Showing papers on "Operator algebra published in 1985"

Nonlocal (parafermion) currents in two-dimensional conformal quantum field theory and self-dual critical points in Z/sub N/-symmetric statistical systems

Abstract: A two-dimensional exactly solvable model of a conformal quantum field theory is developed which is self-dual and has Z/sub N/ symmetry. The operator algebra, the correlation functions, and the anomalous dimensions of all fields are calculated for this model, which describes self-dual critical points in Z/sub N/-symmetric statistical systems.

Perturbations of Banach algebras

Abstract: Preliminaries.- I. Perturbations of multiplications and onto-isomorphisms.- II. Into-isomorphisms.- III. Isometries in semisimple, commutative Banach algebras.- IV. Perturbations of operator algebras.- V. Stability.

Boson Quantum Field Models

Abstract: Quantum fields, from a mathematical point of view, are highly singular. These fields are believed to describe the interactions of elementary particles. For the interaction of electrons with light (photons), the quantum field description is exact within the limits of experimental accuracy (5 significant figures). For these reasons, i.e. the mathematical difficulties and the importance to physics, the problem of formulating the mathematical foundations of quantum field theory has attracted the attention of both mathematicians and physicists over a period of several decades. On the side of the physicists, the most striking achievements were the calculation in the late 1940’s and early 1950’s of the Lamb shift and the anomalous magnetic moment of the electron together with the development of the renormalization method on which these calculations were based. Of the mathematicians, J. von Neumann was the first to realize that new mathematical theories would be required to formulate quantum field theory correctly and this realization was one of the motives for developing the theory of operator algebras.

Abstract theory of abelian operator algebras: an application of forcing

Abstract: The abstract abelian operator theory is developed from a general standpoint, using the method of forcing and Boolean-valued models. 1. Introduction. One aspect of the study of operator algebras is the description of the algebraic structure of algebras of operators, and representation of abstract algebras on a Hilbert space. This "algebraization" of the theory of algebras of operators is well understood in the case of bounded normal operators. The theory of von Neumann algebras (or the more general C *-algebras) is based on Stone's characterization of abelian (commutative) algebras of bounded operators in (13). Stone's theory describes such algebras axiomatically, in algebraic terms, without reference to Hilbert space, and develops a function calculus and the spectral theory for the abstract algebras. Moreover, the algebras are algebraically isomorphic to the algebra C(X) of all (complex-valued) continuous functions on an extremally disconnected compact Hausdorff space X. The functional representation of abelian von Neumann algebras has been used to extend Stone's work to abelian algebras of unbounded normal operators. This has been done for instance by Fell and Kelley in (2); a detailed account of spectral theory based on such an approach is presented by Kadison and Ringrose in (5). For a given abelian von Neumann algebra _, one defines an algebra _ of normal (not necessarily bounded) operators affiliated with -d. If C(X) is the functional algebra isomorphic to X, then -is isomorphic to the algebra of all normal functions on X. This provides both the spectral theory and a Borel function calculus for unbounded normal operators.

Concepts of conditional expectations in quantum theory

Abstract: A concept of conditional expectations in quantum theory is established with interrelations to previously introduced concepts of the Cycon–Hellwig conditional expectations and a posteriori states, which are analogous to the existing interrelations in the classical probability theory among conditional expectations related to random variables, those related to σ subalgebras and conditional probability distributions. These three concepts are shown to have satisfactory statistical interpretation in the quantum measuring processes. For the above purpose, we introduce an integration with respect to functions with values in the states of operator algebras and positive operator valued measures such that the resulting indefinite integrals are completely positive map valued measures. Eventually, it is proved that in the von Neumann algebraic formulation, the Cycon–Hellwig conditional expectations always exist as completely positive map valued measures.

Realistic physical origin of the quantum observable operator algebra in the frame of the causal stochastic interpretation of quantum mechanics: The relativistic spin-zero case

Abstract: The deduction by Guerra and Marra of the usual quantum operator algebra from a canonical variable Hamiltonian treatment of Nelson's hydrodynamical stochastic description of real nonrelativistic Schroedinger waves is extended to the causal stochastic interpretation given by Guerra and Ruggiero and by Vigier of relativistic Klein-Gordon waves. A specific representation shows that the Poisson brackets for canonical hydrodynamical observables become ''averages'' of quantum observables in the given state. Stochastic quantization thus justifies the standard procedure of replacing the classical particle (or field) observables with operators according to the scheme p/sub ..mu../..-->..-ihpartial/sub ..mu../ and L/sub munu/..-->..-ih(x/sub ..mu../partial/sub ..nu../-x/sub ..nu../partial/sub ..mu../ ).

Multiplication operators and traces of commutators

Abstract: Some new sufficient conditions are found on operators A, B in a Hilbert space, under which trace (AB−BA)=0.

