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Operator algebra

About: Operator algebra is a research topic. Over the lifetime, 5783 publications have been published within this topic receiving 165303 citations.


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Book
31 Dec 2002
TL;DR: The theory of Partial O*-Algebras has been studied extensively in the literature, see as mentioned in this paper for a detailed overview of some of the most important works on the subject.
Abstract: Foreword. Introduction. I: Theory of Partial O*-Algebras. 1. Unbounded Linear Operators in Hilbert Spaces. 2. Partial O*-Algebras. 3.Commutative Partial O*-Algebras. 4. Topologies on Partial O*-Algebras. 5. Tomita Takesaki Theory in Partial O*-Algebras. II: Theory of Partial *-Algebras. 6. Partial *-Algebras. 7. *-Representations of Partial *-Algebras. 8. Well-behaved X>*-Representations. 9. Biweights on Partial *-Algebras. 10. Quasi *-Algebras of Operators in Rigged Hilbert Spaces. 11. Physical Applications. Outcome. Bibliography. Index.

184 citations

Book
01 Jan 1991
TL;DR: In this article, the authors formulate the absence theorem of phase transitions in its most general form within the C* setting and present a new axiomatic treatment of the construction of time evolutions and KMS states.
Abstract: This book is concerned with the theory of unbounded derivations in C*-algebras, a subject whose study was motivated by questions in quantum physics and statistical mechanics, and to which the author has made a considerable contribution. This is an active area of research, and one of the most ambitious aims of the theory is to develop quantum statistical mechanics within the framework of the C*-theory. The presentation, which is based on lectures given in Newcastle upon Tyne and Copenhagen, concentrates on topics involving quantum statistical mechanics and differentiations on manifolds. One of the goals is to formulate the absence theorem of phase transitions in its most general form within the C* setting. For the first time, he globally constructs, within that setting, derivations for a fairly wide class of interacting models, and presents a new axiomatic treatment of the construction of time evolutions and KMS states.

182 citations

Journal ArticleDOI
TL;DR: In this article, a functional integral representation of the ground states of quantum spin chains is presented with the help of functional integral analysis of the system's equilibrium states, including the possibility of dimerization, conditions for the existence of a spectral gap, and a dichotomy analogous to one found by Affleck and Lieb.
Abstract: A number of interesting features of the ground states of quantum spin chains are analyzed with the help of a functional integral representation of the system's equilibrium states. Methods of general applicability are introduced in the context of the SU(2S+1)-invariant quantum spin-S chains with the interaction −P(o), whereP(o) is the projection onto the singlet state of a pair of nearest neighbor spins. The phenomena discussed here include: the absence of Neel order, the possibility of dimerization, conditions for the existence of a spectral gap, and a dichotomy analogous to one found by Affleck and Lieb, stating that the systems exhibit either slow decay of correlations or translation symmetry breaking. Our representation elucidates the relation, evidence for which was found earlier, of the −P(o) spin-S systems with the Potts and the Fortuin-Kasteleyn random-cluster models in one more dimension. The method reveals the geometric aspects of the listed phenomena, and gives a precise sense to a picture of the ground state in which the spins are grouped into random clusters of zero total spin. E.g., within such structure the dichotomy is implied by a topological argument, and the alternatives correspond to whether, or not, the clusters are of finite mean length.

182 citations

Journal ArticleDOI
TL;DR: A new theory is proposed to accurately simulate quantum dynamics in systems of identical particles based on the second quantization formalism of many-body quantum theory, which unifies the multilayer multiconfiguration time-dependent Hartree theory for both distinguishable and indistinguishable particles.
Abstract: A new theory is proposed to accurately simulate quantum dynamics in systems of identical particles. It is based on the second quantization formalism of many-body quantum theory, in which the Fock space is represented by occupation-number states. Within this representation the overall Fock space can be formally decomposed into smaller subspaces, and the wave function can be expressed as a multilayer multiconfiguration Hartree expansion involving subvectors in these subspaces. The theory unifies the multilayer multiconfiguration time-dependent Hartree theory for both distinguishable and indistinguishable particles. Specific formulations are given for systems of identical fermions, bosons, and combinations thereof. Practical implementations are discussed, especially for the case of fermions, to include the operator algebra that enforces the symmetry of identical particles. The theory is illustrated by a numerical example on vibrationally coupled electron transport.

181 citations

Journal ArticleDOI
TL;DR: In this article, the structure of the tensor product representation of the quantum groupSLq(2,C) was investigated by using the 2-dimensional quantum plane as a building block.
Abstract: We investigate the structure of the tensor product representation of the quantum groupSLq(2,C) by using the 2-dimensional quantum plane as a building block. Two types of 4-dimensional spaces are constructed applying the methods used in twistor theory. We show that the 4-dimensional real representation ofSLq(2,C) generates a consistent non-commutative algebra, and thus it provides a quantum deformation of Minkowski space. The transformation of this 4-dimensional space gives the quantum Lorentz groupSOq(3, 1).

181 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202337
202277
2021125
2020141
2019173
2018169