Showing papers on "Operator algebra published in 1998"

Introduction to Quantum Groups

Abstract: We give an elementary introduction to the theory of algebraic and topological quantum groups (in the spirit of S. L. Woronowicz). In particular, we recall the basic facts from Hopf (*-) algebra theory, theory of compact (matrix) quantum groups and the theory of their actions on compact quantum spaces. We also provide the most important examples, including the classification of quantum SL(2)-groups, their real forms and quantum spheres. We also consider quantum SLq(N)-groups and quantum Lorentz groups.

Restrictions imposed by superconformal invariance on quantum field theories

Abstract: We derive unitarity restrictions on the scaling dimensions of primary operators in a superconformal quantum field theory, in d=3,4,5,6.

Quantum symmetries on operator algebras

Abstract: 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.

Operator Algebras and Conformal Field Theory III. Fusion of positive energy representations of LSU(N) using bounded operators

Abstract: I Positive energy representations of LSU(N) 477 II Local loop groups and their von Neumann algebras 491 III The basic ordinary di€erential equation 505 IV Vector and dual vector primary ®elds 513 V Connes fusion of positive energy representations 525 References 536

Notes on Compact Quantum Groups

Abstract: Compact quantum groups have been studied by several authors and from different points of view. The difference lies mainly in the choice of the axioms. In the end, the main results turn out to be the same. Nevertheless, the starting point has a strong influence on how the main results are obtained and on showing that certain examples satisfy these axioms. In these notes, we follow the approach of Woronowicz and we treat the compact quantum groups in the C ∗ -algebra framework. This is a natural choice when compact quantum groups are seen as a special case of locally compact quantum groups. A deep understanding of compact quantum groups in this setting is very important for understanding the problems that arise in developing a theory for locally compact quantum groups. We start with a discussion on locally compact quantum groups in general but mainly to motivate the choice of the axioms for the compact quantum groups. Then we develop the theory. We give the main examples and we show how they fit into this framework. The paper is an expository paper. It does not contain many new results although some of the proofs are certainly new and more elegant than the existing ones. Moreover, we have chosen to give a rather complete and self-contained treatment so that the paper can also serve as an introductory paper for non-specialists. Different aspects can be learned from these notes and a great deal of insight can be obtained. � Research Assistent of the National Fund for Scientific Research (Belgium)

Quantum theory of geometry: III. Non-commutativity of Riemannian structures

Abstract: The basic framework for a systematic construction of a quantum theory of Riemannian geometry was introduced recently. The quantum versions of Riemannian structures - such as triad and area operators - exhibit a non-commutativity. At first sight, this feature is surprising because it implies that the framework does not admit a triad representation. To better understand this property and to reconcile it with intuition, we analyse its origin in detail. In particular, a careful study of the underlying phase space is made and the feature is traced back to the classical theory; there is no anomaly associated with quantization. We also indicate why the uncertainties associated with this non-commutativity become negligible in the semiclassical regime.

Operator Algebras and Conformal Field Theory

Abstract: We report on a programme to understand unitary conformal field theory (CFT) from the point of view of operator algebras. The earlier stages of this research were carried out with Jones, following his suggestion that there might be a deeper “subfactor” explanation of the coincidence between certain braid group representations that had turned up in subfactors, statistical mechanics, and conformal field theory. (Most of our joint work appears in Section 10.) The classical additive theory of operator algebras, due to Murray and von Neumann, provides a framework for studying unitary Lie group representations, although in specific examples almost all the hard work involves a quite separate analysis of intertwining operators and differential equations. Analogously, the more recent multiplicative theory provides a powerful tool for studying the unitary representations of certain infinite–dimensional groups, such as loop groups or Diff S1. It must again be complemented by a detailed analysis of certain intertwining operators, the primary fields, and their associated differential equations.

Continuous Time and Consistent Histories

Abstract: We discuss the use of histories labelled by a continuous time in the approach to consistent-histories quantum theory in which propositions about the history of the system are represented by projection operators on a Hilbert space. This extends earlier work by two of us [C. J. Isham and N. Linden, J. Math. Phys. 36, 5392–5408 (1995)] where we showed how a continuous time parameter leads to a history algebra that is isomorphic to the canonical algebra of a quantum field theory. We describe how the appropriate representation of the history algebra may be chosen by requiring the existence of projection operators that represent propositions about the time average of the energy. We also show that the history description of quantum mechanics contains an operator corresponding to velocity that is quite distinct from the momentum operator. Finally, the discussion is extended to give a preliminary account of quantum field theory in this approach to the consistent histories formalism.

