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The anomalous Nambu-Goldstone (NG) theorem which is found as a violation ofcounting law of the number of NG bosons of the normal NG theorem innonrelativistic and Lorentz-symmetry-violated relativistic theories is studiedin detail, with emphasis on its mathematical aspect from Lie algebras, geometryto number theory. The basis of counting law of NG bosons in the anomalous NGtheorem is examined by Lie algebras (local) and Lie groups (global). Aquasi-Heisenberg algebra is found generically in various symmetry breakingschema of the anomalous NG theorem, and it indicates that it causes aviolation/modification of the Heisenberg uncertainty relation in an NG sectorwhich can be experimentally confirmed. The formalism of effective potential ispresented for understanding the mechanism of anomalous NG theorem with the aidof our result of Lie algebras. After an investigation on a bosonic kaoncondensation model with a finite chemical potential as an explicitLorentz-symmetry-breaking parameter, a model Lagrangian approach on theanomalous NG theorem is given for our general discussion. Not only thecondition of the counting law of true NG bosons, but also the mechanism togenerate a mass of massive NG boson is also found by our examination on thekaon condensation model. Furthermore, the generation of a massive mode in theNG sector is understood by the quantum uncertainty relation of the Heisenbergalgebra, obtained from a symmetry breaking of a Lie algebra, which realizes inthe effective potential of the kaon condensation model. Hence the relationbetween 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 arepresented.

The Anomalous Nambu-Goldstone Theorem in Relativistic/Nonrelativistic Quantum Field Theory