- A proposal made already earlier is that scattering diagrams as analogs of twistor diagrams are constructible as tree diagrams for CDs connected by free particle lines. Loop contributions are not even well-defined in zero energy ontology (ZEO) and are in conflict with number theoretic vision. The coupling constant evolution would be discrete and associated with the scale of CDs (p-adic coupling constant evolution) and with the hierarchy of extensions of rationals defining the hierarchy of adelic physics.
- The reduction of the scattering amplitudes to tree diagrams is in conflict with unitarity in 4-D situation. The imaginary part of the scattering amplitude would have discontinuity proportional to the scattering rate only for many-particle states with light-like total momenta. Scattering rates would vanish identically for the physical momenta for many-particle states.
In TGD framework the states would be however massless in 8-D sense. Massless pole corresponds now to a continuum for M4 mass squared and one would obtain the unitary cuts from a pole at P2=0! Scattering rates would be non-vanishing only for many-particle states having light-like 8-momentum, which would pose a powerful condition on the construction of many-particle states. This strong form of conformal symmetry has highly non-trivial implications concerning color confinement.
- The key idea is number theoretical discretization in terms of "cognitive representations" as space-time time points with M8-coordinates in an extension of rationals and therefore shared by both real and various p-adic sectors of the adele. Discretization realizes measurement resolution, which becomes an inherent aspect of physics rather than something forced by observed as outsider. This fixes the space-time surface completely as a zero locus of real or imaginary part of octonionic polynomial.
This must imply the reduction of "world of classical worlds" (WCW) corresponding to a fixed number of points in the extension of rationals to a finite-dimensional discretized space with maximal symmetries and Kähler structure.
The simplest identification for the reduced WCW would be as complex Grassmannian - a more general identification would be as a flag manifold. More complex options can of course be considered. The Yangian symmetries of the twistor Grassmann approach known to act as diffeomorphisms respecting the positivity of Grassmannian and emerging also in its TGD variant would have an interpretation as general coordinate invariance for the reduced WCW. This would give a completely unexpected connection with supersymmetric gauge theories and TGD.
- M8 picture implies the analog of SUSY realized in terms of polynomials of super-octonions whereas H picture suggests that supersymmetry is broken in the sense that many-fermion states as analogs of components of super-field at partonic 2-surfaces are not local. This requires breaking of SUSY. At M8 level the breaking could be due to the reduction of Galois group to its subgroup G/H, where H is normal subgroup leaving the point of cognitive representation defining space-time surface invariant. As a consequence, local many-fermion composite in M8 would be mapped to a non-local one in H by M8-H correspondence.
For a summary of earlier postings see Latest progress in TGD.
An answer to FB comment. There are two diagrammatics: Feynman diagrams and twistor diagrams.
ReplyDeletea) Virtual state is mathematically an auxiliary notion related to Feynman diagrammatics coding for the perturbation theory. Virtual particles in Feynman diagrammatics are off-mass-shell.
b) In standard twistor diagrammatics one of course obtains counterparts of loop diagrams. Loops are replaced with diagrams in which particles can have complex four-momentum which however light-like: on-mass-shell in this sense.
BCFW recursion formula provides a powerful manner to calculate the loop corrections recursively. Grassmannian approach gives additional insights to the calculation and possible interpretation. Grassmannian spaces Gr(k,n): k planes in n-D space are in central role.
The problem is that the twistor counterparts of non-planar diagrams are not yet understood. Second problem is that physical particles are not massless in 4-D sense.
In TGD framework twistor approach approach generalizes.
a) There are no loop diagrams: ZEO does not allow to define them and they would spoil the number theoretical vision which allows only scattering amplitudes which are rational functions of data about external particles. Coupling constant evolution is now discrete.
b) This is nice but in conflict with unitarity if momenta were 4-D. But momenta are 8-D and the problem disappears! There is single pole at zero mass but in 8-D sense and also many-particle states have vanishing mass in 8-D sense: this gives all the cuts in 4-D mass squared for all many-particle state. For states not satisfying this condition scattering rates vanish, they do not exist in observable sense! This this was certainly the most significant new discovery in the recent article. BCFW recursion formula for the calculation of amplitudes trivializes and one obtains only tree diagrams. No recursion is needed. Lowest order answer is exact. Finite number of steps are needed for the calculation and these steps are well-understood in 4-D case.
c) To calculate the amplitudes one must be able to explicitly formulate the twistorialization in 8-D case for amplitudes. I have made explicit proposals but have no clear understanding yet. In fact, BCFW makes sense also in higher dimensions unlike Grassmannian approach and it might be that the one can calculate the tree diagrams in TGD framework using 8-D BCFW at M^8 level and then transform the results to M^4xCP_2.
What I said above does yet contain anything about Grassmannians.
a) The mysterious Grassmannians Gr(k,n) might have beautiful intepretation in TGD: they could correspond at M^8 level to reduced WCWs which is natural notion at M^4xCP_2 level obtained by fixing the numbers of external particles in diagrams and performing number theoretical discretization for the space-time surface in terms of cognitive representation consisting of finite number of space-time points.
b) Grassmannian residue integration is somewhat frustrating procedure: it gives as outcome the amplitude as sum of contributions from a finite number of residues. Why this work when outcome is given by something at finite number of points of Grassmannian?!
In M^8 picture in TGD cognitive representations at space-time level as finite sets of points of space-time determining it completely as zero locus of certain polynomials would actualy give WCW coordinates of the space-time surface in finite resolution which is a characteristic of classical and quantum states. The residue integrals in twistor diagrams would be the manner to associate space-time surface to a given twistor diagram by fixing the cognitive representation. It would be highly unique: perhaps modulo the action of Galois group of extension of rationals. This would realize quantum classical correspondence by assigning to quantum state a space-time surface and at the same time the scattering amplitude.