The truth about quantization of Planck constants

The development of ideas about quantization of Planck constants has been a trial-and-error process which only a person like me possessing an exceptionally fuzzy and pathologically associative brain function could go through. I however try to be honest and I can convinve the reader that there is no danger that reading the scandalous truth could infect the brain of reader with the same intolerable fuzziness.

The moment of truth

1. The big idea was that the claims for the Bohr quantization of planetary radii with a gigantic value of Planck constant (called cautiously gravitational Planck constant at that stage) could be understood as a genuine quantization of Planck constant for dark matter serving as a template for ordinary matter. Recall that Nottale had already proposed effective quantization based on hydrodynamics. As a warm up exercise I went on to propose a rather complex and wrong formula for the gravitational Planck constant in terms of Beraha numbers B_n= 4cos²(π/n) assignable to quantum phases q=exp(iπ/n). The inspiration came from Jones inclusions which I believed to relate closely to the quantization.

2. Soon I realized that the formula for Planck constant does not really work. Anyonic arguments based on the idea that Riemann surface like coverings of M⁴ are involved, led to an extremely simple formula for h as h=n×h₀.

3. It became also clear that as far Lie algebras of symmetries are considered, there are actually two Planck constants corresponding to Lie-algebra commutators associated with M⁴_+/- and CP₂ degrees of freedom and that these Planck constants would correspond to Jones inclusions of Clifford subalgebras identifiable in terms of gamma matrix algebras for the world of classical worlds consisting of 3-surfaces in H. These inclusions are characterized by subgroups G_a of SU(2) subset SL(2,C) and G_b of SU(2) subset SU(3). These discrete groups define orbifold coverings of M⁴_+/- by G_b related CP₂ points and vice versa so that a generalization of the notion of imbedding space emerges.

   It also became clear that one must glue various copies of imbedding space along M⁴_+/- in the case that G_b is same for two copies and vice versa. This generalization was easy to discover since also the p-adic variants of the imbedding space are glued together in same manner along common rational and algebraic points making it possible to fuse real and various p-adic physics to single coherent whole. Now this fusion has reached a rather concrete form and involves a lot of fascinating number theory, in particular Riemann Zeta.

   To be honest, I tried to cheat here;-). It took some time to realize that strictly speaking it is M⁴_+/- rather than M⁴ whose coverings are involved but I could argue that this is just an inaccurate use of language. Sincerely, in the middle of a flood of ideas rusing through your head you do not notice this kind of details.

4. It became also clear that the covariant metric of M⁴_+/- must be proportional to n_b² and that of CP₂ proportional to n_a². Here n_a is the order of maximal cyclic subgroup of G_a and n_b the order of maximal cyclic subgroup of G_b.

5. By an anyonic argument Planck constants are given by h(M⁴_+/-)=n_a h₀. For some time I however believed that h(M⁴_+/-)=n_b h₀ so that Schrödinger equation would be invariant under phase transition changing the Planck constants in an obvious contradiction with the the quantization of planetary orbits requiring gigantic Planck constant(!).

6. There was also the characteristic fuzziness in the identification of the observed Planck constant. As one might guess, I made first the wrong identification as h(M⁴_+/-) although I had realized from the beginning that only the ratio n_a/n_b appears in the Kähler action and that the natural interpretation for the nonlinearity is a universal geometric coding of radiative corrections to the induced geometry of the space-time sheet via the nonlinear dependence on the induced metric. Therefore the only sensible conclusion would have been that the observed Planck constant is given by h_eff/h₀= n_a/n_b! This predicts that Planck constant can in principle have all rational values: both larger and smaller than the value for the ordinary matter.

7. In the middle of this flux of good and bad ideas emerged also the hypothesis that the preferred values of n_a and n_b correspond to integers defining n-polygons constructible using ruler and compass alone. The motivation was that quantum phases are expressible in this case using only iterated square rooting of rationals so that number theoretically (and thus cognitively) simple levels of hierarchy of algebraic extensions of p-adic number fields expected to be abundant in cosmos would be in question. One would have n_F= 2^k ∏_s;F_s, where F_s =2^2s +1 are Fermat primes. The known Fermat primes are 3,5,17,257, and 2¹⁶+1. This would mean that the preferred values of h_eff are given as ratios of these integers. In living matter the fractal hierarchy n_F=2^11k seems to be favored and the number 2¹¹ corresponds to a fundamental dimensionless constant in TGD.

And now a desperate attempt to defend myself

After having revealed this scandalous multiple blundering process which has lasted two years I still dare to hope that reader has not made her final conclusions and is willing to listen my excuses. I sincerely hope that some examples about how the quantization of Planck constants could manifest itself in physics anomalies might induce merciful feelings also in the readers who identify themselves as serious scientists.

1. The evidence for Bohr quantization of planetary orbits can be interpreted in terms of a huge value of h_eff= GMm/v₀, where M and m are the masses of, say, Sun and planet (for Nottale's orginal paper explaining quantization hydrodynamically see astro-ph/0310036). The ruler-compass hypothesis means very strong constraints on the ratios of planetary masses satisfied with an accuracy of 3 per cent and also on solar mass satisfied if the fraction of non-dark matter is around 4 per cent. A dramatic prediction is that in this phase the value of n_a is gigantic meaning that dark matter obeys spatial symmetry corresponding to either the cyclic group Z_na or group obtained by adding planar reflection to it. Ring like structures of dark matter analogous to ring like structures analogous to benzene ring in chemistry perhaps assignable to planetary orbits suggest themselves (see this).

