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A topos foundation for theories of physics: II. Daseinisation and the liberation of quantum theory
 Journal of Mathematical Physics
"... 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 when contemplating quantum theories of space and time. Our basic contention is that constructing a t ..."
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Cited by 11 (4 self)
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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 when contemplating quantum theories of space and time. 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 arises when the topos is the category of sets. Other types of theory employ a different topos. In this paper, we study in depth the topos representation of the propositional language, PL(S), for the case of quantum theory. In doing so, we make a direct link with, and clarify, the earlier work on applying topos theory to quantum physics. The key step is a process we term ‘daseinisation ’ by which a projection operator is mapped to a subobject of the spectral presheaf—the topos quantum analogue of a classical state space. In the second part of the paper we change gear with the introduction of the more sophisticated local language L(S). From this point forward, throughout the rest of the series of papers, our attention will be devoted almost entirely to this language. In the present paper, we use L(S) to study ‘truth objects ’ in the topos. These are objects in the topos that play the role of states: a necessary development as the spectral presheaf has no global elements, and hence there are no microstates in the sense of classical physics. Truth objects therefore play a crucial role in our formalism.
A Topos for Algebraic Quantum Theory
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 2009
"... 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 ..."
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Cited by 9 (1 self)
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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 T (A) in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*algebra A. According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum �(A) in T (A), which in our approach plays the role of the quantum phase space of the system. Thus we associate a locale (which is the topostheoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a C*algebra (which is the noncommutative notion of a space). In this setting, states on A become probability measures (more precisely, valuations) on �, and selfadjoint elements of A define continuous functions (more precisely, locale maps) from � to Scott’s interval domain. Noting that open subsets of �(A) correspond to propositions about the system, the pairing map that assigns a (generalized) truth value to a state and a proposition assumes an extremely simple categorical form. Formulated in this way, the quantum theory defined by A is essentially turned into a classical theory, internal to the topos T (A). These results were inspired by the topostheoretic approach to quantum physics proposed by Butterfield and Isham, as recently generalized by Döring and Isham.
Intuitionistic quantum logic of an nlevel system
, 2009
"... A decade ago, Isham and Butterfield proposed a topostheoretic approach to quantum mechanics, which meanwhile has been extended by Döring and Isham so as to provide a new mathematical foundation for all of physics. Last year, three of the present authors redeveloped and refined these ideas by combin ..."
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Cited by 1 (1 self)
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A decade ago, Isham and Butterfield proposed a topostheoretic approach to quantum mechanics, which meanwhile has been extended by Döring and Isham so as to provide a new mathematical foundation for all of physics. Last year, three of the present authors redeveloped and refined these ideas by combining the C*algebraic approach to quantum theory with the socalled internal language of topos theory (see arXiv:0709.4364). The goal of the present paper is to illustrate our abstract setup through the concrete example of the C*algebra Mn(C) of complex n × n matrices. This leads to an explicit expression for the pointfree quantum phase space Σn and the associated logical structure and Gelfand transform of an nlevel system. We also determine the pertinent nonprobabilisitic stateproposition pairing (or valuation) and give a very natural topostheoretic reformulation of the Kochen–Specker Theorem. In our approach, the nondistributive lattice P(Mn(C)) of projections in Mn(C) (which forms the basis of the traditional quantum logic of Birkhoff and von Neumann) is replaced by a specific distributive lattice O(Σn) of functions from the poset C(Mn(C))
Topos theory and ‘neorealist’ quantum theory
, 2007
"... basis for the formulation of physical theories. In this article, we give a brief introduction to this approach, emphasising the logical aspects. Each topos serves as a ‘mathematical universe ’ with an internal logic, which is used to assign truthvalues to all propositions about a physical system. W ..."
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Cited by 1 (1 self)
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basis for the formulation of physical theories. In this article, we give a brief introduction to this approach, emphasising the logical aspects. Each topos serves as a ‘mathematical universe ’ with an internal logic, which is used to assign truthvalues to all propositions about a physical system. We show in detail how this works for (algebraic) quantum theory. 1
The principle of general tovariance
, 2008
"... We tentatively propose two guiding principles for the construction of theories of physics, which should be satisfied by a possible future theory of quantum gravity. These principles are inspired by those that led Einstein to his theory of general relativity, viz. his principle of general covariance ..."
