An algebraic characterization of the contractions of the Poincaré group permits a proper construction of a non-relativistic limit of its tachyonic representation. We arrive at a consistent, nonstandard representation of the Galilei group in configuration space. This representation is here proven to be equivalent to one in momentum space which, although already classified, was disregarded long ago due to supposedly unphysical properties. The corresponding quantum (and classical) theory shares with the relativistic one its fundamentals, and serves as a toy model to better comprehend the unusual behavior of the tachyonic representation. For instance, we see that evolution takes place in a spatial coordinate rather than time, as for relativistic tachyons, but the modulus of the three-momentum is the same for all Galilean observers, leading to a peculiar dispersion relation for a Galilean system. Furthermore, we demonstrate that the physical Hilbert space excludes states with arbitrarily narrow spatial support, ruling out the existence of point-like configurations.
Aldaya et al. (Thu,) studied this question.