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147
The octonions
 Bull. Amer. Math. Soc
, 2002
"... Abstract. The octonions are the largest of the four normed division algebras. While somewhat neglected due to their nonassociativity, they stand at the crossroads of many interesting fields of mathematics. Here we describe them and their relation to Clifford algebras and spinors, Bott periodicity, p ..."
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Cited by 237 (6 self)
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Abstract. The octonions are the largest of the four normed division algebras. While somewhat neglected due to their nonassociativity, they stand at the crossroads of many interesting fields of mathematics. Here we describe them and their relation to Clifford algebras and spinors, Bott periodicity, projective and Lorentzian geometry, Jordan algebras, and the exceptional Lie groups. We also touch upon their applications in quantum logic, special relativity and supersymmetry. 1.
Nonconventional ergodic averages and nilmanifolds
"... Abstract. We study the L2convergence of two types of ergodic averages. The first is the average of a product of functions evaluated at return times along arithmetic progressions, such as the expressions appearing in Furstenberg’s proof of Szemerédi’s Theorem. The second average is taken along cube ..."
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Cited by 156 (16 self)
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Abstract. We study the L2convergence of two types of ergodic averages. The first is the average of a product of functions evaluated at return times along arithmetic progressions, such as the expressions appearing in Furstenberg’s proof of Szemerédi’s Theorem. The second average is taken along cubes whose sizes tend to +∞. For each average, we show that it is sufficient to prove the convergence for special systems, the characteristic factors. We build these factors in a general way, independent of the type of the average. To each of these factors we associate a natural group of transformations and give them the structure of a nilmanifold. From the second convergence result we derive a combinatorial interpretation for the arithmetic structure inside a set of integers of positive upper density. 1.
Modular localization and Wigner particles
 Rev. Math. Phys
"... Dedicated to Huzihiro Araki on the occasion of his seventieth birthday Abstract. We propose a framework for the free field construction of algebras of local observables which uses as an input the BisognanoWichmann relations and a representation of the Poincaré group on the oneparticle Hilbert spac ..."
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Cited by 47 (8 self)
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Dedicated to Huzihiro Araki on the occasion of his seventieth birthday Abstract. We propose a framework for the free field construction of algebras of local observables which uses as an input the BisognanoWichmann relations and a representation of the Poincaré group on the oneparticle Hilbert space. The abstract real Hilbert subspace version of the TomitaTakesaki theory enables us to bypass some limitations of the Wigner formalism by introducing an intrinsic spacetime localization. Our approach works also for continuous spin representations to which we associate a net of von Neumann algebras on spacelike cones with the ReehSchlieder property. The positivity of the energy in the representation turns out to be equivalent to the isotony of the net, in the spirit of Borchers theorem. Our procedure extends to other spacetimes homogeneous under a group of geometric transformations as in the case of conformal symmetries and de Sitter spacetime. 1.
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 32 (4 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.
Semantic Vector Products: Some Initial Investigations
"... Semantic vector models have proven their worth in a number of natural language applications whose goals can be accomplished by modelling individual semantic concepts and measuring similarities between them. By comparison, the area of semantic compositionality in these models has so far remained unde ..."
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Cited by 30 (0 self)
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Semantic vector models have proven their worth in a number of natural language applications whose goals can be accomplished by modelling individual semantic concepts and measuring similarities between them. By comparison, the area of semantic compositionality in these models has so far remained underdeveloped. This will be a crucial hurdle for semantic vector models: in order to play a fuller part in the modelling of human language, these models will need some way of modelling the way in which single concepts are put together to form more complex conceptual structures. This paper explores some of the opportunities for using vector product operations to model compositional phenomena in natural language. These vector operations
Nonsingular transformations and spectral analysis of measures
 Bull. Soc. Math. France
, 1991
"... RÉSUMÉ. — Ce travail approfondit les interactions qui existent entre l’analyse harmonique des mesures et l’étude spectrale des systèmes dynamiques non singuliers. Il est centre ́ sur l’étude de sousgroupes remarquables du cercle, groupes de valeurs propres, groupes de quasiinvariance des mesu ..."
