Revisiting nonrelativistic limits of field theory: Antiparticles

Different subtleties and problems associated with a nonrelativistic limit of the field theory to the Schroedinger theory are discussed. In this paper, we revisit different cases of the nonrelativistic limit of a real and complex scalar field in the level of the Lagrangian and the equation of motion. We develop the nonrelativistic limit of the Dirac equation and action in the way that the nonrelativistic limit of spin-$\frac{1}{2}$ wave functions of particles and antiparticles appear simultaneously. We study the effect of a potential like $U(\phi)\propto \phi^4$ which can be attributed to axion dark matter field in this limit. We develop a formalism for studying the nonrelativistic limit of antiparticles in the quantum mechanics. We discussed the non-local approach for the nonrelativistic limit and its problems.