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February 2006 Extended statistical modeling under symmetry; the link toward quantum mechanics
Inge S. Helland
Ann. Statist. 34(1): 42-77 (February 2006). DOI: 10.1214/009053605000000868


We derive essential elements of quantum mechanics from a parametric structure extending that of traditional mathematical statistics. The basic setting is a set $\mathcal{A}$ of incompatible experiments, and a transformation group G on the cartesian product Π of the parameter spaces of these experiments. The set of possible parameters is constrained to lie in a subspace of Π, an orbit or a set of orbits of G. Each possible model is then connected to a parametric Hilbert space. The spaces of different experiments are linked unitarily, thus defining a common Hilbert space H. A state is equivalent to a question together with an answer: the choice of an experiment $a\in\mathcal{A}$ plus a value for the corresponding parameter. Finally, probabilities are introduced through Born’s formula, which is derived from a recent version of Gleason’s theorem. This then leads to the usual formalism of elementary quantum mechanics in important special cases. The theory is illustrated by the example of a quantum particle with spin.


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Inge S. Helland. "Extended statistical modeling under symmetry; the link toward quantum mechanics." Ann. Statist. 34 (1) 42 - 77, February 2006.


Published: February 2006
First available in Project Euclid: 2 May 2006

zbMATH: 1091.62002
MathSciNet: MR2275234
Digital Object Identifier: 10.1214/009053605000000868

Primary: 62A01
Secondary: 62B15 , 81P10

Keywords: Born’s formula , complementarity , complete sufficient statistics , Gleason’s theorem , group representation , Hilbert space , model reduction , quantum mechanics , quantum theory , symmetry , Transition probability

Rights: Copyright © 2006 Institute of Mathematical Statistics


Vol.34 • No. 1 • February 2006
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