dorsal/arxiv
View SchemaGeometric phases, gauge symmetries and ray representation
| Authors | Kazuo Fujikawa |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0605081 |
| URL | https://arxiv.org/abs/quant-ph/0605081 |
| DOI | 10.1142/S0217751X06033799 |
| Journal | Int.J.Mod.Phys. A21 (2006) 5333-5358 |
Abstract
The conventional formulation of the non-adiabatic (Aharonov-Anandan) phase is based on the equivalence class $\{e^{i\alpha(t)}\psi(t,\vec{x})\}$ which is not a symmetry of the Schr\"{o}dinger equation. This equivalence class when understood as defining generalized rays in the Hilbert space is not generally consistent with the superposition principle in interference and polarization phenomena. The hidden local gauge symmetry, which arises from the arbitrariness of the choice of coordinates in the functional space, is then proposed as a basic gauge symmetry in the non-adiabatic phase. This re-formulation reproduces all the successful aspects of the non-adiabatic phase in a manner manifestly consistent with the conventional notion of rays and the superposition principle. The hidden local symmetry is thus identified as the natural origin of the gauge symmetry in both of the adiabatic and non-adiabatic phases in the absence of gauge fields, and it allows a unified treatment of all the geometric phases. The non-adiabatic phase may well be regarded as a special case of the adiabatic phase in this re-formulation, contrary to the customary understanding of the adiabatic phase as a special case of the non-adiabatic phase. Some explicit examples of geometric phases are discussed to illustrate this re-formulation.
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"abstract": "The conventional formulation of the non-adiabatic (Aharonov-Anandan) phase is\nbased on the equivalence class $\\{e^{i\\alpha(t)}\\psi(t,\\vec{x})\\}$ which is not\na symmetry of the Schr\\\"{o}dinger equation. This equivalence class when\nunderstood as defining generalized rays in the Hilbert space is not generally\nconsistent with the superposition principle in interference and polarization\nphenomena. The hidden local gauge symmetry, which arises from the arbitrariness\nof the choice of coordinates in the functional space, is then proposed as a\nbasic gauge symmetry in the non-adiabatic phase. This re-formulation reproduces\nall the successful aspects of the non-adiabatic phase in a manner manifestly\nconsistent with the conventional notion of rays and the superposition\nprinciple. The hidden local symmetry is thus identified as the natural origin\nof the gauge symmetry in both of the adiabatic and non-adiabatic phases in the\nabsence of gauge fields, and it allows a unified treatment of all the geometric\nphases. The non-adiabatic phase may well be regarded as a special case of the\nadiabatic phase in this re-formulation, contrary to the customary understanding\nof the adiabatic phase as a special case of the non-adiabatic phase. Some\nexplicit examples of geometric phases are discussed to illustrate this\nre-formulation.",
"arxiv_id": "quant-ph/0605081",
"authors": [
"Kazuo Fujikawa"
],
"categories": [
"quant-ph",
"hep-th"
],
"doi": "10.1142/S0217751X06033799",
"journal_ref": "Int.J.Mod.Phys. A21 (2006) 5333-5358",
"title": "Geometric phases, gauge symmetries and ray representation",
"url": "https://arxiv.org/abs/quant-ph/0605081"
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