Showing papers on "Operator algebra published in 1984"
TL;DR: Based on the conformal algebra approach, a general technique for the calculation of multipoint correlation functions in 2D statistical models at the critical point is given in this article, where particular conformal operator algebras are found for operators of the 2D q-component Potts model.
1,317 citations
TL;DR: In this paper, the massless quantum field theories describing the critical points in two-dimensional statistical systems were studied and it was shown that the local fields forming the operator algebra can be classified according to irreducible representations of the Virasoro algebra.
Abstract: We study the massless quantum field theories describing the critical points in two dimensional statistical systems. These theories are invariant with respect to the infinite dimensional group of conformal (analytic) transformations. It is shown that the local fields forming the operator algebra can be classified according to the irreducible representations of the Virasoro algebra. Exactly solvable theories associated with degenerate representations are analized. In these theories the anomalous dimensions are known exactly and the correlation functions satisfy the system of linear differential equations.
350 citations
TL;DR: The conformal algebra for operators of the Z3 model at the phase transition point is built in this article, where critical exponents are found in this approach as solutions of simple algebraic equations.
192 citations
TL;DR: In this paper, the semi-crossed product Z+XαC0(S) is shown to be semiprime, semisimple, and strongly semisimplic.
91 citations
TL;DR: In this article, the factorial state space of a C∗-algebra A is shown to be w ∗-compact if and only if A is unital, liminal and has Hausdorff primitive ideal space.
58 citations
TL;DR: A noncommutative generalization of the central limit theorem for even completely positive mappings between two CCR-algebras is proved in this paper, and quasi-free completely-positive mappings are found to be the generalizations of the gaussian distributed random variables.
35 citations
20 citations
TL;DR: In this article, the authors make a distinction between bounded and unbounded operators in quantum mechanics, aided by knowledge from the field of the theory of operators in Hilbert space, and prove the self-adjointness of the momentum operator in generalized coordinates.
Abstract: The aim of this paper is to contribute to the clarification of concepts usually found in books on quantum mechanics, aided by knowledge from the field of the theory of operators in Hilbert space. Frequently the basic distinction between bounded and unbounded operators is not established in books on quantum mechanics. It is repeatedly overlooked that the condition for an unbounded operator to be symmetric (Hermitian) is not sufficient to make it self-adjoint. To make things worse, nearly all operators in quantum mechanics are unbounded. Often one finds statements such as: For any linear operator A we can write a Hermitian operator HA=(A+A+)/2, where Hermitian is thought to mean self-adjoint. Along these lines, self-adjointness of the momentum operator in generalized coordinates, taken from that expression, is questioned. In particular, the redescription in terms of spherical polar coordinates and its implications for the eventual loss of self-adjointness of the momenta conjugate to them are studied.
16 citations
TL;DR: In this article, a dilation theory for a class of contraction operators acting on a separable, infinite dimensional, complex Hilbert space was developed, in which the algebra of bounded linear operators on ~ is denoted by Y ( ~ ).
Abstract: 1. This note is a continuation of our earlier paper [-3], in which we developed a dilation theory for a certain class of contraction operators acting on a separable, infinite dimensional, complex Hilbert space ~ . The notation and terminology in what follows is taken from [3]. For the convenience of the reader we recall a few pertinent definitions. The algebra of bounded linear operators on ~ is denoted by Y ( ~ ) . If T~Se(2C), the ultraweakly closed algebra generated by T and l~e is denoted by dr; we recall that d r can be identified with the dual space of the quotient space Q r = ( z c ) / • where (zc) denotes the ideal of trace-class operators in 5~(24 ~) and • is the preannihilator of d r in (z c), under the pairing
15 citations
TL;DR: A survey of the recent work on the infinitesimal generators of one-parameter semigroups of positivity preserving maps on operator algebras, in the presence of compact symmetry groups or flows can be found in this paper.
Abstract: A survey of the recent work on the infinitesimal generators of one-parameter semigroups of positivity preserving maps on operator algebras, in the presence of compact symmetry groups or flows.
TL;DR: In this paper, the concept of asymptotic operators is introduced and their mathematical properties in the weak, the uniform and the strong operator topologies are studied in a series of theorems, lemmas and corollaries.
Abstract: The asymptotic behaviour of quantum mechanical states at large times has recently been discussed by Wan and McLean. This paper deals with the corresponding behaviour of quantum mechanical operators. The concept of asymptotic operators is introduced and their mathematical properties in the weak, the uniform and the strong operator topologies studied. Results are presented in a series of theorems, lemmas and corollaries.
TL;DR: In this article, the problem of characterising those quantum logics which can be identified with the lattice of projections in a JBW-algebra or a von Neumann algebra is considered.
Abstract: The problem of characterising those quantum logics which can be identified with the lattice of projections in a JBW-algebra or a von Neumann algebra is considered. For quantum logics which satisfy the countable chain condition and which have no TypeI2 part, a characterisation in terms of geometric properties of the quantum state space is given.
TL;DR: In this article, the authors considered the quantum systems of interacting Bose particles confined to a bounded region Λ of the configuration spaces ℝv and obtained bounds on exponentials of local number operators for any temperature and activity.
Abstract: We consider the quantum systems of interacting Bose particles confined to a bounded region Λ of the configuration spaces ℝv. For a class of superstable interactions we obtain bounds on exponentials of local number operators for any temperature and activity. The method we use is the Wiener integral formalism in statistical mechanics. As a consequence any thermodynamic limit states are entire analytic and locally normal in the CCR algebra. In some cases these are modular states.
TL;DR: The theory of operator algebra and von Neumann algebras on complex Hilbert spaces is of increasing importance to many branches of mathematics, for example, integration theory, operator theory, algebraic topology, and particularly mathematical physics and quantum mechanics.
