Completeness and the complex numbers Export

@article{citeulike:4633656, 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, to prove our results. These dagger-limits can be used to characterize the properties of the dagger-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 dagger-category with dagger-limits and a simple tensor unit, in which the self-adjoint scalars are Dedekind-complete, the scalars are valued in the complex numbers.}, archivePrefix = {arXiv}, author = {Vicary, Jamie}, citeulike-article-id = {4633656}, citeulike-linkout-0 = {http://arxiv.org/abs/0807.2927v2}, citeulike-linkout-1 = {http://arxiv.org/pdf/0807.2927v2}, day = {21}, eprint = {0807.2927v2}, keywords = {category-theory, mathematics, quantum}, month = {May}, posted-at = {2009-05-25 20:57:13}, priority = {3}, title = {Completeness and the complex numbers}, url = {http://arxiv.org/abs/0807.2927v2}, year = {2009} }

TY - JOUR ID - citeulike:4633656 L3 - citeulike-article-id:4633656 N2 - 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, to prove our results. These dagger-limits can be used to characterize the properties of the dagger-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 dagger-category with dagger-limits and a simple tensor unit, in which the self-adjoint scalars are Dedekind-complete, the scalars are valued in the complex numbers. TI - Completeness and the complex numbers KW - category-theory KW - mathematics KW - quantum AU - Vicary, Jamie PY - 2009/May/21/ UR - http://arxiv.org/abs/0807.2927v2 ER -