(untitled) (Communications in Computational Physics, 2018)
We present a general framework of the theory of spontaneous symmetry breaking in non-relativistic systems. We discuss a spontaneous symmetry breaking in a system with general global symmetry given by a Lie group G. The Nambu-Goldstone boson and Higgs boson are represented explicitly by local fields by means of the basis of the Lie algebra of G. An application to superconductivity is discussed. We evaluate the Green's functions of the Nambu-Goldstone and Higgs bosons in superconductors. We show that the Nambu-Goldstone and Higgs modes exhibit interesting behaviors in multi-component superconductors.