Gravitons as Goldstone Bosons

A Lagrangian model is constructed in which gravitons appear as Goldstone bosons associated with a breaking of Lorentz symmetry. It is assumed that a fermion field interacts with itself by a tensor-tensor coupling. The invariance of the vacuum state under Lorentz transformations is violated by assigning a nonzero vacuum expectation value to the tensor. Interactions between the fermions and this nonzero vacuum tensor are treated by a Hartree-Fock procedure; other interactions are treated by perturbation theory. The divergent integrals that appear are handled by a cutoff whose magnitude, expressed as a length, is of the order of ${10}^{-33\mathrm{}}$ cm. When the $S$ matrix for fermion-fermion scattering is calculated in first approximation, "gravitons" make their appearance as poles at momentum transfer $k^2=0\mathrm{}$. These gravitons also appear as solutions of the Bethe-Salpeter equation for bound fermion-antifermion states. The model reproduces the results of the linear approximation to gravitational theory.