How could Planck length be actually equal to much larger CP2 radius?!

The following argument stating that Planck length l_P equals to CP₂ radius R: l_P=R and Newton's constant can be identified G= R²/ℏ_eff. This idea looking non-sensical at first glance was inspired by an FB discussion with Stephen Paul King.

First some background.

1. I believed for long time that Planck length l_P would be CP₂ length scale R squared multiplied by a numerical constant of order 10^-3.5. Quantum criticality would have fixed the value of l_P and therefore G=l_P²/ℏ.
   
   
2. Twistor lift of TGD led to the conclusion that that Planck length l_P is essentially the radius of twistor sphere of M⁴ so that in TGD the situation seemed to be settled since l_P would be purely geometric parameter rather than genuine coupling constant. But it is not! One should be able to understand why the ratio l_P/R but here quantum criticality, which should determine only the values of genuine coupling parameters, does not seem to help.
   

   Remark: M⁴ has twistor space as the usual conformal sense with metric determined only apart from a conformal factor and in geometric sense as M⁴× S²: these two twistor spaces are part of double fibering.
   
   

Could CP₂ radius R be the radius of M⁴ twistor sphere, and could one say that Planck length l_P is actually equal to R: l_P=R? One might get G= l_P²/ℏ from G= R²/ℏ_eff!

1. It is indeed important to notice that one has G=l_P²/ℏ. ℏ is in TGD replaced with a spectrum of ℏ_eff=nℏ₀, where ℏ= 6ℏ₀ is a good guess. At flux tubes mediating gravitational interactions one has
   

   ℏ_eff=ℏ_gr= GMm/v₀ ,
   

   where v₀ is a parameter with dimensions of velocity. I recently proposed a concrete physical interpretation for v₀ (see this). The value v₀=2^-12 is suggestive on basis of the proposed applications but the parameter can in principle depend on the system considered.
   
   

2. Could one consider the possibility that twistor sphere radius for M⁴ has CP₂ radius R: l_P= R after all? This would allow to circumvent introduction of Planck length as new fundamental length and would mean a partial return to the original picture. One would l_P= R and G= R²/ℏ_eff. ℏ_eff/ℏ would be of 10⁷-10⁸!
   
   

The problem is that ℏ_eff varies in large limits so that also G would vary. This does not seem to make sense at all. Or does it?!

To get some perspective, consider first the phase transition replacing hbar and more generally hbar_eff,i with hbar_eff,f=h_gr .

1. Fine structure constant is what matters in electrodynamics. For a pair of interacting systems with charges Z₁ and Z₂ one has coupling strength Z₁Z₂e²/4πℏ= Z₁Z₂α, α≈ 1/137.
   
   
2. One can also define gravitational fine structure constant α_gr. Only α_gr should matter in quantum gravitational scattering amplitudes. α_gr wold be given by
   

   α_gr= GMm/4πℏ_gr= v₀/4π .
   

   v₀/4π would appear as a small expansion parameter in the scattering amplitudes. This in fact suggests that v₀ is analogous to α and a universal coupling constant which could however be subject to discrete number theoretic coupling constant evolution.
   
   

3. The proposed physical interpretation is that a phase transition hbar_eff,i→ hbar_eff,f=h_gr at the flux tubes mediating gravitational interaction between M and m occurs if the perturbation series in α_gr=GMm/4π/hbar fails to converge (Mm∼ m_Pl² is the naive first guess for this value). Nature would be theoretician friendly and increase h_eff and reducing α_gr so that perturbation series converges again.
   

   Number theoretically this means the increase of algebraic complexity as the dimension n=h_eff/h₀ of the extension of rationals involved increases fron n_i to n_f and the number n sheets in the covering defined by space-time surfaces increases correspondingly. Also the scale of the sheets would increase by the ratio n_f/n_i.
   

   This phase transition can also occur for gauge interactions. For electromagnetism the criterion is that Z₁Z₂α is so large that perturbation theory fails. The replacement hbar→ Z₁Z₂e²/v₀ makes v₀/4π the coupling constant strength. The phase transition could occur for atoms having Z≥ 137, which are indeed problematic for Dirac equation. For color interactions the criterion would mean that v₀/4π becomes coupling strength of color interactions when α_s is above some critical value. Hadronization would naturally correspond to the emergence of this phase.
   

   One can raise interesting questions. Is v₀ (presumably depending on the extension of rationals) a completely universal coupling strength characterizing any quantum critical system independent of the interaction making it critical? Can for instance gravitation and electromagnetism are mediated by the same flux tubes? I have assumed that this is not the case. It it could be the case, one could have for GMm<m_Pl² a situtation in which effective coupling strength is of form (GmMm/Z₁Z₂e²) (v₀/4π).
   

The possibility of the proposed phase transition has rather dramatic implications for both quantum and classical gravitation.

1. Consider first quantum gravitation. v₀ does not depend on the value of G at all!The dependence of G on ℏ_eff could be therefore allowed and one could have l_P= R. At quantum level scattering amplitudes would not depend on G but on v₀. I was happy of having found small expansion parameter v₀ but did not realize the enormous importance of the independence on G!
   

   Quantum gravitation would be like any gauge interaction with dimensionless coupling, which is even small! This might relate closely to the speculated TGD counterpart of AdS/CFT duality between gauge theories and gravitational theories.
   
   

2. But what about classical gravitation? Here G should appear. What could the proportionality of classical gravitational force on 1/ℏ_eff mean? The invariance of Newton's equation
   

   dv/dt =-GM r/r³
   

   under h_eff→ xh_eff would be achieved by scaling v→ v/x and t→ t/x. Note that these transformations have general coordinate invariant meaning as transformations of coordinates of M⁴ in M⁴×CP₂. This scaling means the zooming up of size of space-time sheet by x, which indeed is expected to happen in
   h_eff→ xh_eff!
   
   

What is so intriguing that this connects to an old problem that I pondered a lot during the period 1980-1990 as I attempted to construct to the field equations for Kähler action approximate spherically symmetric stationary solutions. The naive arguments based on the asymptotic behavior of the solution ansatz suggested that the one should have G= R²/ℏ. For a long time indeed assumed R=l_P but p-adic mass calculations and work with cosmic strings forced to conclude that this cannot be the case. The mystery was how G= R²/ℏ could be normalized to G=l_P²/ℏ: the solution of the mystery is ℏ→ ℏ_eff as I have now - decades later - realized!

See the article About the physical interpretation of the velocity parameter in the formula for the gravitational Planck constant or the new chapter About the Nottale's formula for h_gr and the possibility that Planck length l_P and CP₂ length R are identical giving G= R²/ℏ_eff of "Physics in many-sheeted space-time".

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

Articles and other material related to TGD.