Friday, May 19, 2006

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

Gb subset of SU(2) subset of SU(3)

in CP2 degrees of freedom and

Ga 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 Gb orbit of CP2 points and vice versa. Thus space-time sheet becomes N(Ga) fold covering of CP2 and N(Gb)-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

σH =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 (An, D2n, 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?.

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