Still a little correction to the quantization of Planck constants

The Jones inclusions for the sub-algebras of infinite-dimensional Clifford algebras for world of classical worlds, or more technically configuration space of 3-surfaces, should by quantum classical correspondence have classical space-time correlates. Indeed Jones inclusions N into M based on groups G_a× G_b acting as invariance group of elements of N have precise classical correlate at space-time level. The bundle projections H→ H/G_a× G_b for singular G_a× G_b bundle represent geometric duals for Jones inclusions. The gauge fixing assigning to each point of H/G_a× G_b point of H would correspond to Jones inclusion.

The generalized imbedding space is obtained by gluing all these copies of H with singular bundle structure together. If G_a is common group then gluing occurs isometrically along M⁴_+/- factor and if G_b is common group then same occurs isometrically along common CP₂ factors. More precisely, the Cartesian product of these factors with with fixed points in the second factors is shared by the two copies of H. The common points of M⁴_+/- (CP₂) factors correspond to the singular orbifold points of bundle remaining invariant under the two groups G_a (G_b). For G_a:s acting as plane rotations and reflections the set of singular points is time-like plane corresponding to the choice of rest system and a unique quantization axis for angular momentum so that Jones hierarchy has interpretation in terms of quantum measurement theory. If G_a corresponds to the symmetries of tedrahedron or icosahedron (exceptional group E₆ or E₈ by McKay correspondence) then the set of singular points reduces to a time-like line and the choice of quantization axes is not unique since maximal cyclic subgroup can perform rotations around any 3- (5-) symmetry axis of tedrahedron (icosahedron).

The quantization of Planck constants h(M⁴_+/-)=n_ah₀ and h(CP₂)=n_bh₀ and its geometric counterpart at imbedding space level is now also reasonably well understood. The original naive argument led to the guess that M⁴ and CP₂ covariant metrics are scaled by n_b² and n_a² respectively. In the case of CP₂ metric this scaling however implies a gigantic size of CP₂ for dark matter in astrophysical length scales: effectively imbedding space would look like 8-D Minkowski space. This looks weird. Also a mathematical problem emerges in the attempt to glue imbedding spaces with same group G_b isometrically along the common CP₂ factors since the sizes of CP₂:s are not same and gluing can be only partial and would be discontinuous. For M⁴_+/- factors with different scaling factors of metric non-compactness allows to circumvent the problem.

The resolution of the problem was trivial. Kähler action is invariant under over-all scaling of the metric of H so that one can perform the 1/n_a² scaling for H metric besides the naive scalings and this implies that M⁴_+/- metric is scaled by (n_b/n_a)² and CP₂ metric remains invariant (as is natural since projective space is in question). Ordinary Planck constant has a purely geometric interpretation. The quantitative predictions of the existing scenario are not affected by the more precise view about situation since quantum dynamics does not depend on the overall scaling of H metric.

For more details see the chapter Does TGD Predict the Spectrum of Planck Constants? or the chapter TGD and Astrophysics.