Statistical geometry in quantum mechanics

Abstract: A statistical model M is a family of probability distributions, characterised by a set of continuous parameters known as the parameter space. This possesses natural geometrical properties induced by the embedding of the family of probability distributions into the space of all square–integrable functions. More precisely, by consideration of the square–root density function we can regard M as a submanifold of the unit sphere S in a real Hilbert space H . Therefore, H embodies the ‘state space’ of the probability distributions, and the geometry of the given statistical model can be described in terms of the embedding of M in S . The geometry in question is characterised by a natural Riemannian metric (the Fisher–Rao metric), thus allowing us to formulate the principles of classical statistical inference in a natural geometric setting. In particular, we focus attention on the variance lower bounds for statistical estimation, and establish generalisations of the classical Cramer–Rao and Bhattacharyya inequalities, described in terms of the geometry of the underlying real Hilbert space. As a comprehensive illustration of the utility of the geometric framework, the statistical model M is then specialised to the case of a submanifold of the state space of a quantum mechanical system. This is pursued by introducing a compatible complex structure on the underlying real Hilbert space, which allows the operations of ordinary quantum mechanics to be reinterpreted in the language of real Hilbert space geometry. The application of generalised variance bounds in the case of quantum statistical estimation leads to a set of higher order corrections to the Heisenberg uncertainty relations for canonically conjugate observables.

Broken conformal invariance and spectrum of anomalous dimensions in QCD

Abstract: Employing the operator algebra of the conformal group and the conformal Ward identities, we derive the constraints for the anomalies of dilatation and special conformal transformations of the local twist-2 operators in Quantum Chromodynamics. We calculate these anomalies in the leading order of perturbation theory in the minimal subtraction scheme. From the conformal consistency relation we derive then the off-diagonal part of the anomalous dimension matrix of the conformally covariant operators in the two-loop approximation of the coupling constant in terms of these quantities. We deduce corresponding off-diagonal parts of the Efremov-Radyushkin-Brodsky-Lepage kernels responsible for the evolution of the exclusive distribution amplitudes and non-forward parton distributions in the next-to-leading order in the flavour singlet channel for the chiral-even parity-odd and -even sectors as well as for the chiral-odd one. We also give the analytical solution of the corresponding evolution equations exploiting the conformal partial wave expansion.

Quantum Kac-Moody Algebras and Vertex Representations

Abstract: We introduce an affinization of the quantum Kac-Moody algebra associated to a symmet- ric generalized Cartan matrix. Based on the affinization, we construct a representation of the quantum Kac-Moody algebra by vertex operators from bosonic fields. We also obtain a combinatorial identity about Hall-Littlewood polynomials. Mathematics Subject Classifications ( 1991): Primary: 17Bxx; Secondary: 05Exx.

An Algebraic Characterization of Boundary Representations

Abstract: We show that boundary representations of an operator algebra may be characterized as those (irreducible) completely contractive representations that determine Hilbert modules that are simultaneously orthogonally projective and orthogonally injective. As a corollary, we conclude that if an operator algebra is an admissable subalgebra of its C * —envelope, in the sense of Arveson, then it has a completely isometric representation such that the associated Hilbert module is simultaneously orthogonally projective and orthogonally injective.

Translational Shape Invariance and the Inherent Potential Algebra

Abstract: For all quantum-mechanical potentials that are known to be exactly solvable, there are two different, and seemingly independent methods of solution. The first approach is the potential algebra of symmetry groups; the second is supersymmetric quantum mechanics, applied to shape-invariant potentials, which comprise the set of known exactly solvable potentials. Using the underlying algebraic structures of Natanzon potentials, of which the translational shape-invariant potentials are a special subset, we demonstrate the equivalence of the two methods of solution. In addition, we show that, while the algebra for the general Natanzon potential is so(2,2), the subgroup so(2,1) suffices for the shape invariant subset. Finally, we show that the known set of exactly solvable potentials in fact constitutes the full set of such potentials.

Linear algebra for quantum theory

Abstract: Elements of Set Theory. Linear Spaces. Binary Product Spaces. Axioms of Quantum Theory Formulated as a Trace Algebra. References. Appendices. Index.

