Quantum Mechanics from an Equivalence Principle

We show that requiring diffeomorphic equivalence for one-dimensional stationary states implies that the reduced action S0 satisfies the quantum Hamilton-Jacobi equation with the Planck constant playing the role of a covariantizing parameter. The construction shows the existence of a fundamental initial condition which is strictly related to the Möbius symmetry of the Legendre transform and to its involutive character. The universal nature of the initial condition implies the Schrödinger equation in any dimension. While general relativity is based on a simple fundamental principle, similar geometrical structures do not seem to underlie quantum mechanics. In this letter we show that requiring that for any one-dimensional stationary state there is always a coordinate choice q̃ in which W(q) ≡ V (q) − E corresponds to W̃(q̃) = 0, implies that the reduced action satisfies the quantum Hamilton-Jacobi equation with the Planck constant playing the role of covariantizing parameter. Diffeomorphic equivalence implies that S0 is never a constant, rather there is the crucial initial condition S0 = ± i 2 h̄ ln q for W = 0, implying that the Legendre transform T0 = q∂qS0 − S0 is defined for any state. Our construction is deeply related to the Möbius symmetry of the Legendre transform and to its dual (involutive) character. The universal nature of the initial condition implies the Schrödinger equation in any dimension. In ref. [1] we introduced the prepotential F in quantum mechanics defined by ψD = F (ψ), where ψ and ψD are two linearly independent solutions of the Schrödinger equation. We showed that the space coordinate can be regarded as the Legendre transform of F with respect to the probability density. Thus, the space coordinate and F obtain equal status from the point of view of describing the system by the Legendre transform. This means that the wave function itself carries information on the geometry of a physical system. Let us begin by noting the Möbius symmetry of the Legendre transform. Let us define the function T0(p) by q = ∂pT0, (1) where q is the coordinate. The Legendre transform of T0 S0 = p∂pT0 − T0, (2) satisfies p = ∂qS0. (3) The Legendre transform has a crucial symmetry under q̃ = Aq +B Cq +D , p̃ = (Cq +D)p, (4)