814 research outputs found
Infinite Permutation Groups and the Origin of Quantum Mechanics
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
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
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 -spaces and embeddings in noncommutative -spaces
We define a notion of nonassociative -space associated to a
-algebra (Jordan von Neumann algebra) equipped with a normal
faithful state . In the particular case of -algebras
underlying von Neumann algebras, we connect these spaces to a complex
interpolation theorem of Ricard and Xu on noncommutative -spaces.
We also make the link with the nonassociative -spaces of Iochum
associated to -algebras and the investigation of contractively
complemented subspaces of noncommutative -spaces. More precisely,
we show that our nonassociative -spaces contain isometrically the
-spaces of Iochum and that all tracial nonassociative
-spaces from -factors arise as positively
contractively complemented subspaces of noncommutative -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
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 , a central limit theorem for the star-transpositions 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
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
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 is never model complete if
its direct integral decomposition contains factors
such that embeds into an ultrapower of
. The proof in the case of factors uses an
explicit construction based on random matrices and quantum expanders.Comment: 38 pages, comments are welcom
The spinor bundle on loop space
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
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)
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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