What is it about?
Quantum mechanics is the most precisely tested theory in the history of science. Nevertheless, its mathematical rules are hard to stomach. The rules do not have a conceptual explanation from prior principles humans are intuitively familiar with. They work remarkably well, but nobody "really" knows "why". These rules are traditionally given in terms of quantum systems and the relationships between them, which in total make up the so-called category of Hilbert spaces. This research shows just what is special about the category of Hilbert spaces among all other categories. It tells you how to recognize when any given category is in fact that of Hilbert spaces.
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Why is it important?
Knowing when a given category is in fact that of Hilbert spaces is a first step in "understanding" the mathematical rules of quantum theory. It lets us work with descriptions of quantum systems in a different way, especially if we could physically justify or interpret the characteristic properties of the category of Hilbert spaces. The characteristic properties found in this research are remarkable because they are entirely algebraic: they explain "continuous" things like probabilities and complex numbers, in terms of purely "non-continuous" things like combining two objects in an independent way.
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This page is a summary of: Axioms for the category of Hilbert spaces, Proceedings of the National Academy of Sciences, February 2022, Proceedings of the National Academy of Sciences,
DOI: 10.1073/pnas.2117024119.
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