ρ2

An enlarged gauge group acts nonlinearly on the class of nonlinear Schrödinger equa-tions introduced by the author in joint work with Doebner. Here the equations and the group action are displayed in the presence of an external electromagnetic field. All the gauge-invariants are listed for the coupled nonlinear “Schrödinger-Maxwell” theory. Time-dependent gauge parameters result in additional terms of the type in-troduced by Kostin and Bialynicki-Birula and Mycielski, but Maxwell’s equations for the (non-quantized) gauge-invariant electric and magnetic fields remain linear. 1 Nonlinear Schrödinger Equations In earlier work Doebner and I introduced a family of nonlinear Schrödinger equations in order to interpret quantum-mechanically certain representations of infinite-dimensional algebras and groups [1–3]. We proposed these equations as candidates for describing quantum systems with dissipation. Let us now set them up in a slightly different but mathematically and physically useful form [4,5]. Put ρ = ψψ, j ̂ = 1 2i [ψ∇ψ − (∇ψ)ψ] , (1.1) where ψ(x, t) is a square-integrable, time-dependent wave function, ρ is the associated probability density in position as a function of time, and j ̂ is a (non-gauge invariant) probability flux density. Introduce the real nonlinear functional