Classical anharmonic oscillator with U(1) symmetry: why must there be dynamics in a gauge theory?

Gauge symmetry and, consequently, gauge theories, which emerge naturally in the context of the electromagnetic theory, in their non-Abelian version encompass electroweak and strong interactions; this has consolidated quantum field theories as a way of systematically describing the processes of fundamental particles. The success of this proposal is due to the fact that, by including a gauge field as the mediator of an interaction, a universal formulation may be adopted to describe the diverse fundamental interactions. In this contribution, we pick out the example of classical harmonic/anharmonic two-dimensional oscillators to endeavour a presentation of gauge theories in an introductory way, compatible with the level of knowledge of an undergraduate course. To relate this discussion to a field-theory scenario, we start off from the scalar electrodynamics and make use of a dimensional reduction technique to get a Lagrangian of a classical mechanical system. In this scenario, we motivate and justify why, besides ensuring local invariance, gauge fields must correspond to propagating degrees of freedom.

Keywords:
Classical anharmonic oscillator; Gauge symmetry; Quantum electrodynamics; Gauge theory