Infinite-dimensional Categorical Quantum Mechanics

TL;DR

This paper introduces a non-standard analysis-based category for infinite-dimensional quantum mechanics, enabling rigorous treatment of operators and observables in separable Hilbert spaces.

Contribution

It develops a new categorical framework for infinite-dimensional quantum mechanics using non-standard analysis, including the construction of a suitable category and the definition of key quantum structures.

Findings

Category $^ extstar ext{Hilb}$ is compact closed with partial traces.

Standard bounded operators embed into $^ extstar ext{Hilb}$.

Position and momentum observables form a strongly complementary pair.

Abstract

We use non-standard analysis to define a category $^\star\!\operatorname{Hilb}$ suitable for categorical quantum mechanics in arbitrary separable Hilbert spaces, and we show that standard bounded operators can be suitably embedded in it. We show the existence of unital special commutative $\dagger$-Frobenius algebras, and we conclude $^\star\!\operatorname{Hilb}$ to be compact closed, with partial traces and a Hilbert-Schmidt inner product on morphisms. We exemplify our techniques on the textbook case of 1-dimensional wavefunctions with periodic boundary conditions: we show the momentum and position observables to be well defined, and to give rise to a strongly complementary pair of unital commutative $\dagger$-Frobenius algebras.