Showing papers on "Operator algebra published in 1982"
TL;DR: In this paper, some nonperturbative constraints on supersymmetry breaking are derived and it is demonstrated that dynamical supersymmetric breaking does not occur in certain interesting classes of theories.
1,980 citations
TL;DR: In this paper, the authors considered the triangle inequalities in operator algebras and showed that triangle inequalities can be obtained in linear and multilinear algebraes.
Abstract: (1982). Triangle inequalities in operator algebras. Linear and Multilinear Algebra: Vol. 11, No. 2, pp. 167-178.
81 citations
TL;DR: In this paper, a general criterion for the absence or presence of anomalies in the quantum non-local charge of the non-linear σ-model on a riemannian symmetric space is presented.
78 citations
46 citations
41 citations
TL;DR: In this article, a quantum set theory is constructed from the null set by the familiar quantum techniques of tensor product and antisymmetrization, where rank and cardinality operators are analogous to Schroedinger coordinates of the theory, in that they are multiplication or Q-type operators.
Abstract: The mathematical language presently used for quantum physics is a high-level language. As a lowest-level or basic language I construct a quantum set theory in three stages: (1) Classical set theory, formulated as a Clifford algebra of “S numbers” generated by a single monadic operation, “bracing,” Br = {…}. (2) Indefinite set theory, a modification of set theory dealing with the modal logical concept of possibility. (3) Quantum set theory. The quantum set is constructed from the null set by the familiar quantum techniques of tensor product and antisymmetrization. There are both a Clifford and a Grassmann algebra with sets as basis elements. Rank and cardinality operators are analogous to Schroedinger coordinates of the theory, in that they are multiplication or “Q-type” operators. “P-type” operators analogous to Schroedinger momenta, in that they transform theQ-type quantities, are bracing (Br), Clifford multiplication by a setX, and the creator ofX, represented by Grassmann multiplicationc(X) by the setX. Br and its adjoint Br* form a Bose-Einstein canonical pair, andc(X) and its adjointc(X)* form a Fermi-Dirac or anticanonical pair. Many coefficient number systems can be employed in this quantization. I use the integers for a discrete quantum theory, with the usual complex quantum theory as limit. Quantum set theory may be applied to a quantum time space and a quantum automaton.
32 citations
01 Jan 1982
TL;DR: The structure of the predual of an ultra-weakly closed operator algebra can be very revealing of internal structural properties of the algebra, and has been studied in the theory of von Neumann algebras as discussed by the authors.
Abstract: The structure of the predual of an ultraweakly closed operator algebra can be very revealing of internal structural properties of the algebra. This relationship has been most important in the theory of von Neumann algebras, and has recently become significant in the study of more general ultraweakly closed algebras.
29 citations
TL;DR: In this paper, a dynamical semi-group a = {a, r>o as a u-weakly continuous one-parameter semi group of normal positive maps on a von Neumann algebra M is defined.
25 citations
TL;DR: In this paper, a quantum theory of a free massless spin-3/2 field on Einstein spaces (Rab=Λgab) is formulated in an algebraic framework.
Abstract: A quantum theory of a free massless spin-3/2 field on Einstein spaces (R
ab=Λgab) is formulated in an algebraic framework. Attention is confined to the structure of the quantum operator algebra. In particular, the issue of positivity of the anticommutator is investigated and found to depend on whether or not the space-time admits “rero-frequency” neutrino solutions. Using methods developed for the purpose, a class of space-time that does not admit neutrino “zero modes” is characterized. An appendix introduces a useful technique for obtaining an initial value formulation of spinor field equations.
24 citations
TL;DR: In this article, the theory of sequential quantum processes has been extended to Liouville space via the use of non-Hermitian projection operators in order to treat the evolution of the quantum density operator and to enable physically important matrix elements of the density operator to be calculated.
Abstract: The theory of sequential quantum processes has been extended to Liouville space via the use of non-Hermitian projection operators in order to treat the evolution of the quantum density operator and to enable physically important matrix elements of the density operator to be calculated. The formal relationship of master equation methods to the theory of sequential quantum processes is established, and a new set of coupled master equations is derived. Special choices of projection operators lead to further simplification of the results. The Markoff approximation is also examined.
TL;DR: In this article, it is shown how quantum theory can be expressed as a probability theory on Hilbert space, treated as a measure space, allowing the description of both bounded and unbounded observables as measurable functions.
Abstract: It is shown how quantum theory (QT) can be expressed as a probability theory on Hilbert space, treated as a measure space. The approach generalizes the work of Bach and clarifies the ’’generalized trace’’ of Langerholc. It permits the description of both bounded and unbounded observables as measurable functions. Dynamical evolution can be described in terms of stochastic processes.
