From gauging nonrelativistic translations to N body dynamics
Jul, 2000
Citations per year
Abstract:
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.Note:
- LaTeX, 38 pages, 1 picture included
- particle: nonrelativistic
- gravitation
- dimension: 3
- field equations: solution
- Hamiltonian formalism
- conservation law
- quantization: 1
- Schroedinger equation
References(48)
Figures(0)
- [1]
- [2]
- [3]
- [3]
- [4]
- [5]
- [6]
- [7]
- [8]
- [9]
- [10]
- [11]
- [12]
- [13]
- [14]
- [15]
- [16]
- [17]
- [18]
- [19]
- [20]
- [21]
- [22]
- [23]
- [23]