Lie symmetries and ghost-free representations of the Pais-Uhlenbeck model

TL;DR

The paper addresses ghost instabilities in higher time-derivative dynamics by using the Pais-Uhlenbeck oscillator as a testbed. It combines Lie symmetry analysis with a Bi-Hamiltonian framework to construct dynamics-preserving Poisson brackets and alternative Hamiltonians, including positive-definite forms, in suitable parameter regimes. It then develops two-dimensional first-order representations of PU dynamics and analyzes which transformed branches preserve the original flow, highlighting ghost-free possibilities and their limitations. The study also shows that adding interaction terms generally breaks the Bi-Hamiltonian structure, thereby clarifying the scope and applicability of the symmetry-based approach for stabilizing higher-derivative theories with potential relevance to quantum field theory and modified gravity.

Abstract

We investigate the Pais-Uhlenbeck (PU) model, a paradigmatic example of a higher time-derivative theory, by identifying the Lie symmetries of its associated fourth-order dynamical equation. Exploiting these symmetries in conjunction with the model's Bi-Hamiltonian structure, we construct distinct Poisson bracket formulations that preserve the system's dynamics. Amongst other possibilities, this allow us to recast the PU model in a positive definite manner, offering a solution to the long-standing problem of ghost instabilities. Furthermore, we systematically explore a family of transformations that reduce the PU model to equivalent first-order, higher-dimensional systems. Finally we examine the impact on those transformations by adding interaction terms of potential form to the PU model and demonstrate how they usually break the Bi-Hamiltonian structure. Our approach yields a unified framework for interpreting and stabilising higher time-derivative dynamics through a symmetry analysis in some parameter regime.