  An enlarged group G of nonlinear transformations, modeled on the general
linear group GL(2,R), leads to a beautiful, apparently unremarked symmetry
between the wave function's phase and the logarithm of its amplitude. Equations
Doebner and I earlier proposed are embedded in a wider, natural family of
nonlinear time-evolution equations, on which G acts as a gauge group (leaving
physical observations invariant). There exist G-invariant quantities that
reduce to the usual probability density and flux for linearizable quantum
theories in a particular gauge.
