The timelike sigma-model case is well-defined and stable when the vector norm is fixed by a constraint however, when it is determined by minimizing a potential there is necessarily a tachyonic ghost, and therefore an instability. There are precisely three kinetic terms that are not manifestly unstable: a sigma model $(partial_mu A_nu)^2$, the Maxwell Lagrangian $F_$, and a scalar Lagrangian $(partial_mu A^mu)^2$.
Models with generic kinetic terms are plagued either by ghosts or by tachyons, and are therefore physically unacceptable. Gresham, Heywood Tam Abstract: We investigate the stability of theories in which Lorentz invariance is spontaneously broken by fixed-norm vector "aether" fields. Instabilities in the Aether Authors: Sean M. Tim Dulaney, Moira Gresham, Heywood Tam and I have been thinking about these issues for a while, and we've just come out with two papers presenting what we've worked out. But the ideas are still relatively new, and there are still questions about whether such models are fundamentally well-defined. A lot of work has gone into exploring the possible consequences and experimental constraints on the idea of an aether field pervading the universe (see reviews by Ted Jacobson or David Mattingly, or Alan Kostelecky's web page ).
But the name "aether" is too good to abandon, so we've re-purposed it for modern use. Instead of a light-carrying medium, we are interested in the possibility of a Lorentz-violating vector field - some four-dimensional vector that has a fixed non-zero length and points in some direction at every event in spacetime. So it's interesting to contemplate the possibility that there really is some kind of field in the universe that defines an absolute standard of rest, within the modern context of low-energy effective field theories. But precision, even when exquisite, is never absolute, and important discoveries are often lurking in the margins. Indeed, if you had asked Hendrik Antoon Lorentz himself about the meaning of the famous Lorentz transformations he invented, he would not have said "they relate physical quantities measured in different inertial frames" he would have said "they relate quantities as measured in some moving reference frame to their true values in the rest frame of the aether." We know a lot more about field theory as well as about relativity these days, so we don't need to invoke a concept like the aether to explain the propagation of light, and the idea that there is no special preferred frame of rest has been experimentally tested to exquisite precision. The idea was eventually superseded by special relativity, although (as with most interesting ideas) some adherents gave up only reluctantly. This idea was basically scotched by the Michelson-Morley experiment, which showed that the speed of light was unaffected by the motion of the Earth around the Sun. If that idea had been true, one would imagine there would be a unique frame of reference in which the aether was at rest, while it was moving in other frames consequently, the speed of light would depend on one's motion through the aether. Prior to Einstein, physicists believed that light waves, like water waves, were ripples in a medium: instead of the ocean, they posited the existence of the luminiferious aether, some form of substance which supported the propagation of electromagnetic waves.
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