Showing papers on "Operator algebra published in 1970"

Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics. i. mapping theorems and ordering of functions of noncommuting operators.

Abstract: A new calculus for functions of noncommuting operators is developed, based on the notion of mapping of functions of operators onto $c$-number functions. The class of linear mappings, each member of which is characterized by an entire analytic function of two complex variables, is studied in detail. Closed-form solutions for such mappings and for the inverse mappings are obtained and various properties of these mappings are studied. It is shown that the most commonly occurring rules of association between operators and $c$-numbers (the Weyl, the normal, the antinormal, the standard, and the antistandard rules) belong to this class and are, in fact, the simplest ones in a clearly defined sense. It is shown further that the problem of expressing an operator in an ordered form according to some prescribed rule is equivalent to an appropriate mapping of the operator on a $c$-number space. The theory provides a systematic technique for the solution of numerous quantum-mechanical problems that were treated in the past by ad hoc methods, and it furnishes a new approach to many others. This is illustrated by a number of examples relating to mappings and ordering of operators.

Algebraic and analytic aspects of operator algebras

Abstract: An algebraic prelude Continuity of automorphisms and derivations $C^*$-algebra axiomatics and basic results Derivations of $C^*$-algebras Homogeneous $C^*$-algebras CCR-algebras $W^*$ and $AW^*$-algebras Miscellany Mappings preserving invertible elements Nonassociativity Bibliography.

Quantum field theory on lightlike slabs

Abstract: Restricting the support of relativistic quantum fields to lightlike hyperplanes (e.g. x0+x3)=const) we find examples of such fields to exist as well-defined self-adjoint operators with properties however that differ vastly from those of fields on the usual spacelike surfaces. We show that on a lightlike hyperplane 1) the free-field algebra is irreducible (instead of Abelian, and in contrast to what one would expect of data on a characteristic surface) and 2) fields with different masses become unitarily equivalent (whereas they are inequivalent on spacelike planes). Furthermore the field algebra restricted to the space-time slab between two parallel lightlike planes is always irreducible (while there are counterexamples for spacelike slabs). We establish this directly for generalized free fields and rederive it for Wightman fields in gereral.

Nets of C *-algebras and classification of states

Abstract: The concept of locality in quantum physics leads to mathematical structures in which the basic object is an operator algebra with a net of distinguished subalgebras (the “local” subalgebras). Such nets provide a classification of the states of this algebra in equivalence classes determined by local or asymptotic properties. The corresponding equivalence relations are natural generalizations of the (more stringent) standard quasiequivalence relation (they are also useful for classifying states by their properties with respect to automorphism groups). After discussing general nets from this point of view we investigate in the last section more specialized nets (funnels of von Neumann algebras) with special emphasis on their locally normal states.

Causality in Hilbert Space

Abstract: A new formalism, termed the resolution space, is presented within which the theory of causal systems may be unified and extended. The resulting formalism, which is defined as a Hilbert space together with a resolution of the identity, readily includes the commonly encountered function and sequence space causality concepts yet is sufficiently straightforward to allow the various aspects of network and system theory which are dependent on the time parameter to be studied in an operator theoretic context without the detailed structure of a function space. Specific results include additive and multiplicative decomposition theorems for causal operators which naturally extend the “realizable part” and “spectral” decompositions of classical system theory and an integral representation theorem for linear operators on a resolution space. The general theory is illustrated with a number of examples concerning passive “networks”, those including an operator theoretic approach to the passive synthesis problem over an arbitrary resolution space.

Boson Expansions for an Exactly Soluble Model of Interacting Fermions with SU(3) Symmetry

Abstract: An exactly soluble three‐level model for a system of fermions with Hamiltonian formed from generators of the group SU(3) is studied. A basis for the representation of which the ground state is a member is constructed. It is demonstrated that in this representation, the generators can each be expanded in a series as functions of a pair of ``kinematical'' boson operators; the series are uniquely determined to satisfy the operator algebra and the invariants of the representation by a method of Marumori. It is seen that the lowest anharmonic approximation to the Hamiltonian and other operators yields excellent numerical agreement with exact results for all regimes of interaction strength considered. An alternative description of the system in terms of a dynamically more meaningful boson, called the ``physical'' boson, is shown to be appropriate for relatively weak coupling where one has a near harmonic spectrum.

