Remarks on the conformal transformations
  Conformal transformations are obtained by demanding that the form of the metric change by a conformal factor. Nevertheless, this transformation of the metric is not taken into account when a variation of the action is performed. The basic purpose of this paper is to take the transformation of the metric into the variation of the action. When this is done, we obtain now that even massive particles are invariant under the conformal transformations. 
L. C. T. Guillen