Quantum Theory From Five Reasonable Axioms

TL;DR

This paper derives quantum theory from five reasonable axioms, clarifying its structure and distinguishing it from classical probability theory by emphasizing the role of continuous reversible transformations.

Contribution

It introduces five axioms that uniquely characterize quantum theory, providing insights into its mathematical structure and its distinction from classical probability.

Findings

Quantum theory can be derived from five axioms.

Axiom 5 distinguishes quantum from classical probability.

The trace rule and complex numbers are justified by the axioms.

Abstract

The usual formulation of quantum theory is based on rather obscure axioms (employing complex Hilbert spaces, Hermitean operators, and the trace rule for calculating probabilities). In this paper it is shown that quantum theory can be derived from five very reasonable axioms. The first four of these are obviously consistent with both quantum theory and classical probability theory. Axiom 5 (which requires that there exists continuous reversible transformations between pure states) rules out classical probability theory. If Axiom 5 (or even just the word "continuous" from Axiom 5) is dropped then we obtain classical probability theory instead. This work provides some insight into the reasons quantum theory is the way it is. For example, it explains the need for complex numbers and where the trace formula comes from. We also gain insight into the relationship between quantum theory and classical probability theory.