Gauge Transformations in Quantum Mechanics and the Unification of Nonlinear Schr'odinger Equations

摘要

Beginning with ordinary quantum mechanics for spinless particles, togetherwith the hypothesis that all experimental measurements consist of positionalmeasurements at different times, we characterize directly a class of nonlinearquantum theories physically equivalent to linear quantum mechanics throughnonlinear gauge transformations. We show that under two physically-motivatedassumptions, these transformations are uniquely determined: they are exactlythe group of time-dependent, nonlinear gauge transformations introducedpreviously for a family of nonlinear Schr\"odinger equations. The generalequation in this family, including terms considered by Kostin, byBialynicki-Birula and Mycielski, and by Doebner and Goldin, with time-dependentcoefficients, can be obtained from the linear Schr\"odinger equation throughgauge transformation and a subsequent process we call gauge generalization. Wethus unify, on fundamental grounds, a rather diverse set of nonlineartime-evolutions in quantum mechanics.