dorsal/arxiv
View SchemaGauge Transformations in Quantum Mechanics and the Unification of Nonlinear Schr\"odinger Equations
| Authors | H. -D. Doebner, G. A. Goldin, P. Nattermann |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9709036 |
| URL | https://arxiv.org/abs/quant-ph/9709036 |
| DOI | 10.1063/1.532786 |
| Journal | J.Math.Phys. 40 (1999) 49-63 |
Abstract
Beginning with ordinary quantum mechanics for spinless particles, together with the hypothesis that all experimental measurements consist of positional measurements at different times, we characterize directly a class of nonlinear quantum theories physically equivalent to linear quantum mechanics through nonlinear gauge transformations. We show that under two physically-motivated assumptions, these transformations are uniquely determined: they are exactly the group of time-dependent, nonlinear gauge transformations introduced previously for a family of nonlinear Schr\"odinger equations. The general equation in this family, including terms considered by Kostin, by Bialynicki-Birula and Mycielski, and by Doebner and Goldin, with time-dependent coefficients, can be obtained from the linear Schr\"odinger equation through gauge transformation and a subsequent process we call gauge generalization. We thus unify, on fundamental grounds, a rather diverse set of nonlinear time-evolutions in quantum mechanics.
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"abstract": "Beginning with ordinary quantum mechanics for spinless particles, together\nwith the hypothesis that all experimental measurements consist of positional\nmeasurements at different times, we characterize directly a class of nonlinear\nquantum theories physically equivalent to linear quantum mechanics through\nnonlinear gauge transformations. We show that under two physically-motivated\nassumptions, these transformations are uniquely determined: they are exactly\nthe group of time-dependent, nonlinear gauge transformations introduced\npreviously for a family of nonlinear Schr\\\"odinger equations. The general\nequation in this family, including terms considered by Kostin, by\nBialynicki-Birula and Mycielski, and by Doebner and Goldin, with time-dependent\ncoefficients, can be obtained from the linear Schr\\\"odinger equation through\ngauge transformation and a subsequent process we call gauge generalization. We\nthus unify, on fundamental grounds, a rather diverse set of nonlinear\ntime-evolutions in quantum mechanics.",
"arxiv_id": "quant-ph/9709036",
"authors": [
"H. -D. Doebner",
"G. A. Goldin",
"P. Nattermann"
],
"categories": [
"quant-ph"
],
"doi": "10.1063/1.532786",
"journal_ref": "J.Math.Phys. 40 (1999) 49-63",
"title": "Gauge Transformations in Quantum Mechanics and the Unification of Nonlinear Schr\\\"odinger Equations",
"url": "https://arxiv.org/abs/quant-ph/9709036"
},
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