## The Annals of Statistics

### Extended statistical modeling under symmetry; the link toward quantum mechanics

Inge S. Helland

#### Abstract

We derive essential elements of quantum mechanics from a parametric structure extending that of traditional mathematical statistics. The basic setting is a set 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 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

#### 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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