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We present explicitly a relation between the Nambu-Goldstone boson and theorder parameter in non-relativistic systems with spontaneous symmetry breaking.We show that the Nambu-Goldstone bosons are characterized by transformationproperty of the order parameter under symmetry transformation of a system. Wegive an explicit formula for the Nambu-Goldstone boson for a general Lie group$G$, and then the number of the Nambu-Goldstone boson is derivedstraightforwardly from the symmetry of the order parameter, i.e. the type ofsymmetry breaking. We show that the Ward-Takahashi identity is modified in thepresence of the Nambu-Goldstone boson, where the generalized Ward-Takahashiidentity includes the coupling (the vertex function) between fermions andNambu-Goldstone bosons. The closed equation for the Green's functions ofNambu-Goldstone bosons is derived by introducing the fermion-Nambu-Goldstoneboson vertex function. Examples are given for $G=SU(2)$ (ferromagnetic), $U(1)$(superconductor) and $SU(3)$ symmetry breaking.

Nambu-Goldstone bosons characterized by the order parameter in spontaneous symmetry breaking