  We develop in a systematic fashion the idea of gauging 1d-space translations
with fixed Newtonian time for nonrelativistic matter (particles and fields). By
starting with a nonrelativistic free theory we obtain its minimal gauge
invariant extension by introducing two gauge fields with a Maxwellian self
interaction. We fix the gauge so that the residual symmetry group is the
Galilei group and construct a representation of the extended Galilei algebra.
The reduced N-particle Lagrangian describes geodesic motion in a
(N-1)-dimensional (Pseudo-) Riemannian space. The singularity of the metric for
negative gauge coupling leads in classical dynamics to the formation of
geometric bags in the case of two or three particles. The ordering problem
within the quantization scheme for $N$-particles is solved by canonical
quantization of a pseudoclassical Schroedinger theory obtained by adding to the
continuum generalization of the point-particle Lagrangian an appropriate
quantum correction. We solve the two-particle bound state problem for both
signs of the gauge coupling. At the end we speculate on the possible physical
relevance of the new interaction induced by the gauge fields.