The Arveson frequency response and systems theory

Abstract: A frequency-domain theory which is applicable to all linear systems and a class of non-linear systems is formulated in a Hilbert resolution space setting. Our approach is predicated on an ‘analytic’ characterization of certain non-selfadjoint operator algebras due to Arveson. Although originally used as an analytic tool, Arveson, in fact, formulated an operator-valued frequency response function for the elements of these algebras which, in particular, include the causal operators defined on a Hilbert resolution space. The resultant Arveson frequency response (AFR) is well defined for all linear and non-linear systems on a Hilbert resolution space and in the linear case it is characterized by an analyticity theory which is identical to that of the classical time-invariant frequency response concept. Moreover, it yields computationally viable formulae for the most widely studied classes of linear systems including the time-varying ABCD systems. Finally, the classical frequency response concepts and their ti...

Joint browder spectra and tensor products

Abstract: On montre que les differentes notions de spectres de Browder conjoints coincident pour une classe speciale d'operateurs

Banach-power-associative algebras and J-B algebras

Abstract: It is proved that the set of observables of a quantum system, stable under linear combinations and square, and complete with respect to a compatible norm topology, is a J-B algebra. This means that the Jordan identity can be replaced by the power-associativity in the definition of a J-B algebra On demontre que l'ensemble des observables d'un systeme quantique, stable par combinaisons lineaires et carre, et complet relativement a une topologie definie par une norme compatible, est une J-B algebre. Cela signifie que, dans la definition d'une J-B algebre, l'identite de Jordan peut etre remplacee par la propriete de puissance associative

Joint numerical ranges for unbounded normal operators

Abstract: For bounded operators, the theory of the joint numerical range has been developed by various authors [ 1,2,3,4,5 ]. Especially, the properties of commuting normal n-tuples are discussed in detail. Our purpose here is to show that many results in the above references still hold in the case of unbounded normal operators (see Theorem 2.3, Corrollary 3.5, Theorem 4.1, Theorem 4.2). Besides, the operator algebras are closely related to the theory of joint spectrum and joint numerical ranges in the boundedcase (cf. [ 1,3 ]). How about unbounded operators? It seems that one must consider unbounded operator algebras. Some work has been done in this direction for the joint spectrum of unbounded normal operators [ 9 ]. In the last section of this paper, we provide some intimate relations between the joint numerical range and the unbounded operator algebras for unbounded normal operators.

Nonuniversity in a model Hamiltonian with four-spin interactions

Abstract: We introduce a model Hamiltonian with six types of four-spin interaction terms (coupling constants {Гi}) that couple four Ising-like lattices in two dimensions. The model reduces to the asymmetric Ashkin-Teller or eight-vertex models by appropriate choices of {Гi}. The decoupling point is studied using first order perturbation theory in {Гi}. We obtain an apparently nonuniversal critical surface. The stability of a first-order formula is analyzed, in three particular cases, using Kadannoff's operator algebra and renormalization-group flow equations. Also, the model has three nonuniversal decoupling planes on which coupling only occurs between pairs of lattices. In each plane there are four multicritical lines with an additional marginal operator, which is responsible for a probable nonuniversality when we move out of the plane. This follows from the equivalence between the eight-vertex and Ashkin-Teller models with the Gaussian model.

Raman-Nath equation and classical diffusion on one-dimensional random chains. Technical report No. 47, 1 October 1984-30 September 1985

Abstract: In a recent Letter, Heinrichs discussed an exact analytical result for the long-time classical diffusion of particles on random chains with the nearest-neighbor transfer rates having Gaussian distributions about fixed systematic (constant) rates. He has shown for static randomness, for both symmetric (S) and asymmetric transition rates (A), that the equations satisfied by the configuration-averaged probability of finding the particle at x at time t, p(x,t) can be expressed in the form of a differential, finite-difference recursive relation. These equations have appeared in many areas of physics, particularly in the quantum theory of free-electron lasers and quantum optics. In this reprint we present an alternative method for solving Eq. (4) based on the introduction of raising and lowering operators and performing simple operator algebra. This method(3) has been successfully applied to a wide class of differential-recursive equations known as Raman-Nath (RN) equations of which Eq. (4) is a particular example. Although we reach the same conclusions as in Ref. 1 and derive no new physical results, the method offers definite advantages of simplicity which should be of interest to researchers in this field.

Generalized FS×W versus Dyson’s classification of irreducible representations

Abstract: We study the characterization of the 13 cases obtained in the classification of the irreducible linear–antilinear representations of semigroups in a finite‐dimensional vector space X over an algebraically closed field K with a conjugation j (generalized Frobenius–Schur–Wigner, or FS×W, classification). It has already been shown that each case can be characterized by various equivalent properties, some of which can be endowed with a physical interpretation. We show here that, whenever K is the complex field C, each case can be characterized by the structure (in the sense specified by the Weyl theorem on the structure of the matrix algebras and their commutators) of a pair of suitable operator algebras over the real field R. This characterization coincides with the one given by Dyson for each case of his classification of symmetry groups. Thus, the latter classification is recovered under more general assumptions and in a generalized framework, and its one‐to‐one correspondence with the generalized FS×W cla...

Best approximation and quasitriangular algebras

Abstract: On cherche les meilleurs approximants dans une algebre quasitriangulaire donnee pour un operateur donne