Structure of nonlinear gauge transformations

Nonlinear Doebner-Goldin [Phys. Rev. A 54, 3764 (1996)] gauge transformations (NGT’s) defined in terms of a wave function $\psi \mathrm{}\left(x\right)\mathrm{}$ do not form a group. To get a group property one has to consider transformations that act differently on different branches of the complex argument function and the knowledge of the value of $\psi \mathrm{}\left(x\right)\mathrm{}$ is not sufficient for a well-defined NGT. NGT’s that are well defined in terms of $\psi \mathrm{}\left(x\right)\mathrm{}$ form a semigroup parametrized by a real number γ and a nonzero λ that is either an integer or $-1<~\lambda <~1.$ An extension of NGT’s to projectors and general density matrices leads to NGT’s with complex γ. Both the linearity of evolution and Hermiticity of density matrices are gauge-dependent properties.