Spontaneous breaking of spacetime symmetries and the inverse Higgs effect

It has long been known that when spacetime symmetry is spontaneously broken, some of the broken generators may not give rise to independent gapless Nambu-Goldstone excitations. We provide two complementary viewpoints of this phenomenon. On the one hand, we show that the corresponding field degrees of freedom have the same symmetry transformation properties as massive matter fields. The inverse Higgs constraints, sometimes employed to eliminate these modes from the theory, are reinterpreted as giving a field parametrization that makes these transformation properties manifest. On the other hand, relations among classical symmetry transformations generally lead to identities for the associated Noether currents that allow saturation of the Ward-Takahashi identities for all the broken symmetries with fewer gapless excitations than suggested by the mere counting of broken generators.