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01 May 1982
TL;DR: In this paper, the role of $K$-theory in non-commutative algebraic topology is discussed and a selection of problems, edited by Edward G. Salinas, are discussed.
Abstract: Index theory, bordism, and $K$-homology by P. Baum and R. G. Douglas The radial Fourier-Stieltjes algebra of free groups by J. M. Cohen and L. De Michele Reducible topological Markov chains via $K_0$-theory and Ext by D. Handelman Miscenko's work on Novikov's conjecture by W. C. Hsiang and H. D. Rees Pseudo-differential operators and differential structures by J. Kaminker $L^2$-index on elliptic operators on locally symmetric spaces of finite volume by H. Moscovici $K$-theory for actions of the circle group on $C^\ast$-algebras by W. L. Paschke Connes' analogue for crossed products of the Thom isomorphism by M. A. Rieffel The role of $K$-theory in non-commutative algebraic topology by J. Rosenberg Some remarks on the classification of essentially $n$-normal operators by N. Salinas A selection of problems, edited by Edward G. Effros.
TL;DR: In this article, the collective coordinate operator X( t) and the field operator x(x, t) are decomposed into a new set of Heisenberg operators and their conjugates p t and Jr t, respectively.
Abstract: The operator structure of the collective coordinate associated with extended objects in quantum field theory is discussed in the context of renormalized perturbation theory. The analysis of a quantum field theory with extended objects must take into account the presence of these position operators. Two methods have been proposed so far. One is the collective coordinate method,4) in which the Heisenberg field operator and its canonical conjugate are decomposed into a new set of Heisenberg operators, namely the collective coordinate operator X( t) and field operator x(x, t) and their conjugates p( t) and Jr(x, t), respectively. This decomposition is accompanied by certain constraints which define X( t) and p( t). It is required that X( t)--> X( t) + a induces the space translation of the Heisenberg operators and that X( t) and x(x, t) are independent as Heisenberg operators. The other method expresses the Heisenberg operators in terms of the physical operators or asymptotic fields which construct the physical Hilbert space. 3 ) This expression is called the dynamical map. In this method it has been shown that the set of physical operators consists of two mutually commuting sets (q, p) and ((l, (l t), where q is the quantum mechanical position-operator (quantum coordinate) and p is its canonical con jugate while (l and (l t stand for the annihilation and creation operators of particle-like modes respectively.3),5) Thus, the Hilbert space is found to be a
TL;DR: In this article, the authors consider two types of dynamical systems employed in the foundations of quantum theory, C*-systems and W*-Systems, and show that both of them are type III.
TL;DR: In this paper, the Dirac operator is assigned to a geometric structure called aco-Riemannian metric, which is related to the geometric structure of quantum mechanics, e.g., the symplectic structure of the projective space of Hilbert space.
Abstract: A central idea of modern geometric analysis is the assignment of a geometric structure, usually called thesymbol, to a differential operator. It is known that this operation is closely related to quantum mechanics. For a class of linear operators, including the Dirac operator, a geometric structure, called aco-Riemannian metric, is assigned to such symbols. Certain other topics related to the geometric structure of quantum mechanics, e.g., the symplectic structure of the projective space of Hilbert space, are briefly treated.
TL;DR: In this paper, it was shown that a general quantum measurement of commuting observables can be represented by a local transition map, a special type of positive linear map on a von Neumann algebra.
Abstract: On the basis of four physically motivated assumptions, it is shown that a general quantum measurement of commuting observables can be represented by a “local transition map,” a special type of positive linear map on a von Neumann algebra. In the case that the algebra is the bounded operators on a Hilbert space, these local transition maps share two properties of von Neumann-type measurements: they decrease “matrix elements” of states and increase their entropy. It is also shown that local transition maps have all the properties of a conditional expectation of a von Neumann algebra onto a subalgebra except that their range is not restricted to the subalgebra. The notion of locality arises from requiring that a quantum measurement may be treated classically when restricted to the commutative algebra generated by the measured observables. The formalism established applies to observables with arbitrary spectrum. In the case that the spectrum is continuous we have only “incomplete” measurements.
01 Jan 1982
TL;DR: In this paper, it was shown that a Boolean algebra of projections is G-σ-complete if and only if it coincides with the range of a spectral measure of class G (Theorem 2).
