Ghost-free theory with third-order time derivatives

TL;DR

This paper develops a ghost-free higher-derivative theory with third-order time derivatives in Lagrangian mechanics, clarifying conditions for stability and second-order reducibility, advancing understanding of higher-derivative physical theories.

Contribution

It introduces conditions for constructing ghost-free third-order derivative theories, extending previous second-order frameworks and ensuring bounded Hamiltonian and second-order equations of motion.

Findings

Identifies necessary and sufficient conditions for ghost-free third-order derivative theories.

Demonstrates Hamiltonian boundedness under specific ghost-free conditions.

Shows that equations of motion can be reduced to second-order systems.

Abstract

As the first step to extend our understanding of higher-derivative theories, within the framework of analytic mechanics of point particles, we construct a ghost-free theory involving third-order time derivatives in Lagrangian. While eliminating linear momentum terms in the Hamiltonian is necessary and sufficient to kill the ghosts associated with higher derivatives for Lagrangian with at most second-order derivatives, we find that this is necessary but not sufficient for the Lagrangian with higher than second-order derivatives. We clarify a set of ghost-free conditions under which we show that the Hamiltonian is bounded, and that equations of motion are reducible into a second-order system.