  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.