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We tentatively propose two guiding principles for the construction of theories of physics, which should be satisfied by a possible future theory of quantum gravity. These principles are inspired by those that led Einstein to his theory of general relativity, viz. his principle of general covariance and his equivalence principle, as well as by the two mysterious dogmas of Bohr’s interpretation of quantum mechanics, i.e. his doctrine of classical concepts and his principle of complementarity. An appropriate mathematical language for combining these ideas is topos theory, a framework earlier proposed for physics by Isham and collaborators. Our principle of general tovariance states that any mathematical structure appearing in the laws of physics must be definable in an arbitrary topos (with natural numbers object) and must be preserved under socalled geometric morphisms. This principle identifies geometric logic as the mathematical language of physics and restricts the constructions and theorems to those valid in intuitionism: neither Aristotle’s principle of the excluded third nor Zermelo’s Axiom of Choice may be invoked. Subsequently, our equivalence principle states that any algebra of observables (initially defined in the topos Sets) is empirically equivalent to a commutative one in some other topos.
Intuitionistic quantum logic of an nlevel system
, 902
"... A decade ago, Isham and Butterfield proposed a topostheoretic approach to quantum mechanics, which meanwhile has been extended by Döring and Isham so as to provide a new mathematical foundation for all of physics. Last year, three of the present authors redeveloped and refined these ideas by combin ..."
Abstract
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A decade ago, Isham and Butterfield proposed a topostheoretic approach to quantum mechanics, which meanwhile has been extended by Döring and Isham so as to provide a new mathematical foundation for all of physics. Last year, three of the present authors redeveloped and refined these ideas by combining the C*algebraic approach to quantum theory with the socalled internal language of topos theory (see arXiv:0709.4364). The goal of the present paper is to illustrate our abstract setup through the concrete example of the C*algebra Mn(C) of complex n × n matrices. This leads to an explicit expression for the pointfree quantum phase space Σn and the associated logical structure and Gelfand transform of an nlevel system. We also determine the pertinent nonprobabilisitic stateproposition pairing (or valuation) and give a very natural topostheoretic reformulation of the Kochen–Specker Theorem. In our approach, the nondistributive lattice P(Mn(C)) of projections in Mn(C) (which forms the basis of the traditional quantum logic of Birkhoff and von Neumann) is replaced by a specific distributive lattice O(Σn) of functions from the poset C(Mn(C))
London SW7 2BZ
, 2007
"... This paper is the fourth in a series whose goal is to develop a fundamentally new way of building theories of physics. The motivation comes from a desire to address certain deep issues that arise in the quantum theory of gravity. Our basic contention is that constructing a theory of physics is equiv ..."
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This paper is the fourth in a series whose goal is to develop a fundamentally new way of building theories of physics. The motivation comes from a desire to address certain deep issues that arise in the quantum theory of gravity. 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 arises when the topos is the category of sets. Other types of theory employ a different topos. The previous papers in this series are concerned with implementing this programme for a single system. In the present paper, we turn to considering a collection of systems: in particular, we are interested in the relation between the topos representation for a composite system, and the representations for its constituents. We also study this problem for the disjoint sum of two systems. Our approach to these matters is to construct a category of systems and to find a topos representation of the entire category.
Topos Mediated Gravity: Toward the Categorical Resolution of the Cosmological Constant
, 2009
"... According to Döring and Isham the spectral topos corresponds to any quantum system. The description of a system in the topos becomes similar to this given by classical theory. However, the logic of the emergent theory is rather intuitionistic than classical. According to the recent proposition by th ..."
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According to Döring and Isham the spectral topos corresponds to any quantum system. The description of a system in the topos becomes similar to this given by classical theory. However, the logic of the emergent theory is rather intuitionistic than classical. According to the recent proposition by the author, topoi can modify local smooth spacetime structure. A way how to add gravity into the spectral topos of a system is presented. Supposing that a quantum system modifies the local spacetime structure and interacts with a gravitational field via the spectral topos, a natural pattern for nongravitating quantum lowest energy modes of the system, appears. Moreover, a theory of gravity and systems should be symmetric with respect to the 2group of automorphisms of the category of systems. Thus, any quantum system modifies the spacetime structure locally, and their lowest modes have vanishing contributions to the cosmological constant. Nullifying the contributions to the cosmological constant is deeply related with a fundamental higher symmetry group of gravity. This is the preliminary version of the paper submitted to FP.