Axioms for the category of Hilbert spaces

TL;DR

This paper establishes purely categorical axioms that characterize the category of Hilbert spaces and continuous linear functions, addressing foundational questions in quantum theory without relying on analytical structures.

Contribution

It introduces new categorical axioms that precisely characterize the category of Hilbert spaces, advancing the mathematical foundations of quantum theory.

Findings

Axioms guarantee category equivalence to Hilbert spaces

Purely categorical approach without analytical assumptions

Addresses foundational questions in quantum reconstruction

Abstract

We provide axioms that guarantee a category is equivalent to that of continuous linear functions between Hilbert spaces. The axioms are purely categorical and do not presuppose any analytical structure. This addresses a question about the mathematical foundations of quantum theory raised in reconstruction programmes such as those of von Neumann, Mackey, Jauch, Piron, Abramsky, and Coecke.