Completeness of dagger-categories and the complex numbers
arXiv:0807.2927 · doi:10.1063/1.3549117
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.
39 pages. Accepted for publication in the Journal of Mathematical Physics
References in corpus (5)
Cited by in corpus (11)
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- The Abstract Structure of Quantum Algorithms
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- Dagger categories and the complex numbers: Axioms for the category of finite-dimensional Hilbert spaces and linear contractions
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