Quantum Hall effect and Hierarchy of Planck Constants

I have already earlier proposed the explanation of FQHE, anyons, and fractionization of quantum numbers in terms of hierarchy of Planck constants realized as a generalization of the imbedding space H = M×CP2 to a book like structure. The book like structure applies separately to CP2 and to causal diamonds (CD ⊂ M) defined as intersections of future and past directed light-cones. The pages of the Big Book correspond to singular coverings and factor spaces of CD (CP2) glued along 2-D subspace of CD (CP2) and are labeled by the values of Planck constants assignable to CD and CP2 and appearing in Lie algebra commutation relations. The observed Planck constant h̄, whose square defines the scale of M metric corresponds to the ratio of these Planck constants. The key observation is that fractional filling factor results if h̄ is scaled up by a rational number. In this chapter I try to formulate more precisely this idea. The outcome is a rather detailed view about anyons on one hand, and about the Kähler structure of the generalized imbedding space on the other hand. 1. Fundamental role is played by the assumption that the Kähler gauge potential of CP2 contains a gauge part with no physical implications in the context of gauge theories but contributing to physics in TGD framework since U(1) gauge transformations are representations of symplectic transformations of CP2. Also in the case of CD it makes also sense to speak about Kähler gauge potential. The gauge part codes for Planck constants of CD and CP2 and leads to the identification of anyons as states associated with partonic 2surfaces surrounding the tip of CD and fractionization of quantum numbers. Explicit formulas relating fractionized charges to the coefficients characterizing the gauge parts of Kähler gauge potentials of CD and CP2 are proposed based on some empirical input. 2. One important implication is that Poincare and Lorentz invariance are broken inside given CD although they remain exact symmetries at the level of the geometry of world of classical worlds (WCW). The interpretation is as a breaking of symmetries forced by the selection of quantization axis. 3. Anyons would basically correspond to matter at 2-dimensional ”partonic” surfaces of macroscopic size surrounding the tip of the lightcone boundary of CD and could be regarded as gigantic elementary particle states with very large quantum numbers and by charge fractionization confined around the tip of CD. Charge fractionization