Tree like structure of the extended imbedding space

The quantization of hbar in multiples of integer n characterizing the quantum phase q=exp(iπ/n) in M⁴ and CP₂ degreees of freedom separately means also separate scalings of covariant metrics by n₂ in these degrees of freedom. The question is how these copies of imbedding spaces are glued together. The gluing of different p-adic variants of imbedding spaces along rationals and general physical picture suggest how the gluing operation must be carried out.

Two imbedding spaces with different scalings factors of metrics are glued directly together only if either M⁴ or CP₂ scaling factor is same and only along M⁴ or CP₂. This gives a kind of evolutionary tree (actually in rather precise sense as the quantum model for evolutionary leaps as phase transitions increasing hbar(M⁴) demonstrates!). In this tree vertices represent given M⁴ (CP₂) and lines represent CP₂:s (M⁴:s) with different values of hbar(CP₂) (hbar(M⁴)) emanating from it much like lines from from a vertex of Feynman diagram.

1. In the phase transition between different hbar(M⁴):s the projection of the 3-surface to M⁴ becomes single point so that a cross section of CP₂ type extremal representing elementary particle is in question. Elementary particles could thus leak between different M⁴:s easily and this could occur in large hbar(M⁴) phases in living matter and perhaps even in quantum Hall effect. Wormhole contacts which have point-like M⁴ projection would allow topological condensation of space-time sheets with given hbar(M⁴) at those with different hbar(M⁴) in accordance with the heuristic picture.

2. In the phase transition different between CP₂:s the CP₂ projection of 3-surface becomes point so that the transition can occur in regions of space-time sheet with 1-D CP₂ projection. The regions of a connected space-time surface corresponding to different values of hbar (CP₂) can be glued together. For instance, the gluing could take place along surface X³=S²× T (T corresponds time axis) analogous to black hole horizon. CP₂ projection would be single point at the surface. The contribution from the radial dependence of CP₂ coordinates to the induced metric giving ds²= ds²(X³)+g_rrdr² at X³ implies a radial gravitational acceleration and one can say that a gravitational flux is transferred between different imbedding spaces.

   Planetary Bohr orbitology predicting that only 6 per cent of matter in solar system is visible suggests that star and planetary interiors are regions with large value of CP₂ Planck constant and that only a small fraction of the gravitational flux flows along space-time sheets carrying visible matter. In the approximation that visible matter corresponds to layer of thickness Δ R at the outer surface of constant density star or planet of radius R, one obtains the estimate Δ R=.12R for the thickness of this layer: convective zone corresponds to Δ R=.3R. For Earth one would have Δ R≈ 70 km which corresponds to the maximal thickness of the crust. Also flux tubes connecting ordinary matter carrying gravitational flux leaving space-time sheet with a given hbar (CP₂) at three-dimensional regions and returning back at the second end are possible. These flux tubes could mediate dark gravitational force also between objects consisting of ordinary matter.

Concerning the mathematical description of this process, the selection of origin of M⁴ or CP₂ as a preferred point is somewhat disturbing. In the case of M⁴ the problem disappears since configuration space is union over the configuration spaces associated with future and past light cones of M⁴: CH= CH⁺U CH^-, CH^+/-= U_{m in M⁴} CH^+/-_m. In the case of CP₂ the same interpretation is necessary in order to not lose SU(3) invariance so that one would have CH^+/-= U_{h in H} CH^+/-_h. A somewhat analogous but simpler book like structure results in the fusion of different p-adic variants of H along common rationals (and perhaps also common algebraics in the extensions).

For details see the chapter Does TGD Predict the Spectrum of Planck Constants of "TGD: an Overview".