Spontaneous Breakdown of the Lorentz Invariance and the Nambu-Goldstone Theorem

In a class of three-dimensional abelian gauge theory the Lorentz invariance is spontaneously broken by dynamical generation of a magnetic field. An originally topologically massive photon becomes gapless, i.e. p_0=0 at {\vec p}=0. Indeed, the photon is the Nambu-Goldstone boson associated with the spontaneous breaking of the Lorentz invariance. Although symmetry generated by two Lorentz boost generators is broken, there seems to appear only one physical Nambu-Goldstone boson, namely a photon. We argue that the Ward identities in the Nambu-Goldstone theorem are saturated by the photon pole.