fields in curved space, the equivalence of conformal frames is not clear. At quantum level, this equivalence is not proven due to the lack of a definitive quantum gravity theory: in fact, when the metric $g_{ij}$ is quantized, inequivalent quantum theories can be found [@GrumillerKummerVassilevich03; @AshtekarCorichi03]. One can consider the semiclassical regime in which gravity is classical and matter fields are quantized: again, one would expect the conformal frames to be inequivalent because the conformal transformations can be seen as Legendre transformations [@MagnanoSokolowski94], similar to the Legendre transformations of classical mechanics of point particles which switches from the canonical Lagrangian coordinates $q$ to the variables $\left\{  q,p\right\}  $ of the Hamiltonian formalism. Now, it is well known that Hamiltonians that are classically equivalent become inequivalent when quantized, producing different energy spectra and scattering amplitudes [@CalogeroDegasperis04; @GlassScanio77]. However, the conformal equivalence between Jordan and Einstein frame seems to hold to some extent at the semi-classical level [@Flanagan04b]. Again, only a particular kind of physics has been considered and one cannot make statements about all possible physical situations.

It is important to point out a very basic argument among particle physicists that relies on the equivalence theorem of Lagrangian field theory. It 