dorsal/arxiv
View SchemaTime-Dependent Hilbert Spaces, Geometric Phases, and General Covariance in Quantum Mechanics
| Authors | Ali Mostafazadeh |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0306200 |
| URL | https://arxiv.org/abs/quant-ph/0306200 |
| DOI | 10.1016/j.physleta.2003.12.008 |
| Journal | Phys.Lett. A320 (2004) 375-382 |
Abstract
We investigate consequences of allowing the Hilbert space of a quantum system to have a time-dependent metric. For a given possibly nonstationary quantum system, we show that the requirement of having a unitary Schreodinger time-evolution identifies the metric with a positive-definite (Ermakov-Lewis) dynamical invariant of the system. Therefore the geometric phases are determined by the metric. We construct a unitary map relating a given time-independent Hilbert space to the time-dependent Hilbert space defined by a positive-definite dynamical invariant. This map defines a transformation that changes the metric of the Hilbert space but leaves the Hamiltonian of the system invariant. We propose to identify this phenomenon with a quantum mechanical analogue of the principle of general covariance of General Relativity. We comment on the implications of this principle for geometrically equivalent quantum systems and investigate the underlying symmetry group.
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"abstract": "We investigate consequences of allowing the Hilbert space of a quantum system\nto have a time-dependent metric. For a given possibly nonstationary quantum\nsystem, we show that the requirement of having a unitary Schreodinger\ntime-evolution identifies the metric with a positive-definite (Ermakov-Lewis)\ndynamical invariant of the system. Therefore the geometric phases are\ndetermined by the metric. We construct a unitary map relating a given\ntime-independent Hilbert space to the time-dependent Hilbert space defined by a\npositive-definite dynamical invariant. This map defines a transformation that\nchanges the metric of the Hilbert space but leaves the Hamiltonian of the\nsystem invariant. We propose to identify this phenomenon with a quantum\nmechanical analogue of the principle of general covariance of General\nRelativity. We comment on the implications of this principle for geometrically\nequivalent quantum systems and investigate the underlying symmetry group.",
"arxiv_id": "quant-ph/0306200",
"authors": [
"Ali Mostafazadeh"
],
"categories": [
"quant-ph",
"gr-qc",
"hep-th",
"math-ph",
"math.MP"
],
"doi": "10.1016/j.physleta.2003.12.008",
"journal_ref": "Phys.Lett. A320 (2004) 375-382",
"title": "Time-Dependent Hilbert Spaces, Geometric Phases, and General Covariance in Quantum Mechanics",
"url": "https://arxiv.org/abs/quant-ph/0306200"
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