Sigh of relief

As I have already told, TGD leads naturally to the idea that mere Dirac action with coupling to gauge potentials defines QFT limit of TGD: the reason is that only fermions appear as fundamental particles in TGD framework. This action generates YM action radiatively and predicts all coupling strengths. Somewhat amusingly, this kind of possibility -something extremely beautiful and incredibly simple for me at least- has not been noticed previously (to my best knowledge).

The realization of this idea requires the TGD based notions of zero energy ontology and geometric realization of finite measurement resolution. These concepts indeed lead to a highly unique physical realization of cutoffs in momentum and hyperbolic angle. The reason for two cutoffs is that the momentum integral is Minkowskian: Wick rotation would lead to a mathematical catastrophe in TGD framework and would not be in spirit with Lorentz invariance.

The UV cutoff for hyperbolic angle measured in the rest system of a particle emitting virtual particle is the basic ad hoc element of the model. It is not clear whether it can be deduced within the framework defined by QFT limit or whether only basic quantum TGD predict it. Quantum criticality suggests that the cutoff must be such that a large number of p-adic length scales contributes to the loop integrals so that one obtains interesting coupling constant evolution. In standard QFT the absence of this kind of cutoff would mean that too many length scales involved so that one ends up with divergences. In TGD framework this would mean vanishing gauge boson propagators and a trivial theory. If cutoff is too strong, gauge boson couplings are predicted to be very weak and this is also unsatisfactory situation.

Situation is very much analogous to thermodynamical criticality or criticality for spin glass phase. Above critical temperature there is chaos and below it complete order. This vision leads to a model for the cutoff which gives excellent hopes of coupling constant evolution consistent with standard model. The condition that the values of the fine structure constant at electron and intermediate gauge boson mass scales are reproduced correctly fixes the two parameters of the model and the value of the second parameter is consistent with p-adicization and number theoretical vision. Also the behavior in UV is reasonable and the predictions for other bare couplings follow and are sensible. Gauge boson loops allow to understand the non-abelian aspects of coupling constant evolution and asymptotic freedom.

I have been fighting with the precise calculation of the normalization factor of the bosonic propagator determined by fermionic loop integral. I had to work really hardly to end up with formulas consistent with the calculation at the limit of vanishing momentum squared- one factor of two was missing and one exponent n=1 had changed to n=2 and it was really maddening exercise to identify the mistakes! For the original model for which it was assumed that the measurement resolution for the time scale of CDs is maximal and thus of order CP₂ time scale, the range of the allowed hyperbolic angles was predicted to be extremely narrow in longer length scales so that loops effectively disappeared. As a consequence, the coupling constant evolution became essentially trivial except immediately above UV scale. The attempts to save this model were futile. The model for which time scale resolution corresponds to a fraction of p-adic time scale characterizing the sub-CD in question rather than smallest possible sub-CD, loop integrals receive contributions from all scales and a realistic coupling constant evolution is obtained. But as already mentioned, the quantitative expression for cutoff should be deduced from quantum TGD proper or perhaps from the consistency with the results produced by subtraction procedures in standard QFT.

I have not yet corrected the errors in the formulas in previous postings and encourage the interested reader to can consult the new chapter Quantum Field Theory Limit of TGD from Bosonic Emergence of "Towards M-matrix".