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

A Bicycle Built for Two: The Galilean and U(1) Gauge Invariance of the Schr\"odinger Field