Equivalence Principle, Planck Length and Quantum Hamilton–Jacobi Equation

The Quantum Stationary HJ Equation (QSHJE) that we derived from the equivalence principle, gives rise to initial conditions which cannot be seen in the Schrödinger equation. Existence of the classical limit leads to a dependence of the integration constant l = l1 + il2 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. Let us consider a one–dimensional stationary system of energy E and potential V and set W ≡ V (q)−E. In [1] it has been formulated the following equivalence principle For each pair Wa,Wb, there is a transformation q −→ q = v(q), such that W(q) −→ W(q) = W(q). (1) Implementation of this principle uniquely leads to the Quantum Stationary HJ Equation (QSHJE) [1] 1 2m ( ∂S0(q) ∂q )2 + V (q)− E + h̄ 2 4m {S0, q} = 0, (2) where S0 is the Hamilton’s characteristic function also called reduced action. In this equation the Planck constant plays the role of covariantizing parameter. The fact that a fundamental constant follows from the equivalence principle suggests that other fundamental constants as well may be related to such a principle. We have seen in [1] that the implementation of the equivalence principle implied a cocycle condition which in turn determines the structure of the quantum potential. A property of the formulation is that unlike in Bohm’s theory [2][3], the quantum potential Q(q) = h̄ 4m {S0, q}, (3) like S0, is never trivial. This reflects in the fact that a general solution of the Schrödinger equation will have the form ψ = 1