Article

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

04/2005; DOI:10.1214/009053605000000868
Source: arXiv

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 $\Pi$ of the parameter spaces of these experiments.
The set of possible parameters is constrained to lie in a subspace of $\Pi$, 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
elementary quantum mechanics in important special cases. The theory is
illustrated by the example of a quantum particle with spin.

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5 Apr 2013

### Keywords

Born's formula

cartesian product $\Pi$

common Hilbert space $\mathbf{H}$

corresponding parameter

elementary quantum mechanics

Gleason's theorem

incompatible experiments

orbits

parameter spaces

parametric Hilbert space

possible parameters

quantum mechanics

quantum particle

special cases

subspace