Noncommutative geometry naturally emerges in low energy physics of Landau models as a consequence of level projection. In this work, we proactively utilize the level projection as an effective tool to generate fuzzy geometry. The level projection is specifically applied to the relativistic Landau models. In the first half of the paper, a detail analysis of the relativistic Landau problems on a sphere is presented, where a concise expression of the Dirac–Landau operator eigenstates is obtained based on algebraic methods. We establish [Formula: see text] “gauge” transformation between the relativistic Landau model and the Pauli–Schrödinger nonrelativistic quantum mechanics. After the [Formula: see text] transformation, the Dirac operator and the angular momentum operators are found to satisfy the [Formula: see text] algebra. In the second half, the fuzzy geometries generated from the relativistic Landau levels are elucidated, where unique properties of the relativistic fuzzy geometries are clarified. We consider mass deformation of the relativistic Landau models and demonstrate its geometrical effects to fuzzy geometry. Super fuzzy geometry is also constructed from a supersymmetric quantum mechanics as the square of the Dirac–Landau operator. Finally, we apply the level projection method to real graphene system to generate valley fuzzy spheres.
A relativistic basis of the quantum theory.—II