The Equivalence Principle of Quantum Mechanics: Uniqueness Theorem

TL;DR

This paper demonstrates that the equivalence principle in quantum mechanics uniquely leads to the Schrödinger equation and related transformations, establishing a fundamental connection between classical and quantum formalisms through symmetry and bracket structures.

Contribution

It proves the uniqueness of the quantum Hamilton-Jacobi solution derived from the equivalence principle and introduces a new map and brackets within the quantum canonical framework.

Findings

The equivalence principle uniquely determines the Schrödinger equation.

A map exists that reduces any system to a free system with zero energy.

Canonical and Schrödinger equations are expressed using N=2 SYM brackets.

Abstract

Recently we showed that the postulated diffeomorphic equivalence of states implies quantum mechanics. This approach takes the canonical variables to be dependent by the relation p=\partial_q S_0 and exploits a basic GL(2,C)-symmetry which underlies the canonical formalism. In particular, we looked for the special transformations leading to the free system with vanishing energy. Furthermore, we saw that while on the one hand the equivalence principle cannot be consistently implemented in classical mechanics, on the other it naturally led to the quantum analogue of the Hamilton-Jacobi equation, thus implying the Schroedinger equation. In this letter we show that actually the principle uniquely leads to this solution. Furthermore, we find the map reducing any system to the free one with vanishing energy and derive the transformations on S_0 leaving the wave function invariant. We also express the canonical and Schroedinger equations by means of the brackets recently introduced in the framework of N=2 SYM. These brackets are the analogue of the Poisson brackets with the canonical variables taken as dependent.