Temporally, spatially, or lightlike modulated vacua in Lorentz invariant theories

We study the vacuum structure of a class of Lorentz-invariant field theories where the vacuum expectation values are not constant but are (phase) modulated. The vacua are classified into spatial, temporal, and lightlike modulation types according to the patterns of spontaneous breaking of translational symmetry. The conditions for having temporal or lightlike modulated vacua imply severe constraints on the models. We utilize the notion of generalized Nambu-Goldstone modes which appear in the modulated vacua. Finally, we examine fluctuation modes around these vacua and discuss their dynamics and the absence of ghosts.