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arXiv:1112.2329

To the Many-Hilbert-Space Theory of Quantum Measurements

Zameddin I. İsmailov, Elif Otkun Çevik

Published Dec 11, 2011• 0 citations
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Abstract

In this work, a connection between some spectral properties of direct integral of operators in the direct integral of Hilbert spaces and their coordinate operators has been investigated.
DOI
10.48550/arxiv.1112.2329

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In the Hilbert space formulation of quantum mechanics, ideal measurements of physical variables are discussed using the spectral theory of Hermitian operators and the corresponding projector-valued measures (PVMs). However, more general types of measurements require the treatment in terms of positive-operator-valued measures (POVMs). In the Hamiltonian formulation of quantum mechanics, canonical coordinates are related to PVM. In this paper the results of an analysis of various aspects of applications of POVMs in the Hamiltonian formulation are reported. Several properties of state parameters and quantum observables given by POVMs or represented in an overcomplete basis, including the general Hamiltonian treatment of the Neumark extension, are presented. An analysis of the phase operator, given by the corresponding POVMs, in the Hilbert space and the Hamiltonian frameworks is also given.
In this paper the results from various areas of mathematical research which are necessary for a consistent unification of the Dirac and von Neumann formulations of quantum mechanics are collected and presented as a single synthesis. For this purpose, direct integral decompositions of Hilbert space must be introduced into Dirac's formulation of spectral theory and representation theory; true unit vectors in the direct integral decomposition spaces replace unnormalizable vectors of infinite length. It then becomes clear that families of modified Dirac projection operators are simply related to the Radon-Nikodym derivative of von Neumann spectral measures. In terms of these mathematical preliminaries a second paper will present the more physical aspects of the resulting unified formulation of quantum mechanics.
Abstract We investigate the isomorphism between the formulations of quantum mechanics in a finite-dimensional complex Hilbert space and in a real Kähler space. This investigation establishes a direct correspondence between Hermitian operators in the complex Hilbert space and real operators in the Kähler space. In this real formulation, we note the emergent property of spectral degeneracy. Moreover, we show that the structure of the Kähler space provides a natural description of composite systems in real quantum mechanics.