  A new kind of gauge theory is introduced, where the minimal coupling and
corresponding covariant derivatives are defined in the space of functions
pertaining to the functional Schroedinger picture of a given field theory.
While, for simplicity, we study the example of a U(1) symmetry, this kind of
gauge theory can accommodate other symmetries as well. We consider the
resulting relativistic nonlinear extension of quantum mechanics and show that
it incorporates gravity in the (0+1)-dimensional limit, similar to recently
studied Schroedinger-Newton equations. Gravity is encoded here into a universal
nonlinear extension of quantum theory. A probabilistic interpretation (Born's
rule) holds, provided the underlying model is scale free.
  Keywords: nonlinear functional Schroedinger equation, gauge symmetry,
Newtonian gravity.
