Abstract
In this and the following paper, a new approach for the justification of ensembles in statistical mechanics is given. The essential physical idea is that a measurement is an average of values arising from disjoint regions in three-space. This idea is given a mathematical basis in terms of a class of operators called “local operators,” and the first paper is devoted primarily to the development of the properties of local operators. In particular, a complete characterization of the bounded local operators on ℒ2 spaces of finite measure is given. Two results of importance for statistical mechanics are also derived. First, it is shown that the observables of quantum mechanics are local operators. Second, it is shown that the expectation value of an observable for a pure state can be written formally as an ensemble average. In the following paper, these results are used to develop a new approach for the justification of statistical ensembles.
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This work was supported in part by research grants from the National Science Foundation and the U.S. Public Health Service. The material of this paper is contained in a doctoral dissertation submitted by the author to the University of Oregon (1969).
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Keizer, J.E. A new approach for the justification of ensembles in quantum statistical mechanics — I. J Stat Phys 2, 213–232 (1970). https://doi.org/10.1007/BF01030743
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DOI: https://doi.org/10.1007/BF01030743