Extended statistical modeling under symmetry; the link toward quantum mechanics

The Annals of Statistics · 2006 · Volume 34 · Issue 1

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.

Keywords

62A01, 62B15, 81P10, Born’s formula, complementarity, complete sufficient statistics, Gleason’s theorem, group representation, Hilbert space, model reduction, quantum mechanics, quantum theory, symmetry, Transition probability

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