Towards a Paraconsistent Quantum Set Theory
TL;DR: In this paper, a connection between quantum set theory and topos quantum theory was established by Ozawa, Takeuti and Titani, who studied algebraic valued set-theoretic structures whose truth values correspond to clopen subobjects of the spectral presheaf of an orthomodular lattice of projections onto a given Hilbert space.
Abstract: In this paper, we will attempt to establish a connection between quantum set theory, as developed by Ozawa, Takeuti and Titani, and topos quantum theory, as developed by Isham, Butterfield and Doring, amongst others. Towards this end, we will study algebraic valued set-theoretic structures whose truth values correspond to the clopen subobjects of the spectral presheaf of an orthomodular lattice of projections onto a given Hilbert space. In particular, we will attempt to recreate, in these new structures, Takeuti's original isomorphism between the set of all Dedekind real numbers in a suitably constructed model of set theory and the set of all self adjoint operators on a chosen Hilbert space.
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TL;DR: In this paper, the authors provide a unifying framework that allows us to better understand the relationship between different Q-worlds, and define a general method for transferring concepts and results between TQT and QST, thereby significantly increasing the expressive power of both approaches.
Abstract: Quantum set theory (QST) and topos quantum theory (TQT) are two long running projects in the mathematical foundations of quantum mechanics that share a great deal of conceptual and technical affinity. Most pertinently, both approaches attempt to resolve some of the conceptual difficulties surrounding quantum mechanics by reformulating parts of the theory inside of non-classical 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.
6 citations
09 Mar 2015
TL;DR: In this article, it was shown that the assignment of a complete orthomodular lattice L to a spectral presheaf is contravariant functorial, and that the clopen subobjects form a complete bi-Heyting algebra.
Abstract: With each orthomodular lattice L we associate a spectral presheaf \(\underline{\varSigma }^{L}\), generalising the Stone space of a Boolean algebra, and show that (a) the assignment \(L\mapsto \underline{\varSigma }^{L}\) is contravariantly functorial, (b) \(\underline{\varSigma }^{L}\) is a complete invariant of L, and (c) for complete orthomodular lattices there is a generalisation of Stone representation in the sense that L is mapped into the clopen subobjects of the spectral presheaf \(\underline{\varSigma }^{L}\). The clopen subobjects form a complete bi-Heyting algebra, and by taking suitable equivalence classes of clopen subobjects, one can regain a complete orthomodular lattice isomorphic to L. We interpret our results in the light of quantum logic and in the light of the topos approach to quantum theory.
6 citations
TL;DR: A unifying framework is provided that allows to better understand the relationship between different Q-worlds, and a general method for transferring concepts and results between TQT and QST is defined, thereby significantly increasing the expressive power of both approaches.
Abstract: Quantum set theory (QST) and topos quantum theory (TQT) are two long running projects in the mathematical foundations of quantum mechanics that share a great deal of conceptual and technical affinity. Most pertinently, both approaches attempt to resolve some of the conceptual difficulties surrounding quantum mechanics by reformulating parts of the theory inside of non-classical 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.
6 citations
Cites background from "Towards a Paraconsistent Quantum Se..."
...Specifically, Eva [10] suggests the following definition....
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...1 V (Subcl(Σ)) Eva [10] suggests the possibility of connecting TQT and QST via the set-theoretic structure V (Subcl(Σ)), where Subcl(Σ) is equipped with the negation ∗ rather than the Heyting negation....
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...Eva [10] notes that defining [S]∗ = [S∗] implies that E and L are isomorphic not just as complete lattices, but also as complete ortholattices....
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...Until now, the tantalising prospect of unifying these two (kinds of) Q-worlds, non-distributive set theoretic QST and distributive topos-theoretic TQT, within a single formal setting has gone almost completely unexplored (the prospect was first tentatively suggested by Eva [10])....
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...Eva [10] establishes the following basic properties of the ∗ negation....
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TL;DR: In this article, a new truth value assignment for bounded quantifiers that satisfies De Morgan's Laws was proposed, and it was shown that this assignment has a lower bound determined by the commutator, a projection-valued degree of commutativity, of constants in the formula.
4 citations
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TL;DR: In this paper, the Kochen-Specker theorem is replaced by the notion of a sieves of operators, which is a generalization of the theory of presheaves.
