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The SheafTheoretic Structure Of NonLocality and Contextuality
, 2011
"... Locality and noncontextuality are intuitively appealing features of classical physics, which are contradicted by quantum mechanics. The goal of the classic nogo theorems by Bell, KochenSpecker, et al. is to show that nonlocality and contextuality are necessary features of any theory whose predic ..."
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Locality and noncontextuality are intuitively appealing features of classical physics, which are contradicted by quantum mechanics. The goal of the classic nogo theorems by Bell, KochenSpecker, et al. is to show that nonlocality and contextuality are necessary features of any theory whose predictions agree with those of quantum mechanics. We use the mathematics of sheaf theory to analyze the structure of nonlocality and contextuality in a very general setting. Starting from a simple experimental scenario, and the kind of probabilistic models familiar from discussions of Bell’s theorem, we show that there is a very direct, compelling formalization of these notions in sheaftheoretic terms. Moreover, on the basis of this formulation, we show that the phenomena of nonlocality and contextuality can be characterized precisely in terms of obstructions to the existence of global sections. We give linear algebraic methods for computing these obstructions, and use these methods to obtain a number of new insights into nonlocality and contextuality. For example, we distinguish a proper hierarchy of strengths of nogo theorems, and show that three leading examples — due to Bell, Hardy, and Greenberger, Horne and Zeilinger, respectively — occupy successively higher levels of this hierarchy. We show how our abstract setting can be represented in quantum mechanics. In doing so, we uncover a strengthening of the usual nosignalling theorem, which shows that quantum mechanics obeys nosignalling for arbitrary families of commuting observables, not just those represented on different factors of a tensor product.
Algebraic Topology Foundations of Supersymmetry and Symmetry Breaking in Quantum Field Theory and Quantum Gravity: A Review
, 2009
"... A novel Algebraic Topology approach to Supersymmetry (SUSY) and Symmetry Breaking in Quantum Field and Quantum Gravity theories is presented with a view to developing a wide range of physical applications. These include: controlled nuclear fusion and other nuclear reaction studies in quantum chromod ..."
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A novel Algebraic Topology approach to Supersymmetry (SUSY) and Symmetry Breaking in Quantum Field and Quantum Gravity theories is presented with a view to developing a wide range of physical applications. These include: controlled nuclear fusion and other nuclear reaction studies in quantum chromodynamics, nonlinear physics at high energy densities, dynamic JahnTeller effects, superfluidity, high temperature superconductors, multiple scattering by molecular systems, molecular or atomic paracrystal structures, nanomaterials, ferromagnetism in glassy materials, spin glasses, quantum phase transitions and supergravity. This approach requires a unified conceptual framework that utilizes extended symmetries and quantum groupoid, algebroid and functorial representations of non–Abelian higher dimensional structures pertinent to quantized spacetime topology and state space geometry of quantum operator algebras. Fourier transforms, generalized Fourier–Stieltjes transforms, and duality relations link, respectively, the quantum groups and quantum groupoids with their dual algebraic structures; quantum double constructions are also discussed in this context in relation to quasitriangular, quasiHopf algebras, bialgebroids, GrassmannHopf algebras and Higher Dimensional Algebra. On the one hand, this quantum
The Space of Measurement Outcomes as a Spectral Invariant for NonCommutative Algebras
"... Abstract The recently developed technique of Bohrification associates to a (unital) C*algebra A 1. the Kripke model, a presheaf topos, of its classical contexts; 2. in this Kripke model a commutative C*algebra, called the Bohrification of A; 3. the spectrum of the Bohrification as a locale interna ..."
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Abstract The recently developed technique of Bohrification associates to a (unital) C*algebra A 1. the Kripke model, a presheaf topos, of its classical contexts; 2. in this Kripke model a commutative C*algebra, called the Bohrification of A; 3. the spectrum of the Bohrification as a locale internal in the Kripke model. We propose this locale, the ‘state space’, as a (n intuitionistic) logic of the physical system whose observable algebra is A. We compute a site which externally captures this locale and find that externally its points may be identified with partial measurement outcomes. This prompts us to compare Scottcontinuity on the poset of contexts and continuity with respect to the C*algebra as two ways to mathematically identify measurement outcomes with the same physical interpretation. Finally, we consider the notnotsheafification of the Kripke model on classical contexts and obtain a space of measurement outcomes which for commutative C*algebras coincides with the spectrum. The construction is
A Unified SheafTheoretic Account Of NonLocality and Contextuality
, 2011
"... A number of landmark results in the foundations of quantum mechanics show that quantum systems exhibit behaviour that defies explanation in classical terms, and that cannot be accounted for in such terms even by postulating “hidden variables” as additional unobserved factors. Much has been written o ..."
