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arXiv · 0902.2523

Split-Quaternionic Hopf Map, Quantum Hall Effect, and Twistor Theory

Kazuki Hasebe
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Abstract

Introducing a non-compact version of the Hopf map, we demonstrate remarkable close relations between quantum Hall effect and twistor theory. We first construct quantum Hall effect on a hyperboloid based on the noncompact 2nd Hopf map of split-quaternions. We analyze a hyperbolic one-particle mechanics, and explore many-body problem, where a many-body groundstate wavefunction and membrane-like excitations are derived explicitly. In the lowest Landau level, the symmetry is enhanced from $SO(3, 2)$ to the $SU(2, 2)$ conformal symmetry. We point out that the quantum Hall effect naturally realizes the philosophy of twistor theory. In particular, the emergence mechanism of fuzzy space-time is discussed somehow in detail.

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Kazuki Hasebe. 2010-02-18. Split-Quaternionic Hopf Map, Quantum Hall Effect, and Twistor Theory. https://doi.org/10.1103/physrevd.81.041702

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