dorsal/arxiv
View SchemaFibre bundle formulation of nonrelativistic quantum mechanics. II. Equations of motion and observables
| Authors | Bozhidar Z. Iliev |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9804062 |
| URL | https://arxiv.org/abs/quant-ph/9804062 |
| DOI | 10.1088/0305-4470/34/23/309 |
| Journal | J.Phys.A34:4919-4934,2001 |
Abstract
We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one but it is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it a pure state of some quantum system is described by a state section (along paths) of a (Hilbert) fibre bundle. Its evolution is determined through the bundle (analogue of the) Schr\"odinger equation. Now the dynamical variables and the density operator are described via bundle morphisms (along paths). The mentioned quantities are connected by a number of relations derived in this work. In the second part of this investigation we derive several forms of the bundle (analogue of the) Schr\"odinger equation governing the time evolution of state sections. We prove that up to a constant the matrix-bundle Hamiltonian, entering in the bundle analogue of the matrix form of the conventional Schr\"odinger equation, coincides with the matrix of coefficients of the evolution transport. This allows to interpret the Hamiltonian as a gauge field. Here we also apply the bundle approach to the description of observables. It is shown that to any observable there corresponds a unique Hermitian bundle morphism (along paths) and vice versa.
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"abstract": "We propose a new systematic fibre bundle formulation of nonrelativistic\nquantum mechanics. The new form of the theory is equivalent to the usual one\nbut it is in harmony with the modern trends in theoretical physics and\npotentially admits new generalizations in different directions. In it a pure\nstate of some quantum system is described by a state section (along paths) of a\n(Hilbert) fibre bundle. Its evolution is determined through the bundle\n(analogue of the) Schr\\\"odinger equation. Now the dynamical variables and the\ndensity operator are described via bundle morphisms (along paths). The\nmentioned quantities are connected by a number of relations derived in this\nwork.\n In the second part of this investigation we derive several forms of the\nbundle (analogue of the) Schr\\\"odinger equation governing the time evolution of\nstate sections. We prove that up to a constant the matrix-bundle Hamiltonian,\nentering in the bundle analogue of the matrix form of the conventional\nSchr\\\"odinger equation, coincides with the matrix of coefficients of the\nevolution transport. This allows to interpret the Hamiltonian as a gauge field.\nHere we also apply the bundle approach to the description of observables. It is\nshown that to any observable there corresponds a unique Hermitian bundle\nmorphism (along paths) and vice versa.",
"arxiv_id": "quant-ph/9804062",
"authors": [
"Bozhidar Z. Iliev"
],
"categories": [
"quant-ph",
"gr-qc",
"hep-th"
],
"doi": "10.1088/0305-4470/34/23/309",
"journal_ref": "J.Phys.A34:4919-4934,2001",
"title": "Fibre bundle formulation of nonrelativistic quantum mechanics. II. Equations of motion and observables",
"url": "https://arxiv.org/abs/quant-ph/9804062"
},
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