We consider the gauging of space translations with time-dependent gauge
functions. Using fixed time gauge of relativistic theory, we consider the
gauge-invariant model describing the motion of nonrelativistic particles. When
we use gauge-invariant nonrelativistic velocities as independent variables the
translation gauge fields enter the equations through a d\times (d+1) matrix of
vielbein fields and their Abelian field strengths, which can be identified with
the torsion tensors of teleparallel formulation of relativity theory. We
consider the planar case (d=2) in some detail, with the assumption that the
action for the dreibein fields is given by the translational Chern-Simons term.
We fix the asymptotic transformations in such a way that the space part of the
metric becomes asymptotically Euclidean. The residual symmetries are (local in
time) translations and rigid rotations. We describe the effective interaction
of the d=2 N-particle problem and discuss its classical solution for N=2. The
phase space Hamiltonian H describing two-body interactions satisfies a
nonlinear equation H={\cal H}(\vec x,\vec p;H) which implies, after
quantization, a nonstandard form of the Schr\"odinger equation with energy
dependent fractional angular momentum eigenvalues. Quantum solutions of the
two-body problem are discussed. The bound states with discrete energy levels
correspond to a confined classical motion (for the planar distance between two
particles r\le r_0) and the scattering states with continuum energy correspond
to the classical motion for r>r_0. We extend our considerations by introducing
an external constant magnetic field and, for N=2, provide the classical and
quantum solutions in the confined and unconfined regimes.Comment: LaTeX, 38 pages, 1 picture include