nLab spectral presheaf

Contents

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Operator algebra

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

Introduction

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field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

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renormalization

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Operator algebra

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Noncommutative geometry

Physics

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Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Contents

Idea

Given a C*-algebra AA, not necessarily commutative, write ComSub(A)ComSub(A) for its poset of commutative subalgebras, whose morphisms are the inclusion maps. By Gelfand duality, the presheaf topos PSh(ComSub(A))PSh(ComSub(A)) over this contains a canonical object, namely the presheaf

Σ:CΣ C \Sigma \;\colon\; C \mapsto \Sigma_C

which maps a commutative C*-algebra CAC \hookrightarrow A to (the point set underlying) its Gelfand spectrum Σ C\Sigma_C. This is called the spectral presheaf of AA (Isham-Döring 07)

Properties

The Kochen-Specker theorem of quantum mechanics is equivalent to the statement that for HH a (complex) Hilbert space of dimension greater than 2, then the spectral presheaf of the algebra of bounded operators (H)\mathcal{B}(H) (the quantum observables) has no global element (Butterfield-Hamilton-Isham 98). (This observation motivates the topos-theoretic development in (Isham-Döring 07)).

See at Kochen-Specker theorem and at Bohr topos for more on this.

References

The term “spectral presheaf” was introduced in

Last revised on October 3, 2013 at 10:31:24. See the history of this page for a list of all contributions to it.