Gauge Transformations in Quantum Mechanics and the Unification of Nonlinear Schrödinger Equations

Physics – Quantum Physics

Scientific paper

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RevTeX, 20 pages, no figures. also available on http://www.pt.tu-clausthal.de/preprints/asi-tpa/021-96.html

Scientific paper

10.1063/1.532786

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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