Frobenius Structures Over Hilbert C*-Modules

Abstract

We study the monoidal dagger category of Hilbert C*-modules over a commutative C*-algebra from the perspective of categorical quantum mechanics. The dual objects are the finitely presented projective Hilbert C*modules. Special dagger Frobenius structures correspond to bundles of uniformly finite-dimensional C*-algebras. A monoid is dagger Frobenius over the base if and only if it is dagger Frobenius over its centre and the centre is dagger Frobenius over the base. We characterise the commutative dagger Frobenius structures as finite coverings, and give nontrivial examples of both commutative and central dagger Frobenius structures. Subobjects of the tensor unit correspond to clopen subsets of the Gelfand spectrum of the C*-algebra, and we discuss dagger kernels.

Quantum Field Theory in Categorical Quantum Mechanics

“…This makes the unitary representation U t t∈ 1 ωuv Z 2ω+1 above well-defined, at the cost of introducing some energy level degeneracy between some infinitesimally close momentum values. 9 9 The exact infinitesimal extent of this degeneracy depends, rather interestingly, on both m and the ratio ω uv /ω ir .…”

“…When using non-standard analysis [13], those infinities are no longer an issue and it is perfectly sensible to treat quantum fields as vectors in the infinite tensor product of simple harmonic oscillators over momentum space. That is the approach taken here, building upon previous work on the non-standard approach [7,8] to infinite dimensional categorical quantum mechanics [1,2,4,9].…”

Space in Monoidal Categories

“…We start with a brief section detailing our leading example, the category of Hilbert modules. For more information we refer to Appendix B and [18,11]. Definition 1.…”

“…If X = 1 then a Hilbert module is simply a Hilbert space. More generally, for any t ∈ X there is a monoidal functor Hilb C 0 (X) → Hilb [11,Proposition 2.5]. In the other direction, there is a monoidal functor C 0 (X, −) : Hilb → Hilb C 0 (X) that sends a Hilbert space H to the Hilbert module…”

“…• How does our work relate to approaches to causality using discarding maps [7,17]? One might hope to establish a connection using the CP*-construction [5,11] for dagger categories. Though Hilb C 0 (X) is not a dagger category, its so-called adjointable morphisms form a dagger subcategory [11].…”