Equivalence Principle, Planck Length and Quantum Hamilton-Jacobi Equation

TL;DR

This paper derives a quantum Hamilton-Jacobi equation from the equivalence principle, revealing new initial conditions and a trajectory-based quantum mechanics framework that incorporates the Planck length and intrinsic potential energy.

Contribution

It introduces a novel quantum Hamilton-Jacobi equation derived from the equivalence principle, providing a trajectory representation with intrinsic potential energy and dependence on the Planck length.

Findings

The QSHJE yields initial conditions not visible in Schrödinger's equation.

Solutions provide a trajectory-based quantum mechanics without wave guides.

The quantum potential acts as an intrinsic, non-vanishing potential energy.

Abstract

The Quantum Stationary HJ Equation (QSHJE) that we derived from the equivalence principle, gives rise to initial conditions which cannot be seen in the Schroedinger equation. Existence of the classical limit leads to a dependence of the integration constant $\ell=\ell_1+i\ell_2$ on the Planck length. Solutions of the QSHJE provide a trajectory representation of quantum mechanics which, unlike Bohm's theory, has a non-trivial action even for bound states and no wave guide is present. The quantum potential turns out to be an intrinsic potential energy of the particle which, similarly to the relativistic rest energy, is never vanishing.