Generalization of conformal Hamada operators

Abstract

The six-derivative conformal scalar operator was originally found by Hamada in its critical dimension of spacetime, $d=6$. We generalize this construction to arbitrary dimensions $d$ by adding new terms cubic in gravitational curvatures and by changing its coefficients of expansion in various curvature terms. The consequences of global scale-invariance and of infinitesimal local conformal transformations are derived for the form of this generalized operator. The system of linear equations for coefficients is solved giving explicitly the conformal Hamada operator in any $d$. Some singularities in construction for dimensions $d=2$ and $d=4$ are noticed. We also prove a general theorem that a scalar conformal operator with $n$ derivatives in $d=n-2$ dimensions is impossible to construct. Finally, we compare our explicit construction with the one that uses conformal covariant derivatives and conformal curvature tensors. We present new results for operators built with different orders of conformal covariant derivatives.