dorsal/arxiv
View SchemaExtended statistical modeling under symmetry; the link toward quantum mechanics
| Authors | Inge S. Helland |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0503214 |
| URL | https://arxiv.org/abs/quant-ph/0503214 |
| DOI | 10.1214/009053605000000868 |
| Journal | Annals of Statistics 2006, Vol. 34, No. 1, 42-77 |
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 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.
{
"annotation_id": "c177a9ea-e5dd-4a51-bdf7-9b20d029aca3",
"date_created": "2026-03-02T18:02:16.354000Z",
"date_modified": "2026-03-02T18:02:16.354000Z",
"file_hash": "5a83a962a4d055c0b596177a4021fe769866d208047e7a165d9738a84a122cdd",
"private": false,
"record": {
"abstract": "We derive essential elements of quantum mechanics from a parametric structure\nextending that of traditional mathematical statistics. The basic setting is a\nset $\\mathcal{A}$ of incompatible experiments, and a transformation group $G$\non the cartesian product $\\Pi$ of the parameter spaces of these experiments.\nThe set of possible parameters is constrained to lie in a subspace of $\\Pi$, an\norbit or a set of orbits of $G$. Each possible model is then connected to a\nparametric Hilbert space. The spaces of different experiments are linked\nunitarily, thus defining a common Hilbert space $\\mathbf{H}$. A state is\nequivalent to a question together with an answer: the choice of an experiment\n$a\\in\\mathcal{A}$ plus a value for the corresponding parameter. Finally,\nprobabilities are introduced through Born\u0027s formula, which is derived from a\nrecent version of Gleason\u0027s theorem. This then leads to the usual formalism of\nelementary quantum mechanics in important special cases. The theory is\nillustrated by the example of a quantum particle with spin.",
"arxiv_id": "quant-ph/0503214",
"authors": [
"Inge S. Helland"
],
"categories": [
"quant-ph"
],
"doi": "10.1214/009053605000000868",
"journal_ref": "Annals of Statistics 2006, Vol. 34, No. 1, 42-77",
"title": "Extended statistical modeling under symmetry; the link toward quantum mechanics",
"url": "https://arxiv.org/abs/quant-ph/0503214"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "ffa8bf73-8a35-49dd-aecd-7730fe73c900",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}