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Author: Ola Bratteli
Publisher: Springer Science & Business Media
ISBN: 9783540170938
Category : Mathematics
Languages : en
Pages : 528
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Book Description
This is the first of two volumes presenting the theory of operator algebras with applications to quantum statistical mechanics. The authors' approach to the operator theory is to a large extent governed by the dictates of the physical applications. The book is self-contained and most proofs are presented in detail, which makes it a useful text for students with a knowledge of basic functional analysis. The introductory chapter surveys the history and justification of algebraic techniques in statistical physics and outlines the applications that have been made. The second edition contains new and improved results. The principal changes include: A more comprehensive discussion of dissipative operators and analytic elements; the positive resolution of the question of whether maximal orthogonal probability measure on the state space of C-algebra were automatically maximal along all the probability measures on the space.
Author: Ola Bratteli
Publisher: Springer Science & Business Media
ISBN: 9783540170938
Category : Mathematics
Languages : en
Pages : 528
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Book Description
This is the first of two volumes presenting the theory of operator algebras with applications to quantum statistical mechanics. The authors' approach to the operator theory is to a large extent governed by the dictates of the physical applications. The book is self-contained and most proofs are presented in detail, which makes it a useful text for students with a knowledge of basic functional analysis. The introductory chapter surveys the history and justification of algebraic techniques in statistical physics and outlines the applications that have been made. The second edition contains new and improved results. The principal changes include: A more comprehensive discussion of dissipative operators and analytic elements; the positive resolution of the question of whether maximal orthogonal probability measure on the state space of C-algebra were automatically maximal along all the probability measures on the space.
Author: David Emrys Evans
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 854
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Book Description
In the last 20 years, the study of operator algebras has developed from a branch of functional analysis to a central field of mathematics with applications and connections with different areas in both pure mathematics (foliations, index theory, K-theory, cyclic homology, affine Kac--Moody algebras, quantum groups, low dimensional topology) and mathematical physics (integrable theories, statistical mechanics, conformal field theories and the string theories of elementary particles). The theory of operator algebras was initiated by von Neumann and Murray as a tool for studying group representations and as a framework for quantum mechanics, and has since kept in touch with its roots in physics as a framework for quantum statistical mechanics and the formalism of algebraic quantum field theory. However, in 1981, the study of operator algebras took a new turn with the introduction by Vaughan Jones of subfactor theory and remarkable connections were found with knot theory, 3-manifolds, quantum groups and integrable systems in statistical mechanics and conformal field theory. The purpose of this book, one of the first in the area, is to look at these combinatorial-algebraic developments from the perspective of operator algebras; to bring the reader to the frontline of research with the minimum of prerequisites from classical theory.
Author: Shoichiro Sakai
Publisher: Springer Science & Business Media
ISBN: 3642619932
Category : Mathematics
Languages : en
Pages : 271
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Book Description
From the reviews: "This book is an excellent and comprehensive survey of the theory of von Neumann algebras. It includes all the fundamental results of the subject, and is a valuable reference for both the beginner and the expert." Mathematical Reviews
Author: Hellmut Baumgärtel
Publisher: De Gruyter Akademie Forschung
ISBN:
Category : Mathematical physics
Languages : en
Pages : 488
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Book Description
For advanced students in mathematics and mathematicians, as well as theoretical physicists, this volume presents the theory of nets of operator algebras, in particular nets connected with a causality condition. Such nets appear in mathematical formulations of quantum statistical mechanics and of quantum field theory. In this volume, the emphasis lies on nets which are linked with the algebraic approach to quantum field theory. Assumes a basic knowledge of functional analysis, in particular in the field of operator algebras. Annotation copyright by Book News, Inc., Portland, OR
Author: K. Schmüdgen
Publisher: Birkhäuser
ISBN: 3034874693
Category : Mathematics
Languages : en
Pages : 381
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Book Description
*-algebras of unbounded operators in Hilbert space, or more generally algebraic systems of unbounded operators, occur in a natural way in unitary representation theory of Lie groups and in the Wightman formulation of quantum field theory. In representation theory they appear as the images of the associated representations of the Lie algebras or of the enveloping algebras on the Garding domain and in quantum field theory they occur as the vector space of field operators or the *-algebra generated by them. Some of the basic tools for the general theory were first introduced and used in these fields. For instance, the notion of the weak (bounded) commutant which plays a fundamental role in thegeneraltheory had already appeared in quantum field theory early in the six ties. Nevertheless, a systematic study of unbounded operator algebras began only at the beginning of the seventies. It was initiated by (in alphabetic order) BORCHERS, LASSNER, POWERS, UHLMANN and VASILIEV. J1'rom the very beginning, and still today, represen tation theory of Lie groups and Lie algebras and quantum field theory have been primary sources of motivation and also of examples. However, the general theory of unbounded operator algebras has also had points of contact with several other disciplines. In particu lar, the theory of locally convex spaces, the theory of von Neumann algebras, distri bution theory, single operator theory, the momcnt problem and its non-commutative generalizations and noncommutative probability theory, all have interacted with our subject.