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A number of landmark results in the foundations of quantum mechanics show that quantum systems exhibit behaviour that defies explanation in classical terms, and that cannot be accounted for in such terms even by postulating “hidden variables” as additional unobserved factors. Much has been written on these matters, but there is surprisingly little unanimity even on basic definitions or the interrelationships among the various concepts and results. We use the mathematical language of sheaves and monads to give a very general and mathematically robust description of the behaviour of systems in which one or more measurements can be selected, and one or more outcomes observed. We say that an empirical model is extendable if it can be extended consistently to all sets of measurements, regardless of compatibility. A hiddenvariable model is factorizable if, for each value of the hidden variable, it factors as a product of distributions on the basic measurements. We prove that an empirical model is extendable if and only if there is a factorizable hiddenvariable model which realizes it. From this we are able to prove generalized versions of wellknown NoGo theorems. At the conceptual level, our equivalence result says that the existence of incompatible measurements is the essential ingredient in nonlocal and contextual behavior in quantum mechanics.
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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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))
Macroscopic observables and the Born rule
, 2008
"... Dedicated to the memory of Bernd Kuckert (1968–2008) We clarify the role of the Born rule in the Copenhagen Interpretation of quantum mechanics by deriving it from Bohr’s doctrine of classical concepts, translated into the following mathematical statement: a quantum system described by a noncommutat ..."
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Dedicated to the memory of Bernd Kuckert (1968–2008) We clarify the role of the Born rule in the Copenhagen Interpretation of quantum mechanics by deriving it from Bohr’s doctrine of classical concepts, translated into the following mathematical statement: a quantum system described by a noncommutative C ∗algebra of observables is empirically accessible only through associated commutative C ∗algebras. The Born probabilities emerge as the relative frequencies of outcomes in long runs of measurements on a quantum system; it is not necessary to adopt the frequency interpretation of singlecase probabilities (which will be the subject of a sequel paper). Our derivation of the Born rule uses ideas from a program begun by Finkelstein (1965) and Hartle (1968), intending to remove the Born rule as a separate postulate of quantum mechanics. Mathematically speaking, our approach refines previous elaborations of this program notably the one due to Farhi, Goldstone, and Gutmann (1989) as completed by Van Wesep (2006) in replacing infinite tensor products of Hilbert spaces by continuous fields of C ∗algebras. In combination with our interpretational context, this technical improvement circumvents valid criticisms that earlier derivations of the Born rule have provoked, especially to the effect that such derivations were mathematically flawed as well as circular. Furthermore, instead of relying on the controversial eigenvectoreigenvalue link in quantum theory, our derivation just assumes that pure states in classical physics have the usual interpretation as truthmakers that assign sharp values to observables.
Macroscopic observables and the Born rule I. Long
, 2008
"... Dedicated to the memory of Bernd Kuckert (1968–2008) We clarify the role of the Born rule in the Copenhagen Interpretation of quantum mechanics by deriving it from Bohr’s doctrine of classical concepts, translated into the following mathematical statement: a quantum system described by a noncommutat ..."
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Dedicated to the memory of Bernd Kuckert (1968–2008) We clarify the role of the Born rule in the Copenhagen Interpretation of quantum mechanics by deriving it from Bohr’s doctrine of classical concepts, translated into the following mathematical statement: a quantum system described by a noncommutative C∗algebra of observables is empirically accessible only through associated commutative C∗algebras. The Born probabilities emerge as the relative frequencies of outcomes in long runs of measurements on a quantum system; it is not necessary to adopt the frequency interpretation of singlecase probabilities (which will be the subject of a sequel paper). Our derivation of the Born rule uses ideas from a program begun by Finkelstein (1965) and Hartle (1968), intending to remove the Born rule as a separate postulate of quantum mechanics. Mathematically speaking, our approach refines previous elaborations of this program notably the one due to Farhi, Goldstone, and Gutmann (1989) as completed by Van Wesep (2006) in replacing infinite tensor products of Hilbert spaces by continuous fields of C∗algebras. In combination with our interpretational context, this technical improvement circumvents valid criticisms that earlier derivations of the Born rule have provoked, especially to the effect that such derivations were mathematically flawed as well as circular. Furthermore, instead of relying on the controversial eigenvectoreigenvalue link in quantum theory, our derivation just assumes that pure states in classical physics have the usual interpretation as truthmakers that assign sharp values to observables. 1