Gerald Goldin

Nonlinear gauge transformations and the “hydrodynamical” formulation of quantum mechanics

The group of nonlinear gauge transformations introduced by H.-D. Doebner and the author acts on the Hilbert manifold of quantum-mechanical wave functions. A nonlinear gauge transformation alters the wave functions, changes the time-evolution equation, and modifies the expressions for observables, in such a way as to leave all of the outcomes of physical measurements invariant. The nonlinearities thus introduced into the Schrödinger equation involve functionals suggestive of Madelung’s “hydrodynamical” formulation, and the resulting family of nonlinear equations may be wholly re-expressed in terms of such hydrodynamical variables. After reviewing these earlier results briefly, I shall discuss how this leads to a renewed interest in the role of nodal sets (zeroes of wave functions) in both linear and nonlinear quantum mechanics, and report on some preliminary explorations.