### Magnetic bodies and hierarchy of Planck constants

The proposed hierarchy of Planck constants is given as h(M

^{4}

_{+/-})=n

_{a}h

_{0}and h(CP

_{2})=n

_{b}h

_{0}, 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 CP

_{2}by G

_{a}related M

^{4}

_{+/-}points and G

_{b}covering of M

^{4}

_{+/-}by G

_{b}related CP

_{2}points. Fixed points correspond to orbifold points. The covariant metric of M

^{4}

_{+/-}(CP

_{2}) is proportional to n

_{b}

^{2}(n

_{a}

^{2}). Different copies of imbedding space are glued together along common M

^{4}

_{+/-}or CP

_{2}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})h

_{0}.

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_{2n}which 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/v_{0}, v_{0}=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.

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