**Minimal value of h**

_{eff}from the ratio of Planck mass and CP_{2}mass?
Could one understand and perhaps even predict the minimal value h_{0}of h_{eff}? Here number theory and the notion of n-particle Planck constant h_{eff}(n) suggested by Yangian symmetry could serve as a guidelines.

- Hitherto I have found no convincing empirical argument fixing the value of r=ℏ/ℏ
_{0}: this is true for both single particle and 2-particle case.The value h

_{0}=h/6 as a maximal value of h_{0}is suggested by the findings of Randell Mills and by the idea that spin and color must be representable as Galois symmetries so that the Galois group must contain Z_{6}=Z_{2}× Z_{3}. Smaller values of h_{0}cannot be however excluded. - A possible manner to understand the value r geometrically would be following. It has been assumed that CP
_{2}radius R defines a fundamental length scale in TGD and Planck length squared l_{P}^{2}= ℏ G =x^{-2}× 10^{-6}R^{2}defines a secondary length scale. For Planck mass squared one has m_{Pl}^{2}= m(CP_{2},ℏ)^{2}× 10^{6}x^{2}, m(CP_{2},ℏ)^{2}= ℏ/R^{2}. The estimate for x from p-adic mass calculations gives x≈ 4.2. It is assumed that CP_{2}length is fundamental and Planck length is a derived quantity.But what if one assumes that Planck length identifiable as CP

_{2}radius is fundamental and CP_{2}mass corresponds the minimal value h_{0}of h_{eff}(2)? That the mass formula is quadratic and mass is assignable to wormhole contact connecting two space-time sheets suggests in the Yangian framework that h_{eff}(2) is the correct Planck constant to consider.

_{2}length scale is deduced indirectly from p-adic mass calculation for electron mass assuming h

_{eff}=h and using Uncertainty Principle. This obviously leaves the possibility that R= l

_{P}apart from a numerical constant near unity, if the value of h

_{eff}to be used in the mass calculations is actually h

_{0}= (l

_{P}/R)

^{2}ℏ. This would fix the value of ℏ

_{0}uniquely.

The earlier interpretation makes sense if R(CP_{2}) is interpreted as a dark length scale obtained scaling up l_{P} by ℏ/ℏ_{0}. Also the ordinary particles would be dark.

h_{0} would be very small and α_{K}(ℏ_{0})= (ℏ/ℏ_{0})α_{K} would be very large so that the perturbation theory for it would not converge. This would be the reason for why ℏ and in some cases some smaller values of h_{eff} such as ℏ/2 and ℏ/4 seem to be realized.

For R=l_{P} Nottale formula remains unchanged for the identification M^{2}_{P}= ℏ/R^{2} (note that one could consider also ℏ_{0}/R^{2} used in p-adic mass calculations).

**Various options**

Number theoretical arguments allow to deduce precise value for the ratio ℏ/ℏ_{0}. Accepting the Yangian inspired picture, one can consider two options for what one means with ℏ.

- ℏ refers to the single particle Planck constant ℏ
_{eff}(1) natural for point-like particles. - ℏ refers to h
_{eff}(2). This option is suggested by the proportionality M^{2}∝ ℏ in string models due to the proportionality M^{2}∝ℏ/G in string models. At a deeper level, one has M^{2}∝ L_{0}, where L_{0}is a scaling generator and its spectrum has scale given by ℏ.Since M

^{2}is a p-adic thermal expectation of L_{0}in the TGD framework, the situation is the same. This also due the fact that one has In TGD framework, the basic building bricks of particles are indeed pairs of wormhole throats.

_{eff}→ kh

_{eff}.

** Option 1**: Masses are scaled by k and Compton lengths are unaffected.

** Option 2**: Compton lengths are scaled by k and masses are unaffected.

The interpretation of M_{P}^{2}= (ℏ/ℏ_{0}) M^{2}(CP_{2}) assumes Option 1
whereas the new proposal would correspond to Option 2 actually assumed in various applications.

The interpretation of M_{P}^{2}= (ℏ/ℏ_{0}) M^{2}(CP_{2}) assumes Option 1 whereas the new proposal would correspond to Option 2 actually assumed in various applications.

For Option 1 m_{Pl}^{2}= (ℏ_{eff}/ℏ) M^{2}(CP_{2}). The value of M^{2}(CP_{2})= ℏ/R^{2} is deduced from the p-adic mass calculation for electron mass. One would have R^{2} ≈ (ℏ_{eff}/ℏ) l_{P}^{2} with ℏ_{eff}/ℏ = 2.54× 10^{7}. One could say that the real Planck length corresponds to R.

**Quantum-classical correspondence favours Option 2)**

In an attempt to select between these two options, one can take space-time picture as a guideline. The study of the imbeddings of the space-time surfaces with spherically symmetric metric carried out for almost 4 decades ago suggested that CP_{2} radius R could naturally correspond to Planck length l_{P}. The argument is described in detail in Appendix and shows that the l_{P}=R option with h_{eff}=h used in the classical theory to determine α_{K} appearing in the mass formula is the most natural.

