nLab continuous field of C*-algebras

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

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

Bundles

bundles

Contents

Idea

A continous field of C *C^\ast-algebras is a kind of topological bundle of C *C^\ast-algebras: A topological space XX and a C*-algebra A xA_x for each point xX x \in X, such that these algebras vary continuously, in some sense, as xx varies in XX. When XX is thought of as a topological groupoid with only identity morphisms, this may be understaood as a special case of Fell bundles over XX.

Applications

In strict deformation quantization

In C* algebraic deformation quantization continuous fields of C *C^\ast-algebras over subspaces of the standard interval (tyically {1,12,13,,0}[0,1]\{1 , \frac{1}{2}, \frac{1}{3}, \cdots, 0\} \hookrightarrow [0,1]) which at =0\hbar = 0 are Poisson algebras constitute non-perturbative deformation quantizations of this Poisson algebra (hence of the phase space of some physical system that it represents).

References

The notion originates with:

An efficient reformulation is due to:

See also:

Last revised on January 15, 2024 at 03:53:29. See the history of this page for a list of all contributions to it.