  In analogy with crystalline solids around us, Wilczek recently proposed the
idea of "time crystals" as phases that spontaneously break the continuous time
translation into a discrete subgroup. The proposal stimulated further studies
and vigorous debates whether it can be realized in a physical system. However,
a precise definition of the time crystal is needed to resolve the issue. Here
we first present a definition of time crystals based on the time-dependent
correlation functions of the order parameter. We then prove a no-go theorem
that rules out the possibility of time crystals defined as such, in the ground
state or in the canonical ensemble of a general Hamiltonian, which consists of
not-too-long-range interactions.
