Revised vision about quantum astrophysics
I had deduced earlier a formula for the quantized Planck constant from the requirement that it represents algebraic homomorphism. Two options for which Planck constants were inverses of each other were possible. As usual, I chose the wrong one! The development of a detailed model for fractional quantum Hall effect fixed the choice on basis of physical arguments. The next task is to go through all applications and make the needed modifications. I started from Quantum Astrophysics. A glue below the abstract.
The vision that the quantum dynamics for dark matter is behind the formation of the visible structures suggests that the formation of the astrophysical structures could be understood as a consequence of gravitational Bohr rules. The origin of these rules has remained a little bit mysterious until the discovery that the hierarchy of Planck constants relates very closely to anyons and fractionization of quantum numbers.
 Key element is the notion of partonic 2surface, which for large values of Planck constant can have astrophysical size. This surface contains dark matter in anyonic many particle state if it surrounds the tip of so called causal diamond (the intersection of future and past directed lightcones). Also flux tubes surrounding the orbits of planets and other astrophysical objects containing dark matter would be connected by radial flux tubes to central anyonic 2surface so that the entire system would be anyonic and quantum coherent in astrophysical scale. Visible matter is condensed around these dark matter structures.
 Since spacetimes are 4surfaces in H=M^{4}×CP_{2} (or rather, its generalization to a book like structure), gravitational Bohr rules can be formulated in a manner which is general coordinate invariant and Lorentz invariant.
 The value of the parameter v_{0} appearing in gravitational Planck constant varies and this leads to a weakened form of Equivalence Principle stating that v_{0} is same for given connected anyonic 2surface, which can have very complex topology. In the case of solar system inner planets would be connected to an anyonic surface assignable to Sun and outer planets with different value of v_{0} to an anyonic surface assignable to Sun and inner planets as a whole. If one accepts rulerandcompass hypothesis for allowed values of Planck constant very powerful predictions follow.
This general conceptual framework is applied to build simple models in some concrete examples.
 Concerning Bohr orbitology in astrophysical length scales, the basic observation is that in the case of a straight cosmic string creating a gravitational potential of form v_{1}^{2}/r Bohr quantization does not pose any conditions on the radii of the circular orbits so that a continuous mass distribution is possible. This situation is obviously exceptional. If one however accepts the TGD based vision that the very early cosmology was cosmic string dominated and that elementary p"/public_html/articles/ were generated in the decay of cosmic strings, this situation might have prevailed at very early times. If so, the differentiation of a continuous density of ordinary matter to form the observed astrophysical structures would correspond to an approach to a stationary situation governed by Bohr rules for dark matter and in the first approximation one could neglect the intermediate stages.
 This general picture is applied by considering some simple models for astrophysical systems involving planar structures. There are several universal predictions. Velocity spectrum is universal and only the Bohr radii depend on the choice of mass distribution. The inclusion of cosmic string implies that the system associated with the central mass is finite. Quite generally dark parts of astrophysical objects have shell like structure like atoms as do also ring like structures.
 pAdic length scale hypothesis provides a manner to obtain a realistic model for the central objects meaning a structure consisting of shells coming as half octaves of the basic radius: this obviously relates to TitiusBode law. Also a simple model for planetary rings is obtained. Bohr orbits do not follow cosmic expansion which is obtained only in the average sense if phase transitions reducing the value of basic parameter v_{0} occur at preferred values of cosmic time. This explains why v_{0} has different values and also the decomposition of planetary system to outer and inner planets with different values of v_{0}.
TGD Universe is quantum critical and quantum criticality corresponds very naturally to what has been identified as the transition region to quantum chaos.
 The basic formulation of quantum TGD is consistent with what has been learned from the properties of quantum chaotic systems and quantum chaotic scattering. Wave functions are concentrated around Bohr orbits in the limit of quantum chaos, which is just what dark matter picture assumes.
 The model for the emission and detection of dark gravitons allows to conclude that the transition to chaos via generation of subharmonics of fundamental frequency spoiling the original exact periodicity corresponds to a sequence of phase transitions in which Planck constant transforms from integer to a rational number whose denominator increases as chaos is approached. This gives a precise characterization for the phase transitions leading to quantum chaos in general.
 In this framework the chaotic motion of astrophysical object becomes the counterpart of quantum chaotic scattering and the description in terms of classical chaos is predicted to fail. By Equivalence Principle the value of the mass of the object does not matter at all so that the motion of sufficiently light objects in solar system might be understandable only as quantum chaotic scattering. The motion of gravitationally unbound comets and rings of Saturn and Jupiter and the collisions of galactic structures known to exhibit the presence of cartwheel like structures define possible applications.
The description of gravitational radiation provides a stringent test for the idea about dark matter hierarchy with arbitrary large values of Planck constants. In accordance with quantum classical correspondence, one can take the consistency with classical formulas as a constraint allowing to deduce information about how dark gravitons interact with ordinary matter. The standard facts about gravitational radiation are discussed first and then TGD based view about the situation is sketched.
For details and background see the updated chapter Quantum Astrophysics.
