the same symmetries. This is not the case for the conformal group and then the requirement that the theory is invariant under conformal transformations is an additional assumption not related to the gauge invariance. Furthermore, we are concerned with Lagrangians which means that we assume the same kinematics on which these Lagrangians set up a dynamics. In general, kinematics has other symmetries with respect to dynamics and these symmetries are not related to those of the metric because no field equation relates kinematic quantities to metric. Kinematic symmetries are constrained only by the Ricci identity which gives the constraint and propagation equations.

The issue of “which is the physical frame” has been debated for a long time and it emerges as soon as some authors argue in favor of one frame against the other, and others support the idea that the two frames are physically equivalent. In the latter case, authors claim that the issue is a pseudo-problem. The final result is that there is a good deal of confusion in the literature.

Fierz was the first to pose the problem [@GrumillerKummerVassilevich03], but the main argument is due to Dicke, who discussed the conformal transformation for Brans-Dicke theory [@Dicke62]. The 