Weyl's unified field theory: Perspectives for a new approach

We revisit Weyl's unified field theory, which arose in 1918, shortly after general relativity was discovered. As is well known, in order to extend the program of geometrization of physics started by Einstein to include the electromagnetic field, H. Weyl developed a new geometry which constitutes a kind of generalization of Riemannian geometry. However, despite its mathematical elegance and beauty, a serious objection was made by Einstein, who considered Weyl's theory not suitable as a physical theory since it seemed to lead to the prediction of a not yet observed effect, the so-called "second clock effect". In this paper, our aim is to discuss Weyl's proposal anew and examine its consistency and completeness as a physical theory. Finally, we propose new directions and possible conceptual changes in the original work. We believe that in this way we could perhaps be led to a rich and interesting new modified gravity theory.