Photon as a Vector Goldstone Boson: Nonlinear $\sigma$ Model for QED

We show that QED in the Coulomb gauge can be considered as a low energy linear approximation of a non-linear $\sigma$-type model where the photon emerges as a vector Goldstone boson related to the spontaneous breakdown of Lorentz symmetry down to its spatial rotation subgroup at some high scale $M$. Starting with a general massive vector field theory one naturally arrives at this model if the pure spin-1 value for the vector field $A_{\mu }(x)$ provided by the Lorentz condition ${\mathrm{\partial }}_{\mu }{A}_{\mu }(x)=0$ is required. The model coincides with conventional QED in the Coulomb gauge in the limit of M going to infinity and generates a very particular form for the Lorentz and CPT symmetry breaking terms, which are suppressed by powers of $M$.