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

TL;DR

This paper offers a new physical interpretation of the Ermakov-Lewis invariant for time-dependent oscillators, clarifying its constancy and extending understanding to quantum field phenomena in dynamic backgrounds.

Contribution

It introduces a novel physical interpretation of the Ermakov-Lewis invariant and related invariants, addressing conceptual issues in quantum field theory in time-dependent backgrounds.

Findings

Provides a simple physical interpretation of the invariant

Extends the interpretation to related invariants

Addresses conceptual issues in quantum field phenomena

Abstract

It is well known that the time dependent harmonic oscillator 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 external, time dependent, backgrounds like in the case of particle production in an expanding universe and Schwinger effect.