Extended statistical modeling under symmetry; the link toward quantum mechanics

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 $\mathbf{H}$. 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.