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

We addressed quantization phenomena in open systems and confined motion in low-dimensional systems, as well as quantized sources in 3-dimensions. The thesis of the paper is that if we simply cast the Bohr-Sommerfeld (B-S) quantization condition as a U(1) gauge theory, like the gauge field of Chern-Simons gauge theory or as in topological band theory (TBT) of condensed matter physics in terms of Berry connection and Berry curvature to make it self-consistent, then the quantization method in all the physical phenomena treated in this paper are unified in the sense of being traceable to the self-consistent B-S quantization. These are the stationary quantization of due to oscillatory dynamics in compactified 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 through the gauge field. Thus, the self-consistent B-S quantization condition permeates the quantization of integer quantum Hall effect (IQHE), fractional quantum Hall effect (FQHE), the Berezenskii-Kosterlitz-Thouless vortex quantization, Aharonov-Bohm effect, 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 transport of open systems, we introduced a novel phase-space derivation of the quantized conductance of the IQHE based on nonequilibrium quantum transport and lattice Weyl transform approach.