The Annals of Statistics

Extended statistical modeling under symmetry; the link toward quantum mechanics

Inge S. Helland

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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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Ann. Statist., Volume 34, Number 1 (2006), 42-77.

First available in Project Euclid: 2 May 2006

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Zentralblatt MATH identifier

Primary: 62A01: Foundations and philosophical topics
Secondary: 81P10: Logical foundations of quantum mechanics; quantum logic [See also 03G12, 06C15] 62B15: Theory of statistical experiments

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


Helland, Inge S. Extended statistical modeling under symmetry; the link toward quantum mechanics. Ann. Statist. 34 (2006), no. 1, 42--77. doi:10.1214/009053605000000868.

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