  Just like the vector bosons in Abelian and non-Abelian gauge theories,
gravitons can attain mass by spontaneous local symmetry breaking. The question
is whether this can happen in a Lorentz-invariant way. We consider the use of
four scalar fields that break coordinate reparametrization invariance, by
playing the role of preferred flat coordinates x, y, z, and t. In the unbroken
representation, the theory has a (negative) cosmological constant, which is
tuned to zero by the scalars in the broken phase. Massive spin 2 bosons and a
single massive scalar survive. The theory is not renormalizable, so at best it
can be viewed as an effective field theory for massive spin 2 particles. One
may think of applications in cosmology, but a more tantalizing idea is to apply
it to string theory approaches to QCD: if the gluon sector is to be described
by a compactified 26 or 10 dimensional bosonic string theory, then the ideas
considered here could be used to describe the mechanism that removes a massless
or tachyonic scalar and provides mass to the spin 2 glueball states. The
delicate problem of removing indefinite metric and/or negative energy states is
addressed. The scalar particle has negative metric, so that unitarity demands
that only states with an even number of them are allowed. Various ways are
considered to adapt the matter section of the theory such that matter only
couples to positive metric states, and we succeed in suppressing the main
contributions to unitarity-violating amplitudes, but the exact restoration of
unitarity in the spinless sector will continue to be a delicate issue in
theories of this sort.
