  In this paper we develop a formalism for studying the nonrelativistic limit
of relativistic field theories in a systematic way. By introducing a simple,
nonlocal field redefinition, we transform a given relativistic theory,
describing a real, self-interacting scalar field, into an equivalent theory,
describing a complex scalar field that encodes at each time both the original
field and its conjugate momentum. Our low-energy effective theory incorporates
relativistic corrections to the kinetic energy as well as the backreaction of
fast-oscillating terms on the behavior of the dominant, slowly varying
component of the field. Possible applications of our new approach include axion
dark matter, though the methods developed here should be applicable to the
low-energy limits of other field theories as well.
