Resumen de Weyl invariance in metric f(R) gravity
We aim to derive the most general f(R) gravity theory, including the matter, so that it be Weyl invariant. Making use of the mathematical equivalence of these theories with an type of scalar-tensor theory (which includes a scalar degree of freedom, φ) and by imposing the Weyl invariance for the pure gravity (under this label, we understand the part that does not involve fields of matter although it could include kinetic terms linked to φ) as well as for the matter sector, we obtain the fundamental equation that restricts the form of V .= Rφ − f(R) (and, accordingly, of f(R)) so that the resulting action to be Weyl invariant in the Jordan frame. We show that this action is not other than the so-called gravity-dilaton action with one scalar field,Φ, which effective mass is R and Φ dependent. In the Einstein frame, the action becomes the Einstein-Hilbert action with the Ricci scalar being constant due to that the effective mass of scalar field in this frame vanish. So, we can assume that the Ricci scalar, in the Einstein frame, is the true Cosmological Constant. Therefore, is not preposterous to guess that, at least mathematically, all Weyl invariant metric f(R) theory in the Jordan frame is equivalent, at classical level, to the Einstein gravity, in the Einstein frame, with a constant Ricci scalar. At quantum level, as it is known, both theories are not equivalent due to the presence of anomalies in one of the frames.
