Exorcising the Ostrogradsky ghost in coupled systems

TL;DR

This paper explores how adding variables to higher-derivative systems can prevent Ostrogradsky ghosts, providing conditions on the Lagrangian to ensure stability and discussing applications to advanced field theories.

Contribution

It systematically derives conditions on the Lagrangian that eliminate Ostrogradsky ghosts in coupled systems, extending the understanding of higher-derivative theories.

Findings

Identified conditions for primary and secondary constraints to avoid ghosts

Analyzed implications for equations of motion and variable redefinitions

Discussed applications to multi-Galileons and beyond Horndeski theories

Abstract

The Ostrogradsky theorem implies that higher-derivative terms of a single mechanical variable are either trivial or lead to additional, ghost-like degrees of freedom. In this letter we systematically investigate how the introduction of additional variables can remedy this situation. Employing a Lagrangian analysis, we identify conditions on the Lagrangian to ensure the existence of primary and secondary constraints that together imply the absence of Ostrogradsky ghosts. We also show the implications of these conditions for the structure of the equations of motion as well as possible redefinitions of the variables. We discuss applications to analogous higher-derivative field theories such as multi-Galileons and beyond Horndeski.