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

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