  The anomalous Nambu-Goldstone (NG) theorem which is found as a violation of
counting law of the number of NG bosons of the normal NG theorem in
nonrelativistic and Lorentz-symmetry-violated relativistic theories is studied
in detail, with emphasis on its mathematical aspect from Lie algebras, geometry
to number theory. The basis of counting law of NG bosons in the anomalous NG
theorem is examined by Lie algebras (local) and Lie groups (global). A
quasi-Heisenberg algebra is found generically in various symmetry breaking
schema of the anomalous NG theorem, and it indicates that it causes a
violation/modification of the Heisenberg uncertainty relation in an NG sector
which can be experimentally confirmed. The formalism of effective potential is
presented for understanding the mechanism of anomalous NG theorem with the aid
of our result of Lie algebras. After an investigation on a bosonic kaon
condensation model with a finite chemical potential as an explicit
Lorentz-symmetry-breaking parameter, a model Lagrangian approach on the
anomalous NG theorem is given for our general discussion. Not only the
condition of the counting law of true NG bosons, but also the mechanism to
generate a mass of massive NG boson is also found by our examination on the
kaon condensation model. Furthermore, the generation of a massive mode in the
NG sector is understood by the quantum uncertainty relation of the Heisenberg
algebra, obtained from a symmetry breaking of a Lie algebra, which realizes in
the effective potential of the kaon condensation model. Hence the relation
between a symmetry breaking scheme, a Heisenberg algebra, a mode-mode coupling,
and the mechanism of mass generation in an NG sector is established. Finally,
some relations between the Riemann hypothesis and the anomalous NG theorem are
presented.
