# Infinite−dimensional Categorical Quantum Mechanics

### Abstract

We use non-standard analysis to define a category \^\backslashstar\backslash!\backslashoperatorname\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 \backslashdagger-Frobenius algebras, and we conclude \^\backslashstar\backslash!\backslashoperatorname\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 \backslashdagger-Frobenius algebras.

ISSN
2075−2180
Journal
Electronic Proceedings in Theoretical Computer Science
Keywords
Categorical Quantum Mechanics‚Non−Standard Analysis
Month
jan
Pages
51–69
Volume
236
Year
2017