Abstract: Any 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 preserves functional relations between them. We constructa new type of valuation which is defined on alloperators, and which respects an appropriate version ofthe functional composition principle. The truth-values assigned to propositions are (i) contextual and(ii) multivalued, where the space of contexts and themultivalued logic for each context come naturally fromthe topos theory of presheaves. The first step in our theory is to demonstrate that theKochen–Specker theorem is equivalent to thestatement that a certain presheaf defined on thecategory of self-adjoint operators has no globalelements. We then show how the use of ideas drawn from the theory ofpresheaves leads to the definition of a generalizedvaluation in quantum theory whose values are sieves ofoperators. In particular, we show how each quantum state leads to such a generalized valuation. Akey ingredient throughout is the idea that, in asituation where no normal truth-value can be given to aproposition asserting that the value of a physical quantity A lies in a subset \(\Delta \subseteq \mathbb{R}\), it is nevertheless possible toascribe a partial truth-value which is determined by theset of all coarse-grained propositions that assert thatsome function f(A) lies in f(Δ), and that are true in a normalsense. The set of all such coarse-grainings forms asieve on the category of self-adjoint operators, and ishence fundamentally related to the theory ofpresheaves.
238 citations
Posted Content•
TL;DR: In this paper, the authors show that the Kochen-Specker theorem is equivalent to the statement that a certain presheaf defined on the category of self-adjoint operators has no global elements.
Abstract: The Kochen-Specker theorem asserts the impossibility of assigning values to quantum quantities in a way that preserves functional relations between them. We construct a new type of valuation which is defined on all operators, and which respects an appropriate version of the functional composition principle. The truth-values assigned to propositions are (i) contextual; and (ii) multi-valued, where the space of contexts and the multi-valued logic for each context come naturally from the topos theory of presheaves.
The first step in our theory is to demonstrate that the Kochen-Specker theorem is equivalent to the statement that a certain presheaf defined on the category of self-adjoint operators has no global elements. We then show how the use of ideas drawn from the theory of presheaves leads to the definition of a generalized valuation in quantum theory whose values are sieves of operators. In particular, we show how each quantum state leads to such a generalized valuation.
180 citations
"Towards a Paraconsistent Quantum Se..." refers methods in this paper
...In this paper, we will attempt to establish a connection between quantum set theory, as developed by Ozawa, Takeuti and Titani (see, for example, [14], [13], [10]), and topos quantum theory, as developed by Isham, Butterfield and Döring, amongst others (see, for example, [8], [6], [3])....
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Book•
21 Aug 1978
156 citations
"Towards a Paraconsistent Quantum Se..." refers background or methods in this paper
...In this paper, we will attempt to establish a connection between quantum set theory, as developed by Ozawa, Takeuti and Titani (see, for example, [14], [13], [10]), and topos quantum theory, as developed by Isham, Butterfield and Döring, amongst others (see, for example, [8], [6], [3])....
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...In [13], it was shown that if B is an algebra of mutually compatible projection operators on some Hilbert space H, then R(B) is isomorphic to the set of all self adjoint operators on H whose spectral projections all lie in B....
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TL;DR: In this paper, it was shown that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. But the problem of quantum topos is different from that of quantum quantum physics.
Abstract: The goal of this paper is to summarise the first steps in developing a fundamentally new way of constructing theories of physics. The motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time. In doing so we provide a new answer to Heidegger's timeless question ``What is a thing?''.
Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. Classical physics uses the topos of sets. Other theories involve a different topos. For the types of theory discussed in this paper, a key goal is to represent any physical quantity $A$ with an arrow $\breve{A}_\phi:\Si_\phi\map\R_\phi$ where $\Si_\phi$ and $\R_\phi$ are two special objects (the `state-object' and `quantity-value object') in the appropriate topos, $\tau_\phi$.
We discuss two different types of language that can be attached to a system, $S$. The first, $\PL{S}$, is a propositional language; the second, $Ł{S}$, is a higher-order, typed language. Both languages provide deductive systems with an intuitionistic logic. With the aid of $\PL{S}$ we expand and develop some of the earlier work (By CJI and collaborators.) on topos theory and quantum physics. A key step is a process we term `daseinisation' by which a projection operator is mapped to a sub-object of the spectral presheaf $\Sig$--the topos quantum analogue of a classical state space. The topos concerned is $\SetH{}$: the category of contravariant set-valued functors on the category (partially ordered set) $\V{}$ of commutative sub-algebras of the algebra of bounded operators on the quantum Hilbert space $\Hi$.
152 citations