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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.
- 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.
- 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)+grrdr2
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+/-=
Um 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+/-= Uh 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.
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