Healthy degenerate theories with higher derivatives (1603.09355v2)

Abstract: In the context of classical mechanics, we study the conditions under which higher-order derivative theories can evade the so-called Ostrogradsky instability. More precisely, we consider general Lagrangians with second order time derivatives, of the form $L(\ddot\phi^a,\dot\phi^a,\phi^a;\dot q^i,q^i)$ with $a = 1,\cdots, n$ and $i = 1,\cdots, m$. For $n=1$, assuming that the $q^i$'s form a nondegenerate subsystem, we confirm that the degeneracy of the kinetic matrix eliminates the Ostrogradsky instability. The degeneracy implies, in the Hamiltonian formulation of the theory, the existence of a primary constraint, which generates a secondary constraint, thus eliminating the Ostrogradsky ghost. For $n>1$, we show that, in addition to the degeneracy of the kinetic matrix, one needs to impose extra conditions to ensure the presence of a sufficient number of secondary constraints that can eliminate all the Ostrogradsky ghosts. When these conditions that ensure the disappearance of the Ostrogradsky instability are satisfied, we show that the Euler-Lagrange equations, which involve a priori higher order derivatives, can be reduced to a second order system.

Critical Analysis of "Healthy Degenerate Theories with Higher Derivatives"

The paper "Healthy Degenerate Theories with Higher Derivatives" by Hayato Motohashi et al. presents an incisive analysis of classical mechanics in the context of theoretical physics, focusing on evading Ostrogradsky instability within higher-order derivative theories. These theories traditionally suffer from instabilities due to the linear dependence of their Hamiltonian on momenta, which leads to the introduction of ghost degrees of freedom (DOF).

Analysis of Higher Derivative Theories

The authors examine the role of higher-order derivative terms, specifically those with second-order time derivatives, in Lagrangians of the form $L(\ddot\phi^a,\dot\phi^a,\phi^a;\dot q^i,q^i)$ for $a = 1,\cdots, n$ and $i = 1,\cdots, m$. The paper emphasizes that for $n=1$, the nondegenerate subsystem formed by $q^i$ can suppress Ostrogradsky instability, confirmed by degeneracy in the kinetic matrix which leads to primary constraints in the Hamiltonian formulation. These constraints propagate to secondary ones, effectively eliminating Ostrogradsky ghosts. The authors extend their analysis to cases where $n>1$, stipulating additional conditions to ensure sufficient secondary constraints for complete elimination of the ghosts.

Extension to Multiple Special Variables

A significant portion of the paper is devoted to theories with multiple special variables. Here, the authors reveal that degeneracy alone is insufficient to suppress all Ostrogradsky ghosts due to the complexity introduced by multiple variables. The paper identifies necessary criteria—degens of a sufficient degree and additional antisymmetric conditions—for such theories to have stable DOF. When these are satisfied, the authors demonstrate how to reduce the ostensibly higher derivative Euler-Lagrange equations to second-order systems. This is crucial for practical applications in theoretical physics where higher-order derivatives can impose computational and interpretative challenges.

Practical Implications and Future Directions

The implications of this research are profound, especially in the domain of modified gravity theories and other advanced theoretical frameworks where classical mechanics principles are applied to scalar-tensor theories. The paper's findings hold potential value for researchers attempting to construct stable, higher-order theories without destabilizing DOF. The robust framework introduced by the authors provides a roadmap for future investigations into multi-variable and field theories, offering insights into how degeneracy and constraints interplay to produce stable, ghost-free formulations.

Conclusion

In conclusion, this paper contributes significantly to the theoretical understanding of higher derivatives in classical mechanics, providing essential criteria and methods to establish healthy degenerate theories. The advanced Hamiltonian and Lagrangian analyses presented could serve as foundational tools for both current and future theoretical inquiries, seeking to develop consistent theories that transcend conventional constraints posed by Ostrogradsky instability. Future studies might look to extend these findings to quantum field theories and explore computationally feasible implementations thereof.