Abstract: Complete and σ-complete Boolean algebras of projections in a complex Banach space were studied first by Bade [1] (see also [3; XVII.3]). The purpose of this paper is to find the appropriate extensions of several of his results to the more general case of G-complete and G-σ-complete Boolean algebras of projections, where G is a total linear manifold in the dual of the underlying Banach space. We shall prove e.g. that a Boolean algebra of projections is G-σ-complete if and only if it coincides with the range of a spectral measure of class G (Theorem 2), and we shall give a sufficient condition ensuring that the uniformly closed operator algebra generated by a G-σ-complete Boolean algebra B of projections coincides with the first commutant of B (Theorem 3). The new techniques will include the application of certain weak topologies and some of the duality theory of paired linear spaces as well as an idea due to Palmer [5].
TL;DR: In this paper, the commutator of the pair current with the pair tunneling Hamiltonian has been identified with pair number operator, in the pseudoangularmomentum description.
Abstract: The operator algebra associated with Josephson tunneling is reexamined, with special attention given to the commutator of the pair current with the pair tunneling Hamiltonian. This commutator has been identified with the pair number operator, in the pseudoangular-momentum description. I find this identification to be erroneous. The commutator in question vanishes identically because of a symmetry in the underlying microscopic theory. Consequently the conventional Josephson-Anderson number-phase formalism provides a complete description of pair tunneling.
01 Jan 1982
TL;DR: The relation between the symbols of Gohberg-Krupnik and of Duduchava-Dynin is explained by means of a connection between local Fourier and Mellin transforms as mentioned in this paper.
Abstract: The elements of the closed operator algebra ∑p (Γρ) generated by singular integral operators with piecewise continuous coefficients on a closed piecewise Ljapunov contour can be written as the sum of a singular integral operator, countably many generalized Mellin convolutions, and a compact operator. The relation between the symbols of Gohberg-Krupnik and of Duduchava-Dynin is explained by means of a connection between local Fourier and Mellin transforms.
01 Jan 1982
TL;DR: In this paper, a Hilbert space and the corresponding Schroedinger equation have been introduced into the theory of non-associative quantum mechanics, where the underlying Lie algebra A is assumed to be flexible and Lie-admissible.
Abstract: Within an algebraic framework of non-associative quantum mechanics, a Hilbert space and the corresponding Schroedinger equation have been introduced into the theory as follows. The underlying non-associative operator algebra A is assumed to be flexible and Lie-admissible so that it is compatible with the Heisenberg equation of motion and quantization. A Hilbert space can be introduced into the theory as a faithful representation of the associated Lie algebra A/sup -/ of the Lie-admissible algebra A. However, if A/sup -/ is the standard Heisenberg algebra, and if the representation is irreducible, then it is shown that A must be associative, reproducing the standard associative quantum mechanics. Some discussions to circumvent this No-Go theorem are discussed. Especiallyyu if we are interested in the formulation of infinite component wave equations, then this difficulty can be avoided.
TL;DR: In this paper, expressions for unitary operators representing a thin prism, an ideal thin lens, a plane grating and a zone-plate have been derived, and the operator algebra is then applied to a thick lens and a pair of thin lenses separated by a distance.
Abstract: Expressions for the unitary operators representing a thin prism, an ideal thin lens, a plane grating and a zone-plate have been derived. The operator algebra is then applied to a thick lens and a pair of thin lenses separated by a distance.
01 Jan 1982
TL;DR: In this article, the notions of spectral localization which are well-known for operators are generalized to an arbitrary Banach algebra and several improvements and corrections of the existing results for operators were obtained.
Abstract: The notions of the theory of spectral localization which are well-known for operators are generalized to an arbitrary Banach algebra. In this setting several improvements and corrections of the existing results for operators are obtained.
01 Jan 1982
TL;DR: In this paper, the authors reviewed some applications of the Tomita-Take4 saki theory in QFT, where the vacuum vector is cyclic and separating for R(O) in a Eaag-Kastler theory.
Abstract: In this lecture we have reviewed some applications of the Tomita-Take4 saki theory [ I ] in QFT. Let O c ~ and R(O) be the von Neumann algebra associated to O in a Eaag-Kastler theory, me assume (Reeh-Schlieder theorem) that the vacuum vector ~ is cyclic and separating for R(O), therefore one constructs [I ] a positive , non-singular selfadjoint operator A O (the modular operator) and an anti-unitary involution Jo (the modular conjugation) such that
TL;DR: In this paper, the debole compattezza delle applicazioni lineari positive dalla parte autoaggiunta di un'algebra di von Neumann su uno spazio di Banach parzialmente ordinato, separabile, normale.
Abstract: Si dimostra la debole compattezza delle applicazioni lineari positive dalla parte autoaggiunta di un'algebra di von Neumann su uno spazio di Banach parzialmente ordinato, separabile, normale.