Stable Self-Interacting Pais–uhlenbeck Oscillator

Abstract

It is shown that the interacting Pais-Uhlenbeck oscillator necessarily leads to a description with a Hamiltonian that contains positive and negative energies associated with two oscillators. Descriptions with a positive definite Hamiltonians, considered by some authors, can hold only for a free Pais-Uhlenbeck oscillator. We demonstrate that the solutions of a self-interacting Pais-Uhlenbeck oscillator are stable on islands in the parameter space, as already observed in the literature. If we slightly modify the system, by considering a sine interaction term, and/or by taking unequal masses of the two oscillators, then the system is stable on the continents that extend from zero to infinity in the parameter space. Therefore, the Pais-Uhlenbeck oscillator is quite acceptable physical system.

Deformation quantization of the Pais–Uhlenbeck fourth order oscillator

“…This ⋆-product is known as the Moyal product, and will be defined as an involution in (7). For a two dimensional case (n = 1) this deformed product reads…”

“…The Pais-Uhlenbeck fourth order linear oscillator, originally introduced in [1], is perhaps the simplest example and definitely the best known higher derivative mechanical system and, in particular, it has served as a toy model to understand several important issues related to Ostrogradsky instabilities emerging naturally in higher order field theories [2,3,4,5,6,7,8]. Recently, the Pais-Uhlenbeck oscillator has been used as a guide to study higher order structures associated to supersymmetric field theory [9], P T -symmetric Hamiltonian mechanics [10], and geometric models within the scalar field cosmology context [11].…”

A non-trivial zero length limit of the Nambu–Goto string

“…As shown in Refs. [19][20][21][22], there exist interacting second order systems that are unconditionally stable. Moreover, as pointed out by Woodard [23], the presence of a sufficient number of gauge constraints can stabilize the system.…”

Higher order gravities and the Strong Equivalence Principle

“…The modified gravity theories remain dynamical metric theories, automatically satisfying the Weak Equivalence Principle (WEP) if matter only couples minimally to the metric in the action. 3 General covariance ensures the on-shell covariant divergenceless of the matter energy-momentum tensor.…”

“…Nevertheless, see also refs. [3,4]. 2 We are only going to consider metric theories based on a general-covariant action principle.…”