Weyl versus Conformal Invariance in Quantum Field Theory

We argue that conformal invariance in flat spacetime implies Weyl invariance in a general curved background metric for all unitary theories in spacetime dimensions $d\le 10$. We also study possible curvature corrections to the Weyl transformations of operators, and show that these are absent for operators of sufficiently low dimensionality and spin. We find possible `anomalous’ Weyl transformations proportional to the Weyl (Cotton) tensor for $d>3$ ($d=3$). The arguments are based on algebraic consistency conditions similar to the Wess-Zumino consistency conditions that classify possible local anomalies. The arguments can be straightforwardly extended to larger operator dimensions and higher $d$ with additional algebraic complexity.