Magnetic bodies and hierarchy of Planck constants

The proposed hierarchy of Planck constants is given as h(M⁴_+/-)=n_ah₀ and h(CP₂)=n_bh₀, where the integers n_a and n_b correspond to orders of the maximal cyclic subgroups of groups G_a subset SU(2) subset SL(2,C) (Lorentz group) and G_b subset SU(2) subset SU(3) (color group). G_a defines covering of CP₂ by G_a related M⁴_+/- points and G_b covering of M⁴_+/- by G_b related CP₂ points. Fixed points correspond to orbifold points. The covariant metric of M⁴_+/- (CP₂) is proportional to n_b² (n_a²). Different copies of imbedding space are glued together along common M⁴_+/- or CP₂ factors to form an infinite tree with each noding containing infinitely many branches. The effective Planck constant appearing in Schrödinger equation is given by h_eff= (n_a/n_b)h₀.

The question concerns the interpretation of G_a. The first observation that apart from two exceptions, which correspond to symmetries of tedrahedron and dodecahedron, the group G_a acts in plane. G_a =Z_n consists of rotations by multiples of 2π/n and for G_a =D_2n rotations by 2π/2n combined with reflection. The orbit of D_2nwhich could be genuinely 3-dimensional decomposing to discrete orbits of Z_2n above and below plane. For small values of n say, n=5 and 6 orbits correspond to cycles and a highly attractive idea is that 5- and 6-cycles appearing in the fundamental bio-chemistry correspond to these orbits. Electron pairs are associated with 5- and 6-rings and the hypothesis would be that these pairs are in dark phase with n_a=5 or 6. Graphene which is a one-atom thick hexagonal lattice could be also an example of (conduction) electronic dark matter with n_a=6.

For some of the proposed physical applications the order n_a/n_b is very large. In particular, the Bohr quantization of planetary orbits, which stimulated the idea about dark matter as a large Planck constant phase, requires n_a/n_b= GMm/v₀, v₀=2^-11 so that the values are gigantic. A possible interpretation is in terms of a dark (gravi)magnetic body assignable to the system playing a key role in TGD inspired quantum biology. Topological quantization of magnetic field means a decomposition of the (gravi)magnetic field to flux tubes represented as space-time sheets. In the case of (say) dipole field the rotational and mirror symmetry with respect to the dipole axis would break down to Z_n or D_2n. This would also correspond to a transition to a phase with quantum group symmetry. Of course, the field body could contain also electric part consisting of radial flux tubes and obeying same symmetry. If star and each planet interact via a field body connecting only them, one could understand why the gravitational Planck constant depends on both M and m rather than being something universal.

Particles of dark matter would reside at the flux tubes but would be delocalized (exist simultaneously at several flux tubes) and belonging to irreducible representations of G_a. What looks weird is that one would have an exact macroscopic or even astroscopic symmetry at the level of generalized imbedding space. Visible matter would reflect this symmetry approximately. This representation would make sense also at the level of biochemistry and predict that magnetic properties of 5- and 6-cycles are of special significance for biochemistry. Same should hold true for graphene.

The chapter Does TGD Predict the Spectrum of Planck Constants? of "Towards S-matrix" contains more details.