On quantum Hall effect, Kosterlitz-Thouless phase transition, Dirac magnetic monopole, and Bohr-Sommerfeld quantization

Abstract: We addressed quantization phenomena in transport and vortex/precession-motion of low-dimensional systems, stationary quantization of confined motion in phase space due to oscillatory dynamics or compacti fication of space and time for steady-state systems (e.g., particle in a box or torus, Brillouin zone, and Matsubara time zone or Matsubara quantized frequencies), and the quantization of sources. We discuss how the self-consistent Bohr-Sommerfeld quantization condition permeates the relationships between the quantization of integer Hall effect, fractional quantum Hall effect, the Berezenskii-Kosterlitz-Thouless vortex quantization, the Dirac magnetic monopole, the Haldane phase, contact resistance in closed mesoscopic circuits of quantum physics, and in the monodromy (holonomy) of completely integrable Hamiltonian systems of quantum geometry. In quantum transport of open systems, quantization occurs in fundamental units of quantum conductance, other closed systems in quantum units dictated by Planck's constant, and for sources in units of discrete vortex charge and Dirac magnetic monopole charge. The thesis of the paper is that if we simply cast the B-S quantization condition as a U(1) gauge theory, like the gauge field of the topological quantum field theory (TQFT) via the Chern-Simons gauge theory, or specifically as in topological band theory (TBT) of condensed matter physics in terms of Berry connection and curvature to make it self-consistent, then all the quantization method in all the physical phenomena treated in this paper are unified.