The abstract of the article is here.
Some comments about article are in order.
- The emphasis of the article is on the rules satisfied by resonance frequencies. The vanishing of the sum of rotation frequencies with integer weights is one manner to end up with the quantization rules and allowing classical interpretation in terms of KAM theory. In TGD approach a genuine quantization of dark matter based on hierarchy of Planck constants is the explanation for the rules. Integer multiples of basic frqeuency scale implied by resonance conditions fit nicely with the number theoretical quantization rules requiring rationals.
- These resonance rules follow automatically from Bohr quantization for systems for which the values of various physical parameters are simple rationals with a suitable choice of units. This is of course true for hydrogen atom type systems, harmonic oscillator, etc.. Also known exoplanets demonstrate this quantization many of them with orbital radius corresponding to the ground state. In TGD framework space-time surface are preferred extremals of Kähler action (not simply absolute minima as assumed initially) so that Bohr orbitology is coded into the basic definition of the theory and classical theory is genuine part of quantum theory. Quantum criticality is second essential element: in accordance with general ideas about quantum chaos it implies that dark matter wave functions are concentrated around classical Bohr orbits.
- In TGD framework the quantization for gravitational Planck constant hbargr= GMm/v0, where v0 =2-11 is the most preferred value, is also essential element. The form of gravitational Planck constant follows from Equivalence Principle. The dependence on both M and m is difficult to understand in the context of standard quantum field theory but makes perfect sense if it characterizes the gravitational "field body" mediating gravitational interaction between two systems. In principle each interaction corresponds to its own "field body" and hbar. This requires a profound generalization of the notion of imbedding space.
- Also certain harmonics and subharmonics of v0 appear and rational valued spectrum is the most general one. Number theoretically simple ruler-and-compass rationals (corresponding quantum phases exp(i2π/n) are expressible in terms of square root operations applied to rationals) allow to understand quantization in solar system. The quantization of hbargr leads also to number-theoretic predictions for mass ratios of planets as ruler-and-compass rationals and typically holding true with accuracy better than 10 per cent.
- Titius-Bode type rules (not necessarily only powers of 2) are discussed in the article as a manner to satisfy the resonance conditions. Titius-Bode rules have also interpretation in terms of p-adic length scale hypothesis favoring powers of 2, and quantization rules starting from continuous distribution of matter around planets lead naturally to this kind of rules. These rules conform reasonably well with the Bohr type quantization rules for hydrogen atom but the predictions of hydrogen atom quantization are more accurate. It is possible that also inside Sun similar onionlike hierarchy in powers of two holds true.
- Both general coordinate invariance and Poincare invariance of is a serious problem for the Bohr rule based quantization in General Relativity context. In TGD framework space-times are 4-surfaces so that Minkowski coordinates for M4×CP2 define preferred coordinates. An important prediction is that astrophysical quantum states cannot participate in cosmological expansion except by quantum transitions in which gravitational Planck constant changes. Masreliez observed for years ago that planetary system seems to shrink and the explanation is that the distances in this approach correpond to the radial coordinate of Robertson-Walker metric rather than genuine distance. This picture allows also to understand different value of hgr for inner and outer planets as signature of past dynamics of solar system. Also the acceleration of cosmic expansion can be understood as associated with the criticality associated with quantum phase transition increasing hgr in cosmological length scale (large voids).
- Khodolenko proposes Exclusion Principle for planetary systems. It is not needed in the picture where macroscopic quantization is genuine physical effect and associated with dark matter. Visible matter condensed around dark matter and makes the presence of it "visible" via approximate Bohr rules.