## Scattering amplitudes in positive Grassmannian: TGD perspective
The quite recent but not yet published proposal of Hamed and his former student Trnka has gained a lot of attention. There is a popular article in Quanta Magazine about their work. There is a video talk by Jaroslav Trnka about positive Grassmannian (the topic is actually touched at the end of the talk but it gives an excellent view about the situation) and a video talk by Nima Arkani-Hamed. One can also find the slides of Trnka . For beginners like me the article of Henriette Envang and Yu-tin Huang serves as an enjoyable concretization of the general ideas.
The basic claim is that the Grassmannian amplitudes reduce to volumes of positive Grassmannians determined by external particle data and realized as polytopes in Grassmannians such that their facets correspond to logarithmic singularities of a volume form in one-one correspondence with the singularities of the scattering amplitude. Furthermore, t the factorization of the scattering amplitude at singularities corresponds to the singularities at facets. Scattering amplitudes would characterize therefore purely geometric objects. The crucial Yangian symmetry would correspond to diffeomorphisms preserving the positivity property. Unitarity and locality would be implied by the volume interpretation. Nima concludes that unitarity and locality, gauge symmetries, space-time, and even quantum mechanics emerge. One can however quite well argue that its the positive Grassmannian property and volume interpretation which emerge. In particular, the existence of twistor structure possible in Minkowskian signature only in M The talks inspired the consideration of the possible Grassmannian formulation in TGD framework in more detail and to ask whether positivity might have some deeper meaning in TGD framework.
The vision about what BCFW approach to generalized Feynman diagframs could mean has been fluctuating wildly during last months. The Grassmannian formalism for scattering amplitudes is expected to generalize for generalized Feynman diagrams: the basic modification is due to the possible presence of CP - Fermions of internal lines are massless in real sense and have unphysical helicity. Wormhole contacts carrying fermion and antifermion at their opposite throats correspond to basic building bricks of bosons. For fermions second throat is empty. The residue integral over the momenta of internal lines replaces fermionic propagator with its inverse and replaces 4-D momentum integration with integration over light-cone using the standard Lorentz invariant integration measure.
- 4-fermion vertex defines the fundamental vertex and contains constant proportional to length squared. This is definitely a problem. If all 4-fermion vertices contain one bosonic wormhole contact, one can replace regard the verties effectively as BFF or BBB vertices. The four-fermion coupling constant L
^{2}having dimensions length squared can be replaced with 1/p^{2}for the bosonic line: this is the new ingredient allowing to overcome the difficulties assignable to four-fermion vertex. - For QFT type BFCW BFF and BBB vertices would be an outcome of bosonic emergence (bosons as wormhole contacts) and 4-fermion vertex is proportional to factor with dimensions of inverse mass squared and naturally identifiable as proportional to the factor 1/p
^{2}assignable to each boson line. This predicts a correct form for the bosonic propagators for which mass squared is in general non-vanishing unlike for fermion lines. The usual BFCW construction would emerge naturally in this picture. There is however a problem: the emergent bosonic propagator diverges or vanishes depending on whether one assumes SUSY at the level of single wormhole throat or not. By the special properties of the analog of N=4 SUSY SUSY generated by right handed neutrino the SUSY cannot be applied to single wormhole throat but only to a pair of wormhole throats. - This as also the fact that physical particles are necessarily pairs of wormhole contacts connected by fermionic strings forces stringy variant of BFCW avoiding the problems caused by non-planar diagrams. Now boson line BFCW cuts are replaced with stringy cuts and loops with stringy loops. By generalizing the earlier QFT twistor Grassmannian rules one ends up with their stringy variants in which super Virasoro generators G, G
^{†}and L bringing in CP_{2}scale appear in propagator lines: most importantly, the fact that G and G^{†}carry fermion number in TGD framework ceases to be a problem since a string world sheet carrying fermion number has 1/G and 1/G^{†}at its ends. The general rules is simple: each line emerging from 4-fermion vertex carries 1/G and 1/G^{†}as vertex factor. Twistorialization applies because all fermion lines are light-like. - A more detailed analysis of the properties of right-handed neutrino demonstrates that modified gamma matrices in the modified Dirac action mix right and left handed neutrinos but that this happens markedly only in very short length scales comparable to CP
_{2}scale. This makes neutrino massive and also strongly suggests that SUSY generated by right-handed neutrino emerges as a symmetry at very short length scales so that spartners would be very massive and effectively absent at low energies. Accepting CP_{2}scale as cutoff in order to avoid divergent gauge boson propagators QFT type BFCW makes sense. The outcome is consistent with conservative expectations about how QFT emerges from string model type description. - The generalization to gravitational sector is not a problem in sub-manifold gravity since M
^{4}- the only space-time geometry with Minkowski signature allowing twistor structure - appears as a Cartesian factor of the imbedding space. A further finding is that CP_{2}and S^{4}are the only Euclidian 4-manifolds allowing twistor space with Kähler structure. Since S^{4}does not allow Kähler structure, CP_{2}is completely unique just like M^{4}. Stringy picture indeed treats gravitons and other elementary particles completely democratically. - The analog of twistorial construction in CP
_{2}degrees of freedom based on the notion of flag manifold and geometric quantization is proposed. Light-likeness in*real*sense poses a powerful constraint analogous to constraints posed by moves in the case of SYMs and if volume of a convex polytope dictated by the external momenta and helicities provides a representation of the scattering amplitude, the tree diagrams would give directly the full volume.
As both Arkani-Hamed and Trnka state "everything is positive". This is highly interesting since p-adicization involves canonical identification, which is well defined only for non-negative reals without further assumptions. This raises the conjecture that positivity is necessary in order to achieve number theoretical universality.
For details see the chapter Some fresh ideas about twistorialization of TGD or the article with the same title. |