Gauging kinematical and internal symmetry groups for extended systems: the Galilean one-time and two-times harmonic oscillators

TL;DR

This paper explores how gauging symmetry groups in extended non-relativistic systems like harmonic oscillators leads to novel interactions and modifications in internal dynamics, revealing internal gauge fields' roles as spin connections and metrics.

Contribution

It introduces a framework for gauging Galilean symmetry groups in extended systems, revealing new interactions and internal geometric structures induced by gauge fields.

Findings

External gauge fields induce unique interactions with the system.

Internal gauge fields modify the dynamics via covariant derivatives.

Yang-Mills fields act as internal spin connections and metrics.

Abstract

The possible external couplings of an extended non-relativistic classical system are characterized by gauging its maximal dynamical symmetry group at the center-of-mass. The Galilean one-time and two-times harmonic oscillators are exploited as models. The following remarkable results are then obtained: 1) a peculiar form of interaction of the system as a whole with the external gauge fields; 2) a modification of the dynamical part of the symmetry transformations, which is needed to take into account the alteration of the dynamics itself, induced by the {\it gauge} fields. In particular, the Yang-Mills fields associated to the internal rotations have the effect of modifying the time derivative of the internal variables in a scheme of minimal coupling (introduction of an internal covariant derivative); 3) given their dynamical effect, the Yang-Mills fields associated to the internal rotations apparently define a sort of Galilean spin connection, while the Yang-Mills fields associated to the quadrupole momentum and to the internal energy have the effect of introducing a sort of dynamically induced internal metric in the relative space.