Completeness of dagger-categories and the complex numbers

TL;DR

This paper introduces a category-theoretical framework demonstrating that certain completeness conditions in physical theories necessarily incorporate the complex numbers, using a new concept called dagger-limits to characterize quantum structures.

Contribution

It develops a novel categorical approach with dagger-limits to characterize the role of complex numbers in quantum theory, linking completeness properties to the structure of the scalars.

Findings

Dagger-limits characterize the dagger-functor on finite-dimensional Hilbert spaces.

In dagger-categories with certain properties, scalars embed into an involutive field.

Completeness conditions imply the necessity of complex numbers in the theory.

Abstract

The complex numbers are an important part of quantum theory, but are difficult to motivate from a theoretical perspective. We describe a simple formal framework for theories of physics, and show that if a theory of physics presented in this manner satisfies certain completeness properties, then it necessarily includes the complex numbers as a mathematical ingredient. Central to our approach are the techniques of category theory, and we introduce a new category-theoretical tool, called the dagger-limit, which governs the way in which systems can be combined to form larger systems. These dagger-limits can be used to characterize the dagger-functor on the category of finite-dimensional Hilbert spaces, and so can be used as an equivalent definition of the inner product. One of our main results is that in a nontrivial monoidal dagger-category with all finite dagger-limits and a simple tensor unit, the semiring of scalars embeds into an involutive field of characteristic 0 and orderable fixed field.