The Representation Theory of Decoherence Functionals in History Quantum Theories

In the first part of this paper the general perspective of history quantum theoriesis reviewed. History quantum theories provide a conceptual and mathematicalframework for formulating quantum theories without a globally definedHamiltonian time evolution and for introducing the concept of space-time eventinto quantum theory. On a mathematical level a history quantum theory ischaracterized by the space of histories, which represent the space-time events, andby the space of decoherence functionals, which represent the quantum mechanicalstates in the history approach. The second part of this paper is devoted to thestudy of the structure of the space of decoherence functionals for some physicallyreasonable spaces of histories in some detail. The temporal reformulation ofstandard Hamiltonian quantum theories suggests to consider the case that thespace of histories is given by (i) the lattice of projection operators on someHilbert space or, slightly more generally, (ii) the set of projection operators insome von Neumann algebra. In the case (i) the conditions are identified underwhich decoherence functionals can be represented by, respectively, trace classoperators, bounded operators, or families of trace class operators on the tensorproduct of the underlying Hilbert space by itself. Moreover, we discuss thenaturally arising representations of decoherence functionals as sesquilinear forms.The paper ends with a discussion of the consequences of the results for thegeneral axiomatic framework of history theories.     

Realization of the three-dimensional quantum Euclidean space by differential operators

The three-dimensional quantum Euclidean space is an example of a non-commutative space that is obtained from Euclidean space by q-deformation. Simultaneously, angular momentum is deformed to , it acts on the q-Euclidean space that becomes a -module algebra this way. In this paper it is shown, that this algebra can be realized by differential operators acting on functions on . On a factorspace of a scalar product can be defined that leads to a Hilbert space, such that the action of the differential operators is defined on a dense set in this Hilbert space and algebraically self-adjoint becomes self-adjoint for the linear operator in the Hilbert space. The self-adjoint coordinates have discrete eigenvalues, the spectrum can be considered as a q-lattice. Received: 27 June 2000 / Published online: 9 August 2000     

On the structure of the Galilean and translationally invariant operator algebra describing intrinsic properties of finite nuclear systems

The structure of the Galilean and translationally invariant operator algebra for finite systems of fermions is investigated. After performing the decomposition of the Fock space into Hilbert spaces for the center-of-mass motion and the intrinsic motion, “intrinsic” field operators are defined and their commutation relations established. These relations deviate in a certain particle number-dependent way from the usual fermion relations. It is shown that the operators corresponding to the intrinsic (e.g. nuclear) observables can be represented in the familiar way, the usual field operators being replaced by the intrinsic ones. In this theory the normal shell model calculations appear as the approximation performed by treating matrix elements of nuclear observables as if the intrinsic field operators were satisfying the exact Fermi commutation relations.     

NON-LINEAR AEROELASTIC ANALYSIS USING THE POINT TRANSFORMATION METHOD, PART 2: HYSTERESIS MODEL

This paper investigates the dynamic response of a two-dimensional aeroelastic system with structural non-linearity represented by hysteresis. The formulations of the point transformation method developed in Part 1 of this study for the aeroelastic system with a freeplay model is extended for a hysteresis model. These formulations can be applied not only to predict the amplitude and frequency of limit cycle oscillations, but also to detect complex aeroelastic responses such as periodic motion with harmonics, period doubling, chaotic motion and the coexistence of stable limit cycles. It is shown that the point transformation technique is the most suitable to analyze the aeroelastic response of systems containing piecewise continuous restoring forces.     

Joint distributions of observables on quantum logics

A joint distribution of a set of observables on a quantum logic in a statem is defined and its properties are derived. It is shown that if the joint distribution exists, then the observables can be represented in the statem by a set of commuting operators on a Hilbert space.     

On the algebra of test functions for field operators

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.     

The field algebra and its positive linear functionals

We show that positive linear functionals on the field algebra are necessarily continuous and can be represented by conical measures. Furthermore extension theorems for continuous linear functionals, defined on a subspace of the field algebra, to positive linear functionals are discussed.Supported in part by National Science Foundation, GP 19479, and a Summer Research Initiation Fellowship from the University of Colorado.     

Continuity of the von Neumann Entropy

A general method for proving continuity of the von Neumann entropy on subsets of positive trace-class operators is considered. This makes it possible to re-derive the known conditions for continuity of the entropy in more general forms and to obtain several new conditions. The method is based on a particular approximation of the von Neumann entropy by an increasing sequence of concave continuous unitary invariant functions defined using decompositions into finite rank operators. The existence of this approximation is a corollary of a general property of the set of quantum states as a convex topological space called the strong stability property. This is considered in the first part of the paper.     

On the algebra of symmetries of Laplace and Dirac operators

We consider a generalization of the classical Laplace operator, which includes the Laplace–Dunkl operator defined in terms of the differential-difference operators associated with finite reflection groups called Dunkl operators. For this Laplace-like operator, we determine a set of symmetries commuting with it, in the form of generalized angular momentum operators, and we present the algebraic relations for the symmetry algebra. In this context, the generalized Dirac operator is then defined as a square root of our Laplace-like operator. We explicitly determine a family of graded operators which commute or anticommute with our Dirac-like operator depending on their degree. The algebra generated by these symmetry operators is shown to be a generalization of the standard angular momentum algebra and the recently defined higher-rank Bannai–Ito algebra.     

