The Annals of Statistics

Extended statistical modeling under symmetry; the link toward quantum mechanics

Inge S. Helland

Full-text: Open access

Abstract

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.

Article information

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

Dates
First available in Project Euclid: 2 May 2006

Permanent link to this document
https://projecteuclid.org/euclid.aos/1146576255

Digital Object Identifier
doi:10.1214/009053605000000868

Mathematical Reviews number (MathSciNet)
MR2275234

Zentralblatt MATH identifier
1091.62002

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

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

Citation

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. https://projecteuclid.org/euclid.aos/1146576255


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