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Zeitschrift für Analysis und ihre Anwendungen

A Theory of Quantum Measurement Based on the CCR Algebra L$^+(\mathcal W)$

Daniel A. Dubin, Juan Sotelo-Campos

Published Feb 28, 1986• 4 citations
Publisher

Abstract

Starting from L ^+(\mathcal W) as the algebra of observables, \mathcal W a space of type \mathcal S , a theory of quantum measurement is devised. It is based on the theory of Davies and Lewis, but adapted to unbounded symmetric operators defined on a common dense domain and no restriction on the spectra.
DOI
10.4171/zaa/176

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A Theory of Quantum Measurement Based on the CCR Algebra L+(W)^+(\mathcal W)+(W)
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.
This research work presents an important Banach space in functional analysis which is known and calledHilbert space. We verified the crucial operations in this space and their applications in physics, particularlyin quantum mechanics. The operations are restricted to the unbounded linear operators densely definedin Hilbert space which is the case of prime interest in physics, precisely in quantum machines. Precisely,we discuss the role of unbounded linear operators in quantum mechanics, particularly, in the study ofHeisenberg uncertainty principle, time-independent Schrodinger equation, Harmonic oscillation, andfinally, the application of Hamilton operator. To make these analyses fruitful, the knowledge of Hilbertspaces was first investigated followed by the spectral theory of unbounded operators, which are claimedto be densely defined in Hilbert space. Consequently, the theory of probability is also employed to studysome systems since the operators used in studying these systems are only dense in H (i.e., they must (orprobably) be in the domain of H defined by L2 ( ) −∞,+∞ ).
The algebraic formulation of quantum statistical mechanics is extended so as to include local unbounded observables. We start, in a usual way, with a C*-algebra U of quasilocal bounded observables in Fock space HF, an arbitrary, locally normal state φ on U, and a corresponding Gel'fand-Naimark-Segal (GNS) representation Rφ of U in a Hilbert space Hφ. We then construct a set QL of local, closed operators whose domains are dense in HF, such that QL includes all the local observables of the system. The representation Rφ is then extended so as to provide a *-homomorphism of HL into the closed, densely defined operators in Hφ. Correspondingly, a number of results previously established for the local bounded observables are extended to the unbounded ones. For appropriate classes of locally normal states, these extended results include the Kubo-Martin-Schwinger boundary conditions, the spatially asymptotic and ergodic properties of space-correlation functions, and the temporally ergodic properties of time-correlation functions. It is also shown that, for locally normal Gibbs states, the time correlations between elements of a specified subset of QL are thermodynamical limits of the corresponding correlations for finite systems.