needed? The notion of light-ray is central for the proposed construction of preferred extremals. Should momentum eigen states be replaced with light ray momentum eigen states with a complete localization in degrees of freedom transversal to light-like momentum? This concept is favored both by the notion of number theoretic braid and by the massless extremals (MEs) representing "topological light rays" as analogs of laser beams and serving as space-time correlates for photons represented as wormhole contacts connecting two parallel MEs. The transversal position of the light ray would bring in tildeμ. This would require a modification of the perturbation theory and the introduction of the ray analog of Feynman propagator. This generalization would be M⁴ counterpart for the highly successful twistor diagrammatics relying on twistor Fourier transform but making sense only for the (2,2) signature of Minkowski space.

3. In perturbation theory one can also consider the crazy idea of restricting the loop momenta to light-like momenta so that the auxiliary M² twistors would not be needed at all. This idea failed but led to a first precise proposal for how Feynman diagrammatics producing unitarity and UV finite S-matrix could emerge from TGD, where only fermions are elementary particles and all coupling constants are in principle predictions of the theory. Emergence would mean that the fundamental action is just the Dirac action with gauge boson couplings and containing no bosonic kinetic term, that the perturbative functional integral over the fermion fields in the construction of the effective action induces bosonic kinetic term radiatively, and that a further perturbative functional integral over the gauge boson fields gives an effective action in which all bosonic n-point functions have emerged from the fermionic dynamics. Physically this would mean that bosons interact only when the wormhole contact representing boson and carrying fermion and antifermion quantum numbers at the opposite light-like wormhole throats decays to a pair of fermion and anti-fermion represented by CP₂ type extremals with single wormhole throat only. Even fermionic propagators would emerge radiatively from the modified Dirac operator in more fundamental description . What is remarkable is that p-adic length scale hypothesis and the notion of finite measurement resolution lead to a precise proposal how UV divergences are tamed in a description taking into account the finite measurement resolution.

To sum up, perhaps the most important outcome of the interaction of twistor approach with TGD is a proposal for precise Feynman rules allowing to construct unitary and UV finite S-matrix. This realizes a 31 year old dream to a surprisingly high degree. Everything would emerge radiatively from the modified Dirac operator and boson-fermion vertices dictated by the charge matrix of the boson coding boson as a fermion-antifermion bilinear.

For a summary of the recent situation concerning TGD and twistors the reader can consult the new chapter Twistors, N=4 Super-Conformal Symmetry, and Quantum TGD of "Towards M-matrix".