Reconsidering the Ostrogradsky theorem: Higher-derivatives Lagrangians, Ghosts and Degeneracy

TL;DR

This paper reviews the Ostrogradsky theorem, explores how degeneracy in higher-derivative Lagrangians can avoid instabilities, and discusses quantum implications and covariant theories where traditional ghost issues are mitigated.

Contribution

It extends the Ostrogradsky theorem to multiple variables, clarifies conditions for avoiding ghosts, and analyzes quantum and covariant cases with bounded Hamiltonians.

Findings

Degenerate higher-derivative Lagrangians can evade Ostrogradsky ghosts.

Higher-derivatives do not necessarily cause instabilities if degeneracy conditions are met.

Quantum analysis reveals how Ostrogradsky instabilities manifest at the quantum level.

Abstract

We review the fate of the Ostrogradsky ghost in higher-order theories. We start by recalling the original Ostrogradsky theorem and illustrate, in the context of classical mechanics, how higher-derivatives Lagrangians lead to unbounded Hamiltonians and then lead to (classical and quantum) instabilities. Then, we extend the Ostrogradsky theorem to higher-derivatives theories of several dynamical variables and show the possibility to evade the Ostrogradsky instability when the Lagrangian is "degenerate", still in the context of classical mechanics. In particular, we explain why higher-derivatives Lagrangians and/or higher-derivatives Euler-Lagrange equations do not necessarily lead to the propagation of an Ostrogradsky ghost. We also study some quantum aspects and illustrate how the Ostrogradsky instability shows up at the quantum level. Finally, we generalize our analysis to the case of higher order covariant theories where, as the Hamiltonian is vanishing and thus bounded, the question of Ostrogradsky instabilities is subtler.