  Whether Jordan's and Einstein's frame descriptions of F(R) theory of gravity
are physically equivalent, is a long standing debate. However, practically none
questioned on true mathematical equivalence, since classical field equations
may be translated from one frame to the other following a transformation
relation. Here we show that neither Noether symmetries, Noether equations, nor
may quantum equations be translated from one to the other. The reason being, -
conformal transformation results in a completely different system, with a
different Lagrangian. Field equations match only due to the presence of
diffeomorphic invariance. Unless a symmetry generator is found which involves
Hamiltonian constraint equation, the mathematical equivalence between the two
frames appears to be vulnerable. In any case, in quantum domain Mathematical
and therefore physical equivalence can't be established.
