The Experts below are selected from a list of 303 Experts worldwide ranked by ideXlab platform
Cecilia Flori - One of the best experts on this subject based on the ideXlab platform.
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A Second Course in Topos Quantum Theory
2018Co-Authors: Cecilia FloriAbstract:Logic of Propositions in Topos Quantum Theory -- Developments on Self Adjoint Operators in Topos Quantum Theory -- Physical quantities interpreted as modal operators -- Group action in Topos Quantum Theory take two -- Quantization in Topos Quantum Theory -- Groethendieck Topoi -- Locales -- Spacetime -- Topos and Logic -- Internalizing Objects in Topos Theory -- What information can be recovered from the abelian subalgebras of a von-Neumann algebra -- Extending the spectral presheaf to non-abelian unital C*-algebras.
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Internalizing Objects in Topos Theory
Lecture Notes in Physics, 2018Co-Authors: Cecilia FloriAbstract:In this chapter we will explain how to define categorical notions internally within a Topos. This internal description of objects is needed to understand the covariant approach to Topos quantum theory explained in the next chapter.
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Extending the Topos Quantum Theory Approach
Lecture Notes in Physics, 2018Co-Authors: Cecilia FloriAbstract:As it has been developed so far, the mathematical formalism of Topos quantum theory only allows for taking into consideration one physical system at a time. This has clearly some limitations, in particular when trying to consider composite systems. Hence it comes natural to try and enlarge the mathematical formalism so that is is possible to take into consideration various physical systems at the same time. This implies considering a Topos somewhat “larger” than the Topos \(\textbf {Sets}^{\mathcal {V}(\mathcal {H})^{\mathrm {op}}}\). In particular what needs to be “enlarged” is the category \( \mathcal {V}(\mathcal {H})\). In fact this category only refers to the physical system with associated von Neumann algebra \(\mathcal {N}\), whose category of abliean subalgebras is given by \(\mathcal {V}(\mathcal {H})\). However we would like to consider all physical systems, each of which, has associated to it a different von Neumann algebra. To account for this, one possibility would be to construct a category in which each element is itself a Topos which represents the mathematical formalism of a physical system. Then one would have to construct a mapping which associates to each physical system its associated Topos. The aim would be to turn this map into a geometric morphism of some sort between topoi, such that it possesses nice properties which would help to better understand composite systems. In the following we will present all work done so far in this direction, which is an exposition of the results obtained in [30]. As it will be clear in due course, there are still many open problems to be addressed.
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a first course in Topos quantum theory
2013Co-Authors: Cecilia FloriAbstract:Introduction.- Philosophical Motivations.- Kochen-Specker Theorem.- Introducing Category Theory.- Functors.- The Category of Functors.- Topos.- Topos of Presheaves.- Topos Analogue of the State Space.- Topos Analogue of Propositions.- Topos Analogues of States.- Truth Values.- Quantity Value Object and Physical Quantities.- Sheaves.- Probabilities in Topos Quantum theory.- Group Action in Topos Quantum Theory.- Topos History Quantum Theory.- Normal Operators.- KMS States.- Future Research.- Topos and Logic.- Worked out Examples.
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Topos of Presheaves
Lecture Notes in Physics, 2013Co-Authors: Cecilia FloriAbstract:For an arbitrary category \(\mathcal{C}\), the category \(Sets^{\mathcal{C}^{op}}\) of presheaves is actually a Topos. This Topos is important since, for a particular choice of \(\mathcal{C}\), it will be the Topos we will utilise to express quantum theory.
Chris Heunen - One of the best experts on this subject based on the ideXlab platform.
