An axiomatic basis for quantum mechanics

TL;DR

This paper derives the Hilbert space formulation of quantum mechanics from basic axioms within generalized probabilistic theories, utilizing geometric and symmetry principles to characterize quantum structures.

Contribution

It introduces a set of axioms that lead to the Hilbert space formulation, connecting geometric and symmetry aspects in quantum theory.

Findings

Hilbert spaces characterized among orthomodular spaces

Axioms lead to the Hilbert space formulation of quantum mechanics

Partial reduction of assumptions via generalized Wigner theorem

Abstract

In this paper we use the framework of generalized probabilistic theories to present two sets of basic assumptions, called axioms, for which we show that they lead to the Hilbert space formulation of quantum mechanics. The key results in this derivation are the co-ordinatization of generalized geometries and a theorem of Sol\'er which characterizes Hilbert spaces among the orthomodular spaces. A generalized Wigner theorem is applied to reduce some of the assumptions of the theorem of Sol\'er to the theory of symmetry in quantum mechanics. Since this reduction is only partial we also point out the remaining open questions.