Monday, October 22, 2007

Connes tensor product and perturbative expansion in terms of generalized braid diagrams

Many steps of progress have occurred in TGD lately.
  1. In a given measurement resolution characterized by the inclusion of HFFs of type II1 Connes tensor product defines an almost universal M-matrix apart from the non-uniqueness due to the facts that one has a direct sum of hyper-finite factors of type II1 (sum over conformal weights at least) and the fact that the included algebra defining the measurement resolution can be represented in a reducible manner. The S-matrices associated with irreducible factors would be unique in a given measurement resolution and the non-uniqueness would make possible non-trivial density matrices and thermodynamics.

  2. Higgs vacuum expectation is proportional to the generalized position dependent eigenvalue of the modified Dirac operator and its minima define naturally number theoretical braids as orbits for the minima of the universal Higgs potential: fusion and decay of braid strands emerge naturally. Thus the old speculation about a generalization of braid diagrams to Feynman diagram likes objects, which I already began to think to be too crazy to be true, finds a very natural realization.

In the previous posting I explained how generalized braid diagrams emerge naturally as orbits of the minima of Higgs defined as a generalized eigenvalue of the modified Dirac operator. I also explained how Connes tensor product relates to a diagrammatic expansion in terms of generalized braid diagrams. Because of the utmost importance of this result I decided to move it from the end of the earlier posting here. Sorry for any inconvenience.

The association of generalized braid diagrams to incoming and outgoing 3-D partonic legs and possibly also vertices of the generalized Feynman diagrams forces to ask whether the generalized braid diagrams could give rise to a counterpart of perturbation theoretical formalism via the functional integral over configuration space degrees of freedom.

The question is how the functional integral over configuration space degrees of freedom relates to the generalized braid diagrams. The basic conjecture motivated also number theoretically is that radiative corrections in this sense sum up to zero for critical values of Kähler coupling strength and Kähler function codes radiative corrections to classical physics via the dependence of the scale of M4 metric on Planck constant. Cancellation occurs only for critical values of Kähler coupling strength αK: for general values of αK cancellation would require separate vanishing of each term in the sum and does not occur.

The natural guess is that finite measurement resolution in the sense of Connes tensor product can be described as a cutoff to the number of generalized braid diagrams. Suppose that the cutoff due to the finite measurement resolution can be described in terms of inclusions and M-matrix can be expressed as a Connes tensor product. Suppose that the improvement of the measurement resolution means the introduction of zero energy states and corresponding light-like 3-surfaces in shorter time scales bringing in increasingly complex 3-topologies.

This would mean following.

  1. One would not have perturbation theory around a given maximum of Kähler function but as a sum over increasingly complex maxima of Kähler function. Radiative corrections in the sense of perturbative functional integral around a given maximum would vanish (so that the expansion in terms of braid topologies would not make sense around single maximum). Radiative corrections would not vanish in the sense of a sum over 3-topologies obtained by adding radiative corrections as zero energy states in shorter time scale.

  2. Connes tensor product with a given measurement resolution would correspond to a restriction on the number of maxima of Kähler function labelled by the braid diagrams. For zero energy states in a given time scale the maxima of Kähler function could be assigned to braids of minimal complexity with braid vertices interpreted in terms of an addition of radiative corrections. Hence a connection with QFT type Feyman diagram expansion would be obtained and the Connes tensor product would have a practical computational realization.

  3. The cutoff in the number of topologies (maxima of Kähler function contributing in a given resolution defining Connes tensor product) would be always finite in accordance with the algebraic universality.

  4. The time scale resolution defined by the temporal distance between the tips of the causal diamond defined by the future and past light-cones applies to the addition of zero energy sub-states and one obtains a direct connection with p-adic length scale evolution of coupling constants since the time scales in question naturally come as negative powers of two. More precisely, p-adic p-adic primes near power of two are very natural since the coupling constant evolution comes in powers of two of fundamental 2-adic length scale.

There are still some questions. Radiative corrections around given 3-topology vanish. Could radiative corrections sum up to zero in an ideal measurement resolution also in 2-D sense so that the initial and final partonic 2-surfaces associated with a partonic 3-surface of minimal complexity would determine the outcome completely? Could the 3-surface of minimal complexity correspond to a trivial diagram so that free theory would result in accordance with asymptotic freedom as measurement resolution becomes ideal?

The answer to these questions seems to be 'No'. In the p-adic sense the ideal limit would correspond to the limit p→ 0 and since only p→ 2 is possible in the discrete length scale evolution defined by primes, the limit is not a free theory. This conforms with the view that CP2 length scale defines the ultimate UV cutoff.

For more details see the chapter Construction of Quantum Theory: Symmetries of "Towards S-matrix".

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