Euclidean linear conformal gravity

Abstract

A Euclidean quantum conformal gravity in a linear approximation is considered. It is assumed that its solution possesses an exact conformal symmetry. The main difficulty, which is a characteristic not only of gravitation theory but of all gauge theories as well, consists in the fact that in all previous formulations the conformal invariance can be stated unambiguously only in the purely gauge sector. In the present new formulation this difficulty is absent. This is obtained by means of new transformation laws for the metric tensor field hmu nu and energy-momentum tensor Tmu nu . Previous transformation laws corresponded to the indecomposable representations of the conformal group and this fact was the origin of the difficulty mentioned. Explicit expressions are obtained for the propagator of the field hmu nu as well as for the three-point functions including hmu nu or Tmu nu and the matter fields. It is shown that the equations of linear conformal gravity are the consequences of the conformal invariance. They are the manifestation of a mathematical fact of the equivalence of the conformal group representations attributed in the approach to the fields hmu nu , Tmu nu and the Weyl tensor.