An application of operator *-algebras to the theory of quantum measurement

Sotelo-Campos, Juan (1987). An application of operator *-algebras to the theory of quantum measurement. PhD thesis The Open University.

DOI: https://doi.org/10.21954/ou.ro.0000f940

Abstract

In this thesis we adapt the Davies and Lewis’ operational approach to quantum probability to the Op*-algebra framework.

In Chapter I we introduce our formalism of quantum mechanics. We define the space of pure states U as a nuclear-Frechet space generated by a self-adjoint operator M whose inverse is nuclear. The algebra of observables is taken to be L+<w>, the adjoint-stable set of linear maps from U into itself. This algebra and its dual under the uniform topology form a dual pair. We derive several algebraic and topological properties of this dual pair and show that it forms a suitable structure for the adaptation of Davies and Lewis’ theory to the Op*-algebra framework.

In Chapter II we set the basis of the Davies and Lewis’ operational approach to quantum probability within the framework developed in chapter I. Here we define the fundamental concepts of an expectation and an instrument and discuss their relationship. We show that instruments are bounded Radon measures and that the compose of two of them is not in general a stable operation on the universe of all instruments. We conclude the chapter discussing the general Robertson-Heisenberg
uncertainty relations.

In Chapter III we build a special class of instruments to measure Q, P and H on S<R>; these are based on Davies’ approximate position measurements. In particular we show that the compose of any two, hence any finite number, of these instruments is an instrument.