Quantum mechanics revisited (v.3)

TL;DR

This paper rigorously revisits the foundations of Quantum Mechanics, proving key axioms using mathematical properties of models, and clarifies the connection between mathematical structures and physical phenomena, including new insights into physical anomalies.

Contribution

It provides a mathematical validation of quantum axioms, linking Hilbert spaces, operators, and the Schrödinger equation without relying on physical hypotheses.

Findings

States can be represented in a Hilbert space

Observables correspond to self-adjoint operators

Measurement results are eigenvalues with standard probabilities

Abstract

The purpose of the paper is to study the foundations of the main axioms of Quantum Mechanics. From a general study of the mathematical properties of the models used in Physics to represent systems, we prove that the states of a system can be represented in a Hilbert space, that a self-adjoint operator is associated to any observable, that the result of a measure must be the eigen value of the operator and appear with the usual probability. Furthermore an equivalent of the Wigner's theorem holds, which leads to the Schr{\"o}dinger equation. These results are based on well known mathematics, and do not involve any specific hypothesis in Physics. They validate and explain the methods currently used, which are made simpler and safer, and open new developments. In the third edition of this paper developments have been added about the estimation of physical anomalies.