  We build an effective field theory (EFT) for quasicrystals -- aperiodic
incommensurate lattice structures -- at finite temperature, entirely based on
symmetry arguments and a well-define action principle. By means of
Schwinger-Keldysh techniques, we derive the full dissipative dynamics of the
system and we recover the experimentally observed diffusion-to-propagation
crossover of the phason mode. From a symmetry point of view, the diffusive
nature of the phason at long wavelengths is due to the fact that the internal
translations, or phason shifts, are symmetries of the system with no associated
Noether currents. The latter feature is compatible with the EFT description
only because of the presence of dissipation (finite temperature) and the lack
of periodic order. Finally, we comment on the similarities with certain
homogeneous holographic models and we formally derive the universal relation
between the pinning frequency of the phonons and the damping and diffusion
constant of the phason.