Representations of local nonrelativistic current algebras

Abstract: In this paper we show how to specify the particle content of a nonrelativistic quantum theory of $N$ identical spinless particles in terms of observables like the particle number density and the flux density of particles. Our approach to this problem is through a study of the irreducible representations of the local, equal-time current algebra. It is shown how these representations define a functional representation of the current algebra, and that the Hamiltonian can be written in terms of the currents in a nonsingular fashion in any irreducible representation.

Geometry of the unit sphere of a c -algebra and its dual

Abstract: In this study we exploit as a main tool a polar decomposition for linear functionals on operator algebras, introduced in 1958 by Sakai, to determine various types of extremal behavior in the unit spheres of C*-algebras and their duals. We discuss exposed points and complex extreme points as well as extreme points. Throughout this paper M, F, and A will be generic symbols for a von Neumann algebra, its predual, and a C*-algebra which may not have a unit, respectively. By a C*-algebra is meant a Banach *-algebra in which ||#*a;|| = \\x\\2 holds for all x. A von Neumann algebra is a C*-algebra of operators on a Hubert space which is closed in the weak operator topology and contains the identity operator. Each von Neumann algebra M is equivalent (as a Banach space) to the dual of the Banach space F of ultra-weakly continuous ( = normal) linear functionals on M. The space F is called the predual of M (and is unique). References for the preceding facts, as well as any others to follow concerning C* -algebras and von Neumann algebras, are the two monographs of Dixmier [3], [4], and the lecture notes of Sakai [13]. We will denote by w, s, uw, us the weak operator, strong operator, ultra-weak and ultra-strong topologies of M respectively, and n refers to the norm topology of a Banach

On the mathematical description of quantized fields

Abstract: We start from Haag's proposal to describe quantum fields at a point, corresponding to the heuristic description by means of their matrix elements (A(x)Φ‖Ψ) between vectors of a dense linear manifoldD of the Hilbert space. We particularize this idea, so that the sesquilinear functional that describes the field at a point may be considered as an element of the sequential completion of a space of operators, endowed with a suitable “D-weak” topology.

Hilbert spaces of functional power series.

Abstract: The paper concerns linear spaces of functional power series over arbitrary linear spaces with a given bilinear form and involution. The domain of convergence, growth and type or convergence radius resp. of functional power series belonging to certain Hilbert spaces are estimated. Sufficient conditions (concerning growth and type) for a functional to belong to a given Hilbert space are determined. The connection of Hilbert spaces of functional power series and Fock spaces is discussed. The paper contains estimates for functional power series and their derivatives in certain multiplets consisting of topological spaces invariant with respect to differentiation on certain domains. The results are applied to quantum field theory, yielding an estimation of the convergence domain and asymptotic behaviour of the generating functionals for the S-matrix on and off mass shell, for the field and for the derivatives of these quantities with respect to elements of the basic space.

Thermal averages in molecular physics by operator methods

Abstract: Operator methods and Schwinger's perturbation expansion are used in the calculation of thermal averages in molecular physics with regard to gas electron diffraction. These are the partition function, averages of powers of the oscillator occupation number operator therm, the moments of the normal coordinate therm and of the exponential of the normal coordinate therm. It is shown that the equations developed are quite general, including anharmonic resonances like Fermi and Darling Dennison resonance in polyatomic molecules. Simple algebra is used. A compact and general formulation of the theory of electron diffraction by gases is given, the calculated averages are proportional to the mean radius r g, the mean amplitude l e and the intensity function therm. Further applications are summarized. The operator algebra is given in the appendix.