Could one understand and perhaps even predict the minimal value h0of heff? Here number theory and the notion of n-particle Planck constant heff(n) suggested by Yangian symmetry could serve as a guidelines.
- Hitherto I have found no convincing empirical argument fixing the value of r=ℏ/ℏ0: this is true for both single particle and 2-particle case.
The value h0=h/6 as a maximal value of h0 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 Z6=Z2× Z3. Smaller values of h0 cannot be however excluded.
- A possible manner to understand the value r geometrically would be following. It has been assumed that CP2 radius R defines a fundamental length scale in TGD and Planck length squared lP2= ℏ G =x-2 × 10-6R2 defines a secondary length scale. For Planck mass squared one has mPl2= m(CP2,ℏ)2× 106x2 , m(CP2,ℏ)2= ℏ/R2. The estimate for x from p-adic mass calculations gives x≈ 4.2. It is assumed that CP2 length is fundamental and Planck length is a derived quantity.
But what if one assumes that Planck length identifiable as CP2 radius is fundamental and CP2 mass corresponds the minimal value h0 of heff(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 heff(2) is the correct Planck constant to consider.
The earlier interpretation makes sense if R(CP2) is interpreted as a dark length scale obtained scaling up lP by ℏ/ℏ0. Also the ordinary particles would be dark.
h0 would be very small and αK(ℏ0)= (ℏ/ℏ0)αK 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 heff such as ℏ/2 and ℏ/4 seem to be realized.
For R=lP Nottale formula remains unchanged for the identification M2P= ℏ/R2 (note that one could consider also ℏ0/R2 used in p-adic mass calculations).
Various options
Number theoretical arguments allow to deduce precise value for the ratio ℏ/ℏ0. Accepting the Yangian inspired picture, one can consider two options for what one means with ℏ.
- ℏ refers to the single particle Planck constant ℏeff(1) natural for point-like particles.
- ℏ refers to heff(2). This option is suggested by the proportionality M2∝ ℏ in string models due to the proportionality M2∝ℏ/G in string models. At a deeper level, one has M2 ∝ L0, where L0 is a scaling generator and its spectrum has scale given by ℏ.
Since M2 is a p-adic thermal expectation of L0 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.
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 MP2= (ℏ/ℏ0) M2(CP2) assumes Option 1 whereas the new proposal would correspond to Option 2 actually assumed in various applications.
The interpretation of MP2= (ℏ/ℏ0) M2(CP2) assumes Option 1 whereas the new proposal would correspond to Option 2 actually assumed in various applications.
For Option 1 mPl2= (ℏeff/ℏ) M2(CP2). The value of M2(CP2)= ℏ/R2 is deduced from the p-adic mass calculation for electron mass. One would have R2 ≈ (ℏeff/ℏ) lP2 with ℏeff/ℏ = 2.54× 107. 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 CP2 radius R could naturally correspond to Planck length lP. The argument is described in detail in Appendix and shows that the lP=R option with heff=h used in the classical theory to determine αK appearing in the mass formula is the most natural.
Deduction of the value of ℏ/ℏ0
Assuming Option 2), the questions are following.
- Could lP=R be true apart from some numerical constant so that CP2 mass M(CP2) would be given by M(CP2)2= ℏ0/lP2, where ℏ0≈ 2.4× 10-7 ℏ (ℏ corresponds to ℏeff(2)) is the minimal value of ℏeff(2). The value of h0 would be fixed by the requirement that classical theory is consistent with quantum theory! It will be assumed that ℏ0 is also the minimal value of ℏeff(1) both ℏeff(2).
- Could ℏ(2)/ℏ0(2)=n0 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 n0=m2?
- 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 Sk, where k is the degree of polynomial.
This condition fixes the value k to k=7 and gives m=k!=7! = 5040 and gives n0= (k!)2= 25401600=2.5401600 × 107. The value of ℏ0(2)/ℏ(2)=m-2 would be rather small as also the value of ℏ0(1)ℏ(1). p-Adic mass calculations lead to the estimate mPl/m(CP2)= m1/2 m(CP2)=4.2× 103, which is not far from m=5040.
- The interpretation of the product structure S7 × S7 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. ℏgr would be 2-particle Planck constant heff(2) to be distinguished from the ordinary Planck constant, which is single particle Planck constant and could be denoted by heff(1).
The normal subgroups of S7 × S7 S7× A7 and A7× A7, S7, A7 and trivial group. A7 is simple group and therefore does not have any normal subgroups expect the trivial one. S7 and A7 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 mPl the fundamental mass unit. Note however that for very small values of αK in long p-adic length scales also the values of heff<h, even h0, are in principle possible.
The large value of αK ∝ 1/ℏeff for Galois groups with order not considerably smaller than m=(7!)2 suggests that very few values of heff(2)<h are realized. Perhaps only S7 × S7 S7× A7 and A7× A7 are allow by perturbation theory. Now however that in the "stringy phase" for which super-conformal invariance holds true, h0 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.
- A7 is the only normal subgroup of S7 and also a simple group and one has S7/A7= Z2. S7× S7 has S7× S7/A7× A7= Z2× Z2 with n=n0/4 and S7× S7/A7× S7= Z2 with n=n0/2. This would allow the values ℏ/2 and ℏ/4 as exotic values of Planck constant.
The atomic energy levels scale like 1/ℏ2 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=n1× n2 = k1k2m2 .
Equivalently, ni would be a multiple of m. One could say that MPl=(ℏ/ℏ0)1/2M(CP2) effectively replaces M(CP2) as a mass unit. At the level of polynomials this would mean that polynomials are composites P○ P0 where P0 is ground state polynomial and has a Galois group with degree n0. Perhaps S7 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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