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We show that the Wigner-Bargmann program of grounding non-relativisticquantum mechanics in the unitary projective representations of the Galileigroup can be extended to include all non-inertial reference frames. The keyconcept is the \emph{Galilean line group}, the group of transformations thatties together all accelerating reference frames, and its representations. Theserepresentations are constructed under the natural constraint that they reduceto the well-known unitary, projective representations of the Galilei group whenthe transformations are restricted to inertial reference frames. Thisconstraint can be accommodated only for a class of representations with asufficiently rich cocycle structure. Unlike the projective representations ofthe Galilei group, these cocycle representations of the Galilean line group donot correspond to central extensions of the group. Rather, they correspond to aclass of non-associative extensions, known as \emph{loop prolongations}, thatare determined by three-cocycles. As an application, we show that the phaseshifts due to the rotation of the earth that have been observed in neutroninterferometry experiments and the rotational effects that lead to simulatedmagnetic fields in optical lattices can be rigorously derived from therepresentations of the loop prolongations of the Galilean line group.

Loop prolongations and three-cocycles in simulated magnetic fields from rotating reference frames