Inverting a normal harmonic oscillator: Physical interpretation and applications

A harmonic oscillator with time-dependent mass $m(t)$ and a time-dependent (squared) frequency $\omega^2(t)$ occurs in the modelling of several physical systems. It is generally believed that systems, with $m(t)>0$ and $\omega^2(t)>0$ (normal oscillator) are stable while systems with $m(t)>0$ and $\omega^2(t)<0$ (inverted oscillator) are unstable. We show that it is possible to represent the \textit{same} physical system either as a normal oscillator or as an inverted oscillator by redefinition of dynamical variables. While we expect the physics to be invariant under such redefinitions, it is not obvious how this invariance actually comes about. We study the relation between these two, normal and inverted, representations of an oscillator in detail both in Heisenberg and Schr\"{o}dinger pictures to clarify several conceptual and technical issues. The situation becomes more involved when the oscillator is coupled to another (semi)classical degree of freedom $C(t)$ and we want to study the back-reaction of the quantum system $q(t)$ on $C(t)$, in the semi-classical approximation. We provide a simple prescription for the back-reaction based on energy conservation and study that the dynamics of the full system in both normal and the inverted oscillator representation. The physics again remains invariant but there are some extra subtleties which we clarify. The implications of these results for quantum field theory in cosmological backgrounds are discussed briefly in an appendix.