Inverting a normal harmonic oscillator: Physical interpretation and applications

Abstract

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 Schrödinger 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.

Title changed; Material reorganised; More discussions added; Published version

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Cited by in corpus (12)

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• Generalized Schwinger effect and particle production in an expanding universe
• Dynamical scaling symmetry and asymptotic quantum correlations for time-dependent scalar fields
• Ladder Operators in Repulsive Harmonic Oscillator with Application to the Schwinger Effect
• Non-trivial time crystal-like ground state for gravitational perturbation in quadratic gravity
• The cosmological constant of emergent spacetime in the Newtonian approximation
• Complex time route to quantum backreaction
• Schwinger-Keldysh path integral formalism for a Quenched Quantum Inverted Oscillator
• Backreaction of particle production on physical gravitons in the de Sitter space