Modular Structures on Trace Class Operators and Applications to Landau Levels

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 state can be built which, in fact, is the Gibbs state for an ensemble of harmonic oscillators. Mathematically, the modular structure is shown to arise as the natural modular structure associated with the…