   An even more dramatic prediction is that the scaling of CP₂ metric by n_a means that it has astrophical size so that in dark sector imbedding space looks like uncompactified M⁸ in human length scales! But this is dark sector and since the mass spectra of elementary particles do not depend at all on the values of Planck constants no obvious contradictions with observations are predicted! Contrary to what super string model suggest, big hyper-space dimensions would not be seen in particle accelerators but in astrophysics.

2. Hierarchy of scaled variants of atomic physics

The binding energy scale of hydrogen atom is proportional to 1/h_eff²= (n_b/n_a)² so that a fractionization of occurs.

1. The findings of Mills about fractionization of energy spectrum of hydrogen atom (hydrino atom) with scaling factor k=2,3,4,5,6,7,9,10 can be understood for k= n_b/n_a. For some time ago I constructed a model for hydrino using q-Laquerre equation and predicting k=2 as a new state. Also k> 2 result approximately. It seems that these states could serve as intermediate states in the transition to a phase with modified Planck constants. In this case effective Planck constant is smaller than its standard value and the sizes of hydrino atoms are smaller. Also zoomed up versions of ordinary atoms with identical chemical properties but sizes scaled up by n_a=n_b are predicted.

2. Exotic atoms with increased sizes and reduced binding energy scale are predicted and the integers n_F are especially interesting. Atomic nucleus can be or ordinary and one obtains N-atoms by putting electron on several sheets of n_a-fold covering of CP₂. This leads to a model for hydrogen bonds and active catalyst sites and for how symbolic level emerges in bio-chemistry. It is also possible that only the valence electrons of ordinary atom are in dark phase (live at different branch of generalized imbedding space) so that only these electronic orbitals are scaled up. This could make possible anomalous conductivity and even super-conductivity.

   For instance, the 5- and 6-rings characteristic for the fundamental bio-molecules (sugars, DNA, important neurotransmitters including those containing four aminoacids having 5- and 6-cycles, hallucinogens) could correspond to n_a= 5 or 6 for free electron pairs characterizing these rings. The mysterious conductivity of DNA (Science (1997), vol. 275, 7. March 1997) could be understood in terms of the delocalization of the aromatic electron pairs associated with the 5- and 6-rings due to n_a² fold scaling of the orbitals making possible overlap between the rings in the DNA ladder (see this).

3. The weird looking properties of graphene (in particular its high conductivity) forming hexagonal carbon atom lattice of thickness of single atom could be understood if one has n_a=n_b=6 for the free electron pairs assignable to Carbon rings. TGD based model for particle massivation implying that conformal weight and thus mass squared is additive for hadron type bound states of partons can also explain neatly why conduction electrons behave as massless particles. The mass of bound state parton is m²- p_T² and transverse momentum squared p_T² can compensate the mass of quark/electron completely. This could also explain why massive quarks seem to behave as massless particles inside hadrons. TGD also predicts that warped vacuum imbedding without gravitational fields can induce anomalous time dilation and large reduction of effective light velocity: this could explain why light velocity for these electrons is c/300 (see this).

3. Anomalous properties of water

   The anomalous properties of water, in particular the chemical formula H_1.5O suggesting itself in attosecond scale, provided first challenge for the proposed model of dark matter (see this). Tedrahedral and icosahedral clusters are characteristic for water and the corresponding subgroups of SU(2) correspond to the exceptional Lie groups E₆ and E₈ via ADE correspondence: these are the only genuinely 3-dimensional discrete subgroups of SU(2). The value of Planck constant would be scaled up by a factor n_a=3 or 5 for these sub-groups and n_a=5 is the minimal value making possible topological quantum computation using braid S-matrix. Icosahedral and dual odecahedral structures are abundant in living matter: viruses being only one example.

4. Mono-atomic elements

   Mono-atomic elements or ORMEs (see this) are transition elements claimed by Hudson to have strange properties. They are claimed to be non-visible in ordinary emission spectroscopy but become visible after a time which is 90 s instead of 15 seconds for ordinary elements in typical case. The ratio of times is 6 and Golden rules suggests that one has n_a=6 at least for valence electrons. Chemistry would not be affected if one has n_b=6 too. The atomic clusters are predicted to have hexagonal symmetry. The other strange properties of these compounds such as claimed super-conductivity suggest that also the phase with n_b=1 is present as indeed required by the general scenario for phase transitions changing the values of Planck constants as a leakage between different sectors of the imbedding space.

5. Dark EEG hierarchy

   In dark phase the energy associated with a photon of given frequency is scaled up by a factor n_a. If n_a is large enough, even EEG photons can have energies above thermal energy. The finding that ELF photons with frequences which are harmonics of cyclotron frequencies for biologically important ions have effects on vertebrate brain supports this idea. This observation leads to a model for a fractal hierarchy of EEGs based on the assumption that the integers n_a=2^11k are especially favored: the motivation is that 2¹² corresponds to a fundamental dimensionless constant in TGD. Even individual narrow peaks in beta and theta bands are predicted correctly (see this).

The chapter Does TGD Predict the Spectrum of Planck Constants? of "Towards S-matrix" explains in detail the recent view about the quantization of Planck constants.