An email to Carlos

Below an email response to Carlos Castro. Actually a detypoed, edited, and decorated version of the original.

Dear Carlos, thank you for telling about the journal.

The enthusiastic burst that I sent to you was probably still quite far away from the recent situation. Many painful drawbacks have occurred during last two weeks but the situation seems to have settled down now.

It turned out that 2-D factorizable S-matrices cannot give rise to realistic S-matrices in ordinary sense. The are however tailor made for describing the scattering of zero energy states (all states in TGD are predicted to possess vanishing net conserved charges: only now I realized that this must be the starting point in the construction of quantum states and S-matrix). The almost triviality of this S-matrix guarantees almost equivalence of zero energy ontology with the ordinary one although there are important and fascinating differences too. One nice thing is that zero energy states can be localized to a finite portion of Minkowski space and one can assign to them densities,etc. This allows also to replace the horribly ugly routed from S-matrix to scattering rates with a real understanding.

Amusingly, this Buddhistic ontology (what you perceive represents different forms of emptiness) is also consistent with the creation and doomsday myths of religions. That perceived world looks stable continuum rather than like sequence of uncorrelated little bangs followed by little crunches tells that the temporal distances between positive and negative energy components are much longer than the time scale of human perception: dark matter and large hbars. This of course applies only on what we can sensorily perceive.

Zero energy states could code for the counterpart of ordinary S-matrix as entanglement coefficients between negative and positive energy components of the states. This is however not necessary if one accepts that theory predicts only scattering probabilities and unitarity results as a kind of thermal average. This kind of coding of S-matrix is already familiar from spherically symmetric potential scattering in wave mechanics. It is absolutely crucial that the condition Tr(S^†S)= Tr(Id)=1 (instead of infinity) applies in the case of hyperfinite type II₁ factors and implies that unitarity condition can be interpreted as a normalization of states formed by entangling negative energy states with positive energy states.

Quantum measurement can induce projections to subfactors of II₁ factor M for both positive and negative energy factors but the result is always a state with S-matrix entangled state in sub-factor so that it there is no hope of achieving a one-dimensional ray in the state space. Quantum state is like a hologram representing quantum dynamics in its structure and containing infinite amounts of information.

Super Virasoro and Super Kac-Moody conditions for zero energy states are extremely natural and dictate the entanglement coefficients in super conformal degrees of freedom to a high degree. k and c for positive and negative energy states are of opposite sign and compensate each other so that k=c=0 results. The entanglement matrix turns out to be a generalized braiding matrix determined by the universal R-matrix associated with the Super Kac Moody algebras: braiding operation emerges naturally in the calculation of the amplitude for creating zero energy state from Fock vacuum. The product of R-matrices describing the braiding emerges as one moves an operator creating negative energy fermion past negative and positive energy fermions until it anticommutes to c-number with some positive energy fermion. This R-matrix is highly analogous to the exponent of L₀ but more complicated so that stringy picture emerges naturally for the scattering amplitudes.

An absolutely essential point is that in TGD framework all particles consist of fermions. Therefore fermion and antifermion numbers are separately conserved although they can arrange themselves to fermions and bosons, even Higgs particles with fermions associated with the throats of wormhole contact. This makes possible braiding connecting positive energy fermions with negative energy antifermions and R is associated with this braiding and there is no need to generalize braiding by allowing branchings as was the original proposal. The Zweig rule of pre QCD era would be much deeper truth than ever thought.

There remains many things to be understood but I think that TGD S-matrix, or actually an infinite hierarchy (or hierarchies) of S-matrices, is now understood at the level of physical interpretation and what comes to the basic mathematical building blocks. The book of Chari and Pressley turned out to be surprisingly easy to read although at first it looks rather unfriendly acquaintance. I wish I had their mathematical brains.

The chapter Construction of Quantum Theory of "Towards S-matrix" represents the detailed construction as it is now (it could still change!).