The existing transformation from a relativistic real scalar field to a complex non-relativistic 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 non-relativistic action up to second order in small parameters. We also show that the transformation can be interpreted as a Bogoliubov transformation between relativistic and non-relativistic creation and annihilation operators and comment on emerging symmetries in the non-relativistic theory.
L. H. Heyen and S. Floerchinger, “Real scalar field, the nonrelativistic limit, and the cosmological
expansion”, Phys. Rev. D 102, 036024 (2020).
Related to Project: C06