Showing papers on "Operator algebra published in 1997"

Quantum theory of geometry: I. Area operators

Abstract: A new functional calculus, developed recently for a fully non-perturbative treatment of quantum gravity, is used to begin a systematic construction of a quantum theory of geometry. Regulated operators corresponding to areas of 2-surfaces are introduced and shown to be self-adjoint on the underlying (kinematical) Hilbert space of states. It is shown that their spectra are purely discrete, indicating that the underlying quantum geometry is far from what the continuum picture might suggest. Indeed, the fundamental excitations of quantum geometry are one dimensional, rather like polymers, and the three-dimensional continuum geometry emerges only on coarse graining. The full Hilbert space admits an orthonormal decomposition into finite-dimensional subspaces which can be interpreted as the spaces of states of spin systems. Using this property, the complete spectrum of the area operators is evaluated. The general framework constructed here will be used in a subsequent paper to discuss three-dimensional geometric operators, e.g. the ones corresponding to volumes of regions.

Quantum Theory of Geometry II: Volume operators

Abstract: A functional calculus on the space of (generalized) connections was recently introduced without any reference to a background metric. It is used to continue the exploration of the quantum Riemannian geometry. Operators corresponding to volume of three-dimensional regions are regularized rigorously. It is shown that there are two natural regularization schemes, each of which leads to a well-defined operator. Both operators can be completely specified by giving their action on states labelled by graphs. The two final results are closely related but differ from one another in that one of the operators is sensitive to the differential structure of graphs at their vertices while the second is sensitive only to the topological characteristics. (The second operator was first introduced by Rovelli and Smolin and De Pietri and Rovelli using a somewhat different framework.) The difference between the two operators can be attributed directly to the standard quantization ambiguity. Underlying assumptions and subtleties of regularization procedures are discussed in detail in both cases because volume operators play an important role in the current discussions of quantum dynamics.

Integrable structure of conformal field theory. ii. q-operator and ddv equation

Abstract: This paper is a direct continuation of [1] where we began the study of the integrable structures in Conformal Field Theory. We show here how to construct the operators ${\bf Q}_{\pm}(\lambda)$ which act in the highest weight Virasoro module and commute for different values of the parameter λ. These operators appear to be the CFT analogs of the Q - matrix of Baxter [2], in particular they satisfy Baxter's famous T- Q equation. We also show that under natural assumptions about analytic properties of the operators as the functions of λ the Baxter's relation allows one to derive the nonlinear integral equations of Destri-de Vega (DDV) [3] for the eigenvalues of the Q-operators. We then use the DDV equation to obtain the asymptotic expansions of the Q - operators at large λ; it is remarkable that unlike the expansions of the T operators of [1], the asymptotic series for Q(λ) contains the “dual” nonlocal Integrals of Motion along with the local ones. We also discuss an intriguing relation between the vacuum eigenvalues of the Q - operators and the stationary transport properties in the boundary sine-Gordon model. On this basis we propose a number of new exact results about finite voltage charge transport through the point contact in the quantum Hall system.

On quantum Galois theory

Abstract: The goals of the present paper are to initiate a program to systematically study and rigorously establish what a physicist might refer to as the “operator content of orbifold models.” To explain what this might mean, and to clarify the title of the paper, we will assume that the reader is familiar with the algebraic formulation of 2-dimensional CFT in the guise of vertex operator algebras (VOA), see [B], [FLM] and [DM] for more information on this point. In the paper [DVVV], several ideas are proposed concerning the structure of a holomorphic orbifold. In other words, if V is a holomorphic VOA and if G is a finite group of automorphisms of V, then the sub VOA V G of G-invariants is itself a VOA and the subject of [DVVV] is very much concerned with speculation on the nature of the V -modules. It turns out to be more useful − at least for purpose of inductive proofs − to take V to be a simple VOA. We will then see that V G is also simple whenever G is a finite group of automorphisms of V. One consequence of our main results is the following:

Modular invariance of trace functions in orbifold theory

Abstract: The goal of the present paper is to provide a mathematically rigorous foundation to certain aspects of rational orbifold conformal field theory, in other words the theory of rational vertex operator algebras and their automorphisms. Under a certain finiteness condition on a rational vertex operator algebra V which holds in all known examples, we determine the precise numbers of g-twisted sectors for any automorphism g of V of finite order. We prove that the trace functions and correlations functions associated with such twisted sectors are holomorphic functions in the upper half-plane and, under suitable conditions, afford a representations of the modular group of the type prescribed in string theory. We establish the rationality of conformal weights and central charge. In addition to conformal field theory itself, where our conclusions are required on physical grounds, there are applications to the generalized Moonshine conjectures of Conway-Norton-Queen and to equivariant elliptic cohomology.

