Novel formulation of Hamilton-Jacobi equation for higher derivative theory and quantum mechanical correspondence

TL;DR

This paper introduces new Hamilton-Jacobi formulations for higher derivative theories that lead to nonlinear quantum mechanics, potentially resolving negative energy issues present in traditional approaches.

Contribution

It presents novel Hamilton-Jacobi equations derived via Caratheodory's method, offering a different perspective from canonical formulations and connecting to nonlinear quantum mechanics.

Findings

New Hamilton-Jacobi formulations for higher derivative theories

Quantum correspondence leads to nonlinear quantum mechanics

Potential resolution of negative energy problems

Abstract

For higher derivative theories, using the approach of Caratheodory's equivalent Lagrangian, we show that there exist novel formulations of Hamilton-Jacobi equations, which are different from the formulations derived from Hamilton's canonical approach. The quantum mechanical correspondences of these novel Hamilton-Jacobi equations lead to nonlinear quantum mechanics, which seem being able to avoid the unbounded negative energy problem in the quantum mechanics of higher derivative theories.