From Gauging Nonrelativistic Translations to N-Body Dynamics

TL;DR

This paper develops a gauge-invariant model for nonrelativistic particle dynamics using teleparallel gravity concepts, deriving novel two-body interaction solutions and exploring quantum bound and scattering states.

Contribution

It introduces a gauge-invariant nonrelativistic framework with teleparallel structures, deriving a nonlinear Hamiltonian and analyzing quantum two-body solutions with external fields.

Findings

Derived a nonlinear two-body Hamiltonian with energy-dependent fractional angular momentum.

Obtained classical and quantum solutions for confined and unconfined regimes.

Extended the model to include external magnetic fields and analyzed their effects.

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.