# A Topos for Algebraic Quantum Theory

@article{Heunen2009ATF,
title={A Topos for Algebraic Quantum Theory},
author={Chris Heunen and Nicolaas P. Landsman and Bas Spitters},
journal={Communications in Mathematical Physics},
year={2009},
volume={291},
pages={63-110}
}
• Published 27 September 2007
• Mathematics
• Communications in Mathematical Physics
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 $${\underline{A… 148 Citations ### Bohrification of operator algebras and quantum logic • Mathematics Synthese • 2011 It is proved that the Heyting algebra thus associated to A arises as a basis for the internal Gelfand spectrum (in the sense of Banaschewski–Mulvey) of the “Bohrification” of A, which is a commutative Rickart C*-algebra in the topos of functors from A to the category of sets. ### Topos-Theoretic Approaches to Quantum Theory Starting from a naive investigation into the nature of experiments on a physical system one can argue that states of the system should pair non-degenerately with physical observables. This duality is ### Topologies on quantum topoi induced by quantization In the present paper, we consider effects of quantization in a topos approach of quantum theory. A quantum system is assumed to be coded in a quantum topos, by which we mean the topos of presheaves ### Concerning Intuitionistic Quantum Logic While traditionally quantum logic is regarded as a non-distributive type of logic, intuitionistic logic may suit the philosophy of quantum mechanics better. Quantum toposophy is the application of ### From Quantum-Mechanical Lattice of Projections to Smooth Structure of$$\mathbb {R}^4R4

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### Intuitionistic Quantum Logic of an n-level System

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### Topos quantum theory on quantization-induced sheaves

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## References

SHOWING 1-10 OF 134 REFERENCES

### Bohrification of operator algebras and quantum logic

• Mathematics
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.

### Intuitionistic Quantum Logic of an n-level System

• Mathematics
Foundations of Physics
• 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

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

• 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

### Kochen–Specker Theorem for von Neumann Algebras

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

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

• 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

### A relativity principle in quantum mechanics

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

### 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,

### Topos Theory as a Framework for Partial Truth

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