  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.
