The Geometric Proca Field in Gauge-Invariant Weyl Theory

We present a detailed study of the geometrization of the Proca field in the so-called Weyl Invariant Theory, shedding new light on the physical interpretation of the Weyl field. We first describe the field equations of the theory. We then obtain a solution for the weak field using a spherically symmetric and static approximate metric. Our analysis revealed that the Weyl field, in the weak field approximation, exhibits a behaviour identical to the Yukawa potential, similar to the Proca field. Furthermore, the obtained metric solution is equivalent to the Einstein-Proca case, demonstrating that the description of the Weyl field in the Weyl Invariant Theory is consistent with Proca theory in the context of General Relativity. Finally, we conclude that the Weyl field can be formally interpreted as a Proca field of geometrical nature.