Necessary and sufficient conditions for integral representations of Wightman functionals at Schwinger points

Given a set of Wightman functions one would like to associate to it a field on Euclidean space admitting a simultaneous diagonalization. We investigate when this can be done in such a way that the Schwinger functions are the expectation values of this commutative field with a bounded metric operator commuting with the field. This requires as a tool the characterization of those linear functionals on the symmetric tensor algebra over a space of test functions which can be represented by complex measures on the corresponding space of distributions.     

Hilbert space sectors for solutions of non-linear relativistic field equations

In the set of Cauchy data corresponding to the solutions of non-linear classical relativistic field equations having locally finite kinetic energy a structure of Hilbert space sectors is introduced. Each sector is invariant under time evolution and a total energy and linear momentum functionals can be defined as global quantities. Within this framework the existence of conserved dynamical charges is proved and the mechanism by which a symmetry can be spontaneously broken is explained.Partially supported by C.N.R. (gruppo G.N.A.F.A.)     

On gauge invariant bosonic strings

Following Witten, an associative product is constructed for fields on open strings, with BRST ghosts, which has the same topological structure as that in the light-cone gauge. The product is shown to be well defined on a certain class of string functionals. The corresponding action is that of Chern-Simons. The relation of this product to that of others is discussed.     

Method for construction of operators in Fock space

It is shown, by providing a general method for the construction that any Fock space linear operator defined on the dense linear manifold spanned by the particle number representation basis can be represented in terms of the annihilation and creation operators. The normal form of the representation is unique.     

Global attractors and invariant measures for non-invertible planar piecewise isometric maps

The authors investigate dynamical behaviors of discrete systems defined by iterating non-invertible planar piecewise isometries, which are piecewisely defined maps that preserve Euclidean distance. After discussing subtleties for these kind of dynamical systems, they have characterized global attractors via invariant measures and via positive continuous functions on phase space. The main result of this Letter is that a compact set A is the global attractor for a piecewise isometry if and only if the Lebesgue measure restricted to A is invariant, while it is not invariant restricted to any measurable set B which contains A and whose Lebesgue measure is strictly larger than that of A.     

Degeneracy in loop variables

The small algebra of loop functionals, defined by Rovelli and Smolin, on the Ashtekar phase space of general relativity is studied. Regarded as coordinates on the phase space, the loop functionals become degenerate at certain points. All the degenerate points are found and the corresponding degeneracy is discussed. The intersection of the set of degenerate points with the real slice of the constraint surface is shown to correspond precisely the Goldberg-Kerr solutions. The evolution of the holonomy group of Ashtekar's connection is examined, and the complexification of the holonomy group is shown to be preserved under it. Thus, an observable of the gravitational field is constructed.     

Gauge-independent quantization of the schrödinger and radiation fields

The quantization of several Schrödinger fields interacting with the electromagnetic field is carried out without reference to a particular gauge. The canonical formalism requires a modification introduced by Dirac and Bergmann for constraints. The Coulomb interaction is separated from the radiation and it gives rise to bound states of atoms and molecules. Particle operators are represented in the usual manner in Fock space, while the radiation field can be described by state functionals. Constraints can be included in the canonical formalism by Lagrange multipliers, leading to results equivalent to those of Dirac and Bergmann.     

Existence and Uniqueness of the Solution of the Non-stationary Boltzmann-Equation for the Electrons in a Collision Dominated Plasma by Means of Operator Semigroups

Based on the semigroup approach a new proof is presented of the existence of a unique solution of the non-stationary Boltzmann-equation for the electron component of a collision dominated plasma. All interactions can be included which yield bounded collision operators. The electric and magnetic fields were permitted to be inhomogeneous in space, and the investigations were performed for a bounded plasma. It has been shown that the Boltzmann-operator is the infinitesimal generator of a strongly continuous operator-semigroup which uniquely determines the nonnegative solution from a given initial function taking into account the given boundary conditions.     

A Wigner-function formulation of finite-state quantum mechanics

For a non-relativistic system with only continous degrees of freedom (no spin, for example), the original Wigner function can be used as an alternative to the density matrix to represent an arbitrary quantum state. Indeed, the quantum mechanics of such systems can be formulated entirely in terms of the Wigner function and other functions on phase space, with no mention of state vectors or operators. In the present paper this Wigner-function formulation is extended to systems having only a finite number of orthogonal states. The “phase space” for such a system is taken to be not continuous but discrete. In the simplest cases it can be pictured as an N×N array of points, where N is the number of orthogonal states. The Wigner function is a real function on this phase space, defined so that its properties are closely analogous to those of the original Wigner function. In this formulation, observables, like states, are represented by real functions on the discrete phase space. The complex numbers still play an important role: they appear in an essential way in the rule for forming composite systems.     

Quantum Linear Gravity in de Sitter Universe on Gupta-Bleuler Vacuum State

Application of Krein space quantization to the linear gravity in de Sitter space-time have constructed on Gupta-Bleuler vacuum state, resulting in removal of infrared divergence and preserving de Sitter covariant. By pursuing this path, the non uniqueness of vacuum expectation value of the product of field operators in curved space-time disappears as well. Then the vacuum expectation value of the product of field operators can be defined properly and uniquely.