Showing papers on "Operator algebra published in 1973"

On the Algebra of Test Functions for Field Operators Decomposition of Linear Functionals into Positive Ones

Abstract: It is shown that every continuous linear functional on the field algebra can be defined by a vector in the Hilbert space of some representation of the algebra. The functionals which can be written as a difference of positive ones are characterized. By an example it is shown that a positive functional on a subalgebra does not always have an extension to a positive functional on the whole algebra.

A linear algebra setting for the rota-mullin theory of polynomials of binomial type

Abstract: The linear algebra and combinatorial aspects of the Rota-Mullin theory of polynomials of binomial type are separated and the former is developed in terms of shift operators on infinite dimensional vector spaces with a view towards application in the calculus of finite differences.

Hypercontractivity for fermions

Abstract: If T is a fermion one‐particle operator, with ‖ T ‖ 2, where C is the fermion von Neumann algebra. The proof is an adaptation to the fermion case of the corresponding proof for bosons given by Nelson. This leads to a generalization of a theorem of Gross, inasmuch as T is not required to be self‐adjoint.

Position operators in relativistic quantum theory: Analysis of algebraic operator relationships

Abstract: This work is the first of a series of three papers examining different aspects of position operators in relativistic quantum theory. In this paper the properties of the position operator X and the velocity operator V are derived for single particle matrix elements in the context of the Poincare generator algebra. Both the physical meaning and the mathematical implications of each property are discussed. The algebraic structure of the extended set of relationships including the Poincare generators, X and V is examined. It is found that this set defines an infinite algebra which is intractable mathematically. The Casimir operators of the Poincare algebra are required to be Casimir operators for X and V, a new condition on V is formulated, and a simple solution for K is constructed. These conditions, together with familiar position operator properties, give the constraints and solutions for the extended algebra.

An algebra of the Yang‐Mills field

Abstract: It is shown how the classical Yang‐Mills (and the Maxwell) field equations can be written in a simple form by introducing differential operators based on the split‐Cayley algebra. As a result, a field algebra accommodating both space‐time symmetry (Lorentz invariance) and internal symmetry evolves. The connection between the view of the Yang‐Mills field as a split‐Cayley algebra and the exceptional Jordan angebra M83, discovered by Jordon, von Neumann, and Wigner, is discussed.

Remarks on reductive operator algebras

Abstract: It is shown that a weakly closed operator algebra with the property that each of its invariant subspaces is reducing and which is either strictly cyclic or has only closed invariant linear manifolds, must be a von Neumann algebra.

Subnormal operators in strictly cyclic operator algebras.

Abstract: to be normal, and, moreover, its spectrum is afinite set. Thus, the commutant of a subnormal operator cannot bestrictly cyclic and separated unless the underlying Hubert space isfinite-dimensional (since the commutant is then abelian and hence theoperator, which is normal, must have simple spectrum). More generally,it is shown that a uniformly closed subalgebra

The class sum operator approach for the point group O and D4

Abstract: The class sum operator approach to the representation theory of the point groups O and D4 is described and illustrated by means of several examples. Modified character tables are given for both groups, together with the class multiplication table for O. The construction of tensor operators within the group algebra of each group is discussed, using a modified version of traditional character analysis, and it is found that no E type tensor operator appears in the D4 group algebra.

Analytic continuation of Moyal and operator algebras

Abstract: We consider the analytic continuation of the commutator[qm, pn for values of m,n ≠ positive integers. This is carried out both in the Moyal algebra and the operator algebra.

Problem of the time operator and collision duration

Abstract: The notion of collision duration and properties of the symmetrical time operator in the nonrelativistic quantum mechanics are investigated. Using the theory of genenalized extensions of symmetrical operators, a conclusion is made: not only self-adjoint but also maximal symmetrical operators can correspond to the observables in the quantum mechanics. (auth)

Orthogonal Operators and Phase Space Distributions in Quantum Optics

Abstract: Electromagnetic fields at optical frequencies are excited by indeterministic sources. Their statistical description is usually given by means of the density operator ρ or a phase-space distribution which is, in general, a quasi-probability distribution. Expansions of the density matrix ρ in terms of a given complete set of orthogonal operators establishes a one-to-one correspondence between the expanded operator ρ and the quasi-probability phase-space distribution that appears in said expansion. The time evolution of the field’s statistics are obtained then as a solution of the equation of motion of the density operator ρ or the differential equation of the corresponding phase-space distribution. The latter method has the advantage of being unburdened by problems of commutativity associated with the ρ operator. With the statistical information vested in the phase-space distribution, in lieu of the density matrix ρ the expected value of an observable F is given by an integral of the phase-space distribution multiplied by a weighting function which is representative of the observable F. This is essentially the method first introduced by Wigner[l] and Moyal[2].