Galilean and U(1)U(1)-gauge symmetry of the Schrödinger field

This paper undertakes a study of the nature of the force associated with the local U(1)U(1)-gauge symmetry of a non-relativistic quantum particle. To ensure invariance under local U(1)U(1) symmetry, a matter field must couple to a gauge field. We show that such a gauge field satisfies Maxwell’s equations, whether the matter field coupled to it is relativistic or non-relativistic. This result suggests that the structure of Maxwell’s equations is determined by gauge symmetry rather than the symmetry transformation properties of space-time. In order to assess the validity of this notion, we examine the transformation properties of the coupled matter and gauge fields under Galilean transformations. Our main technical result is the Galilean invariance of the full equations of motion of the U(1)U(1) gauge field.