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 †-limit, which governs the way in which systems can be combined to form larger systems. These †-limits can be used to characterize the properties of the †-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 †-category with finite †-limits and a simple tensor unit, the semiring of scalars embeds into an involutive field of characteristic 0 and orderable fixed field.
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August 2011
Research Article|
August 25 2011
Completeness of †-categories and the complex numbers
Jamie Vicary
Jamie Vicary
a)
Oxford University Computing Laboratory
, Wolfson Building, Parks Road, Oxford, OX1 3QD, United Kingdom
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Electronic mail: [email protected].
J. Math. Phys. 52, 082104 (2011)
Article history
Received:
January 25 2010
Accepted:
November 13 2010
Citation
Jamie Vicary; Completeness of †-categories and the complex numbers. J. Math. Phys. 1 August 2011; 52 (8): 082104. https://doi.org/10.1063/1.3549117
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