Abstract
The SO(5) Landau model is the mathematical platform of the 4D quantum Hall effect and provide a rare opportunity for a physical realization of the fuzzy four-sphere. We present an integrated analysis of the SO(5) Landau models and the associated matrix geometries through the Landau level projection. With the SO(5) monopole harmonics, we explicitly derive matrix geometry of a four-sphere in any Landau level: In the lowest Landau level the matrix coordinates are given by the generalized SO(5) gamma matrices of the fuzzy four-sphere satisfying the quantum Nambu algebra, while in higher Landau level the matrix geometry becomes a nested fuzzy structure realizing a pure quantum geometry with no counterpart in classical geometry. The internal fuzzy geometry structure is discussed in the view of an SO(4) Pauli-Schrödinger model and the SO(4) Landau model, where we unveil a hidden singular gauge transformation between their background non-Abelian field configurations. Relativistic versions of the SO(5) Landau model are also investigated and relationship to the Berezin-Toeplitz quantization is clarified. We finally discuss the matrix geometry of the Landau models in even higher dimensions.
Highlights
More than forty years ago, Yang proposed an SU (2) generalization [1] of the Dirac’s monopole [2]
In the previous studies [27, 28, 29], we demonstrated that the quantum Nambu geometry appear in the higher dimensional Landau models, and is elegantly intertwined with exotic ideas of differential
We performed a comprehensive study of the SO(5) Landau models and their matrix geometries
Summary
More than forty years ago, Yang proposed an SU (2) generalization [1] of the Dirac’s monopole [2]. The Landau level projection truncates the whole quantum mechanical Hilbert space to a sub-space and provides a physical set-up where the non-commutative geometry naturally appears. Along this line, the fuzzy four-sphere geometry has been discussed in the context of the SO(5) Landau model [7, 14, 15]. Since the total Hilbert space of the Landau model is mathematically well-defined, the truncated subspace of the Landau level necessarily provides a sound formulation of non-commutative geometry Based on this observation, we derived matrix geometries of the SO(3) Landau models [31] and the SO(4) Landau models [32]. Note that the SU (2) (anti-)monopole gauge field does not act to the SU (2)L operators but acts to the SU (2)R operators only (17b), as if the right SU (2) angular momentum acquires additional SU (2) spin degrees of freedom
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