What is quantum mechanics? A minimal formulation

TL;DR

This paper introduces a minimal, self-contained formulation of nonrelativistic quantum mechanics that parallels classical mechanics, relying on Hilbert space and fundamental theorems to derive core principles like the Born rule.

Contribution

It presents a minimal, conceptually clear formulation of quantum mechanics based solely on Hilbert space and fundamental theorems, avoiding external apparatus or agents.

Findings

Derives the Born rule from Hilbert space theorems

Provides a self-contained microscopic theory applicable to any closed system

Shows classical and quantum theories as special cases of a broader microscopic framework

Abstract

This paper presents a minimal formulation of nonrelativistic quantum mechanics, by which is meant a formulation which describes the theory in a succinct, self-contained, clear, unambiguous and of course correct manner. The bulk of the presentation is the so-called \lq microscopic theory' (MIQM), applicable to any closed system $S$ of arbitrary size $N$, using concepts referring to $S$ alone, without resort to external apparatus or external agents. An example of a similar minimal microscopic theory is the standard formulation of classical mechanics, which serves as the template for a minimal quantum theory. The only substantive assumption required is the replacement of the classical Euclidean phase space by Hilbert space in the quantum case, with the attendant all-important phenomenon of quantum incompatibility. Two fundamental theorems of Hilbert space, the Kochen-Specker-Bell theorem and Gleason's theorem, then lead inevitably to the well-known Born probability rule. For both classical and quantum mechanics, questions of physical implementation and experimental verification of the predictions of the theories are the domain of the macroscopic theory, which is argued to be a special case or application of the more general microscopic theory.