Hyper-finite factors and construction of S-matrix

During years I have spent a lot of time and effort to attempts to imagine various options for the construction of S-matrix. Contrary to my original belief, the real problem has not been the lack of my analytic skills but the failure of ordinary QFT based thinking in TGD framework.

Super-conformal symmetries generalized from string model context to TGD framework are symmetries of S-matrix. This is very powerful constraint to S-matrix but useless unless one has precisely defined ontology translated to a rigorous mathematical framework. The zero energy ontology of TGD is now rather well understood but differs dramatically from that of standard quantum field theories. Second deep difference is that path integral formalism is given up and the goal is to construct S-matrix as a generalization of braiding S-matrices with reaction vertices replaced with the replication of number theoretic braids associated with partonic 2-surfaces taking the role of vertices. Also number theoretic universality requiring fusion of real physics and various p-adic physics to single coherent whole is a completely new element.

The most recent vision about S-matrix combines ideas scattered in various chapters of various books and often drowned with details. A very brief summary would be as follows.

1. In TGD framework functional integral formalism is given up. S-matrix should be constructible as a generalization of braiding S-matrix in such a manner that the number theoretic braids assignable to light-like partonic 3-surfaces glued along their ends at 2-dimensional partonic 2-surfaces representing reaction vertices replicate in the vertex. This means a replacement of the free dynamics of point particles of quantum field theories with braiding dynamics associated with partonic 2-surfaces carrying braids and the replacement of particle creation with the creation of partons and replication of braids.

2. The construction of braiding S-matrices assignable to the incoming and outgoing partonic 2-surfaces is not a problem. The problem is to express mathematically what happens in the vertex. Here the observation that the tensor product of hyper-finite factors (HFFs) of type II is HFF of type II is the key observation. Many-parton vertex can be identified as a unitary isomorphism between the tensor product of incoming resp. outgoing HFFs. A reduction to HFF of type II₁ occurs because only a finite-dimensional projection of S-matrix in bosonic degrees of freedom defines a normalizable state. Most importantly, unitarity and non-triviality of S-matrix follows trivially.

3. HFFs of type III could also appear at the level of field operators used to create states but that at the level of quantum states everything reduces to HFFs of type II₁ and their tensor products giving these factors back. If braiding automorphisms reduce to the purely intrinsic unitary automorphisms of HFFs of type III then for certain values of the time parameter of automorphism having interpretation as a scaling parameter these automorphisms are trivial. These time scales could correspond to p-adic time scales so that p-adic length scale hypothesis would emerge at the fundamental level. In this kind of situation the braiding S-matrices associated with the incoming and outgoing partons could be trivial so that everything would reduce to this unitary isomorphism: a counterpart for the elimination of external legs from Feynman diagram in QFT. p-Adic thermodynamics and particle massivation could be also obtained when the time parameter of the automorphism is allowed to be complex as a generalization of thermal QFT.

4. One might hope that all complications related to what happens for space-like 3-surfaces could be eliminated by quantum classical correspondence stating that space-time view about particle reaction is only a space-time correlate for what happens in quantum fluctuating degrees of freedom associated with partonic 2-surfaces. This turns out to be the case only in non-perturbative phase. The reason is that the arguments of n-point function appear as continuous moduli of Kähler function. In non-perturbative phases the dependence of the maximum of Kähler function on the arguments of n-point function cannot be regarded as negligible and Kähler function becomes the key to the understanding of these effects including formation of bound states and color confinement.

5. In this picture light-like 3-surface would take the dual role as a correlate for both state and time evolution of state and this dual role allows to understand why the restriction of time like entanglement to that described by S-matrix must be made. For fixed values of moduli each reaction would correspond to a minimal braid diagram involving exchanges of partons being in one-one correspondence with a maximum of Kähler function. By quantum criticality and the requirement of ideal quantum-classical correspondence only one such diagram would contribute for given values of moduli. Coupling constant evolution would not be however lost: it would be realized as p-adic coupling constant at the level of free states via the log(p) scaling of eigen modes of the modified Dirac operator.

6. A completely unexpected prediction deserving a special emphasis is that number theoretic braids replicate in vertices. This is of course the braid counterpart for the introduction of annihilation and creation of particles in the transition from free QFT to an interacting one. This means classical replication of the number theoretic information carried by them. This allows to interpret one of the TGD inspired models of genetic code in terms of number theoretic braids representing at deeper level the information carried by DNA. This picture provides also further support for the proposal that DNA acts as topological quantum computer utilizing braids associated with partonic light-like 3-surfaces (which can have arbitrary size). In the reverse direction one must conclude that even elementary particles could be information processing and communicating entities in TGD Universe.

To sum up, my personal feeling is that the constraints identified hitherto might lead to a more or less unique final result and I can only hope that some young analytically blessed brain would bother to transform this picture to concrete calculational recipes.

For more details see the new chapter Hyper-Finite Factors and Construction of S-matrix of "Towards S-matrix".