  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. As
an application, we solve the field equations assuming a
Friedmann-Robertson-Walker universe and a perfect fluid as its source. Although
we have entirely abandoned Weyl's atempt to identify the vector field with the
4-dimensional electromagnetic potentials, which here must be simply viewed as
part of the space-time geometry, we believe that in this way we could perhaps
be led to a rich and interesting new modified gravity theory.
