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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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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.
and
, 2007
"... This paper is the third in a series whose goal is to develop a fundamentally new way of viewing theories of physics. 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. In pape ..."
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This paper is the third in a series whose goal is to develop a fundamentally new way of viewing theories of physics. 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. In paper II, we studied the topos representations of the propositional language PL(S) for the case of quantum theory, and in the present paper we do the same thing for the, more extensive, local language L(S). One of the main achievements is to find a topos representation for selfadjoint operators. This involves showing that, for any physical quantity A, there is an arrow ˘ δo (A) : Σ → IR ≽ , where IR ≽ is the quantityvalue object for this theory. The construction of ˘ δo ( Â) is an extension of the daseinisation of projection operators that was discussed in paper II., of The object IR ≽ is a monoidobject only in the topos, τφ = SetsV(H)op the theory, and to enhance the applicability of the formalism, we apply to IR ≽ a topos analogue of the Grothendieck extension of a monoid to a group. The resulting object, k(IR ≽), is an abelian groupobject in τφ. We also discuss another candidate, IR ↔ , for the quantityvalue object. In this presheaf, both inner and outer daseinisation are used in a symmetric way. Finally, there is a brief discussion of the role of unitary operators in the quantum topos scheme.
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.
1 On Trasformations Between Belief Spaces
"... Abstract: Commutative monoids of belief states have been defined by imposing one or more of the usual axioms and employing a combination rule. Familiar operations such as normalization and the Voorbraak map are surjective homomorphisms. The latter, in particular, takes values in a space of Bayesian ..."
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Abstract: Commutative monoids of belief states have been defined by imposing one or more of the usual axioms and employing a combination rule. Familiar operations such as normalization and the Voorbraak map are surjective homomorphisms. The latter, in particular, takes values in a space of Bayesian states. The pignistic map is not a homomorphism between these same spaces. We demonstrate an impact this may have on robust decision making for frames of cardinality at least 3. We adapt the measure zero reflection property of some maps between probability spaces to define a category of belief states having plausibility zero reflecting functions as morphisms. Our definition encapsulates a generalization of the notion of absolute continuity to the context of belief spaces. We show that the Voorbraak map is a functor valued in this category. 1
Layout Randomization and Nondeterminism
"... In security, layout randomization is a popular, effective attack mitigation technique. Recent work has aimed to explain it rigorously, focusing on deterministic systems. In this paper, we study layout randomization in the presence of nondeterministic choice. We develop a semantic approach based on d ..."
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In security, layout randomization is a popular, effective attack mitigation technique. Recent work has aimed to explain it rigorously, focusing on deterministic systems. In this paper, we study layout randomization in the presence of nondeterministic choice. We develop a semantic approach based on denotational models and simulation relations. This approach abstracts from language details, and helps manage the delicate interaction between probabilities and nondeterminism. Keywords: security, semantics, probabilities, nondeterminism, full abstraction.