Quantum Mechanics from an Equivalence Principle

TL;DR

This paper derives the Schrödinger equation from an equivalence principle applied to coordinate transformations, linking quantum mechanics to a geometric symmetry and introducing the Planck constant as a covariantizing parameter.

Contribution

It presents a novel derivation of quantum mechanics from an equivalence principle and explores the role of GL(2,C) symmetry in the foundational structure of quantum theory.

Findings

Derivation of Schrödinger equation from an equivalence principle

Identification of self-dual states ensuring quantum consistency

Connection between Planck constant and covariantization parameter

Abstract

We postulate that physical states are equivalent under coordinate transformations. We then implement this equivalence principle first in the case of one-dimensional stationary systems showing that it leads to the quantum analogue of the Hamilton-Jacobi equation which in turn implies the Schroedinger equation. In this context the Planck constant plays the role of covariantizing parameter. The construction is deeply related to the GL(2,C)-symmetry of the second-order differential equation associated to the Legendre transformation which selects, in the case of the quantum analogue of the Hamiltonian characteristic function, self-dual states which guarantee its existence for any physical system. The universal nature of the self-dual states implies the Schroedinger equation in any dimension.