Notes on the type classification of von Neumann algebras

• URL: http://arxiv.org/abs/2302.01958v3
• Date: Fri, 29 Mar 2024 15:54:16 GMT
• Title: Notes on the type classification of von Neumann algebras
• Authors: Jonathan Sorce,
• Abstract summary: These notes provide an explanation of the type classification of von Neumann algebras. The goal is to bridge a gap in the literature between resources that are too technical for the non-expert reader.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: These notes provide an explanation of the type classification of von Neumann algebras, which has made many appearances in recent work on entanglement in quantum field theory and quantum gravity. The goal is to bridge a gap in the literature between resources that are too technical for the non-expert reader, and resources that seek to explain the broad intuition of the theory without giving precise definitions. Reading these notes will provide you with: (i) an argument for why "factors" are the fundamental von Neumann algebras that one needs to study; (ii) an intuitive explanation of the type classification of factors in terms of renormalization schemes that turn unnormalizable positive operators into "effective density matrices;" (iii) a mathematical explanation of the different types of renormalization schemes in terms of the allowed traces on a factor; (iv) an intuitive characterization of type I and II factors in terms of their "standard forms;" and (v) a list of some interesting connections between type classification and modular theory, including the argument for why type III$_1$ factors are believed to be the relevant ones in quantum field theory. None of the material is new, but the pedagogy is different from other sources I have read; it is most similar in spirit to the recent work on gravity and the crossed product by Chandrasekaran, Longo, Penington, and Witten.

Related papers

• Pure state entanglement and von Neumann algebras [41.94295877935867]
  We develop the theory of local operations and classical communication (LOCC) for bipartite quantum systems represented by commuting von Neumann algebras. Our theorem implies that, in a bipartite system modeled by commuting factors in Haag duality, a) all states have infinite one-shot entanglement if and only if the local factors are not of type I. In the appendix, we provide a self-contained treatment of majorization on semifinite von Neumann algebras and $sigma$-finite measure spaces.
  arXiv  Detail & Related papers  (2024-09-26T11:13:47Z)
• A Category-theoretical Meta-analysis of Definitions of Disentanglement [97.34033555407403]
  Disentangling the factors of variation in data is a fundamental concept in machine learning. This paper presents a meta-analysis of existing definitions of disentanglement.
  arXiv  Detail & Related papers  (2023-05-11T15:24:20Z)
• Evaluating the Robustness of Interpretability Methods through Explanation Invariance and Equivariance [72.50214227616728]
  Interpretability methods are valuable only if their explanations faithfully describe the explained model. We consider neural networks whose predictions are invariant under a specific symmetry group.
  arXiv  Detail & Related papers  (2023-04-13T17:59:03Z)
• Higher derivative Hamiltonians with benign ghosts from affine Toda lattices [0.0]
  We study the properties of the classical phase spaces for a number of affine Toda lattices theories. We identify several types of scenarios for theories with higher charge Hamiltonians.
  arXiv  Detail & Related papers  (2023-01-26T18:56:38Z)
• Multiparameter Persistent Homology-Generic Structures and Quantum Computing [0.0]
  This article is an application of commutative algebra to the study of persistent homology in topological data analysis. The generic structure of such resolutions and the classifying spaces are studied using results spanning several decades of research.
  arXiv  Detail & Related papers  (2022-10-20T17:30:20Z)
• Quantum teleportation in the commuting operator framework [63.69764116066747]
  We present unbiased teleportation schemes for relative commutants $N'cap M$ of a large class of finite-index inclusions $Nsubseteq M$ of tracial von Neumann algebras. We show that any tight teleportation scheme for $N$ necessarily arises from an orthonormal unitary Pimsner-Popa basis of $M_n(mathbbC)$ over $N'$.
  arXiv  Detail & Related papers  (2022-08-02T00:20:46Z)
• Self-adjoint extension schemes and modern applications to quantum Hamiltonians [55.2480439325792]
  monograph contains revised and enlarged materials from previous lecture notes of undergraduate and graduate courses and seminars delivered by both authors over the last years on a subject that is central both in abstract operator theory and in applications to quantum mechanics. A number of models are discussed, which are receiving today new or renewed interest in mathematical physics, in particular from the point of view of realising certain operators of interests self-adjointly.
  arXiv  Detail & Related papers  (2022-01-25T09:45:16Z)
• A Mathematical Framework for Causally Structured Dilations and its Relation to Quantum Self-Testing [0.0]
  This thesis recast quantum self-testing [MY98,MY04] in operational terms. An input-output process is modelled by a causally structured channel in some fixed theory. implementations are modelled by causally structured dilations formalising hidden side-computations.
  arXiv  Detail & Related papers  (2021-03-03T10:32:34Z)
• The Struggles of Feature-Based Explanations: Shapley Values vs. Minimal Sufficient Subsets [61.66584140190247]
  We show that feature-based explanations pose problems even for explaining trivial models. We show that two popular classes of explainers, Shapley explainers and minimal sufficient subsets explainers, target fundamentally different types of ground-truth explanations.
  arXiv  Detail & Related papers  (2020-09-23T09:45:23Z)
• Quiver Mutations, Seiberg Duality and Machine Learning [3.717544246477156]
  We focus on the case of quiver gauge theories, a problem also of interest in mathematics in the context of cluster algebras. We study how the performance of machine learning depends on several variables, including number of classes and mutation type. In all questions considered, high accuracy and confidence can be achieved.
  arXiv  Detail & Related papers  (2020-06-18T18:01:19Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.