Demystifying the constancy of the Ermakov–Lewis invariant for a time-dependent oscillator

Abstract

It is well known that the time-dependent harmonic oscillator (TDHO) possesses a conserved quantity, usually called Ermakov–Lewis invariant. I provide a simple physical interpretation of this invariant as well as a whole family of related invariants. This interpretation does not seem to have been noticed in the literature before. The procedure also allows one to tackle some key conceptual issues which arise in the study of quantum fields in the external, time-dependent backgrounds like in the case of particle production in an expanding universe and Schwinger effect.