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Cited by 23 (0 self)
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RÉSUMÉ. — Ce travail approfondit les interactions qui existent entre l’analyse harmonique des mesures et l’étude spectrale des systèmes dynamiques non singuliers. Il est centre ́ sur l’étude de sousgroupes remarquables du cercle, groupes de valeurs propres, groupes de quasiinvariance des mesures..., dont les exemples les plus naturels sont définis par des conditions diophantiennes. La conjonction des points de vue permet d’obtenir nombre de résultats nouveaux dans les deux théories, y compris dans des problèmes classique d’analyse de Fourier. ABSTRACT. — This work explores in depth the interactions existing between harmonic analysis of measures and spectral theory of nonsingular dynamical systems. It focuses on the study of some classes of remarkable subgroups of the circle: eigenvalue groups, groups of quasiinvariance of measures..., the most natural examples of which are defined by diophantine conditions. The conjonction of these points of view leads to many new results in both theories, including some classical problems in Fourier analysis. 1.
The BisognanoWichmann theorem for massive theories
 Annales Henri Poincaré 2, 907 (2001), [arXiv: hepth/0101227
"... Abstract. The geometric action of modular groups for wedge regions (BisognanoWichmann property) is derived from the principles of local quantum physics for a large class of Poincaré covariant models in d = 4. As a consequence, the CPT theorem holds for this class. The models must have a complete in ..."
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Cited by 22 (5 self)
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Abstract. The geometric action of modular groups for wedge regions (BisognanoWichmann property) is derived from the principles of local quantum physics for a large class of Poincaré covariant models in d = 4. As a consequence, the CPT theorem holds for this class. The models must have a complete interpretation in terms of massive particles. The corresponding charges need not be localizable in compact regions: The most general case is admitted, namely localization in spacelike cones. Introduction. In local relativistic quantum theory [23], a model is specified in terms of a net of local observable algebras and a representation of the Poincaré group, under which the net is covariant. The BisognanoWichmann theorem [2, 3] intimately connects these two, algebraic and geometric, aspects. Namely, it asserts that under certain conditions modular covariance
Linear logic for generalized quantum mechanics
 In Proc. Workshop on Physics and Computation (PhysComp'92
, 1993
"... Quantum logic is static, describing automata having uncertain states but no state transitions and no Heisenberg uncertainty tradeoff. We cast Girard’s linear logic in the role of a dynamic quantum logic, regarded as an extension of quantum logic with time nonstandardly interpreted over a domain of l ..."
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Cited by 21 (2 self)
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Quantum logic is static, describing automata having uncertain states but no state transitions and no Heisenberg uncertainty tradeoff. We cast Girard’s linear logic in the role of a dynamic quantum logic, regarded as an extension of quantum logic with time nonstandardly interpreted over a domain of linear automata and their dual linear schedules. In this extension the uncertainty tradeoff emerges via the “structure veil. ” When VLSI shrinks to where quantum effects are felt, their computeraided design systems may benefit from such logics of computational behavior having a strong connection to quantum mechanics. 1
Physical traces: Quantum vs. classical information processing
 In Proceedings of Category Theory and Computer Science 2002 (CTCS’02), volume 69 of Electronic Notes in Theoretical Computer Science. Elsevier Science
, 2003
"... a setting that enables qualitative differences between classical and quantum processes to be explored. The key construction is the physical interpretation/realization of the traced monoidal categories of finitedimensional vector spaces with tensor product as monoidal structure and of finite sets an ..."
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Cited by 19 (5 self)
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a setting that enables qualitative differences between classical and quantum processes to be explored. The key construction is the physical interpretation/realization of the traced monoidal categories of finitedimensional vector spaces with tensor product as monoidal structure and of finite sets and relations with Cartesian product as monoidal structure, both of them providing a socalled wavestyle GoI. The developments in this paper reveal that envisioning state update due to quantum measurement as a process provides a powerful tool for developing highlevel approaches to quantum information processing.