In the conformal symmetry, used in some QFT theories, the infinitesimal generators, applying to space-time, are all linear (translations, rotations, boosts, dilatation), except the special conformal transformations, which are not linear. I feel that this non - linearity could cause a conflict with locality, in the following sense:
Let us consider two wave packets 1 and 2, separated by a space-like interval, corresponding to impulsions P1 and P2. If we make a linear transformation, for example a Lorentz transformation, we obtain impulsions P'1 and P'2. Now let us consider the total system 1 + 2; it has an impulsion P, which is equal to P1 + P2 because both systems are not causally related. By this same Lorentz transformation, the total impulsion P becomes P', but because the Lorentz transformation is linear, we have P' = P'1 + P'2, and P' is linear in P'1 and P'2, and is linear in P1 and P2. This is coherent with the fact that there is still no interaction between the systems 1 and 2.
On the other hand, imagine we had taken, for example, a pseudo-Lorentz transformation , which would be, for example, quadratic, then, in the expression of P', we would have had cross-terms of type P1.P2, what would mean an interaction between systems not causally connected, which is incoherent.
So, I need a detailed argumentation, to infirm or confirm my feeling : non-linear special conformal transformations could cause a conflict with locality.