The following argument stating that Planck length lP equals to CP2 radius R: lP=R and Newton's constant can be identified G= R2/ℏeff. This idea looking non-sensical at first glance was inspired by an FB discussion with Stephen Paul King.
First some background.
- I believed for long time that Planck length lP 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 lP and therefore G=lP2/ℏ.
- Twistor lift of TGD led to the conclusion that that Planck length lP is essentially the radius of twistor sphere of M4 so that in TGD the situation seemed to be settled since lP would be purely geometric parameter rather than genuine coupling constant. But it is not! One should be able to understand why the ratio lP/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.
- It is indeed important to notice that one has G=lP2/ℏ. ℏ is in TGD replaced with a spectrum of ℏeff=nℏ0, where ℏ= 6ℏ0 is a good guess. At flux tubes mediating gravitational interactions one has
ℏeff=ℏgr= 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.
- Could one consider the possibility that twistor sphere radius for M4 has CP2 radius R: lP= 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 lP= R and G= R2/ℏeff. ℏeff/ℏ would be of 107-108!
To get some perspective, consider first the phase transition replacing hbar and more generally hbareff,i with hbareff,f=hgr .
- 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.
- One can also define gravitational fine structure constant αgr. Only αgr should matter in quantum gravitational scattering amplitudes. αgr wold be given by
αgr= GMm/4πℏgr= 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.
- The proposed physical interpretation is that a phase transition hbareff,i→ hbareff,f=hgr at the flux tubes mediating gravitational interaction between M and m occurs if the perturbation series in αgr=GMm/4π/hbar fails to converge (Mm∼ mPl2 is the naive first guess for this value). Nature would be theoretician friendly and increase heff and reducing αgr so that perturbation series converges again.
Number theoretically this means the increase of algebraic complexity as the dimension n=heff/h0 of the extension of rationals involved increases fron ni to nf 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 nf/ni.
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 αs 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<mPl2 a situtation in which effective coupling strength is of form (GmMm/Z1Z2e2) (v0/4π).
- Consider first quantum gravitation. v0 does not depend on the value of G at all!The dependence of G on ℏeff could be therefore allowed and one could have lP= 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.
- But what about classical gravitation? Here G should appear. What could the proportionality of classical gravitational force on 1/ℏeff mean? The invariance of Newton's equation
v/dt =-GM r/r3
under heff→ xheff 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
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 hgr and the possibility that Planck length lP and CP2 length R are identical giving G= R2/ℏeff of "Physics in many-sheeted space-time".
For a summary of earlier postings see Latest progress in TGD.