Constraints for conformal field theories on the plane: Reviving the conformal bootstrap

We obtain a number of results by reexamining conformal field theory on the plane, using only elementary mathematics in the process. In particular we rederive the constraints on conformal dimensions found by Vafa, using the crossing symmetry of four-point amplitudes; find stronger, inequality versions of these conditions which constrain the magnitude of the conformal dimensions of primary fields which can appear as intermediate states in a given amplitude; and study the conformal bootstrap perturbatively using the power-series expansions for conformal block functions. This latter approach gives constraints of a quite different type, in particular tree-level constraints on the central charge.