Thursday, November 09, 2006

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 Ga× Gb acting as invariance group of elements of N have precise classical correlate at space-time level. The bundle projections H→ H/Ga× Gb for singular Ga× Gb bundle represent geometric duals for Jones inclusions. The gauge fixing assigning to each point of H/Ga× Gb 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 Ga is common group then gluing occurs isometrically along M4+/- factor and if Gb 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 Ga (Gb). For Ga: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 Ga 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+/-)=nah0 and h(CP2)=nbh0 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 nb2 and na2 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 Gb 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/na2 scaling for H metric besides the naive scalings and this implies that M4+/- metric is scaled by (nb/na)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.


At 4:30 PM, Blogger Kea said...

Hi Matti

I just noticed that new book on The New Physics and The Mind. Cool stuff. And don't worry - my friend and I have figured out how to calculate stuff. Look out for a paper soon!

At 8:07 PM, Blogger Matti Pitkanen said...

I think I have visited your blog. There is a lot to calculate and made precise. All that is known from N=4 super-conformal invariance would be extremely helpful but I am so unanalytic and my brain is too old and too lazy;-).

I have been developing powerpoint representations summarizing TGD and related things. The inspiration to create order to this massive chaos comes from my travel to Hungary. They can be found at my homepage. It would be interesting to known whether anyone understands anything about them.



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