Abstract: Publisher Summary The theory of operator algebras, that is, C*-algebras and von Neumann algebras on complex Hilbert spaces is of increasing importance to many branches of mathematics, for example, integration theory, operator theory, algebraic topology, and particularly mathematical physics and quantum mechanics. Because C*-algebras provide a natural framework for the foundations of quantum mechanics and quantum field theory, it is an important problem to characterize the class of C*-algebras by certain properties, for instance, motivated by physical experiments. Two characterizations of operator algebras in different categories exist: (1) A. Connes' characterization of von Neumann algebras in terms of self-dual homogeneous Hilbert cones and (2) the work of Alfsen and Shultz characterizing the state spaces of C*-algebras using the geometry of compact convex sets and their affine function spaces.
TL;DR: In this paper, a reduction formula for the causal multipoint Green function is proposed for real-time finite-temperature quantum field theory. But the method is applicable to the case in which the field operators form some algebra.
01 Jan 1984
TL;DR: In this article, the topological structure of the collection of all invertible elements of certain operator algebras is investigated, and it is shown that the invertable elements of /A are in the principal component of the identity when assigned the uniform topology.
Abstract: The topological structure of the collection of all invertible elements of certain operator algebras is investigated. The first such algebra A will be a nest subalgebra of a von Neumann algebra. It is shown that, the collection of invertible elements of A with inverse also in A, satisfying certain boundary conditions, are in the principal component of the identity when assigned the strong operator topology. The second /A is a subalgebra of A. When assigned the uniform topology, it is shown that the invertible elements of /A are in the principal component of the identity. The results are applied to a large variety of examples where /A is shown to be extensive.
01 Jan 1984
TL;DR: In this article, a symmetry and reality condition is imposed on non-linear operators T from % to Ψ defined on a dense subspace <3) in % with range in W. They are generally unbounded and have different extensions f defined on subspaces Φ in % containing ty.
Abstract: Let % be a separable complex oo-dimensional Hubert space and let ? be the Fock space of symmetric tensors over %. We consider non-linear operators T from % to Ψ defined on a dense subspace <3) in % with range in W. A symmetry and reality condition is imposed on the operators T under consideration. They are generally unbounded and have different extensions f defined on subspaces Φ in % containing ty. Generalizing a result of Arveson for bounded operators (alias functions from % to ίF), we show that if T is affiliated with a maximal abelian von Neumann algebra in B(%)9 then it follows that there is an extension f of T which is unitarily equivalent to a (non-linear) multiplication operator.
TL;DR: In this paper, it was shown that a symmetric operator commuting with a conjugation has a self-adjoint extension which also commutes with J. This is an analogue of the von Neumann's theorem which states that a semidefinite generator of a continuous action of R on C(H) which extends δ and anti-commutes with α has finite spatial deficiency-indices.
TL;DR: In this paper, a representation of a quantum logic is obtained by means of projection operators on the state space, and geometrical conditions are imposed on a cone in an abstract Banach space which allow us to show that certain projections leaving this cone invariant will form a QL with conditioning.
Abstract: After obtaining a representation of a quantum logic by means of projection operators on the state space, geometrical conditions are imposed on a cone in an abstract Banach space which allow us to show that certain projections leaving this cone invariant will form a quantum logic with conditioning. Several examples are also presented.
TL;DR: In this article, it was shown that the representability problem is one of finding positive extensions to positive linear functionals, a problem that has been addressed by M. G. Krein in the 1930s.
Abstract: The notion of reduced density operators is generalized and placed in the context of abstract operator algebras. In this setting it is shown that the representability problem is one of finding positive extensions to positive linear functionals, a problem that has been addressed by M. G. Krein in the 1930s. An extension of Krein's theorem is given and used in the analysis of the fermion representability problem.
01 Jan 1984
TL;DR: There is a remarkably close relationship between the operator algebra of the Dirac equation and the corresponding operators of the spinorial relativistic rotator (an indecomposable object lying on a mass-spin Regge trajectory) as discussed by the authors.
Abstract: There exists a remarkably close relationship between the operator algebra of the Dirac equation and the corresponding operators of the spinorial relativistic rotator (an indecomposable object lying on a mass-spin Regge trajectory). The analog of the Foldy-Wouthuysen transformation (more generally, the transformation between quasi-Newtonian and Minkowski coordinates) is constructed and explicit results are discussed for the spin and position operators. Zitterbewegung is shown to exist for a system having only positive energies.
TL;DR: In this article, it was shown that the emergence of translation modes in the quantization of some at least nonlinear field theory models implies a specific structure of their state spaces namely this of the direct integral Hilbert space, which follows from the reducibility of the involved quantum field canonical commutation relations algebras.
Abstract: We demonstrate that the emergence of translation modes in the quantization of some at least nonlinear field theory models (like, e.g., φ4 or the sine–Gordon systems) implies a specific structure of their state spaces namely this of the direct integral Hilbert space, which follows from the reducibility of the involved quantum field canonical commutation relations (CCR) algebras. As a special manifestation of this structure, one recovers infinite constituent ‘‘elementary’’ quantum systems living in the commutant of the CCR algebra, which appear as the Schrodinger or the two level ones. The corresponding Hamiltonians are derived. In addition, we propose a modification of the standard infrared Hilbert (photon field) space construction employed in quantum electrodynamics. We demonstrate that, in principle, Fermi (CAR) generators, carrying the spin–charge–momentum labels of Dirac particles, can be defined as operators in the electromagnetic (photon field) Hilbert space. The photon field (CCR) algebra is highly ...