Yang's system of particles and Hecke algebras

Abstract: The graded Hecke algebra has a simple realization as a certain algebra of operators acting on a space of smooth functions. This operator algebra arises from the study of the root system analogue of Yang's system of n particles on the real line with delta function potential. It turns out that the spectral problem for this generalization of Yang's system is related to the problem of finding the spherical tempered representations of the graded Hecke algebra. This observation turns out to be very useful for both these problems. Application of our technique to affine Hecke algebras yields a simple formula for the formal degree of the generic Iwahori spherical discrete series representations.

Deformation quantization and Nambu Mechanics

Abstract: Starting from deformation quantization (star-products), the quantization problem of Nambu Mechanics is investigated. After considering some impossibilities and pushing some analogies with field quantization, a solution to the quantization problem is presented in the novel approach of Zariski quantization of fields (observables, functions, in this case polynomials). This quantization is based on the factorization over ℝ of polynomials in several real variables. We quantize the infinite-dimensional algebra of fields generated by the polynomials by defining a deformation of this algebra which is Abelian, associative and distributive. This procedure is then adapted to derivatives (needed for the Nambu brackets), which ensures the validity of the Fundamental Identity of Nambu Mechanics also at the quantum level. Our construction is in fact more general than the particular case considered here: it can be utilized for quite general defining identities and for much more general star-products.

Operator -theory for groups which act properly and isometrically on Hilbert space

Abstract: Let G be a countable discrete group which acts isometrically and metrically properly on an infinite-dimensional Euclidean space. We calculate the K-theory groups of the C∗-algebras C∗ max(G) and C∗ red(G). Our result is in accordance with the Baum-Connes conjecture.

Volume and quantizations

Abstract: The differential structure of the Ashtekar - Isham quantum configuration space of canonical gravity allows the expression of the operators, representing various geometrical objects, by compact analytic formulae. In this paper such a formula is presented for the Rovelli - Smolin volume operator. This operator is compared with the quantum volume defined by Ashtekar and Lewandowski and a difference is indicated.

Framed vertex operator algebras, codes and the moonshine module

Abstract: For a simple vertex operator algebra whose Virasoro element is a sum of commutative Virasoro elements of central charge 1/2, two codes are introduced and studied. It is proved that such vertex operator algebras are rational. For lattice vertex operator algebras and related ones, decompositions into direct sums of irreducible modules for the product of the Virasoro algebras of central charge 1/2 are explicitly described. As an application, the decomposition of the moonshine vertex operator algebra is obtained for a distinguished system of 48 Virasoro algebras.

Non-commutative Euclidean and Minkowski structures

Abstract: A noncommutative *-algebra that generalizes the canonical commutation relations and that is covariant under the quantum groups SO q (3) or SO q(1, 3) is introduced. The generating elements of this algebra are hermitean and can be identified with coordinates, momenta and angular momenta. In addition a unitary scaling operator is part of the algebra.

Null-vectors in integrable field theory

Abstract: The form factor bootstrap approach allows to construct the space of local fields in the massive restricted sine-Gordon model. This space has to be isomorphic to that of the corresponding minimal model of conformal field theory. We describe the subspaces which correspond to the Verma modules of primary fields in terms of the commutative algebra of local integrals of motion and of a fermion (Neveu–Schwarz or Ramond depending on the particular primary field). The description of null-vectors relies on the relation between form factors and deformed hyper-elliptic integrals. The null-vectors correspond to the deformed exact forms and to the deformed Riemann bilinear identity. In the operator language, the null-vectors are created by the action of two operators ? (linear in the fermion) and ? (quadratic in the fermion). We show that by factorizing out the null-vectors one gets the space of operators with the correct character. In the classical limit, using the operators ? and ? we obtain a new, very compact, description of the KdV hierarchy. We also discuss a beautiful relation with the method of Whitham.

Linear isometries between subspaces of continuous functions

Abstract: We say that a linear subspace A of C0(X) is strongly separating if given any pair of distinct points x1, x2 of the locally compact space X, then there exists f ∈ A such that |f(x1)| 6= |f(x2)|. In this paper we prove that a linear isometry T of A onto such a subspace B of C0(Y ) induces a homeomorphism h between two certain singular subspaces of the Shilov boundaries of B and A, sending the Choquet boundary of B onto the Choquet boundary of A. We also provide an example which shows that the above result is no longer true if we do not assume A to be strongly separating. Furthermore we obtain the following multiplicative representation of T : (Tf)(y) = a(y)f(h(y)) for all y ∈ ∂B and all f ∈ A, where a is a unimodular scalar-valued continuous function on ∂B. These results contain and extend some others by Amir and Arbel, Holsztyński, Myers and Novinger. Some applications to isometries involving commutative Banach algebras without unit are announced.

