The Nature of the Vector and Scalar Potentials and Gauge Invariance in the Context of Gauge Theory

Modern undergraduate textbooks in electricity and magnetism typically focus on a force representation of electrodynamics with an emphasis on Maxwell's Equations and the Lorentz Force Law. The vector potential $\mathbf{A}$ and scalar potential $\Phi$ play a secondary role mainly as quantities used to calculate the electric and magnetic fields. However, quantum mechanics including quantum electrodynamics (QED) and other gauge theories demands a potential ($\Phi$,$\mathbf{A}$) oriented representation where the potentials are the more fundamental quantities. Here, we help bridge that gap by showing that the homogeneous Maxwell's equations together with the Lorentz Force Law can be derived from assuming that the potentials represent potential energy and momentum per unit charge. Furthermore, we enumerate the additional assumptions that are needed to derive the inhomogeneous Maxwell's equations. As part of this work we demonstrate the physical nature and importance of gauge invariance.