Classical and quantum stability of higher-derivative dynamics

TL;DR

This paper investigates the classical and quantum stability of higher-derivative systems, showing that bounded integrals of motion can ensure stability and proposing methods to incorporate interactions without losing this stability.

Contribution

It introduces a general approach to maintain stability in higher-derivative systems at both classical and quantum levels, including explicit constructions and interaction procedures.

Findings

Bounded integrals of motion ensure classical stability.

A quantization method preserves quantum stability.

Explicit examples demonstrate the stability-preserving interactions.

Abstract

We observe that a wide class of higher-derivative systems admits a bounded integral of motion that ensures the classical stability of dynamics, while the canonical energy is unbounded. We use the concept of a Lagrange anchor to demonstrate that the bounded integral of motion is connected with the time-translation invariance. A procedure is suggested for switching on interactions in free higher-derivative systems without breaking their stability. We also demonstrate the quantization technique that keeps the higher-derivative dynamics stable at quantum level. The general construction is illustrated by the examples of the Pais-Uhlenbeck oscillator, higher-derivative scalar field model, and the Podolsky electrodynamics. For all these models, the positive integrals of motion are explicitly constructed and the interactions are included such that keep the system stable.