More precise view about S-matrix

In TGD framework the nontriviality of S-matrix would basically result by a replacement of the ordinary tensor product with Connes tensor product for free fields. Each Jones inclusion defines different Connes tensor product. According to the speculation of Jones, the fusion rules of conformal theories are equivalent with Connes tensor product for tensor products of Kac Moody representations at least. Various conformal field theories would thus represent different Jones inclusions characterizing limitations of the measurer in various kinds of quantum measument situations and characterized by ADE diagrams or extended ADE diagrams. Somewhat ironically, in ideal measurement S-matrix would be trivial! In string model context this approach does not work since one has necessarily c=h=0.

The picture about S-matrix has been developing rapidly and it is now clear that the 27 year old dream is finally realized: I understand S-matrix (or rather S-matrices) in TGD framework. Below comments about results that have emerged during last week.

1. Effective 2-dimensionality and quantum classical correspondence

The requirement that quantum measurement theory in the sense of TGD emerges as part of the construction of S-matrix allows to add details to the master formula and derive also non-trivial predictions. The effective 2-dimensionality of the configuration space metric (only the deformations of 2-D partonic surfaces appear in the line element) suggests that the incoming space-time sheets and the space-time sheet representing vertex intersect only along 2-D partonic space-time sheets. This allows incoming space-time sheets to have classical conserved charges identical with a maximal subset of commuting quantum numbers. The functional integral over 3-surfaces is very much analogous to the 2-dimensional functional integral of Euclidian string models. Vertex 3-surface in turn represents the quantum jump at space-time level.

The 3-surface representing N-vertex can be decomposed into parts representing vertices/propagators of conventional tree diagrams as space-/light like portions. No loops appear. By their strong classical non-determinism CP₂ type extremals representing elementary particles are unique as classical representations of propagation of off-mass-shell particles. For other light like causal determinants, such as those assignable to massless extremals, only massless momentum exchanges are typically possible (not however in 2-particle scattering) so that scattering becomes extremely deterministic. The prediction is that even gravitational interaction could be strong in lower-dimensional regions of the phase space corresponding to light-like momentum exchanges.

2. Reduction of the Connes tensor product to fusion in conformal field theories

There are good reasons to expect that Connes tensor product corresponds to the fusion procedure of conformal field theories so that various conformal field theories at partonic boundary components would characterize various measurement situations.

The realization that various super-conformal algebras at partonic space-time sheets have dual hyper-quaterionic representations and that the restriction of these representations to commuting hyper-complex planes or real lines has strong mathematical and physical justification allows to write formulas for S-matrix in terms of n-point functions of the conformal field theory characterizing Jones inclusion applying to the quantum measurement situation by just replacing complex argument with hyper-complex argument.

The differences between TGD and string models become clear and one can formulate precisely where string models go wrong. In TGD framework breaking of conformal symmetry can occur and mass squared corresponds to a genuine conformal weight rather than a contribution causing the vanishing of conformal weight. Hence the vanishing of conformal parameters (c,h), which was originally believed to make string models unique, is not necessary. Furthermore, four-momentum does not appear in Super Virasoro generators as in string models so that super generators can carry fermion number since the Majorana character of super-generators becomes un-necessary. Hence dimension D=10 or 11 for imbedding space is not necessary.

One can understand why gravitational constant is so weak as compared to gauge couplings. Also particle massivation and p-adic mass scale are coded into the statistical properties of the zitterbewegung of CP₂ type extremals (the projection to Minkowski space is random light-like curve).

3. What went wrong with string models?

Things went wrong with string models at many levels. With my personal background it is not difficult to see what went wrong with string models at the level of mathematical and physical understanding. Basically the wrong view about conformal invariance implied that theory made sense only in critical dimensions. Because this is so important I want to recapitulate: the basic mistake was the idea that mass squared compensates the contributions to conformal weight obtained in the ordinary conformal theory. This led to c=h=0 and to wrong realization of super generators and Majorana conditions and the rest we known. It is not surprising that so powerful constraints led to what looked a highly unique theory.

p-Adic mass calculations made it obvious for me that this view about conformal invariance is wrong. Thermal mass squared in string model sense means breaking of Lorentz invariance: mass squared must result as thermal average of conformal weight. This requires that physical states have non-vanishing conformal weights and that mass squared corresponds to this weight and that momentum does not appear in Super Virasoro generators. Ordinary Euclidian conformal theory for partonic 2-surfaces and its dual for hyper-complex planes of Minkowski space become the basic mathematical tool in the construction of physical states and S-matrix.

There are also many other thins that went wrong: I shared for a long time the belief in the stringy description of particle decay in terms of trouser diagrams until physical arguments forced to interpret this process as a space-time correlate for what happens in double slit experiments. Mathematically this was of course obvious from the beginning since it is just propagation via different routes which happens for induced spinor fields: it is extremely artificial to build vertices using this framework. A direct generalization of Feynman diagrams as singular manifolds obtained by gluing surfaces together along their ends turned out to the only acceptable topological description of particle decay but one can argue that this is mathematically too singular process. It is: and this description indeed turned out to be only effective when the master formula for S-matrix emerged.

Let us return to the consequences of misunderstood conformal invariance. I can well understand the excitement when people realized that perhaps only some little work might reveal the theory of everything. The resulting physics was of course not about this world. It is quite understandable that this sparkled the dream that Kaluza-Klein might save the theory and eventually we had landscape, anthropic principle, and even the brand new vision that the physics that we experience everyday could be anything else and that even at the level of principle it is impossible to predict anything. Now the theory is sold using the exact opposites of the arguments that were used two decades ago.

What is so sad in this that immense amount of high level technical work is being carried out in attempt to get something sensible out of a dead theory.

Conformal field theories represent just the opposite to string model hype. Instead of ad hoc quantization recipes and uncritical introduction of poorly defined notions like spontaneous compactification and effective field theories, rigorous formulation and understanding of the theory has been the primary goal. The formalism generalizes as such to TGD framework and finds a new interpretation in the framework of quantum measurement theory.

Details can be found here, here, and here.