dorsal/arxiv
View SchemaQuantum Mechanics on a Torus
| Authors | Ron Rubin, Andrew Lesniewski |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9807056 |
| URL | https://arxiv.org/abs/quant-ph/9807056 |
Abstract
We present here a canonical description for quantizing classical maps on a torus. We prove theorems analagous to classical theorems on mixing and ergodicity in terms of a quantum Koopman space $ L^2 (A_\hbar},\tau_\hbar) $ obtained as the completion of the algebra of observables $ A_\hbar $ in the norm induced by the following inner product $(A,B) =\tau_{\hbar}(A^{\dagger}B) $, where $\tau_{\hbar}$ is a linear functional on the algebra analogous to the classical ``integral over phase space.'' We also derive explicit formulas connecting this formulation to the $\theta $-torus decomposition of Bargmann space introduced in ref. \QCITE{cite}{}{KLMR}.
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"date_modified": "2026-03-02T18:02:44.583000Z",
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"abstract": "We present here a canonical description for quantizing classical maps on a\ntorus. We prove theorems analagous to classical theorems on mixing and\nergodicity in terms of a quantum Koopman space $ L^2 (A_\\hbar},\\tau_\\hbar) $\nobtained as the completion of the algebra of observables $ A_\\hbar $ in the\nnorm induced by the following inner product $(A,B) =\\tau_{\\hbar}(A^{\\dagger}B)\n$, where $\\tau_{\\hbar}$ is a linear functional on the algebra analogous to the\nclassical ``integral over phase space.\u0027\u0027 We also derive explicit formulas\nconnecting this formulation to the $\\theta $-torus decomposition of Bargmann\nspace introduced in ref. \\QCITE{cite}{}{KLMR}.",
"arxiv_id": "quant-ph/9807056",
"authors": [
"Ron Rubin",
"Andrew Lesniewski"
],
"categories": [
"quant-ph"
],
"title": "Quantum Mechanics on a Torus",
"url": "https://arxiv.org/abs/quant-ph/9807056"
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