Spontaneous breaking of Lorentz invariance

We describe how a stable effective theory in which particles of the same fermion number attract may spontaneously break Lorentz invariance by giving a nonzero fermion number density to the vacuum (and therefore dynamically generating a chemical potential term). This mechanism yields a finite vacuum expectation value $〈\overline{\psi }{\gamma }^{\mu }\psi 〉,$ which we consider in the context of proposed models that require such a breaking of Lorentz invariance in order to yield composite degrees of freedom that act approximately like gauge bosons. We also make general remarks about how the background source provided by $〈\overline{\psi }{\gamma }^{\mu }\psi 〉$ could relate to work on signals of Lorentz violation in electrodynamics.