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

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.

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