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The British Journal for the Philosophy of Science

Topos Theoretic Quantum Realism

Benjamin Eva

Published Apr 15, 2016• 10 citations
Publisher

Abstract

Topos quantum theory (TQT) is standardly portrayed as a kind of ‘neo-realist’ reformulation of quantum mechanics.1 In this article, I study the extent to which TQT can really be characterized as a realist formulation of the theory, and examine the question of whether the kind of realism that is provided by TQT satisfies the philosophical motivations that are usually associated with the search for a realist reformulation of quantum theory. Specifically, I show that the notion of the quantum state is problematic for those who view TQT as a realist reformulation of quantum theory. 1. Introduction2. Topos Quantum Theory 2.1. Phase space2.2. Hilbert space2.3. Beyond Hilbert space2.4. Defining realism2.5. The spectral presheaf2.6. The logic of topos quantum theory3. Interpreting States in Topos Quantum Theory4. Interpreting Truth Values and Clopen Subobjects in Topos Quantum Theory 4.1. Interpreting the truth values4.2. Interpreting Subcl(Σ̲)5. Neo-realism 5.1. The covariant approach6. Conclusion

DOI
10.1093/bjps/axv057

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Topos quantum theory (TQT) is standardly portrayed as a kind of ‘neo-realist’ reformulation of quantum mechanics.1 In this article, I study the extent to which TQT can really be characterized as a realist formulation of the theory, and examine the question of whether the kind of realism that is provided by TQT satisfies the philosophical motivations that are usually associated with the search for a realist reformulation of quantum theory. Specifically, I show that the notion of the quantum state is problematic for those who view TQT as a realist reformulation of quantum theory. 1. Introduction2. Topos Quantum Theory 2.1. Phase space2.2. Hilbert space2.3. Beyond Hilbert space2.4. Defining realism2.5. The spectral presheaf2.6. The logic of topos quantum theory3. Interpreting States in Topos Quantum Theory4. Interpreting Truth Values and Clopen Subobjects in Topos Quantum Theory 4.1. Interpreting the truth values4.2. Interpreting Subcl(Σ̲)5. Neo-realism 5.1. The covariant approach6. Conclusion
The mathematical universe of the quantum topos, which is formulated on the basis of classical Boolean snapshots, delivers a neo-realist description of quantum mechanics that preserves realism. The main contribution of this article is developing formal objectivity in physical theories beyond quantum mechanics in the topos-theory approach. It will be shown that neo-realist responses to non-perturbative structures of quantum field theory do not preserve realism. In this regard, the method of Feynman graphons is applied to reframe the task of describing objectivity in quantum field theory in terms of replacing the standard Hilbert-space/operator-algebra ontology with a new context category built from a certain family of topological Hopf subalgebras of the topological Hopf algebra of renormalization as algebraic/combinatorial data tied to non-perturbative structures. This topological-Hopf-algebra ontology, which is independent of instrumentalist probabilities, enables us to reconstruct gauge field theories on the basis of the mathematical universe of the non-perturbative topos. The non-Boolean logic of the non-perturbative topos cannot be recovered by classical Boolean snapshots, which is in contrast to the quantum-topos reformulation of quantum mechanics. The article formulates a universal version of the non-perturbative topos to show that quantum field theory is a globally and locally neo-realist theory which can be reconstructed independent of the standard Hilbert-space/operator-algebra ontology. Formal objectivity of the universal non-perturbative topos offers a new route to build objective semantics for non-perturbative structures.
Topos quantum theory (TQT) represents a whole new approach to the formalization of non-relativistic quantum theory. It is well known that TQT replaces the orthomodular quantum logic of the traditional Hilbert space formalism with a new intuitionistic logic that arises naturally from the topos theoretic structure of the theory. However, it is less well known that TQT also has a dual logical structure that is paraconsistent. In this paper, we investigate the relationship between these two logical structures and study the implications of this relationship for the definition of modal operators in TQT.
Abstract Quantum set theory (QST) and topos quantum theory (TQT) are two long running projects in the mathematical foundations of quantum mechanics (QM) that share a great deal of conceptual and technical affinity. Most pertinently, both approaches attempt to resolve some of the conceptual difficulties surrounding QM by reformulating parts of the theory inside of nonclassical mathematical universes, albeit with very different internal logics. We call such mathematical universes, together with those mathematical and logical structures within them that are pertinent to the physical interpretation, ‘Q-worlds’. Here, we provide a unifying framework that allows us to (i) better understand the relationship between different Q-worlds, and (ii) define a general method for transferring concepts and results between TQT and QST, thereby significantly increasing the expressive power of both approaches. Along the way, we develop a novel connection to paraconsistent logic and introduce a new class of structures that have significant implications for recent work on paraconsistent set theory.
We discuss some conceptual issues that any approach to quantum gravity has to confront. In particular, it is argued that one has to find a theory that can be interpreted in a realist manner, because theories with an instrumentalist interpretation are problematic for several well-known reasons. Since the Hilbert space formalism almost inevitably forces an instrumentalist interpretation on us, we suggest that a theory of quantum gravity should not be based on the Hilbert space formalism. We briefly sketch the topos approach, which makes use of the internal logic of a topos associated with a quantum system and comes with a natural (neo-)realist interpretation. Finally, we make some remarks on the relation between system logic and metalogic.
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.