  We study the evolution of quantum fluctuations in systems known as time
crystals, hypothetical systems for which the lowest energy state performs a
periodic motion. We first discuss some general properties shared by time
crystals, and deduce the effective field theory parametrizing the evolution of
their fluctuations. We show that these fluctuations fall into categories
analogous to acoustic and optical phonons, encountered in conventional
crystals. The acoustic phonons correspond to gapless Goldstone boson modes
parametrizing the broken time translation invariance of the crystal, whereas
the optical phonons are identified with modes perpendicular to the broken
symmetry of the system, which generically remain gapped. We study how these two
modes decay and interact together, and discuss some observable features that
could be tested in experimental realizations of time crystals.