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Topos Quantum Theory with Short Posets
Order, 2020Co-Authors: John Harding, Chris HeunenAbstract:Topos quantum mechanics, developed by Döring ( 2008 ); Döring and Harding Houston J. Math. 42 (2), 559–568 ( 2016 ); Döring and Isham ( 2008 ); Flori 2013 )); Flori ( 2018 ); Isham and Butterfield J. Theoret. Phys. 37 , 2669–2733 ( 1998 ); Isham and Butterfield J. Theoret. Phys. 38 , 827–859 ( 1999 ); Isham et al. J. Theoret. Phys. 39 , 1413–1436 ( 2000 ); Isham and Butterfield J. Theoret. Phys. 41 , 613–639 ( 2002 ), creates a Topos of presheaves over the poset V ( N ) $\mathcal {V}(\mathcal {N})$ of Abelian von Neumann subalgebras of the von Neumann algebra N $\mathcal {N}$ of bounded operators associated to a physical system, and established several results, including: (a) a connection between the Kochen-Specker theorem and the non-existence of a global section of the spectral presheaf; (b) a version of the spectral theorem for self-adjoint operators; (c) a connection between states of N $\mathcal {N}$ and measures on the spectral presheaf; and (d) a model of dynamics in terms of V ( N ) $\mathcal {V}(\mathcal {N})$ . We consider a modification to this approach using not the whole of the poset V ( N ) $\mathcal {V}(\mathcal {N})$ , but only its elements V ( N ) ∗ $\mathcal {V}(\mathcal {N})^{*}$ of height at most two. This produces a different Topos with different internal logic. However, the core results (a)–(d) established using the full poset V ( N ) $\mathcal {V}(\mathcal {N})$ are also established for the Topos over the smaller poset, and some aspects simplify considerably. Additionally, this smaller poset has appealing aspects reminiscent of projective geometry.
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Topos quantum theory with short posets
Order, 2020Co-Authors: John Harding, Chris HeunenAbstract:Topos quantum mechanics, developed by Isham et. al., creates a Topos of presheaves over the poset V(N) of abelian von Neumann subalgebras of the von Neumann algebra N of bounded operators associated to a physical system, and established several results, including: (a) a connection between the Kochen-Specker theorem and the non-existence of a global section of the spectral presheaf; (b) a version of the spectral theorem for self-adjoint operators; (c) a connection between states of N and measures on the spectral presheaf; and (d) a model of dynamics in terms of V(N). We consider a modification to this approach using not the whole of the poset V(N), but only its elements of height at most two. This produces a different Topos with different internal logic. However, the core results (a)--(d) established using the full poset V(N) are also established for the Topos over the smaller poset, and some aspects simplify considerably. Additionally, this smaller poset has appealing aspects reminiscent of projective geometry.
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A Topos for Algebraic Quantum Theory
Communications in Mathematical Physics, 2009Co-Authors: Chris Heunen, Nicolaas P. Landsman, Bas SpittersAbstract: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 $${\mathcal{T}(A)}$$ in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*-algebra $${\underline{A}}$$ . According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum $${\underline{\Sigma}(\underline{A})}$$ in $${\mathcal{T}(A)}$$ , which in our approach plays the role of the quantum phase space of the system. Thus we associate a locale (which is the Topos-theoretical 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 $${\underline{\Sigma}}$$ , and self-adjoint elements of A define continuous functions (more precisely, locale maps) from $${\underline{\Sigma}}$$ to Scott’s interval domain. Noting that open subsets of $${\underline{\Sigma}(\underline{A})}$$ 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 $${\mathcal{T}(A)}$$ . These results were inspired by the Topos-theoretic approach to quantum physics proposed by Butterfield and Isham, as recently generalized by Döring and Isham.
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A Topos for Algebraic Quantum Theory
Communications in Mathematical Physics, 2009Co-Authors: Chris Heunen, Nicolaas P. Landsman, Bas SpittersAbstract: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. Moti- vated 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 sys- tem. Thus we associate a locale (which is the Topos-theoretical 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 self-adjoint elements of A define continuous functions (more precisely, locale maps) fromto 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 Topos-theoretic approach to quantum physics pro- posed by Butterfield and Isham, as recently generalized by Doring and Isham.
