Photons emerging as Goldstone bosons from spontaneous Lorentz symmetry breaking: The Abelian Nambu model

After imposing current conservation together with the Gauss law as initial conditions on the Abelian Nambu model, we prove that the resulting theory is equivalent to standard QED in the nonlinear gauge $(A_{\mu }{A}^{\mu }-n^2{M}^2)=0$, to all orders in perturbation theory. We show this by writing both models in terms of the same variables, which produce identical Feynman rules for the interactions and propagators. A crucial point is to verify that the Faddeev-Popov ghosts arising from the gauge fixing procedure in the QED sector decouple to all orders. We verify this decoupling by following a method like that employed in Yang-Mills theories when investigating the behavior of axial gauges. The equivalence between the two theories supports the idea that gauge particles can be envisaged as the Goldstone bosons originating from spontaneous Lorentz symmetry breaking.

Figure 1

The ghost propagator: $s(k)={\sigma }^{(1)}\frac{i}{M}\frac{n^2}{(n·k)}$.

Figure 2

The ghost-photon vertex: $V^1(k^{\mu })={\sigma }^{(-)}\frac{i}{2}{k}^{\mu }$.

Figure 3

The ghost-($2m$ photons) vertex: $V^{2m}(k^{\mu })=-{\sigma }^{(-)}\frac{i}{2}(n·k)V_{{\alpha }_1{\beta }_1{\alpha }_2{\beta }_2\dots {\alpha }_m{\beta }_m}$.

Figure 4

General closed loop ghost diagram with $N$ vertices, showing arbitrary insertions of type $V^1$ and type $V^{2m}$. All photons are incoming. The photons arriving to vertices ($N-1$) and $N$ are not shown.