  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.
