1. bookVolume 9 (2009): Issue 6 (December 2009)
Journal Details
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Journal
eISSN
1335-8871
First Published
07 Mar 2008
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6 times per year
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English
access type Open Access

Measurement, Information Channels, and Discretization: Exploring the Links

Published Online: 23 Dec 2009
Volume & Issue: Volume 9 (2009) - Issue 6 (December 2009)
Page range: 134 - 161
Journal Details
License
Format
Journal
eISSN
1335-8871
First Published
07 Mar 2008
Publication timeframe
6 times per year
Languages
English
Measurement, Information Channels, and Discretization: Exploring the Links

The goal of this paper is to present a unified algebraic-analytic framework for (static and dynamic) deterministic measurement theory, which we find to be fully adequate in engineering and natural science applications. The starting point of this paradigm is the notion of a quantity algebra of a measured system and that of a measuring instrument, underlying the causal linkages in classical ‘system + instrument’ interactions. This approach is then further enriched by providing a superimposed data lattice of measurement outcomes, intended to handle the information flow from the measured system to its measurand's designated instrument.

We argue that the language of Banach and von Neumann algebras is ideally suited for the treatment of quantities, encountered in theoretical and experimental science. These algebras and convex spaces of expectation functionals thereon together with information (co)channels between them provide a comprehensive information-theoretic framework for measurement theory. Concrete examples and applications to length and position measurements are also discussed and rigorously framed within the proposed quantity algebra and associated information channel paradigms.

In modeling physical systems, investigators routinely rely on the assumption that state spaces and time domains form a continuum (locally homeomorphic to the real line or its Cartesian powers). But in sharp contrast, measurement and prediction outcomes pertaining to physical systems under consideration tend to be presented in terms of small discrete sets of rational numbers. We investigate this conceptual gap between theoretical and finitary data models from the perspectives of temporal, spatial and algebraic discretization schemes.

The principal innovation in our approach to classical measurement theory is the representation of interactive instrument-based measurement processes in terms of channel-cochannel pairs constructed between dynamical quantity algebras of a target system and its measurand's measuring instrument.

Keywords

Aerts, D., Daubechies, I. (1978). Physical justification for using the tensor product to describe two quantum systems as one joint system. Helvetica Physica Acta, 51, 661-675.Search in Google Scholar

Batitsky, V., Domotor, Z. (2007). When good theories make bad predictions. Synthese, 157, 79-103.10.1007/s11229-006-9033-0Search in Google Scholar

Benatti, F., Cappellini, V. (2005). Continuous limit of discrete sawtooth maps and its algebraic framework. Journal of Mathematical Physics, 46, 1-25.10.1063/1.1917283Search in Google Scholar

Domotor, Z., Batitsky, V. (2008). The analytic versus representational theory of measurement: A philosophy of science perspective. Measurement Science Review, 8 (6), 129-146.Search in Google Scholar

Domotor, Z., Batitsky, V. (2009). An algebraic-analytic framework for measurement theory. Under review for publication in Measurement.Search in Google Scholar

Gelfand, I. M. (1939). On normed rings. Doklady Akad. Nauk U.S.S.R. 23, 430-432.Search in Google Scholar

Kadison, R. V., Ringrose, J. R. (1983). Fundamentals of the Theory of Operator Algebras, Vol. II. Academic Press, San Diego.Search in Google Scholar

Lasota, A., Mackey, M. C. (1994). Chaos, Fractals, and Noise. Stochastic Aspects of Dynamics, 2nd edition. Springer-Verlag, New York.10.1007/978-1-4612-4286-4Search in Google Scholar

Lima, F. M. S., Arun, P. (2006). An accurate formula for the period of a simple pendulum oscillating beyond the small angle regime. American Journal of Physics, 74, 892-895.10.1119/1.2215616Search in Google Scholar

Lindblad, G. (1996). On the existence of quantum subdynamics. Journal of Physics A: Math. Gen., 29, 4197-4207.10.1088/0305-4470/29/14/037Search in Google Scholar

Nassopoulos, G. F. (1999). On a comparison of real with complex involutive complete algebras. Journal of Mathematical Sciences 36, 3755-3765.10.1007/BF02172669Search in Google Scholar

Sakai, S. (1998). C*-Algebras and W*-Algebras. Springer, New York.10.1007/978-3-642-61993-9Search in Google Scholar

Umegaki, H. (1969). Representations and extremal properties of averaging operators, and their applications to information channels. Journal of Mathematical Analysis and Applications, 25, 41-73.10.1016/0022-247X(69)90212-1Search in Google Scholar

Walters, P. (1981). Introduction to Ergodic Theory. Springer, Tokyo.Search in Google Scholar

Yosida, Y. (1980). Functional Analysis, 6th edition. Springer-Verlag, New York.Search in Google Scholar

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