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 ψ(x) 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 ψ(x) is not sufficient for a well-defined NGT. NGT’s that are well defined in terms of ψ(x) form a semigroup parametrized by a real number γ and a nonzero λ that is either an integer or -1<~λ<~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.