I have been updating the texts about the hierarchy of effective Planck constants coming as multiples of ordinary Planck constant.

The original hypothesis was that the hierarchy of Planck constants is real. In this formulation the imbedding space was replaced with its covering space assumed to decompose to a Cartesian product of singular finite-sheeted coverings of M^{4} and CP_{2}.

Few years ago came the realization that it could be only effective but have same practical implications. The basic observation was that the effective hierarchy need not be postulated separately but follows as a prediction from the vacuum degeneracy of Kähler action. In this formulation Planck constant at fundamental level has its standard value and its effective values come as its integer multiples so that one should write hbar_{eff}=n hbar rather than hbar= nhbar_{0} as I have done. For most practical purposes the states in question would behave as if Planck constant were an integer multiple of the ordinary one. In this formulation the singular covering of the imbedding space became only a convenient auxiliary tool. It is no more necessary to assume that the covering reduces to a Cartesian product of singular coverings of M^{4} and CP_{2} but for some reason I kept this assumption.

The formulation based on multi-furcations of space-time surfaces to N branches. For some reason I assumed that they are simultaneously present. This is too restrictive an assumption. The N branches are very much analogous to single particle states and second quantization allowing all 0<n≤ N-particle states for given N rather than only N-particle states looks very natural. As a matter fact, this interpretation was the original one, and led to the very speculative and fuzzy notion of N-atom, which I later more or less gave up. Quantum multi-furcation could be the root concept implying the effective hierarchy of Planck constants, anyons and fractional charges, and related notions- even the notions of N-nuclei, N-atoms, and N-molecules.

I have now reconsidered the model of fractional quantum Hall effect (FQHE) in this picture. The original naive formulation was rather naive and although there were wrong elements involved, I would not accept it as a referee of a journal;-). The crucial phenomenological notion that I missed was composite fermion. I had been too inpatient to learn basic facts summarized in a brilliant manner in Nobel lecture by Horst L. Stromer.

Feeding the notion of composite fermion, one can predict correctly filling fractions. What remains to be explained using the notion of many-sheeted space-time is how composite fermions are realized as bound states of electron and magnetic flux quanta, how fractional charges and fractional braiding statistics emerge, and how it is possible to obtain non-commutative braiding statistics and associated dynamical non-abelian gauge group for which there are indications. Below is the abstract of the updated chapter.

In this chapter I try to formulate more precisely the recent TGD based view about fractional quantum Hall effect (FQHE). This view is much more realistic than the original rough scenario, which neglected the existing rather detailed understanding. The spectrum of ν, and the mechanism producing it is the same as in composite fermion approach. The new elements relate to the not so well-understood aspects of FQHE, namely charge fractionization, the emergence of braid statistics, and non-abelianity of braid statistics.

*The starting point is composite fermion model so that the basic predictions are same. Now magnetic vortices correspond to (Kähler) magnetic flux tubes carrying unit of magnetic flux. The magnetic field inside flux tube would be created by delocalized electron at the boundary of the vortex. One can raise two questions.*

Could the boundary of the macroscopic system carrying anyonic phase have identification as a macroscopic analog of partonic 2-surface serving as a boundary between Minkowskian and Euclidian regions of space-time sheet? If so, the space-time sheet assignable to the macroscopic system in question would have Euclidian signature, and would be analogous to blackhole or to a line of generalized Feynman diagram.

Could the boundary of the vortex be identifiable a light-like boundary separating Minkowskian magnetic flux tube from the Euclidian interior of the macroscopic system and be also analogous to wormhole throat? If so, both macroscopic objects and magnetic vortices would be rather exotic geometric objects not possible in general relativity framework.

*Taking composite model as a starting point one obtains standard predictions for the filling fractions. One should also understand charge fractionalization and fractional braiding statistics. Here the vacuum degeneracy of Kähler action suggests the explanation. Vacuum degeneracy implies that the correspondence between the normal component of the canonical momentum current and normal derivatives of imbedding space coordinates is 1- to-n. These kind of branchings result in multi-furcations induced by variations of the system parameters and the scaling of external magnetic field represents one such variation.*

*At the orbits of wormhole throats, which can have even macroscopic M*^{4}projections, one has 1→ n_{a}correspondence and at the space-like ends of the space-time surface at light-like boundaries of causal diamond one has 1→ n_{b}correspondence. This implies that at partonic 2-surfaces defined as the intersections of these two kinds of 3-surfaces one has 1→ n_{a}× n_{b}correspondence. This correspondence can be described by using a local singular n-fold covering of the imbedding space. Unlike in the original approach, the covering space is only a convenient auxiliary tool rather than fundamental notion.

*The fractionalization of charge can be understood as follows. A delocalization of electron charge to the n sheets of the multi-furcation takes place and single sheet is analogous to a sheet of Riemann surface of function z*^{1/n}and carries fractional charge q=e/n, n=n_{a}n_{b}. Fractionalization applies also to other quantum numbers. One can have also many-electron stats of these states with several delocalized electrons: in this case one obtains more general charge fractionalization: q= ν e.

*Also the fractional braid statistics can be understood. For ordinary statistics rotations of M*^{4}rotate entire partonic 2-surfaces. For braid statistics rotations of M^{4}(and particle exchange) induce a flow braid ends along partonic 2-surface. If the singular local covering is analogous to the Riemann surface of z^{1/n}, the braid rotation by Δ Φ=2π, where Φ corresponds to M^{4}angle, leads to a second branch of multi-furcation and one can give up the usual quantization condition for angular momentum. For the natural angle coordinate φ of the n-branched covering Δ φ=2π corresponds to Δ Φ=n× 2π. If one identifies the sheets of multi-furcation and therefore uses Φ as angle coordinate, single valued angular momentum eigenstates become in general n-valued, angular momentum in braid statistics becomes fractional and one obtains fractional braid statistics for angular momentum.

*How to understand the exceptional values ν=5/2,7/2 of the filling fraction? The non-abelian braid group representations can be interpreted as higher-dimensional projective representations of permutation group: for ordinary statistics only Abelian representations are possible. It seems that the minimum number of braids is n>2 from the condition of non-abelianity of braid group representations. The condition that ordinary statistics is fermionic, gives n>3. The minimum value is n=4 consistent with the fractional charge e/4.*

The model introduces Z

_{4}valued topological quantum number characterizing flux tubes. This also makes possible non-Abelian braid statistics. The interpretation of this quantum number as a Z_{4}valued momentum characterizing the four delocalized states of the flux tube at the sheets of the 4-furcation suggests itself strongly. Topology would corresponds to that of 4-fold covering space of imbedding space serving as a convenient auxiliary tool. The more standard explanation is that Z_{4}=Z_{2}× Z_{2}such that Z_{2}:s correspond to the presence or absence of neutral Majorana fermion in the two Cooper pair like states formed by flux tubes.

What remains to be understood is the emergence of non-abelian gauge group realizing non-Abelian fractional statistics in gauge theory framework. TGD predicts the possibility of dynamical gauge groups and maybe this kind of gauge group indeed emerges. Dynamical gauge groups emerge also for stacks of N branes and the n sheets of multifurcation are analogous to the N sheets in the stack for many-electron states.

For more details see the chapter Quantum Hall effect and the hierarchy of Planck constants of "p-Adic length scale hypothesis and dark matter hierarchy".

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