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a Topos for algebraic quantum theory
arXiv: Quantum Physics, 2007Co-Authors: Chris Heunen, Nicolaas P. Landsman, Bas SpittersAbstract: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 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. According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum S(A) in T(A), which in our approach plays the role of a quantum phase space of the system. Thus we associate a locale (which is the Topos-theoretical 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 S(A), and self-adjoint elements of A define continuous functions (more precisely, locale maps) from S(A) to Scott's interval domain. Noting that open subsets of S(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).
John Harding - One of the best experts on this subject based on the ideXlab platform.
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Topos Quantum Theory with Short Posets
Order, 2020Co-Authors: John Harding, Chris HeunenAbstract:Topos quantum mechanics, developed by Döring ( 2008 ); Döring and Harding Houston J. Math. 42 (2), 559–568 ( 2016 ); Döring and Isham ( 2008 ); Flori 2013 )); Flori ( 2018 ); Isham and Butterfield J. Theoret. Phys. 37 , 2669–2733 ( 1998 ); Isham and Butterfield J. Theoret. Phys. 38 , 827–859 ( 1999 ); Isham et al. J. Theoret. Phys. 39 , 1413–1436 ( 2000 ); Isham and Butterfield J. Theoret. Phys. 41 , 613–639 ( 2002 ), creates a Topos of presheaves over the poset V ( N ) $\mathcal {V}(\mathcal {N})$ of Abelian von Neumann subalgebras of the von Neumann algebra N $\mathcal {N}$ of bounded operators associated to a physical system, and established several results, including: (a) a connection between the Kochen-Specker theorem and the non-existence of a global section of the spectral presheaf; (b) a version of the spectral theorem for self-adjoint operators; (c) a connection between states of N $\mathcal {N}$ and measures on the spectral presheaf; and (d) a model of dynamics in terms of V ( N ) $\mathcal {V}(\mathcal {N})$ . We consider a modification to this approach using not the whole of the poset V ( N ) $\mathcal {V}(\mathcal {N})$ , but only its elements V ( N ) ∗ $\mathcal {V}(\mathcal {N})^{*}$ of height at most two. This produces a different Topos with different internal logic. However, the core results (a)–(d) established using the full poset V ( N ) $\mathcal {V}(\mathcal {N})$ are also established for the Topos over the smaller poset, and some aspects simplify considerably. Additionally, this smaller poset has appealing aspects reminiscent of projective geometry.
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Topos quantum theory with short posets
Order, 2020Co-Authors: John Harding, Chris HeunenAbstract:Topos quantum mechanics, developed by Isham et. al., creates a Topos of presheaves over the poset V(N) of abelian von Neumann subalgebras of the von Neumann algebra N of bounded operators associated to a physical system, and established several results, including: (a) a connection between the Kochen-Specker theorem and the non-existence of a global section of the spectral presheaf; (b) a version of the spectral theorem for self-adjoint operators; (c) a connection between states of N and measures on the spectral presheaf; and (d) a model of dynamics in terms of V(N). We consider a modification to this approach using not the whole of the poset V(N), but only its elements of height at most two. This produces a different Topos with different internal logic. However, the core results (a)--(d) established using the full poset V(N) are also established for the Topos over the smaller poset, and some aspects simplify considerably. Additionally, this smaller poset has appealing aspects reminiscent of projective geometry.
Bas Spitters - One of the best experts on this subject based on the ideXlab platform.
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A Topos for Algebraic Quantum Theory
Communications in Mathematical Physics, 2009Co-Authors: Chris Heunen, Nicolaas P. Landsman, Bas SpittersAbstract: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 $${\mathcal{T}(A)}$$ in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*-algebra $${\underline{A}}$$ . According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum $${\underline{\Sigma}(\underline{A})}$$ in $${\mathcal{T}(A)}$$ , which in our approach plays the role of the quantum phase space of the system. Thus we associate a locale (which is the Topos-theoretical 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 $${\underline{\Sigma}}$$ , and self-adjoint elements of A define continuous functions (more precisely, locale maps) from $${\underline{\Sigma}}$$ to Scott’s interval domain. Noting that open subsets of $${\underline{\Sigma}(\underline{A})}$$ 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 $${\mathcal{T}(A)}$$ . These results were inspired by the Topos-theoretic approach to quantum physics proposed by Butterfield and Isham, as recently generalized by Döring and Isham.
