Axioms for the category of Hilbert spaces

We provide axioms that guarantee a category is equivalent to that of continuous linear functions between Hilbert spaces. The axioms are purely categorical and do not presuppose any analytical structure. This addresses a question about the mathematical foundations of quantum theory raised in reconstruction programmes such as those of von Neumann, Mackey, Jauch, Piron, Abramsky, and Coecke. Quantum mechanics has mathematically been firmly founded on Hilbert spaces and operators between them for nearly a century [21]. There has been continuous inquiry into the special status of this foundation since [18, 5, 12]. How are the mathematical axioms to be interpreted physically? Can the theory be reconstructed from a different framework whose axioms can be interpreted physically? Such reconstruction programmes involve a mathematical reformulation of (a generalisation of) the theory of Hilbert spaces and their operators, such as operator algebras [16], orthomodular lattices [10, 17], and, most recently, categorical quantum mechanics [1, 4]. The latter uses the framework of category theory [15], and emphasises operators more than their underlying Hilbert spaces. It postulates a category with structure that models physical features of quantum theory [9]. The question of how “to justify the use of Hilbert space” [17] then becomes: which axioms guarantee that a category is equivalent to that of continuous linear functions between Hilbert spaces? This article answers that mathematical question. The axioms are purely categorical in nature, and do not presuppose any analytical structure such as continuity, complex numbers, or probabilities. The approach is similar to Lawvere’s categorical characterisation of the theory of sets [14].