On q-Analogues of Bounded Symmetric Domains and Dolbeault Complexes

Abstract: A very well known result by Harish-Chandra claims that any Hermitian symmetric space of non-compact type admits a canonical embedding into a complex vector space $V$. The image of this embedding is a bounded symmetric domain in $V$. This work provides a construction of q-analogues of a polynomial algebra on $V$ and the differential algebra of exterior forms on $V$. A way of producing a q-analogue of the bounded function algebra in a bounded symmetric domain is described. All the constructions are illustrated by detailed calculations in the case of the simplest Hermitian symmetric space $SU(1,1)/U(1)$. The development of these ideas can be found in math.QA/9803110 and math.QA/9809038 .

On a Harish-Chandra homomorphism

Abstract: We prove the graded surjectivity of the Harish-Chandra homomorphism for the space of invariant differential operators on a semisimple Lie algebra. This answers a question of Wallach ( see [15]) and has a number of further consequences ( see [8] and [14]).

Fusion of Positive Energy Representations of LSpin(2n)

Abstract: Building upon the Jones-Wassermann program of studying Conformal Field Theory using operator algebraic tools, and the work of A. Wassermann on the loop group of LSU(n) (Invent. Math. 133 (1998), 467-538), we give a solution to the problem of fusion for the loop group of Spin(2n). Our approach relies on the use of A. Connes' tensor product of bimodules over a von Neumann algebra to define a multiplicative operation (Connes fusion) on the (integrable) positive energy representations of a given level. The notion of bimodules arises by restricting these representations to loops with support contained in an interval I of the circle or its complement. We study the corresponding Grothendieck ring and show that fusion with the vector representation is given by the Verlinde rules. The computation rests on 1) the solution of a 6-parameter family of Knizhnik-Zamolodchikhov equations and the determination of its monodromy, 2) the explicit construction of the primary fields of the theory, which allows to prove that they define operator-valued distributions and 3) the algebraic theory of superselection sectors developed by Doplicher-Haag-Roberts.

A Finite Dimensional Introduction to Operator Algebra

Abstract: This article surveys some recent advances in operator algebra that were inspired by considerations from ring theory, particularly the representation theory of finite dimensional algebras.

On the causal structure of Minkowski spacetime

Abstract: The causal structure of Minkowski spacetime M is discussed, in terms of the notions of causal complementation and causal completion. These geometric notions are relevant for quantum field theory and the theory of the Klein-Gordon equation. Particular attention is given to closed, convex, causally complete subsets of M, and the properties of such sets are discussed. The study of such sets is motivated by potential applications to the theory of local nets of von Neumann algebras. The notion of the envelope of uniqueness of a subset of M, familiar from the theory of the wave equation, is discussed, and some results about the relation of this envelope to the causal completion of the set are presented.

SU(N) quantum Yang-Mills theory in two-dimensions: A Complete solution

Abstract: A closed expression of the Euclidean Wilson-loop functionals is derived for pure Yang–Mills continuum theories with gauge groups SU(N) and U(1) and space-time topologies R1×R1 and R1×S1. (For the U(1) theory, we also consider the S1×S1 topology.) The treatment is rigorous, manifestly gauge invariant, manifestly invariant under area preserving diffeomorphisms and handles all (piecewise analytic) loops in one stroke. Equivalence between the resulting Euclidean theory and and the Hamiltonian framework is then established. Finally, an extension of the Osterwalder–Schrader axioms for gauge theories is proposed. These axioms are satisfied in the present model.

Generalized Rationality and a "Jacobi Identity" for Intertwining Operator Algebras

Abstract: We prove a generalized rationality property and a new identity that we call the ``Jacobi identity'' for intertwining operator algebras. Most of the main properties of genus-zero conformal field theories, including the main properties of vertex operator algebras, modules, intertwining operators, Verlinde algebras, and fusing and braiding matrices, are incorporated into this identity. Together with associativity and commutativity for intertwining operators proved by the author in \cite{H2} and \cite{H3}, the results of the present paper solve completely the problem of finding a natural purely algebraic structure on the direct sum of all inequivalent irreducible modules for a suitable vertex operator algebra. Two equivalent definitions of intertwining operator algebra in terms of this Jacobi identity are given.

Conjugated Operators in Quantum Algorithms

Abstract: This paper addresses the question of understanding quantum algorithms in terms of unitary operators. Many quantum algorithms can be expressed as applications of operators formed by conjugating so-called classical operators. The operators that are used for conjugation are determined by the problem and any additional structure possessed by the Hilbert space that is acted upon. We prove many new commutative laws between these different operators, and we use those to phrase and analyze old and new problems and algorithms. As an example, we review the Abelian subgroup problem. We then introduce the problem of determining a group homomorphism, and we give classical and quantum algorithms for it. We also generalize Deutsch''s problem and improve the previous best algorithms for earlier generalizations of it.