dorsal/arxiv
View SchemaOn the classical limit of the hyperbolic quantum mechanics
| Authors | Andrei Yu. Khrennikov |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0401035 |
| URL | https://arxiv.org/abs/quant-ph/0401035 |
| Journal | Infinite Dimensional Analysis, Quantum Probability and Related Topics 10, 421-438 (2007) |
Abstract
We demonstrated that classical mechanics have, besides the well known quantum deformation, another deformation -- so called hyperbolic quantum mechanics. The classical Poisson bracket can be obtained as the limit $h\to 0$ not only of the ordinary Moyal bracket, but also hyperbolic analogue of the Moyal bracket. Thus there are two different deformations of classical phase-space: complex Hilbert space and hyperbolic Hilbert space (module over a so called hyperbolic algebra -- the two dimensional Clifford algebra). To prove the correspondence principle we use the calculus over the hyperbolic algebra similar to functional superanalysis of Vladimirov-Volovich. Ordinary (complex) and hyperbolic quantum mechanics are characterized by two types of interference perturbation of the classical formula of total probability: ordinary $\cos$-interference and hyperbolic $\cosh$-interference.
{
"annotation_id": "2380d951-26bd-47be-8112-3a54e05ddd67",
"date_created": "2026-03-02T18:02:03.630000Z",
"date_modified": "2026-03-02T18:02:03.630000Z",
"file_hash": "db8e59635027699f631b47df6075f8b1a97f1e5b7a384bdf4704ede4f9416fec",
"private": false,
"record": {
"abstract": "We demonstrated that classical mechanics have, besides the well known quantum\ndeformation, another deformation -- so called hyperbolic quantum mechanics. The\nclassical Poisson bracket can be obtained as the limit $h\\to 0$ not only of the\nordinary Moyal bracket, but also hyperbolic analogue of the Moyal bracket. Thus\nthere are two different deformations of classical phase-space: complex Hilbert\nspace and hyperbolic Hilbert space (module over a so called hyperbolic algebra\n-- the two dimensional Clifford algebra). To prove the correspondence principle\nwe use the calculus over the hyperbolic algebra similar to functional\nsuperanalysis of Vladimirov-Volovich. Ordinary (complex) and hyperbolic quantum\nmechanics are characterized by two types of interference perturbation of the\nclassical formula of total probability: ordinary $\\cos$-interference and\nhyperbolic $\\cosh$-interference.",
"arxiv_id": "quant-ph/0401035",
"authors": [
"Andrei Yu. Khrennikov"
],
"categories": [
"quant-ph"
],
"journal_ref": "Infinite Dimensional Analysis, Quantum Probability and Related\n Topics 10, 421-438 (2007)",
"title": "On the classical limit of the hyperbolic quantum mechanics",
"url": "https://arxiv.org/abs/quant-ph/0401035"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "8863aede-3296-4598-a730-91d6fbfa6c82",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}