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65
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.
A Topos Perspective on StateVector Reduction
"... A preliminary investigation is made of possible applications in quantum theory of the topos formed by the collection of all Msets, where M is a monoid. Earlier results on topos aspects of quantum theory can be rederived in this way. However, the formalism also suggests a new way of constructing a ‘ ..."
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A preliminary investigation is made of possible applications in quantum theory of the topos formed by the collection of all Msets, where M is a monoid. Earlier results on topos aspects of quantum theory can be rederived in this way. However, the formalism also suggests a new way of constructing a ‘neorealist’ interpretation of quantum theory in which the truth values of propositions are determined by the actions of the monoid of strings of finite projection operators. By these means, a novel topos perspective is gained on the concept of statevector reduction. 1
Is it true; or is it false; or somewhere in between? The logic of quantum theory. Contempory Phys
, 2005
"... The paper contains a relatively nontechnical summary of some recent work by the author and Jeremy Butterfield. The goal is to find a way of assigning meaningful truth values to propositions in quantum theory: something that is not possible in the normal, instrumentalist interpretation. The key math ..."
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The paper contains a relatively nontechnical summary of some recent work by the author and Jeremy Butterfield. The goal is to find a way of assigning meaningful truth values to propositions in quantum theory: something that is not possible in the normal, instrumentalist interpretation. The key mathematical tool is presheaf theory where, multivalued, contextual truth values arise naturally. We show how this can be applied to quantum theory, with the ‘contexts ’ chosen to be Boolean subalgebras of the set of all projection operators.
C ∞Smooth Singularities Exposed: Chimeras of the Differential Spacetime Manifold
, 2004
"... The glaringly serious conflict between the principle of general covariance of General Relativity (GR) and the existence of C ∞smooth singularities assailing the differential spacetime manifold on which the classical relativistic field theory of gravity vitally depends, is resolved by using the basi ..."
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The glaringly serious conflict between the principle of general covariance of General Relativity (GR) and the existence of C ∞smooth singularities assailing the differential spacetime manifold on which the classical relativistic field theory of gravity vitally depends, is resolved by using the basic manifold independent and, in extenso, Calculusfree concepts, techniques and results of Abstract Differential Geometry (ADG). As a physical toy model to illustrate these ideas, the ADGtheoretic resolution of both the exterior, but more importantly, of the inner, Schwarzschild singularities of the gravitational field of a point particle is presented, with the resolution of the latter being carried out entirely by finitisticalgebraic and sheaftheoretic means, and in two different ways. First, by regarding it as a localized, ‘static’ pointsingularity, we apply Sorkin’s finitary topological poset discretization scheme in its Gel’fand dual representation in terms of ‘discrete ’ differential incidence algebras [297, 298] and the finitary spacetime sheaves thereof [289]. Then we exercise the ADG machinery on those sheaves in the manner of [249, 250, 251] to show that the vacuum Einstein equations still hold over the classically offensive locus occupied by the pointmass both at the ‘discrete’
Finitary Topos for Locally Finite, Causal and Quantal Vacuum Einstein Gravity, submitted to the
 International Journal of Theoretical Physics
"... ..."
Finitary Čechde Rham Cohomology
 Int. J. Theor. Phys
, 2002
"... much ado without C ∞smoothness ..."
Quantum theory as a statistical theory under symmetry
 In Foundations of Probability and Physics 3
, 2005
"... The aim of the paper is to derive essential elements of quantum mechanics from a parametric structure extending that of traditional mathematical statistics. The main extensions, which also can be motivated from an applied statistics point of view, relate to symmetry, the choice between complementary ..."
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The aim of the paper is to derive essential elements of quantum mechanics from a parametric structure extending that of traditional mathematical statistics. The main extensions, which also can be motivated from an applied statistics point of view, relate to symmetry, the choice between complementary experiments and hence complementary parametric models, and use of the fact that there for simple systems always is a limited experimental basis that is common to all potential experiments. Concepts related to transformation groups together with the statistical concept of sufficiency are used in the construction of the quantummechanical Hilbert space. The Born formula is motivated through recent analysis by Deutsch and Gill, and is shown to imply the formulae of elementary quantum probability / quantum inference theory in the simple case. Planck’s constant, and the Schrödinger equation are also derived from this conceptual framework. The theory is illustrated by one and
Quantum Observables Algebras and Abstract Differential Geometry, preprint
, 2004
"... We construct a sheaf theoretical representation of Quantum Observables Algebras over a base Category equipped with a Grothendieck topology, consisting of epimorphic families of commutative Observables Algebras, playing the role of local arithmetics in measurement situations. This construction makes ..."
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We construct a sheaf theoretical representation of Quantum Observables Algebras over a base Category equipped with a Grothendieck topology, consisting of epimorphic families of commutative Observables Algebras, playing the role of local arithmetics in measurement situations. This construction makes possible the application of the methodology of Abstract Differential Geometry in a Category theoretical environment, and subsequently, the extension of the mechanism of differentials in the Quantum regime. 1
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
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))