Author: Igor Frenkel
Publisher: Academic Press
ISBN: 0080874541
Category : Mathematics
Languages : en
Pages : 563
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Book Description
This work is motivated by and develops connections between several branches of mathematics and physics--the theories of Lie algebras, finite groups and modular functions in mathematics, and string theory in physics. The first part of the book presents a new mathematical theory of vertex operator algebras, the algebraic counterpart of two-dimensional holomorphic conformal quantum field theory. The remaining part constructs the Monster finite simple group as the automorphism group of a very special vertex operator algebra, called the "moonshine module" because of its relevance to "monstrous moonshine."
Author: Huzihiro Araki
Publisher: Springer Science & Business Media
ISBN: 9401728232
Category : Science
Languages : en
Pages : 452
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Book Description
In the past decade, there has been a sudden and vigorous development in a number of research areas in mathematics and mathematical physics, such as theory of operator algebras, knot theory, theory of manifolds, infinite dimensional Lie algebras and quantum groups (as a new topics), etc. on the side of mathematics, quantum field theory and statistical mechanics on the side of mathematical physics. The new development is characterized by very strong relations and interactions between different research areas which were hitherto considered as remotely related. Focussing on these new developments in mathematical physics and theory of operator algebras, the International Oji Seminar on Quantum Analysis was held at the Kansai Seminar House, Kyoto, JAPAN during June 25-29, 1992 by a generous sponsorship of the Japan Society for the Promotion of Science and the Fujihara Foundation of Science, as a workshop of relatively small number of (about 50) invited participants. This was followed by an open Symposium at RIMS, described below by its organizer, A. Kishimoto. The Oji Seminar began with two key-note addresses, one by V.F.R. Jones on Spin Models in Knot Theory and von Neumann Algebras and by A. Jaffe on Where Quantum Field Theory Has Led. Subsequently topics such as Subfactors and Sector Theory, Solvable Models of Statistical Mechanics, Quantum Field Theory, Quantum Groups, and Renormalization Group Ap proach, are discussed. Towards the end, a panel discussion on Where Should Quantum Analysis Go? was held.
Author: Hellmut Baumgärtel
Publisher: De Gruyter Akademie Forschung
ISBN:
Category : Mathematics
Languages : en
Pages : 238
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Book Description
In the course of lectures, held from summer 1993 up to summer 1994 at the Humboldt University of Berlin (SS 93), the Technical University of Berlin (WS 93/94) and the University of Potsdam (SS 94) the author presents basic operatoralgebraic material which is necessary to establish basic concepts of the algebraic quantum field theory as well as to get essential results in this field. The original ansatz of R. Haag (and others) started with the "working hypothesis" of a net of algebras of local observables. The aim of the lectures is to show that the success of this ansatz is strongly connected with deep results in the theory of operator algebras. Some characteristic mutual relations are described. Key words are "Vacuum representations on the four-dimensional Minkowski space and on the unit sphere, DHR-superselection theory, field algebra", and others. The emphasis is to make the material presented clear and readable without missing depth. The hope is to convince the reader of the beauty and stringency of this theory. While the first part of the book describes vacuum representations on the four-dimensional Minkowski space, the second part deals with the so-called DHR-superselection theory, firstly for the automorphism case and secondly for the general case. The constructions of the field algebra and the symmetry group are included. Finally, vacuum representations on the unit sphere S1 are discussed.
Author: Bert Schroer
Publisher:
ISBN:
Category : Operator algebras
Languages : en
Pages : 80
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Book Description
Author: Marcel Bischoff
Publisher: Springer
ISBN: 3319143018
Category : Science
Languages : en
Pages : 103
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Book Description
C* tensor categories are a point of contact where Operator Algebras and Quantum Field Theory meet. They are the underlying unifying concept for homomorphisms of (properly infinite) von Neumann algebras and representations of quantum observables. The present introductory text reviews the basic notions and their cross-relations in different contexts. The focus is on Q-systems that serve as complete invariants, both for subfactors and for extensions of quantum field theory models. It proceeds with various operations on Q-systems (several decompositions, the mirror Q-system, braided product, centre and full centre of Q-systems) some of which are defined only in the presence of a braiding. The last chapter gives a brief exposition of the relevance of the mathematical structures presented in the main body for applications in Quantum Field Theory (in particular two-dimensional Conformal Field Theory, also with boundaries or defects).
