Modular structures on trace class operators and applications to Landau levels

@article{Ali2009ModularSO,
  title={Modular structures on trace class operators and applications to Landau levels},
  author={S. Twareque Ali and Fabio Bagarello and Gilbert Honnouvo},
  journal={Journal of Physics A: Mathematical and Theoretical},
  year={2009},
  volume={43},
  pages={105202}
}
The energy levels, generally known as the Landau levels, which characterize the motion of an electron in a constant magnetic field, are those of the one-dimensional harmonic oscillator, with each level being infinitely degenerate. We show in this paper how the associated von Neumann algebra of observables displays a modular structure in the sense of the Tomita–Takesaki theory, with the algebra and its commutant referring to the two orientations of the magnetic field. A Kubo–Martin–Schwinger… 
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