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.
Publié le : 2006-02-14
Classification:
Born’s formula,
complementarity,
complete sufficient statistics,
Gleason’s theorem,
group representation,
Hilbert space,
model reduction,
quantum mechanics,
quantum theory,
symmetry,
transition probability,
62A01,
81P10,
62B15
@article{1146576255,
author = {Helland, Inge S.},
title = {Extended statistical modeling under symmetry; the link toward quantum mechanics},
journal = {Ann. Statist.},
volume = {34},
number = {1},
year = {2006},
pages = { 42-77},
language = {en},
url = {http://dml.mathdoc.fr/item/1146576255}
}
Helland, Inge S. Extended statistical modeling under symmetry; the link toward quantum mechanics. Ann. Statist., Tome 34 (2006) no. 1, pp. 42-77. http://gdmltest.u-ga.fr/item/1146576255/