Wednesday, October 03, 2018

Atyiah, fine structure constant, and TGD view based view about coupling constant evolution

Atyiah has recently proposed besides a proof of Riemann Hypothesis also an argument claiming to derive the value of the structure constant (see this). The mathematically elegant arguments of Atyiah involve a lot of refined mathematics including notions of Todd exponential and hyper-finite factors of type II (HFFs) assignable naturally to quaternions. The idea that 1/α could result by coupling constant evolution from π looks however rather weird for a physicist.

What makes this interesting from TGD point of view is that in TGD framework coupling constant evolution can be interpreted in terms of inclusions of HFFs with included factor defining measurement resolution. An alternative interpretation is in terms of hierarchy of extensions of rationals with coupling parameters determined by quantum criticality as algebraic numbers in the extension.

In the following I will explain what I understood about Atyiah's approach. My critics includes the arguments represented also in the blogs of Lubos Motl (see this) and Sean Carroll (see this). I will also relate Atyiah's approach to TGD view about coupling evolution. The hasty reader can skip this part although for me it served as an inspiration forcing to think more precisely TGD vision.

There are two TGD based formulations of scattering amplitudes.

  1. The first formulation is at the level of infinite-D "world of classical worlds" (WCW) uses tools like functional integral. The huge super-symplectic symmetries generalizing conformal symmetries raise hopes that this formulation exists mathematically and that it might even allow practical calculations some day. TGD would be an analog of integrable QFT.

  2. Second - surprisingly simple - formulation is based on the analog of micro-canonical ensemble in thermodynamics (quantum TGD can be seen as complex square root of thermodynamics). It relates very closely to TGD analogs of twistorialization and twistor amplitudes.

    During writing I realized that this formulation can be regarded as a generalization of cognitive representations of space-time surfaces based on algebraic discretization making sense for all extensions of rationals to the level of scattering amplitudes. This formulation allows a continuation to p-adic sectors and adelization and adelizability is what leads to the concrete formula - something new - for the evolution of Kähler coupling strength αK forced by the adelizability condition.

    The condition is childishly simple: the exponent of complex action S (more general than that of Kähler function) equals to unity: exp(S)=1 and thus is common to all number fields. This condition allows to avoid the grave mathematical difficulties cause by the requirement that exp(S) exists as a number in the extension of rationals considered. Second necessary condition is the reduction of twistorial scattering amplitudes to tree diagrams implied by quantum criticality.

  3. One can also understand the relationship of the two formulations in terms of M8-H duality. This view allows also to answer to a longstanding question concerning the interpretation of the surprisingly successful p-adic mass calculations: as anticipated, p-adic mass calculations are carried out for a cognitive representation rather than for real world particles and the huge simplification explains their success for preferred p-adic prime characterizing particle as so called ramified prime for the extension of rationals defining the adeles.

  4. I consider also the relationship to a second TGD based formulation of coupling constant evolution in terms of inclusion hierarchies of hyper-finite factors of type II1 (HFFs). I suggest that this hierarchy is generalized so that the finite subgroups of SU(2) are replaced with Galois groups associated with the extensions of rationals. An inclusion of HFFs in which Galois group would act trivially on the elements of the HFFs appearing in the inclusion: kind of Galois confinement would be in question.

See the article TGD view about coupling constant evolution or the chapter of "Towards M-matrix" with same title.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

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