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

TL;DR

This paper analyzes the Pais-Uhlenbeck model using Lie symmetries to develop ghost-free, stable formulations, and explores transformations to simpler systems, addressing longstanding issues in higher-derivative theories.

Contribution

It introduces a symmetry-based framework for stabilizing higher-derivative models and constructs positive definite representations of the Pais-Uhlenbeck system.

Findings

Identified Lie symmetries of the PU model's equations.

Constructed Poisson brackets preserving dynamics.

Recast the PU model in a positive definite form.

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.