Scaling Algebras and Renormalization Group in Algebraic Quantum Field Theory.. II. Instructive Examples

Abstract: The concept of scaling algebra provides a novel framework for the general structural analysis and classification of the short distance properties of algebras of local observables in relativistic quantum field theory. In the present article this method is applied to the simple example of massive free field theory in s=1,2 and 3 spatial dimensions. Not quite unexpectedly, one obtains for s=2,3 in the scaling (short distance) limit the algebra of local observables in massless free field theory. The case s=1 offers, however, some surprises. There the algebra of observables acquires in the scaling limit a non-trivial center and describes charged physical states satisfying Gauss' law. The latter result is of relevance for the interpretation of the Schwinger model at short distances and illustrates the conceptual and computational virtues of the method.

Lecture notes on C*-algebras, Hilbert C*-modules, and quantum mechanics

Abstract: This is a graduate-level introduction to C*-algebras, Hilbert C*-modules, vector bundles, and induced representations of groups and C*-algebras, with applications to quantization theory, phase space localization, and configuration space localization. The reader is supposed to know elementary functional analysis and quantum mechanics.

Modular intersections of von neumann algebras in quantum field theory

Abstract: We show that modular intersections of von Neumann algebras occur naturally in quantum field theory. An example are local observable algebras associated with wedge regions, which have a lightray in common, see also [Bo 2, Wi 3]. Conversely, starting from a set of four algebras lying in a specified modular position relative to each other we construct a net of local observables of a 2+1 dimensional quantum field theory.

Algebra of Screening Operators for the Deformed Wn Algebra

Abstract: We construct a family of intertwining operators (screening operators) between various Fock space modules over the deformed Wn algebra. They are given as integrals involving a product of screening currents and elliptic theta functions. We derive a set of quadratic relations among the screening operators, and use them to construct a Felder-type complex in the case of the deformed W3 algebra.

The k-Fermions as Objects Interpolating Between Fermions and Bosons

Abstract: In the recent years, the theory of deformations, mainly in the spirit of quantum groups and quantum algebras, has been the subject of considerable interest in statistical physics. More precisely, deformed oscillator algebras have proved to be useful in parasiatistics(connected to irreducible representations, of dimensions greater than 1, of the symmetric group), in anyonic statistics(connected to the braid group) that concerns only particles in (one or) two dimensions, and in q-deformed statisticsthat may concern particles in arbitrary dimensions. In particular, the q-deformed statistics deal with: (i) q-bosons (which are bosons obeying a q-deformed Bose-Einstein distribution), (ii) q-fermions (which are fermions obeying a q-deformed Fermi-Dirac distribution), and (iii) quons (with qsuch that q k = 1, where k∈ ℕ \ {0,1}) which are objects, refered to as k-fermions in this work, interpolating between fermions (corresponding to k= 2) and bosons (corresponding to k→ ∞).

Applications of topological*-algebras of unbounded operators

Abstract: In this paper we discuss some physical applications of topological*-algebras of unbounded operators. The first example is a simple system of free bosons. Then we consider other models related to this one. Finally, we discuss the time evolution of two interacting models of matter and bosons. We show that for all these systems it is possible to build up a common mathematical framework where the existence of the thermodynamical limit of the algebraic dynamics can be conveniently analyzed.

Conformal Nets, Maximal Temperature and Models from Free Probability

Abstract: We consider conformal nets on $S^1$ of von Neumann algebras, acting on the full Fock space, arising in free probability. These models are twisted local, but non-local. We extend to the non-local case the general analysis of the modular structure. The local algebras turn out to be $III_1$-factors associated with free groups. We use our set up to show examples exhibiting arbitrarily large maximal temperatures, but failing to satisfy the split property, then clarifying the relation between the latter property and the trace class conditions on $e^{-\b L}$, where $L$ is the conformal Hamiltonian.

Differential calculus and connections on a quantum plane at a cubic root of unity

Abstract: We consider the algebra of N x N matrices as a reduced quantum plane on which a finite-dimensional quantum group H acts. This quantum group is a quotient of U_q(sl(2,C)), q being an N-th root of unity. Most of the time we shall take N=3; in that case \dim(H) = 27. We recall the properties of this action and introduce a differential calculus for this algebra: it is a quotient of the Wess-Zumino complex. The quantum group H also acts on the corresponding differential algebra and we study its decomposition in terms of the representation theory of H. We also investigate the properties of connections, in the sense of non commutative geometry, that are taken as 1-forms belonging to this differential algebra. By tensoring this differential calculus with usual forms over space-time, one can construct generalized connections with covariance properties with respect to the usual Lorentz group and with respect to a finite-dimensional quantum group.