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A Topos for Algebraic Quantum Theory
Communications in Mathematical Physics, 2009Co-Authors: Chris Heunen, Nicolaas P. Landsman, Bas SpittersAbstract: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. Moti- vated 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 sys- tem. Thus we associate a locale (which is the Topos-theoretical 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 self-adjoint elements of A define continuous functions (more precisely, locale maps) fromto 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 Topos-theoretic approach to quantum physics pro- posed by Butterfield and Isham, as recently generalized by Doring and Isham.
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a Topos for algebraic quantum theory
arXiv: Quantum Physics, 2007Co-Authors: Chris Heunen, Nicolaas P. Landsman, Bas SpittersAbstract: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 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. According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum S(A) in T(A), which in our approach plays the role of a quantum phase space of the system. Thus we associate a locale (which is the Topos-theoretical 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 S(A), and self-adjoint elements of A define continuous functions (more precisely, locale maps) from S(A) to Scott's interval domain. Noting that open subsets of S(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).
Gabriele Vezzosi - One of the best experts on this subject based on the ideXlab platform.
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homotopical algebraic geometry i Topos theory
Advances in Mathematics, 2005Co-Authors: Bertrand Toen, Gabriele VezzosiAbstract:This is the rst of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this rst part we investigate a notion of higher Topos. For this, we use S-categories (i.e. simplicially enriched categories) as models for certain kind of 1-categories, and we develop the notions of S-topologies, S-sites and stacks over them. We prove in particular, that for an S-category T endowed with an S-topology, there exists a model category of stacks over T , generalizing the model category structure on simplicial presheaves over a Grothendieck site of A. Joyal and R. Jardine. We also prove some analogs of the relations between topologies and localizing subcategories of the categories of presheaves, by proving that there exists a one-to-one correspodence between S-topologies on an S-category T , and certain left exact Bouseld localizations of the model category of pre-stacks on T . Based on the above results, we study the notion of model Topos introduced by C. Rezk, and we relate it to our model categories of stacks over S-sites. In the second part of the paper, we present a parallel theory where S-categories, S-topologies and S-sites are replaced by model categories, model topologies and model sites. We prove that a canonical way to pass from the theory of stacks over model sites to the theory of stacks over S-sites is provided by the simplicial localization construction of Dwyer and Kan. As an example of application, we propose a denition of etale K-theory of ring spectra, extending the etale K-theory of commutative rings.
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Homotopical algebraic geometry. I. Topos theory.
Advances in Mathematics, 2005Co-Authors: Bertrand Toen, Gabriele VezzosiAbstract:This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts (for part II, see math.AG/0404373). In this first part we investigate a notion of higher Topos. For this, we use S-categories (i.e. simplicially enriched categories) as models for certain kind of \infty-categories, and we develop the notions of S-topologies, S-sites and stacks over them. We prove in particular, that for an S-category T endowed with an S-topology, there exists a model category of stacks over T, generalizing the model category structure on simplicial presheaves over a Grothendieck site of A. Joyal and R. Jardine. We also prove some analogs of the relations between topologies and localizing subcategories of the categories of presheaves, by proving that there exists a one-to-one correspondence between S-topologies on an S-category T, and certain left exact Bousfield localizations of the model category of pre-stacks on T. Based on the above results, we study the notion of model Topos introduced by C. Rezk, and we relate it to our model categories of stacks over S-sites. In the second part of the paper, we present a parallel theory where S-categories, S-topologies and S-sites are replaced by model categories, model topologies and model sites. We prove that a canonical way to pass from the theory of stacks over model sites to the theory of stacks over S-sites is provided by the simplicial localization construction of Dwyer and Kan. We also prove a Giraud's style theorem characterizing model topoi internally. As an example of application, we propose a definition of etale K-theory of ring spectra, extending the etale K-theory of commutative rings.
