dorsal/arxiv
View SchemaA Canonical Quantization of the Baker's Map
| Authors | Ron Rubin, Nathan Salwen |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9807045 |
| URL | https://arxiv.org/abs/quant-ph/9807045 |
| DOI | 10.1006/aphy.1998.5845 |
| Journal | Annals Phys. 269 (1998) 159-181 |
Abstract
We present here a canonical quantization for the baker's map. The method we use is quite different from that used in Balazs and Voros (ref. \QCITE{cite}{}{BV}) and Saraceno (ref. \QCITE{cite}{}{S}). We first construct a natural ``baker covering map'' on the plane $\QTO{mathbb}{\mathbb{R}}^{2}$. We then use as the quantum algebra of observables the subalgebra of operators on $L^{2}(\QTO{mathbb}{\mathbb{R}}) $ generated by $\left\{\exp (2\pi i\hat{x}) ,\exp (2\pi i\hat{p}) \right\} $ . We construct a unitary propagator such that as $\hbar \to 0$ the classical dynamics is returned. For Planck's constant $h=1/N$, we show that the dynamics can be reduced to the dynamics on an $N$-dimensional Hilbert space, and the unitary $N\times N$ matrix propagator is the same as given in ref. \QCITE{cite}{}{BV} except for a small correction of order $h$. This correction is shown to preserve the classical symmetry $x\to 1-x$ and $p\to 1-p$ in the quantum dynamics for periodic boundary conditions.
{
"annotation_id": "fc18206f-9ff9-490a-a64d-809dcf464f4c",
"date_created": "2026-03-02T18:02:44.351000Z",
"date_modified": "2026-03-02T18:02:44.351000Z",
"file_hash": "739bcf43297d8f3a727cc6d417df5db4d247777ce60e37e21f1ef986404913d1",
"private": false,
"record": {
"abstract": "We present here a canonical quantization for the baker\u0027s map. The method we\nuse is quite different from that used in Balazs and Voros (ref.\n\\QCITE{cite}{}{BV}) and Saraceno (ref. \\QCITE{cite}{}{S}). We first construct a\nnatural ``baker covering map\u0027\u0027 on the plane $\\QTO{mathbb}{\\mathbb{R}}^{2}$. We\nthen use as the quantum algebra of observables the subalgebra of operators on\n$L^{2}(\\QTO{mathbb}{\\mathbb{R}}) $ generated by $\\left\\{\\exp (2\\pi i\\hat{x})\n,\\exp (2\\pi i\\hat{p}) \\right\\} $ . We construct a unitary propagator such that\nas $\\hbar \\to 0$ the classical dynamics is returned. For Planck\u0027s constant\n$h=1/N$, we show that the dynamics can be reduced to the dynamics on an\n$N$-dimensional Hilbert space, and the unitary $N\\times N$ matrix propagator is\nthe same as given in ref. \\QCITE{cite}{}{BV} except for a small correction of\norder $h$. This correction is shown to preserve the classical symmetry $x\\to\n1-x$ and $p\\to 1-p$ in the quantum dynamics for periodic boundary conditions.",
"arxiv_id": "quant-ph/9807045",
"authors": [
"Ron Rubin",
"Nathan Salwen"
],
"categories": [
"quant-ph"
],
"doi": "10.1006/aphy.1998.5845",
"journal_ref": "Annals Phys. 269 (1998) 159-181",
"title": "A Canonical Quantization of the Baker\u0027s Map",
"url": "https://arxiv.org/abs/quant-ph/9807045"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "ab94fe98-a7b5-4c45-bf1d-3bda6e738be2",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}