The construction of S-matrix has been one of the eternity projects of TGD. There are many proposals such as the construction based on the quaternionic generalization of twistor Grassmannian approach for cognitive representations involving huge simplification due to the vanishing of loop diagrams but also this approach is indirect. SUSY in TGD sense finally suggests a quite concrete fundamental approach.
See the chapter TGD view about McKay Correspondence, ADE Hierarchy, Inclusions of Hyperfinite Factors, M8-H Duality, SUSY, and Twistors or the article SUSY in TGD Universe.
- The construction would be based on the explicit solution of the super-symmetrized field equations. In principle everything reduces formally to classical partial differential equations for super-space-time surface and super-spinors. One solves preferred extremal as its super-variants which means solving the space-time evolution of multi-spinors defining super-coordinates and in this background one solves super-Dirac equation. This is highly non-trivial but in principle a well-defined procedure. If one gives initial values of various multi-spinor mods at the first light-like boundary of causal diamond (CD), one can deduce super-spinor field at opposite boundary of CD and express it as a superposition of its basic modes with well-defined quark number and other quantum numbers. This gives S-matrix.
- Situation simplifies dramatically for discrete cognitive representation replacing space-time surface with the set of points having imbedding space coordinates in extension of rationals defining the adele. Since finite set of points defining the preferred time scales t=rn as roots of a real polynomial determines the octonionic polynomia, M8-H duality raises the hope that the discretization provided by cognitive representation is exact and improvement in UV/IR resolution means addition of new space-time sheets with smaller/bigger size.
- Partonic 2-surfaces define topological vertices. They are identified as intersections of incoming particle like 4-surfaces as roots of octonionic polynomials with 6-sphere defining analogs of branes in M8 as universal roots of octonionic polynomials and having M4 time t=rn hyperplanes of M4 as their intersections.
Multi-quark-antiquark vertices at partonic 2-surfaces are points of cognitive representation having H-coordinates in an extension of rationals (or at least their pre-images in M8 have this property). Lines defining local multi-quark states fuse and split again into new states in quark number conserving manner. Vertices are super-symmetric in TGD sense and determined as vacuum expectations of the bosonic action and super-Dirac action and analogous to those defined by θ integration in SUSY.
- The counterparts of radiative corrections of QFTs are Wick contraction terms for the fermionic oscillator operators. M8-H duality requires that their contribution from partial multi-derivatives of order higher than the order n of the octonionic polynomial are vanishing. This leads to the conditions having interpretation as conservation of Noether currents of symmetries. As n increases, the number of Wick contractions increases and this gives rise to discrete coupling constant evolution as function of the dimension of extension of rationals defined by the octonionic polynomial.
- No further quantization is needed since super-symmetrization corresponds to second quantization. This is part of the realization of the dream about geometrizing also quantum theory. This should have been realized long time ago also by colleagues since SUSY algebra is Clifford algebra like also oscillator operator algebra.