  Rarely noted paradoxes and their resolution lead to non-Hermitian behaviors
due to boundary terms, even for closed systems and with real potentials. The
role played by these non-Hermiticities on quantum mechanical uncertainty
relations is discussed, especially in multiply-connected spaces. These
subtleties, reflections of topological quantum anomalies (for any
dimensionality, for both Schrodinger and Dirac/Weyl Hamiltonians and for either
continuous or lattice (tight-binding) models) can always be written as global
fluxes of certain generalized current densities Jg. In continuous
nonrelativistic models, these have the forms that had earlier been used by
Chemists, while for Dirac/Weyl or other lattice models they have relativistic
forms only recently worked out. Examples are provided where such
non-Hermiticities have a direct physical significance (for both conventional
and topological materials). In all examples considered, these non-Hermitian
boundary terms turn out to be quantized, the quantization being either of
conventional or of a topological (Quantum Hall Effect (QHE))-type. The latter
claim is substantiated through application to a simple QHE arrangement (2D
Landau system in an external in-plane electric field), where some particular Jg
is related to the well-known edge currents. The non-Hermitian terms play a
subtle role on Berry curvatures in solids and seem to be crucial for the
consistent application of the Modern Theories of Polarization and Orbital
Magnetization. It is emphasized that the above systems can be closed (in
multiply-connected space, so that the boundaries disappear, but the
non-Hermiticity remains), a case in non-Hermitian physics that has not been
discussed. A mapping between the above non-Hermiticity and the many recent
available results in tight-binding Solid State models will lead to enhanced
understanding of quantum behavior at the most fundamental level.
