nLab
Hilb

Context

Functional analysis

Operator algebra

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

HilbHilb

Definition

A category whose objects are Hilbert spaces is typically denoted HilbHilb or similar.

There are different choices of morphisms in use, such as all linear maps or just the short linear maps (linear maps of norm at most 11).

One may regard HilbHilb as a dagger category with morphisms the bounded linear maps between them and the dagger operation assigning adjoint operators. The full subcategory FinHilbFin Hilb of finite-dimensional Hilbert spaces becomes a dagger compact category.

Note that either way, the core (of isomorphisms in the first case, or of unitary isomorphisms in the other case) is the same groupoid, whose morphisms are all invertible linear maps of norm exactly 11.

In any case, the forgetful functor from HilbHilb to Vect is faithful, confirming the intuition that a Hilbert space is a vector space equipped with extra structure. HilbHilb is also a full subcategory of Ban, the category of Banach spaces.

References

A pedagogical description of the monoidal category structure on HilbHilb with an emphasis on their role in quantum mechanics and their relation to nCob:

An axiomatic characterization of the dagger-category of Hilbert spaces, with linear maps between them:

category: category

Last revised on September 16, 2021 at 05:47:44. See the history of this page for a list of all contributions to it.