  A time-varying fine structure constant alpha(t) could give rise to Lorentz-
and CPT-violating changes to the vacuum polarization, which would affect photon
propagation. Such changes to the effective action can violate gauge invariance,
but they are otherwise permitted. However, in the minimal theory of varying
alpha, no such terms are generated at lowest order. At second order, vacuum
polarization can generate an instability--a Lorentz-violating analogue of a
negative photon mass squared -m^2 proportional to alpha [(d alpha/dt) /
alpha]^2 log (Lambda^2), where Lambda is the cutoff for the low-energy
effective theory.
