Does the quantization of Planck constant transform integer quantum Hall effect to fractional quantum Hall effect?

The TGD based model for topological quantum computation inspired the idea that Planck constant might be dynamical and quantized. The work of Nottale (astro-ph/0310036) gave a strong boost to concrete development of the idea and it took year and half to end up with a proposal about how basic quantum TGD could allow quantization Planck constant associated with M⁴ and CP₂ degrees of freedom such that the scaling factor of the metric in M⁴ degrees of freedom corresponds to the scaling of hbar in CP₂ degrees of freedom and vice versa (see the new chapter Does TGD Predict the Spectrum of Planck constants?). The dynamical character of the scaling factors of M⁴ and CP₂ metrics makes sense if space-time and imbedding space, and in fact the entire quantum TGD, emerge from a local version of an infinite-dimensional Clifford algebra existing only in dimension D=8.

The predicted scaling factors of Planck constant correspond to the integers n defining the quantum phases q=exp(iπ/n) characterizing Jones inclusions. A more precise characterization of Jones inclusion is in terms of group

G_b subset of SU(2) subset of SU(3)

in CP₂ degrees of freedom and

G_a subset of SL(2,C)

in M⁴ degrees of freedom. In quantum group phase space-time surfaces have exact symmetry such that to a given point of M⁴ corresponds an entire G_b orbit of CP₂ points and vice versa. Thus space-time sheet becomes N(G_a) fold covering of CP₂ and N(G_b)-fold covering of M⁴. This allows an elegant topological interpretation for the fractionization of quantum numbers. The integer n corresponds to the order of maximal cyclic subgroup of G.

In the scaling hbar₀→ n× hbar₀ of M⁴ Planck constant fine structure constant would scale as

α= (e²/(4πhbar c)→ α/n ,

and the formula for Hall conductance would transform to

σ_H =να → (ν/n)× α .

Fractional quantum Hall effect would be integer quantum Hall effect but with scaled down α. The apparent fractional filling fraction ν= m/n would directly code the quantum phase q=exp(iπ/n) in the case that m obtains all possible values. A complete classification for possible phase transitions yielding fractional quantum Hall effect in terms of finite subgroups G subset of SU(2) subset of SU(3) given by ADE diagrams would emerge (A_n, D_2n, E₆ and E₈ are possible). What would be also nice that CP₂ would make itself directly manifest at the level of condensed matter physics.

For more details see the chapter Topological Quantum Computation in TGD Universe, the chapter Was von Neumann Right After All?, and the chapter Does TGD predict the Spectrum of Planck Constants?.