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

Alexander Ganz (1, 2) , Karim Noui (3, 4, 5)

1 Unipd - Università degli Studi di Padova = University of Padua
2 INFN, Sezione di Padova - Istituto Nazionale di Fisica Nucleare, Sezione di Padova
3 IDP - Institut Denis Poisson
4 APC (UMR_7164) - AstroParticule et Cosmologie
5 LPENS - Laboratoire de physique de l'ENS - ENS Paris

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.

Mots clés: derivative: high; mechanics: classical; stability; ghost; higher-order; Hamiltonian; propagation; covariance

Domaines: Physique des Hautes Energies - Théorie [hep-th]; Relativité Générale et Cosmologie Quantique [gr-qc]

Alexander Ganz, Karim Noui. Reconsidering the Ostrogradsky theorem: Higher-derivatives Lagrangians, Ghosts and Degeneracy. Classical and Quantum Gravity, 2021, 38 (7), pp.075005. ⟨10.1088/1361-6382/abe31d⟩. ⟨hal-02905282⟩