814 research outputs found

    Infinite Permutation Groups and the Origin of Quantum Mechanics

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    We propose an interpretation for the meets and joins in the lattice of experimental propositions of a physical theory, answering a question of Birkhoff and von Neumann in [1]. When the lattice is atomistic, it is isomorphic to the lattice of definably closed sets of a finitary relational structure in First Order Logic. In terms of mapping experimental propositions to subsets of the atomic phase space, the meet corresponds to set intersection, while the join is the definable closure of set union. The relational structure is defined by the action of the lattice automorphism group on the atomic layer. Examining this correspondence between physical theories and infinite group actions, we show that the automorphism group must belong to a family of permutation groups known as geometric Jordan groups. We then use the classification theorem for Jordan groups to argue that the combined requirements of probability and atomicism leave uncountably infinite Steiner 2-systems (of which projective spaces are standard examples) as the sole class of options for generating the lattice of particle Quantum Mechanics.Comment: 23 page

    Borel versions of the Local Lemma and LOCAL algorithms for graphs of finite asymptotic separation index

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    Asymptotic separation index is a parameter that measures how easily a Borel graph can be approximated by its subgraphs with finite components. In contrast to the more classical notion of hyperfiniteness, asymptotic separation index is well-suited for combinatorial applications in the Borel setting. The main result of this paper is a Borel version of the Lov\'asz Local Lemma -- a powerful general-purpose tool in probabilistic combinatorics -- under a finite asymptotic separation index assumption. As a consequence, we show that locally checkable labeling problems that are solvable by efficient randomized distributed algorithms admit Borel solutions on bounded degree Borel graphs with finite asymptotic separation index. From this we derive a number of corollaries, for example a Borel version of Brooks's theorem for graphs with finite asymptotic separation index

    Convergence of Dynamics on Inductive Systems of Banach Spaces

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    Many features of physical systems, both qualitative and quantitative, become sharply defined or tractable only in some limiting situation. Examples are phase transitions in the thermodynamic limit, the emergence of classical mechanics from quantum theory at large action, and continuum quantum field theory arising from renormalization group fixed points. It would seem that few methods can be useful in such diverse applications. However, we here present a flexible modeling tool for the limit of theories: soft inductive limits constituting a generalization of inductive limits of Banach spaces. In this context, general criteria for the convergence of dynamics will be formulated, and these criteria will be shown to apply in the situations mentioned and more.Comment: Comments welcom

    Nonassociative Lp\mathrm{L}^p-spaces and embeddings in noncommutative Lp\mathrm{L}^p-spaces

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    We define a notion of nonassociative Lp\mathrm{L}^p-space associated to a JBW\mathrm{JBW}^*-algebra (Jordan von Neumann algebra) equipped with a normal faithful state φ\varphi. In the particular case of JW\mathrm{JW}^*-algebras underlying von Neumann algebras, we connect these spaces to a complex interpolation theorem of Ricard and Xu on noncommutative Lp\mathrm{L}^p-spaces. We also make the link with the nonassociative Lp\mathrm{L}^p-spaces of Iochum associated to JBW\mathrm{JBW}-algebras and the investigation of contractively complemented subspaces of noncommutative Lp\mathrm{L}^p-spaces. More precisely, we show that our nonassociative Lp\mathrm{L}^p-spaces contain isometrically the Lp\mathrm{L}^p-spaces of Iochum and that all tracial nonassociative Lp\mathrm{L}^p-spaces from JW\mathrm{JW}^*-factors arise as positively contractively complemented subspaces of noncommutative Lp\mathrm{L}^p-spaces.Comment: 24 pages, contains in particular some sections of an old and too long version of arXiv:1909.0039

    Finitary approximations of free probability, involving combinatorial representation theory

