Observables on hypergraphs

Observables on hypergraphs are described by event-valued measures. We first distinguish between finitely additive observables and countably additive ones. We then study the spectrum, compatibility, and functions of observables. Next a relationship between observables and certain functionals on the set of measures M(H) of a hypergraph H is established. We characterize hypergraphs for which every linear functional on M(H) is determined by an observable. We define the concept of an effect and show that observables are related to effect-valued measures. Finally, we define operational transformations from M(H) to itself and show that they can be described as a certain combination of effects.On leave from University of Berne, Institute of Mathematical Statistics, Sidlerstrasse 5, CH-3012 Berne, Switzerland.     

D-algebras

A D-algebra is a generalization of a D-poset in which a partial order is not assumed. However, if a D-algebra is equipped with a natural partial order, then it becomes a D-poset. It is shown that D-algebras and effect algebras are equivalent algebraic structures. This places the partial operation for a D-algebra on an equal footing with the partial operation for an effect algebra. An axiomatic structure called an effect stale-space is introduced. Such spaces provide an operational interpretation for the partial operations and . Finally, a relationship between effect-state spaces and torsion free interval effect algebras is demonstrated.     

Properties of Quantum Languages

The class of quantum languages Q() over an alphabet is the class of languages accepted by quantum automata. We study properties of Q() and compare Q() with the class of regular languages R(). It is shown that Q() is closed under union, intersection, and reversal but is not closed under complementation, concatenation, or Kleene star. It is also shown that Q() and R() are incomparable. Finally, we prove that L Q() if and only if L admits a transition amplitude function satisfying a certain property and a similar characterization is given for R().     

Convex structures and operational quantum mechanics

A general mathematical framework called a convex structure is introduced. This framework generalizes the usual concept of a convex set in a real linear space. A metric is constructed on a convex structure and it is shown that mappings which preserve the structure are contractions. Convex structures which are isomomorphic to convex sets are characterized and for such convex structures it is shown that the metric is induced by a norm and that structure preserving mappings can be extended to bounded linear operators.Convex structures are shown to give an axiomatization of the states of a physical system and the metric is physically motivated. We demonstrate how convex structures give a generalizing and unifying formalism for convex set and operational methods in axiomatic quantum mechanics.     

Quantum Computers

This paper first considers sequential quantum machines (SQMs). The SQMs thatpossess an isometric transition operator and the SQMs that are factorizable orstrongly factorizable are characterized. Quantum Turing machines (QTMs) arestudied next and an alternative proof of the result that characterizes the unitaryevolution of a QTM is given. It is shown that any QTM can be represented interms of two quantum printers which are much simpler than a QTM. UnidirectionalQTMs are studied and it is shown that their corresponding quantum printers areclosely related to each other. A simple method for constructing unidirectionalQTMs is given. Finally, a preliminary development of generalized QTMs andquantum pushdown automata is presented.     

Observables, Calibration, and Effect Algebras

We introduce and study the D-model, which reflects the simplest situation in which one wants to calibrate an observable. We discuss the question of representing the statistics of the D-model in the context of an effect algebra.     

Realistic spin

We present a realistic model in which spin measurements are represented by functions. By employing a simple amplitude density, we derive the usual spin distributions and matrices for the spin-1/2 case. The spin-1 case is also considered. Moreover, we derive the amplitude density itself from deeper principles involving a real-valued influence function.     

Toward a rigorous quantum field theory

This paper outlines a framework that may provide a mathematically rigorous quantum field theory. The framework relies upon the methods of nonstandard analysis. A theory of nonstandard inner product spaces and operators on these spaces is first developed. This theory is then applied to construct nonstandard Fock spaces which extend the standard Fock spaces. Then a rigorous framework for the field operators of quantum field theory is presented. The results are illustrated for the case of Klein-Gordon fields.