Alan Kostelecký and Stuart Samuel showed in 1989 that mechanisms arising in the context of string theory can lead to spontaneous breaking of Lorentz symmetry.
Typically in these models, spontaneous Lorentz violation is caused by the presence of a potential term in the action.
The vacuum value bμ, along with a background metric, give a solution that minimizes the bumblebee potential.
The vacuum value bμ acts as a fixed background field that spontaneously breaks Lorentz symmetry.
It is an example, for the case of a vector, of a coefficient for Lorentz violation as defined in the Standard-Model Extension.
in this example is assumed to have a minimum when This condition is satisfied when the vector field has a vacuum value bμ obeying bμbμ = ±b2.
The value of the constant ±b2 in the potential determines whether the vacuum vector is timelike, lightlike, or spacelike.
The Lagrangian in this case has a Maxwell form for the bumblebee kinetic term, and is given as For this reason, Bμ can be thought of as a generalized vector potential, and interactions with a matter current
A vierbein formalism can be used to introduce local components for the metric, bumblebee, and matter fields at every spacetime point.
The result is a fixed background field in the spacetime frame, which spontaneously breaks particle diffeomorphisms.
Bumblebee models are useful for exploring the effects of spontaneous Lorentz violation in gravitational theories.
The Nambu–Goldstone modes can be thought of as excitations generated by the broken symmetries that stay in the degenerate vacuum of the theory.
In bumblebee models, the excitations generated by the broken diffeomorphisms are contained in both the vector field Bμ and the metric gμν.
Different gauge choices also affect the interpretation of the Nambu–Goldstone modes that arise from spontaneous Lorentz breaking.
In the most general bumblebee models, gauge fixing for the Lorentz transformations and diffeomorphisms can be made so that all of the Nambu–Goldstone modes are contained in the gravitational sector, either in the vierbein or, in some cases, in the metric alone.
One line of investigation is to search for restricted values of the parameters that eliminate the ghosts as propagating modes.
In addition to the Nambu–Goldstone modes, there is a combined excitation in Bμ and gμν that does not stay in the potential minimum.
In the KS model, it can be shown that suitable initial conditions exist that set the massive mode to zero for all time.
In the limit of an infinite mass scale for the massive mode, the KS model is found to be equivalent to Einstein–Maxwell theory in a fixed axial gauge.
[22] The idea that the photon could emerge as Nambu–Goldstone modes in a theory with spontaneous Lorentz violation first arose in the context of special relativity.
In 1951, Paul Dirac considered a vector theory with a Lagrange-multiplier potential as an alternative model giving rise to the charge of the electron.
Twelve years later, in 1963, James Bjorken proposed a model in which collective excitations of a fermion field could lead to composite photons emerging as Nambu–Goldstone modes.
Subsequently, in 1968, Yoichiro Nambu introduced a vector model that did not involve a symmetry-breaking potential.
In the conventional gauge-theory Higgs mechanism, the Nambu–Goldstone modes are reinterpreted as degrees of freedom associated with a massive gauge field.
The possibility that a gravitational Higgs mechanism in bumblebee models could endow the graviton with mass was considered by Kostelecky and Samuel.
In Riemann–Cartan spacetime, covariant derivatives that act on local tensors involve the spin connection.
Since this type of geometry includes torsion, the spin connection provides an additional set of dynamical degrees of freedom that can propagate.
Bumblebee models in Riemann–Cartan spacetime lead to mass terms for the spin connection through spontaneous breaking of local Lorentz symmetry.
The resulting Nambu–Goldstone modes can be reinterpreted, as in a Higgs mechanism, as degrees of freedom that make the spin connection massive.
However, finding suitable kinetic terms for the resulting massive spin connection, free of ghosts and tachyons, remains an open problem.