Perturbations of states in integrable or free field theories are not expected to thermalize due to the constraints imposed by the existence of an extensive number of conserved charges. In this talk, I will show that this is indeed the case when the perturbation probing the system satisfies a simple no-go condition. I will present examples of how this condition applies in 2D CFTs, in higher-dimensional free field theories, and in the transverse field Ising model. At the same time, I will also show how perturbations that do not obey this no-go condition easily thermalize and how the associated operators take a form that evokes the eigenstate thermalization hypothesis.
