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Many of the papers (e.g., this) dealing with nonlinear electrodynamics treat a theory's prediction of vacuum birefringence as undesirable, but don't explain why it would be undesirable. For example:

Starting with the most general non-linear theory derived from an arbitrary Lagrangian depending on two Lorentz invariants of Maxwell’s tensor, L(P,S), he discovered that among all such non-linear theories, the Born-Infeld electrodynamics is the only one ensuring the absence of birefringence, i.e. propagation along a single light-cone, and the absence of shock waves. In this respect the Born-Infeld theory is unique (except for another singular and unphysical Lagrangian L=P/S). A beautiful discussion of these properties can be found in I. Bialynicki-Birula’s paper

The bold italicized emphasis is mine.

Other papers (e.g., this) treat vacuum birefringence in extreme fields as expected and natural.

My question: is vacuum birefringence expected and natural, and Born-Infeld electrodynamics therefore deficient?

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    $\begingroup$ "Expected and natural" from what standpoint? It seems sort of obvious to me that conformance with standard relativity demands all massless objects to move at the same speed in vacuum, and hence forbids vacuum birefringence. So if you want a theory that's compatible with relativity, you can't have birefringence. Does that already answer your question? $\endgroup$ – ACuriousMind Mar 11 at 20:52
  • $\begingroup$ @ACuriousMind Would axions be incompatible with relativity? Maybe one can get around contradictions by saying that the bore of a magnet is not free space. (I am no expert on this, but fellow postdocs worked on an axion experiment.) $\endgroup$ – Pieter Mar 11 at 20:58
  • $\begingroup$ @ACuriousMind, no, it doesn't answer my question. QED seems to require vacuum birefringence. Even absent EM fields, in GR light moves at speed c only with respect to local inertial frames where the speed is measured. $\endgroup$ – S. McGrew Mar 11 at 22:26
  • $\begingroup$ Let's perhaps be clear here what we mean by "vacuum birefringence": Do we mean that in the absence of fields the vacuum may exhibit different propagation speeds for a light wave (which I understand some "birefringent" theories predict), or do we mean that in the presence of strong electromagnetic fields otherwise "empty space" can exhibit such different speeds (which I believe you are referring to when you say that QED "requires" birefringence, since certain perturbative calculation indicate such behavior in strong ambient EM fields)? $\endgroup$ – ACuriousMind Mar 11 at 22:43
  • $\begingroup$ I consider "vacuum" to mean no particles are present. It is still a vacuum if there are EM fields, gravitational fields, or virtual particles present. $\endgroup$ – S. McGrew Mar 11 at 22:51
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I think I've found a partial answer in this paper: “FASTER THAN LIGHT” PHOTONS IN GRAVITATIONAL FIELDS– CAUSALITY, ANOMALIES AND HORIZON", along with numerous papers by Patricio Gaete.

Vacuum birefringence per se is not necessarily bad, as long as it does not allow superluminal motion. Vacuum birefringence results in at least two different light speeds, and therefore two different light cones. As long as both light cones correspond to light speeds less than or equal to c, it's okay.

Vacuum birefringence per se does not necessarily occur in QED since it hasn't yet been detected experimentally. Some folks are looking for evidence of vacuum birefringence in gravitational lensing. It might be possible to observe birefringence via light that has passed close to a magnetar, but it could be very difficult to separate out the effects of a magnetized, rapidly moving plasma in the vicinity of the magnetar. Detection of an electric dipole moment of the electron would suggest that vacuum polarization occurs.

Vacuum birefringence appears to be an area of ongoing theoretical, experimental, and observational activity, with key questions still unanswered. The Born-Infeld Lagrangian is used widely in nonlinear electrodynamics and has a very attractive simplicity. In its basic form, the B-I Lagrangian does not exhibit birefringence; but some generalizations of the B-I Lagrangian do exhibit it.

So, I think the best answer that can be given is that at this point in time the answer is not known. If vacuum birefringence is ever actually observed, the B-I Lagrangian rightfully will be considered deficient.

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