Response under arbitrary groups of point transformations of multipoint-correlation functions of local fluctuating quantities

We derive an identity which determines the infinitesimal response of the cumulant average of the product of $m$ arbitrary local fluctuating quantities (connected $m$-point functions) to changes in its arguments engendered by a member of an arbitrary continuous group of point transformations. We illustrate the use of the identity in a variety of cases, and in particular we show that there is an intimate connection between the covariance of the connected $m$-point functions under the special conformal group and under the group of dilations. By combining our identity with the assumptions of the operator algebra, we find that at the critical point the connected density $m$-point functions are covariant under both groups. We also give another derivation for an expression for the exponent $x=\left(\frac{1}{2}\right)(5-\eta )$, which was previously found by Green and Gunton.