**Deduction of the value of ℏ/ℏ _{0}**

Assuming Option 2), the questions are following.

- Could l
_{P}=R be true apart from some numerical constant so that CP_{2}mass M(CP_{2}) would be given by M(CP_{2})^{2}= ℏ_{0}/l_{P}^{2}, where ℏ_{0}≈ 2.4× 10^{-7}ℏ (ℏ corresponds to ℏ_{eff}(2)) is the minimal value of ℏ_{eff}(2). The value of h_{0}would be fixed by the requirement that classical theory is consistent with quantum theory! It will be assumed that ℏ_{0}is also the minimal value of ℏ_{eff}(1) both ℏ_{eff}(2). - Could ℏ(2)/ℏ
_{0}(2)=n_{0}correspond to the order of the product of identical Galois groups for two Minkowskian space-time sheets connected by the wormhole contact serving as a building brick of elementary particles and be therefore be given as n_{0}=m^{2}?

_{0}=m

^{2}.

- The natural assumption is that Galois symmetry of the ground state is maximal so that m corresponds to the order a maximal Galois group - that is permutation group S
_{k}, where k is the degree of polynomial.This condition fixes the value k to k=7 and gives m=k!=7! = 5040 and gives n

_{0}= (k!)^{2}= 25401600=2.5401600 × 10^{7}. The value of ℏ_{0}(2)/ℏ(2)=m^{-2}would be rather small as also the value of ℏ_{0}(1)ℏ(1). p-Adic mass calculations lead to the estimate m_{Pl}/m(CP_{2})= m^{1/2}m(CP_{2})=4.2× 10^{3, which is not far from m=5040. } - The interpretation of the product structure S
_{7}× S_{7}would be as a failure of irreducibility so that the polynomial decomposes into a product of polynomials - most naturally defined for causally isolated Minkowskian space-time sheets connected by a wormhole contact with Euclidian signature of metric representing a basic building brick of elementary particles.Each sheet would decompose to 7 sheets. ℏ

_{gr}would be 2-particle Planck constant h_{eff}(2) to be distinguished from the ordinary Planck constant, which is single particle Planck constant and could be denoted by h_{eff}(1).The normal subgroups of S

_{7}× S_{7}S_{7}× A_{7}and A_{7}× A_{7}, S_{7}, A_{7}and trivial group. A_{7}is simple group and therefore does not have any normal subgroups expect the trivial one. S_{7}and A_{7}could be regarded as the Galois group of a single space-time sheet assignable to elementary particles. One can consider the possibility that in the gravitational sector all EQs are extensions of this extension so that ℏ becomes effectively the unit of quantization and m_{Pl}the fundamental mass unit. Note however that for very small values of α_{K}in long p-adic length scales also the values of h_{eff}<h, even h_{0}, are in principle possible.The large value of α

_{K}∝ 1/ℏ_{eff}for Galois groups with order not considerably smaller than m=(7!)^{2}suggests that very few values of h_{eff}(2)<h are realized. Perhaps only S_{7}× S_{7}S_{7}× A_{7}and A_{7}× A_{7}are allow by perturbation theory. Now however that in the "stringy phase" for which super-conformal invariance holds true, h_{0}might be realized as required by p-adic mass calculations. The alternative interpretation is that ordinary particles correspond to dark phase with R identified dark scale. - A
_{7}is the only normal subgroup of S_{7}and also a simple group and one has S_{7}/A_{7}= Z_{2}. S_{7}× S^{7}has S_{7}× S^{7}/A_{7}× A_{7}= Z_{2}× Z_{2}with n=n_{0}/4 and S_{7}× S^{7}/A_{7}× S_{7}= Z_{2}with n=n_{0}/2. This would allow the values ℏ/2 and ℏ/4 as exotic values of Planck constant.The atomic energy levels scale like 1/ℏ

^{2}and would be scaled up by factor 4 or 16 for these two options. It is not clear whether ℏ→ ℏ/2 option can explain all findings of Randel Mills in TGD framework, which effectively scale down the principal quantum number n from n to n/2. - The product structure of the Nottale formula suggests
n=n

_{1}× n_{2}= k_{1}k_{2}m^{2}.Equivalently, n

_{i}would be a multiple of m. One could say that M_{Pl}=(ℏ/ℏ_{0})^{1/2}M(CP_{2}) effectively replaces M(CP_{2}) as a mass unit. At the level of polynomials this would mean that polynomials are composites P○ P_{0}where P_{0}is ground state polynomial and has a Galois group with degree n_{0}. Perhaps S_{7}could be called the gravitational or ground state Galois group.

See the article Questions about coupling constant evolution or the chapter with the same title.

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

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