Generalized Quantum Mechanics and Nonlinear Gauge Transformations

There have been several approaches to a nonlinear modification of orthodox quantum mechanics and, in particular, of the Schrödinger equation. As the equations will appear in this paper we cite the nonlinear Schrödinger equations proposed by Bialinycki-Birula and Mycielski [1], 1 $$i\hbar {\partial _t}{\psi _t} = \left( { - \frac{{{\hbar ^2}}}{{2m}}\Delta + V} \right) + {\alpha _1}\ln {\left| {{\psi _t}} \right|^2}{\psi _t}$$ and the one derived by Doebner and Goldin [2–5] from the representation theory of the kinematical algebra for a single particle on ℝ3, $$S({\mathbb{R}^3}) = \mathfrak{X}({\mathbb{R}^3}) \oplus \mathcal{L}{\mathcal{C}^\infty }({\mathbb{R}^3}).$$