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arXiv:1106.5660 · Canadian Journal of Physics

Review of the topos approach to quantum theory

Cecilia Flori

Published Mar 13, 2013• 9 citations
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Abstract

Topos theory has been suggested by Döring and Isham as an alternative mathematical structure with which to formulate, in general, physical theories. However, the motivation for using it to express quantum theory, lies in the desire to solve certain interpretational problems inherent in the standard formulation of the theory. In fact, the topos approach to quantum theory overcomes the instrumentalist–Copenhagen interpretation, rendering the theory more realist. The caveat is that one ends up with a multivalued–intuitionistic logic, rather than a Boolean logic.

DOI
10.1139/cjp-2012-0364

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Topos theory has been suggested by Döring and Isham as an alternative mathematical structure with which to formulate, in general, physical theories. However, the motivation for using it to express quantum theory, lies in the desire to solve certain interpretational problems inherent in the standard formulation of the theory. In fact, the topos approach to quantum theory overcomes the instrumentalist–Copenhagen interpretation, rendering the theory more realist. The caveat is that one ends up with a multivalued–intuitionistic logic, rather than a Boolean logic.
Topos theory has been suggested by Döring and Isham as an alternative mathematical structure with which to formulate physical theories. In particular, it has been used to reformulate standard quantum mechanics in such a way that a novel type of logic is used to represent propositions. In this paper we extend this formulation to include temporally ordered collections of propositions as opposed to single-time propositions. That is to say, we have developed a quantum history formalism in the language of topos theory where truth values can be assigned to temporal propositions. We analyze the extent to which such truth values can be derived from the truth values of the constituent, single-time propositions.
Topos theory has been suggested first by Isham and Butterfield, and then by Isham and Döring, as an alternative mathematical structure within which to formulate physical theories. In particular, it has been used to reformulate standard quantum mechanics in such a way that a novel type of logic is used to represent propositions. In this paper, we extend this formulation to include the notion of a group and group transformation in such a way that we overcome the problem of twisted presheaves. In order to implement this we need to change the type of topos involved, so as to render the notion of continuity of the group action meaningful.
Topos theory has been suggested first by Isham and Butterfield, and then by Isham and Döring, as an alternative mathematical structure within which to formulate physical theories. In particular it has been used to reformulate standard quantum mechanics in such a way that a novel type of logic is used to represent propositions. In this paper we extend this formulation to include the notion of a group and group transformation in such a way that we overcome the problem of twisted presheaves. In order to implement this we need to change the type of topos involved, so as to render the notion of continuity of the group action meaningful.
What is the role of topos theory in the topos models for quantum theory as used by Isham, Butterfield, Döring, Heunen, Landsman, Spitters, and others? In other words, what is the interplay between physical motivation for the models and the mathematical framework used in these models? Concretely, we show that the presheaf topos model of Butterfield, Isham, and Döring resembles classical physics when viewed from the internal language of the presheaf topos, similar to the copresheaf topos model of Heunen, Landsman, and Spitters. Both the presheaf and copresheaf models provide a “quantum logic” in the form of a complete Heyting algebra. Although these algebras are natural from a topos theoretic stance, we seek a physical interpretation for the logical operations. Finally, we investigate dynamics. In particular, we describe how an automorphism on the operator algebra induces a homeomorphism (or isomorphism of locales) on the associated state spaces of the topos models, and how elementary propositions and truth values transform under the action of this homeomorphism. Also with dynamics the focus is on the internal perspective of the topos.
Topos theory has been suggested by Doring and Isham as an alternative mathematical structure with which to formulate physical theories. In particular it has been used to reformulate standard quantum mechanics in such a way that a novel type of logic is used to represent propositions. In this paper we extend this formulation to include temporally-ordered collections of propositions as opposed to single-time propositions. That is to say, we have developed a quantum history formalism in the language of topos theory where truth values can be assigned to temporal propositions. We analyse the extent to which such truth values can be derived from the truth values of the constituent, single-time propositions.
The topos approach to the formulation of physical theories includes a new form of quantum logic. We present this topos quantum logic, including some new results, and compare it to standard quantum logic, all with an eye to conceptual issues. In particular, we show that topos quantum logic is distributive, multi-valued, contextual and intuitionistic. It incorporates superposition without being based on linear structures, has a built-in form of coarse-graining which automatically avoids interpretational problems usually associated with the conjunction of propositions about incompatible physical quantities, and provides a material implication that is lacking from standard quantum logic. Importantly, topos quantum logic comes with a clear geometrical underpinning. The representation of pure states and truth-value assignments are discussed. It is briefly shown how mixed states fit into this approach.