# A Topos for Algebraic Quantum Theory

@article{Heunen2009ATF, title={A Topos for Algebraic Quantum Theory}, author={C. Heunen and N. P. Landsman and Bas Spitters}, journal={Communications in Mathematical Physics}, year={2009}, volume={291}, pages={63-110} }

The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr’s idea that the empirical content of quantum physics is accessible only through classical physics, we show how a noncommutative C*-algebra of observables A induces a topos $${\mathcal{T}(A)}$$ in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*-algebra $${\underline{A… Expand

#### Supplemental Presentations

Presentation Slides

#### Paper Mentions

#### 125 Citations

Bohrification of operator algebras and quantum logic

- Mathematics, Physics
- Synthese
- 2011

It is proved that the Heyting algebra thus associated to A arises as a basis for the internal Gelfand spectrum (in the sense of Banaschewski–Mulvey) of the “Bohrification” of A, which is a commutative Rickart C*-algebra in the topos of functors from A to the category of sets. Expand

Topos-Theoretic Approaches to Quantum Theory

- Mathematics
- 2012

Starting from a naive investigation into the nature of experiments on a physical system one can argue that states of the system should pair non-degenerately with physical observables. This duality is… Expand

Topologies on quantum topoi induced by quantization

- Mathematics
- 2013

In the present paper, we consider effects of quantization in a topos approach of quantum theory. A quantum system is assumed to be coded in a quantum topos, by which we mean the topos of presheaves… Expand

Concerning Intuitionistic Quantum Logic

- 2018

While traditionally quantum logic is regarded as a non-distributive type of logic, intuitionistic logic may suit the philosophy of quantum mechanics better. Quantum toposophy is the application of… Expand

Topos-Theoretic Approaches to Quantum Theory Part III Essay

- 2012

Starting from a naive investigation into the nature of experiments on a physical system one can argue that states of the system should pair nondegenerately with physical observables. This duality is… Expand

Intuitionistic Quantum Logic of an n-level System

- Physics, Mathematics
- 2009

AbstractA decade ago, Isham and Butterfield proposed a topos-theoretic approach to quantum mechanics, which meanwhile has been extended by Döring and Isham so as to provide a new mathematical… Expand

Topos quantum theory on quantization-induced sheaves

- Physics, Mathematics
- 2014

In this paper, we construct a sheaf-based topos quantum theory. It is well known that a topos quantum theory can be constructed on the topos of presheaves on the category of commutative von Neumann… Expand

From Quantum-Mechanical Lattice of Projections to Smooth Structure of \(\mathbb {R}^4\)

- Mathematics
- 2017

Mathematical formalism of quantum mechanics provides an interesting way of thinking not only about real numbers in general, but also about the phenomenon of exotic smoothness. The main point of… Expand

A Comparison of Two Topos-Theoretic Approaches to Quantum Theory

- Mathematics, Physics
- 2010

The aim of this paper is to compare the two topos-theoretic approaches to quantum mechanics that may be found in the literature to date. The first approach, which we will call the contravariant… Expand

Fibred contextual quantum physics

- Computer Science, Mathematics
- 2014

A contextual quantum mechanics via the geometric mathematics to propose a quantum contextuality adaptable in every topos, corresponding to the belief that the quantum world must only be seen from the classical viewpoints a la Bohr. Expand

#### References

SHOWING 1-10 OF 204 REFERENCES

Bohrification of operator algebras and quantum logic

- Mathematics, Physics
- Synthese
- 2011

It is proved that the Heyting algebra thus associated to A arises as a basis for the internal Gelfand spectrum (in the sense of Banaschewski–Mulvey) of the “Bohrification” of A, which is a commutative Rickart C*-algebra in the topos of functors from A to the category of sets. Expand

Intuitionistic Quantum Logic of an n-level System

- Physics, Mathematics
- 2009

AbstractA decade ago, Isham and Butterfield proposed a topos-theoretic approach to quantum mechanics, which meanwhile has been extended by Döring and Isham so as to provide a new mathematical… Expand

A topos foundation for theories of physics: II. Daseinisation and the liberation of quantum theory

- Mathematics, Physics
- 2008

This paper is the second in a series whose goal is to develop a fundamentally new way of constructing theories of physics. The motivation comes from a desire to address certain deep issues that arise… Expand

Quantum Logic in Algebraic Approach

- Physics
- 2001

Yet much is still to be done to establish quantum logic as an indispensable tool, an area of research which not only o!ers new insights into established quantum physics but which progressively… Expand

Between classical and quantum

- Mathematics, Physics
- 2005

The relationship between classical and quantum theory is of central importance to the philosophy of physics, and any interpretation of quantum mechanics has to clarify it. Our discussion of this… Expand

Kochen–Specker Theorem for von Neumann Algebras

- Mathematics, Physics
- 2005

The Kochen–Specker theorem has been discussed intensely ever since its original proof in 1967. It is one of the central no-go theorems of quantum theory, showing the non-existence of a certain kind… Expand

Topos Perspective on the Kochen-Specker Theorem: I. Quantum States as Generalized Valuations

- Physics, Mathematics
- 1998

AbstractAny attempt to construct a realistinterpretation of quantum theory founders on theKochen–Specker theorem, which asserts theimpossibility of assigning values to quantum quantitiesin a way that… Expand

A relativity principle in quantum mechanics

- Mathematics
- 1977

Takeuti has studied models of axiomatic set theory in which the “truth values” are elements of a complete Boolean algebra of projections on closed subspaces of a Hilbert space, and has found that the… Expand

The spectral theory of commutative C*-algebras: The constructive Gelfand-Mazur theorem

- Mathematics
- 2000

It is shown, for a commutative C*-algebra in any Grothendieck topos E, that the locale MFn A of multiplicative linear functionals on A is isomorphic to the locale Max A of maximal ideals of A,… Expand

Topos Theory as a Framework for Partial Truth

- Mathematics
- 2000

This paper develops some ideas from previous work (coauthored, mostly with C.J.Isham). In that work, the main proposal is to assign as the value of a physical quantity in quantum theory (or classical… Expand