What does one really mean with gravitational Planck constant?

There are important questions related to the notion of gravitational Planck constant, to the identification of gravitational constant, and to the general structure of magnetic body. What gravitational Planck constant really is? What the formula for gravitational constant in terms of CP₂ length defining Planck length in TGD does really mean, and is it realistic? What space-time surface as covering space does really mean?

What does one mean with space-time as covering space?

The central idea is that space-time corresponds to n-fold covering for h_eff=n× h₀. It is not however quite clear what this statement does mean.

1. How the many-sheeted space-time corresponds to the space-time of QFT and GRT? QFT-GRT limit of TGD is defined by identifying the gauge potentials as sums of induced gauge potentials over the space-time sheets. Magnetic field is sum over its values for different space-time sheets. For single sheet the field would be extremely small in the present case as will be found.
   
   
2. A central notion associated with the hierarchy of effective Planck constants h_eff/h₀=n giving as a special case ℏ_gr= GMm/v₀ assigned to the flux tubes mediating gravitational interactions. The most general view is that the space-time itself can be regarded as n-sheeted covering space. A more restricted view is that space-time surface can be regarded as n-sheeted covering of M⁴. But why not n-sheeted covering of CP₂? And why not having n=n₁× n₂ such that one has n₁-sheeted covering of CP₂ and n₂-sheeted covering of M⁴ as I indeed proposed for more than decade ago but gave up this notion later and consider only coverings of M⁴? There is indeed nothing preventing the more general coverings.
   
   
3. n=n₁× n₂ covering can be illustrated for an electric engineer by considering a coil in very thin 3 dimensional slab having thickness L. The small vertical direction would serve and as analog of CP₂. The remaining 2 large dimensions would serve as analog for M⁴. One could try to construct a coil with n loops in the vertical direction direction but for very large n one would encounter problems since loops would overlap because the thickness of the wire would be larger than available room L/n. There would be some maximum value of n, call it n_max.
   

   One could overcome this limit by using the decomposition n=n₁× n₂ existing if n is prime. In this case one could decompose the coil into n₁ parallel coils in plane having n₂≥ n_max loops in the vertical direction. This provided n₂ is small enough to avoid problems due to finite thickness of the coil. For n prime this does not work but one can of also select n₂ to be maximal and allow the last coil to have less than n₂ loops.
   

   An interesting possibility is that that preferred extremal property implies the decomposition n_gr=n₁× n₂ with nearly maximal value of n₂, which can vary in some limits. Of course, one of the n₂-coverings of M⁴ could be in-complete in the case that n_gr is prime or not divisible by nearly maximal value of n₂. We do not live in ideal Universe, and one can even imagine that the copies of M⁴ covering are not exact copies but that n₂ can vary.
   
   

4. In the case of M⁴× CP₂ space-time sheet would replace single loop of the coil, and the procedure would be very similar. A highly interesting question is whether preferred extremal property favours the option in which one has as analog of n₁ coils n₁ full copies of n₂-fold coverings of M⁴ at different positions in M⁴ and thus defining an n₁ covering of CP₂ in M⁴ direction. These positions of copies need not be close to each other but one could still have quantum coherence and this would be essential in TGD inspired quantum biology.
   

   Number theoretic vision suggests that the sheets could be related by discrete isometries of CP₂ possibly representing the action of Galois group of the extension of rationals defining the adele and since the group is finite sub-group of CP₂, the number of sheets would be finite.
   

   The finite sub-groups of SU(3) are analogous to the finite sub-groups of SU(2) and if they action is genuinely 3-D they correspond to the symmetries of Platonic solids (tetrahedron,cube,octahedron, icosahedron, dodecahedron). Otherwise one obtains symmetries of polygons and the order of group can be arbitrary large. Similar phenomenon is expected now. In fact the values of n₂ could be quantized in terms of dimensions of discrete coset spaces associated with discrete sub-groups of SU(3). This would give rise to a large variation of n₂ and could perhaps explain the large variation of G identified as G= R²(CP₂)/n₂ suggested by the fountain effect of superfluidity.
   
   

5. There are indeed two kinds of values of n: the small values n=h_em/h₀=n_em assigned with flux tubes mediating em interaction and appearing already in condensed matter physics and large values n=h_gr/h₀=n_gr associated with gravitational flux tubes. The small values of n would be naturally associated with coverings of CP₂. The large values n_gr=n₁× n₂ would correspond n₁-fold coverings of CP₂ consisting of complete n₂-fold coverings of M⁴. Note that in this picture one can formally define constants ℏ(M⁴)= n₁ℏ₀ and ℏ(CP₂)= n₂ℏ₀ as proposed for more than decade ago.
   
   

Planck length as CP₂ radius and identification of gravitational constant G

There is also a puzzle related to the identification of gravitational Planck constant. In TGD framework the only theoretically reasonable identification of Planck length is as CP₂ length R(CP₂), which is roughly 10^3.5 times longer than Planck length. Otherwise one must introduce the usual Planck length as separate fundamental length. The proposal was that gravitational constant would be defined as G =R²(CP₂)/ℏ_gr, ℏ_gr≈ 10⁷ℏ. The G indeed varies in un-expectedly wide limits and the fountain effect of superfluidity suggests that the variation can be surprisingly large.

There are however problems.

1. Arbitrary small values of G=R²(CP₂)/ℏ_gr are possible for the values of ℏ_gr appearing in the applications: the values of order n_gr ∼ 10¹³ are encountered in the biological applications. The value range of G is however experimentally rather limited. Something clearly goes wrong with the proposed formula.
   
   
2. Schwartschild radius r_S= 2GM = 2R²(CP₂)M/ℏ_gr would decrease with ℏ_gr. One would expect just the opposite since fundamental quantal length scales should scale like ℏ_gr.
   
   
3. What about Nottale formula ℏ_gr= GMm/v₀? Should one require self-consistency and substitute G= R²(CP₂)/ℏ_gr to it to obtain ℏ_gr=(R²(CP₂)Mm/v₀)^1/2. This formula leads to physically un-acceptable predictions, and I have used in all applications G=G_N corresponding to n_gr∼ 10⁷ as the ratio of squares of CP₂ length and ordinary Planck length.
   
   

Could one interpret the almost constancy of G by assuming that it corresponds to ℏ(CP₂)= n₂ℏ₀, n₂≈ 10⁷ and nearly maximal except possibly in some special situations? For n_gr=n₁× n₂ the covering corresponding to ℏ_gr would be n₁-fold covering of CP₂ formed from n₁ n₂-fold coverings of M⁴. For n_gr=n₁× n₂ the covering would decompose to n₁ disjoint M⁴ coverings and this would also guarantee that the definition of r_S remains the standard one since only the number of M⁴ coverings increases.

If n₂ corresponds to the order of finite subgroup G of SU(3) or number of elements in a coset space G/H of G (itself sub-group for normal sub-group H), one would have very limited number of values of n₂, and it might be possible to understand the fountain effect of superfluidity from the symmetries of CP₂, which would take a role similar to the symmetries associated with Platonic solids. In fact, the smaller value of G in fountain effect would suggest that n₂ in this case is larger than for G_N so that n₂ for G_N would not be maximal.

See the article TGD View about Quasars or the 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.

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

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