From gauging nonrelativistic translations to N-body dynamics

We consider the gauging of space translations with time-dependent gauge functions. Using a 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 x (d + 1) matrix of vieibein 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 Chein-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 = H (x, p; H) which implies, after quantization, a nonstandard form of the Schrodinger 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 less than or equal to 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. (C) 2001 Academic Press.