Nonlinear QED and physical Lorentz invariance

The spontaneous breakdown of 4-dimensional Lorentz invariance in the framework of QED with the nonlinear vector potential constraint $A_{\mu }^2=M^2$ (where $M$ is a proposed scale of the Lorentz violation) is shown to manifest itself only as some noncovariant gauge choice in the otherwise gauge invariant (and Lorentz invariant) electromagnetic theory. All the contributions to the photon-photon, photon-fermion, and fermion-fermion interactions violating the physical Lorentz invariance happen to be exactly cancelled with each other in the manner observed by Nambu long ago for the simplest tree-order diagrams—the fact which we extend now to the one-loop approximation and for both the timelike ($M^2>0$) and spacelike ($M^2<0$) Lorentz violations. The way to reach the physical breaking of the Lorentz invariance in the pure QED case (and beyond) treated in the flat Minkowskian space-time is also discussed in some detail.

Figure 1

The LIV contribution to the Compton effect in the tree approximation (in the order of $\mathrm{O}(\frac{e}{M})$).

Figure 2

The LIV contribution to the $e-e^{\prime }$ scattering in the one-loop approximation (in the order of $\mathrm{O}(\frac{e^3}{M})$).