How the hierarchy of Planck constants might relate to the almost vacuum degeneracy for twistor lift of TGD?
The twistor lift of classical TGD is attractive physically but it is still unclear whether it satisfies all constraints. The basic implication of twistor lift would be the understanding of gravitational and cosmological constants. Cosmological constant removes the infinite vacuum degeneracy of Kähler action but because of the extreme smallness of cosmological constant Λ playing the role of inverse of gauge coupling strength, the situation for nearly vacuum extremals of Kähler action in the recent cosmology is non-perturbative.
What is remarkable that twistor lift is possible only in zero energy ontology (ZEO) since the volume term would be infinite by infinite volume of space-time surface in ordinary ontology: by the finite size of causal diamond (CD) the space-time volume is however finite in ZEO. Furthermore, the condition that the destructive interference does not cancel vacuum functional implies Bohr quantization for the action in ZEO. The scale of CD corresponds naturally to the length scale LΛ= (8π/Λ)1/2 defined by the cosmological constant.
One motivation for introducing the hierarchy of Planck constants was that the phase transition increasing Planck constant makes possible perturbation theory in strongly interacting system. Nature itself would take care about the converge of the perturbation theory by scaling Kähler coupling strength αK to αK/n, n=heff/h. This hierarchy might allow to construct gravitational perturbation theory as has been proposed already earlier. This would for gravitation to be quantum coherent in astrophysical and even cosmological scales.
The following picture emerges.
- Either Lλ= (8π/Λ)1/2 or the length L characterizing vacuum energy density as ρvac=hbar/L4 or both can obey p-adic length scale hypothesis as analogs of coupling constant parameters. The third option makes sense if the ratio R/lP of CP2 radius and Planck length is power of two: it can be indeed chosen to be R/lP=212 within measurement uncertainties. L(now) corresponds to the p-adic length scale L(k)=∝ 2k/2 for k=175, size scale of neuron and axon.
- A microscopic explanation for the vacuum energy realizing strong form of holography is in terms of vacuum energy for radial flux tubes emanating from the source of gravitational field. The independence of energy from the value of heff=n implies analog of Uncertainty Principle: the product Nn for the number N of flux tubes and the value of n defining the number of sheets of the covering associated with heff=n× h is constant. This picture suggests that holography is realized in biology in terms of pixels whose size scale is characterized by L rather than Planck length.
- Vacuum energy is explained both in terms of Kähler magnetic energy of flux tubes carrying dark matter and of the vacuum energy associated with cosmological constant. The two explanations could be understood as two limits of the theory in which either homological non-trivial and trivial flux tubes dominate. Assuming quantum criticality in the sense that these two phases can tranform to each other, one obtains a prediction for the cosmological constant, string tension, magnetic field, and thickness of the critical flux tubes.
- An especially interesting result is that in the recent cosmology the size scale of a large neuron would be fundamental physical length scale determined by cosmological constant. This gives additional boost to the idea that biology and fundamental physics could relate closely to each other: the size scale of neuron would not be an accident but "determined in stars" and even beyond them!
See the article How the hierarchy of Planck constants might relate to the almost vacuum degeneracy for twistor lift of TGD?. For background see the chapter From Principles to Diagrams or the article with the same title.
For a summary of earlier postings see Latest progress in TGD.