Observables in W*‐Algebraic Quantum Mechanics

Abstract

The problem of the conceptual and formal characterization of observables in quantum mechanics is of central importance. The very name 'observable' rests on the view that one deals with observable, i.e. measurable quantities.This view gave way to a more subtle analysis, involving a distinction between observables and their measurement. Observables are those quantities which characterize a system intrinsically. They are not measurable in a naive sense. Often their value is accessible but only indirectly and through extended theoretical reasoning. A careful examination of classical mechanics shows that even there say the position of a particle cannot be determined. Measurements never tell if a particle's site has rational or irrational coordinates, they give merely a n interval for it. I n quantum mechanics the matter is still more intricate.Independently of the difficulties associated with measurements, one can define observables with regard to the kinematical group of a quantum system (e.g. the Galilei group), namely as 'suitably' transforming operators under an action of this group. For example, operators for position and momentum can be characterized by their transformation properties under Galilei symmetries in both classical and quantum mechanics without appealing to the correspondence principle. Outside this particular situation one gets into difficulties, since the word 'suitably' has then no specified meaning.

Between Classical and Quantum

“…This observation is part of a general theory of macroscopic observables in the setting of von Neumann algebras (Primas, 1983;Rieckers, 1984;Amann, 1986Amann, , 1987Morchio & Strocchi, 1987;Bona, 1988Bona, , 1989Unnerstall, 1990aUnnerstall, , 1990bBreuer, 1994;Atmanspacher, Amann, & Müller-Herold, 1999), which complements the purely C * -algebraic approach of Raggio & Werner (1989, Duffield &Werner (1992a,b,c), andDuffield, Roos, &Werner (1992) explained so far. 289 In our opinion, the latter has the advantage that conceptually the passage to the limit N → ∞ (and thereby the idealization of a large system as an infinite one) is very satisfactory, especially in our reformulation in terms of continuous fields of C * -algebras.…”

From quantum to quantum via decoherence

“…In a recent paper [10] such an algebraic framework which enables a general discussion of environmentally induced classical properties in quantum systems has been proposed. It is worth noting that the idea of using the same algebraic description of both quantum and classical mechanics was suggested in [11]. In this approach observables of any physical system are represented by self-adjoint elements of some operator algebra M, the so-called von Neumann algebra, acting in a Hilbert space associated with the system.…”