  Conformal Field Theories (CFTs) have rich dynamics in heavy states. We
describe the constraints due to spontaneously broken boost and dilatation
symmetries in such states. The spontaneously broken boost symmetries require
the existence of new low-lying primaries whose scaling dimension gap, we argue,
scales as $O(1)$. We demonstrate these ideas in various states, including
fluid, superfluid, mean field theory, and Fermi surface states. We end with
some remarks about the large charge limit in 2d and discuss a theory of a
single compact boson with an arbitrary conformal anomaly.
