Real scalar field, the nonrelativistic limit, and the cosmological expansion

Abstract

The existing transformation from a relativistic real scalar field to a complex nonrelativistic scalar field by Namjoo, Guth, and Kaiser is generalized from Minkowski space to a more general background metric. In that case the transformation is not purely algebraic any more but determined by a differential equation. We apply the generalized transformation to a real scalar with φ4 interaction on an Friedmann-Lemaître-Robertson-Walker cosmologically expanding background and calculate the resulting nonrelativistic action up to second order in small parameters. We also show that the transformation can be interpreted as a Bogoliubov transformation between relativistic and nonrelativistic creation and annihilation operators and comment on emerging symmetries in the nonrelativistic theory.