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arXiv · 0805.0432

Categorical formulation of quantum algebras

Jamie Vicary
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Abstract

We describe how dagger-Frobenius monoids give the correct categorical description of certain kinds of finite-dimensional 'quantum algebras'. We develop the concept of an involution monoid, and use it to construct a correspondence between finite-dimensional C*-algebras and certain types of dagger-Frobenius monoids in the category of Hilbert spaces. Using this technology, we recast the spectral theorems for commutative C*-algebras and for normal operators into an explicitly categorical language, and we examine the case that the results of measurements do not form finite sets, but rather objects in a finite Boolean topos. We describe the relevance of these results for topological quantum field theory.

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Jamie Vicary. 2010-06-22. Categorical formulation of quantum algebras. https://doi.org/10.1007/s00220-010-1138-0

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