Quantum Theory and the Category of Complex Numbers

Jamie Vicary

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, to prove our results. 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. The technical statement of our main theorem is that in a nontrivial monoidal †-category with †-limits and a simple tensor unit, in which the self-adjoint scalars are Dedekind-complete, the scalars are valued in the complex numbers. 1

Chris Heunen, Jamie Vicary

Jamie Vicary

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.

Peter Nonnenmann

M. D. Sheppeard

Complementary observables in quantum mechanics may be viewed as Frobenius structures in a dagger monoidal category, such as the category of finite dimensional Hilbert spaces over the complex numbers. On the other hand, their properties crucially depend on the discrete Fourier transform and its associated quantum torus, requiring only the finite fields that underlie mutually unbiased bases. In axiomatic topos theory, the complex numbers are difficult to describe and should not be invoked unnecessarily. This paper surveys some fundamentals of quantum arithmetic using finite field complementary observables, with a view considering more general axiom systems.