On local scale invariance and the questionable theoretical basis of the
  conformal transformations' issue
  Here we follow the mainstream of thinking about physical equivalence of
different representations of a theory, regarded as the consequence of
invariance of the laws of physics -- represented by an action principle and the
derived motion equations -- under given transformations; be it coordinate,
gauge or conformal transformations. Accordingly the conformal transformations'
issue is discussed by invoking the assumed invariance of the laws of physics --
in particular the laws of gravity -- under conformal transformations of the
metric. It is shown that Brans-Dicke and scalar-tensor theories are not
well-suited to address physical equivalence of the conformal frames since the
corresponding laws of gravity are not invariant under the conformal
transformations or Weyl rescalings. The search for conformal symmetry leads us
to explore the physical consequences of Weyl-invariant theories of gravity,
that represent a natural arena where to discuss on physical equivalence of the
conformally related representations. We show that conformal invariance of the
action of a (supposedly conformal invariant) theory and of the derived motion
equations is not enough to ensure actual Weyl invariance. It is required, also,
that the underlying geometrical structure of the background spacetime be, at
least, Weyl-integrable. Otherwise, if assume (as usual) spacetimes of
(pseudo)Riemannian geometrical structure, the resulting -- apparently conformal
invariant -- theory is anomalous in that, only massless matter fields can be
consistently coupled. Gauge freedom, a distinctive feature of actually
Weyl-invariant theories of gravity, leads to very unusual consequences.