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    This thesis contributes to two theories which approximate free probability by finitary combinatorial structures. The first is finite free probability, which is concerned with expected characteristic polynomials of various random matrices and was initiated by Marcus, Spielman, and Srivastava in 2015. An alternate approach to some of their results for sums and products of randomly rotated matrices is presented, using techniques from combinatorial representation theory. Those techniques are then applied to the commutators of such matrices, uncovering the non-trivial but tractable combinatorics of immanants and Schur polynomials. The second is the connection between symmetric groups and random matrices, specifically the asymptotics of star-transpositions in the infinite symmetric group and the gaussian unitary ensemble (GUE). For a continuous family of factor representations of SS_{\infty}, a central limit theorem for the star-transpositions (1,n)(1,n) is derived from the insight of Gohm-K\"{o}stler that they form an exchangeable sequence of noncommutative random variables. Then, the central limit law is described by a random matrix model which continuously deforms the well-known traceless GUE by taking its gaussian entries from noncommutative operator algebras with canonical commutation relations (CCR). This random matrix model generalizes results of K\"{o}stler and Nica from 2021, which in turn generalized a result of Biane from 1995

    Geometric Phases Characterise Operator Algebras and Missing Information

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    We show how geometric phases may be used to fully describe quantum systems, with or without gravity, by providing knowledge about the geometry and topology of its Hilbert space. We find a direct relation between geometric phases and von Neumann algebras. In particular, we show that a vanishing geometric phase implies the existence of a well-defined trace functional on the algebra. We discuss how this is realised within the AdS/CFT correspondence for the eternal black hole. On the other hand, a non-vanishing geometric phase indicates missing information for a local observer, associated to reference frames covering only parts of the quantum system considered. We illustrate this with several examples, ranging from a single spin in a magnetic field to Virasoro Berry phases and the geometric phase associated to the eternal black hole in AdS spacetime. For the latter, a non-vanishing geometric phase is tied to the presence of a centre in the associated von Neumann algebra.Comment: 45 page

    Quantum Expanders and Quantifier Reduction for Tracial von Neumann Algebras

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    We provide a complete characterization of theories of tracial von Neumann algebras that admit quantifier elimination. We also show that the theory of a separable tracial von Neumann algebra N\mathcal{N} is never model complete if its direct integral decomposition contains II1\mathrm{II}_1 factors M\mathcal{M} such that M2(M)M_2(\mathcal{M}) embeds into an ultrapower of M\mathcal{M}. The proof in the case of II1\mathrm{II}_1 factors uses an explicit construction based on random matrices and quantum expanders.Comment: 38 pages, comments are welcom

    The spinor bundle on loop space

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    We give a construction of the spinor bundle of the loop space of a string manifold together with its fusion product, inspired by ideas from Stolz and Teichner. The spinor bundle is a super bimodule bundle for a bundle of Clifford von Neumann algebras over the free path space, and the fusion product is defined using Connes fusion of such bimodules. As the main result, we prove that a spinor bundle with fusion product on a manifold X exists if and only X is string.Comment: 86 pages; Some minor corrections; added 2 figures; divested material on super bundle gerbes to Appendix

    Embedded Finite Models beyond Restricted Quantifier Collapse

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    We revisit evaluation of logical formulas that allow both uninterpreted relations, constrained to be finite, as well as interpreted vocabulary over an infinite domain: denoted in the past as embedded finite model theory. We extend the analysis of "collapse results": the ability to eliminate first-order quantifiers over the infinite domain in favor of quantification over the finite structure. We investigate several weakenings of collapse, one allowing higher-order quantification over the finite structure, another allowing expansion of the theory. We also provide results comparing collapse for unary signatures with general signatures, and new analyses of collapse for natural decidable theories

    小尺度トポロジーと大尺度トポロジーの統一的枠組み (集合論的および幾何学的トポロジーと関連分野への応用)

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    本稿では,超準解析を用いて小尺度と大尺度のトポロジーを統一的に扱う試みについて,進捗を報告する.まず,位相空間,一様空間,有界型空間,粗空間といった既存の空間概念を同時に一般化する概念として,空間的集合及び空間的写像の概念を導入する.次に,超準的な集合のクラスΠ₁[st]とΣ₁[st]の定義を確認した後,小尺度構造がΠ₁[st]—定義可能な空間的集合に,大尺度構造がΣ₁[st]—定義可能な空間的集合に,それぞれ正確に対応することを示す.位相空間に対するMcCordホモロジーのアイデアを借用し,空間的集合に対するホモロジー論を定義する.そして緩振動写像が大尺度空間のホモロジー群から小尺度空間のホモロジー群への準同型を誘導することを示す
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