Chapter 5 Algebras of Quantum Observables

Vasily Evgenevich Tarasov

Vasily Evgenevich Tarasov

Juan Sotelo-Campos

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+, 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; 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.

Anatolij Dvurečenskij

Shahn Majid

This is the first of two physically-oriented chapters in which we take a break from R-matrices and such topics and explore instead a completely different setting in which Hopf algebras arise naturally. This is the setting of logic, quantum mechanics and probability theory, in all of which contexts the coproduct Δ of the Hopf algebra plays the role of ‘sharing out’ possibilities. The theory has potential applications in computer science as well. In Boolean algebra and quantum mechanics, the product in the algebra corresponds in some sense to deduction of facts (‘putting two and two together’), and supplementing this structure with a coproduct provides a kind of reverse of this process by sharing out possible explanations of a fact. This provides the general theme of the present chapter. The chapter can also be viewed as background material for the next chapter, where we present a number of concrete models of quantum-mechanical Hopf algebras of observables. The simplest example of a combinatorial Hopf algebra is the shuffle algebra associated to an alphabet of symbols. Here the coproduct of a word is the formal sum of all pairs of words which could be shuffled to give the original word . Here, a shuffle of two words takes symbols from each of the words and interleaves them randomly while preserving the order of each word (this is exactly what you do when shuffling a pack of cards).

Keith McFarlane

This thesis is the outcome of research conducted into the quantization of observables defined over a Riemannian configuration manifold, and is naturally divided into three chapters: \n \nThe first chapter is essentially concerned with the development from physical bases, of the concepts of classical and quantum “global measurability”, which, when combined with the requirement that all quantizable momenta shall be either classically or quantum mechanically globally measurable, result in the exclusion form the class of quantizable momenta of all those observables which are not quantizable in accordance with G. W. Mackey’s natural procedure. The refinement of exact global measurability is then introduced and the physically important class of “killing” momenta are found to be exactly classically and quantum mechanically globally measurable and moreover quantizable by means of Mackey’s scheme. \n \nThe second chapter seeks to analyse the algebraic structure of the Mackey-quantizable momenta, so as to compare and contrast his geometric scheme with the various algebraic schemes which have frequently been proposed. As an extension of this work, a natural geometric quantization is proposed for the more general class of observables “linear in momentum”, the set of quantizable such entities determines, and its algebraic properties discussed. It is concluded that, if Mackey’s procedure is, as we argue, correct and exhaustive of the quantizable momenta, then algebraic rules of quantization do not, in general, obtain among the momenta defined over a Riemannian space, ‘though in the case of momenta, such as the Killing momenta, reflecting the symmetry of the configuration manifold, such laws not only exist but are moreover of great practical importance. \n \nThe third chapter is concerned with the quantization of the observables “multilinear in momentum” and in particular with the delimitation of the possible operators of formal quantization as circumscribed by the requirement of operator symmetry, and the question of the essential self-adjointness of local observables associated with the multilinear momenta. Finally some possible means of determining and exact quantization are discussed and contrasted and a tentative selection made of a particular scheme which is then illustrated by concrete example on the real line with the usual metric. \n \nWe conclude this preface with a brief indication of the layout of the thesis: Each chapter or associated group of appendices has been treated for the purpose of all textual numbering and referencing systems, as a separate unit, and is prefixed by a detailed list of contents, and concluded by the appropriate group of reference, so that all pointers of a chapter or group of appendices refer, unless otherwise indicated, to the text of that chapter or group of appendices. Finally note that a bibliography has been included which will serve as a collected list of references to papers and books employed.