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

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