Conformal transformation in gravity

We construct and study the general action of the theory which is conformally equivalent to General Relativity with (or without) cosmological constant. The theory under consideration depends on the metric and also on the scalar field, whereas an extra scalar degree of freedom is compensated by some additional conformal-like symmetry. The action of the theory contains one arbitrary function and the form of the symmetry transformation depends on the form of this function. For some special version of the general metric-dilaton theory without conformal-like symmetry the conformal transformation leads to an additional duality symmetry which links the regimes with strong and weak interactions. On quantum level the conformal-like symmetry can be disturbed by an anomaly. In order to discuss the form of the violation of the conformal symmetry we consider the possible form of the one-loop divergences. So we obtain an infinite family of higher derivative conformal invariant metric-dilaton actions which are distinguished by the form of an arbitrary function in the starting second-derivative action. It turns out that the one-loop counterterms are conformal invariant if the equivalence with General Relativity holds on quantum level.