The first question is what one means with S-matrix in ZEO. I have considered several proposals for the counterparts of S-matrix. In the original U-matrix, M-matrix and S-matrix were introduced but it seems that U-matrix is not needed.
What does unitarity mean in the twistorial approach?
- The first question is whether the unitary matrix is between zero energy states or whether it characterizes zero energy states themselves as time-like entanglement coefficients between positive and negative energy parts of zero energy states associated with the ends of CD. One can argue that the first option is not sensible since positive and negative energy parts of zero energy states are strongly correlated rather than forming a tensor product: the S-matrix would in fact characterize this correlation partially.
The latter option is simpler and is natural in the proposed identification of conscious entity - self - as a generalized Zeno effect, that is as a sequence of repeated state function reductions at either boundary of CD shifting also the boundary of CD farther away from the second boundary so that the temporal distance between the tips of CD increases. Each shift of this kind is a step in which superposition of states with different distances of upper boundary from lower boundary results followed by a localization fixing the active boundary and inducing unitary transformation for the states at the original boundary.
- The proposal is that the the proper object of study for given CD is M-matrix. M-matrix is a product for a hermitian square root of diagonalized density matrix ρ with positive elements and unitary S-matrix S : M= ρ1/2S. Density matrix ρ could be interpreted in this approach as a non-trivial Hilbert space metric. Unitarity conditions are replaced with the conditions MM†= ρ and M†M=ρ. For the single step in the sequence of reductions at active boundary of CD one has M→ MS (Δ T) so that one has S→ SS(Δ T). S(Δ T) depends on the time interval Δ T measured as the increase in the proper time distance between the tips of CD assignable to the step.
- In accordance with the idea that scattering diagrams is a representation for a computation, suppose that the deformations of space-time surfaces defining a given topological diagram as a maximum of the exponent of Kähler function, are the basic objects. They would define different quantum phases of a larger quantum theory regarded as a square root of thermodynamics in ZEO and analogous to those appearing also in QFTs. Unitarity would hold true for each phase separately.
The topological diagrams would not play the role of Feynman diagrams in unitarity conditions although their vertices would be analogous to those appearing in Feynman diagrams. This would reduce the unitarity conditions to those for fermionic states at partonic 2-surfaces at the ends of CDs, actually at the ends of fermionic lines assigned to the boundaries of string world sheets.
- The unitarity conditions be interpreted stating the orthonormality of the basis of zero energy states assignable with given topological diagram. Since 3-surfaces as points of WCW appearing as argument of WCW spinor field are pairs consisting of 3-surfaces at the opposite boundaries of CD, unitarity condition would state the orthonormality of modes of WCW spinor field. If might be even that no mathematically well-defined inner product assignable to either boundary of CD exists since it does not conform with the view provided by WCW geometry. Perhaps this approach might help in identifying the correct form of S-matrix.
- If only tree diagrams constructed using 4-fermion twistorial vertex are allowed, the unitarity relations would be analogous to those obtained using only tree diagrams. They should express the discontinuity for T in S=1+iT along unitary cut as Disc(T)= TT†. T and T† would be T-matrix and its time reversal.
- The correlation between the structure of the fermionic scattering diagram and topological scattering diagrams poses very strong restrictions on allowed scattering reactions for given topological scattering diagram. One can of course have many-fermion states at partonic 2-surfaces and this would allow arbitrarily high fermion numbers but physical intuition suggests that for given partonic 2-surface (throat of wormhole contact) the fermion number is only 0, 1, or perhaps 2 in the case of supersymmetry possibly generated by right-handed neutrino.
The number of fundamental fermions both in initial and final states would be finite for this option. In quantum field theory with only masive particles the total energy in the final state poses upper bound on the number of particles in the final state. When massless particles are allowed there is no upper bound. Now the complexity of partonic 2-surface poses an upper bound on fermions.
This would dramatically simplify the unitarity conditions but might also make impossible to satisfy them. The finite number of conditions would be in spirit with the general philosophy behind the notion of hyper-finite factor. The larger the number of fundamental fermions associated with the state, the higher the complexity of the topological diagram. This would conform with the idea about QCC. One can make non-trivial conclusions about the total energy at which the phase transitions changing the topology of space-time surface defined by a topological diagram must take place.
For background see chapter Some questions related to the twistor lift of TGD or the article with the same title.