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.

我们在高阶理论中回顾了Ostrogradsky幽灵的命运。我们首先回顾原始的Ostrogradsky定理，并在古典力学的背景下说明高导数拉格朗日法如何导致无穷的哈密顿量，然后导致（经典和量子）不稳定性。然后，我们将Ostrogradsky定理扩展到几个动力学变量的高阶导数理论，并显示了在拉格朗日算式为“简并”的情况下仍可在经典力学的情况下规避Ostrogradsky不稳定性的可能性。特别是，我们解释了为什么高导数的拉格朗日方程和/或高导数的Euler-Lagrange方程不一定导致Ostrogradsky幽灵的传播。我们还研究了一些量子方面，并说明了Ostrogradsky不稳定性是如何在量子水平上出现的。最后，