One of the (many different, somewhat independent) routes to gauge theory is to start from a global symmetry of some kind and "gauge" it, which involves promoting it to a local symmetry and then introducing one or many massless gauge boson(s) to enforce consistency of the theory. This procedure can be done for the case of a phase symmetry of a fermion field to yield QED, more general invariance of a collection of fermions under a unitary group to yield Yang-Mills, and for translational symmetries to give general relativity. In general the gauge bosons need to couple to the conserved (Noether) currents of the global symmetry that is being gauged, which tells us that the photon couples to charge and that the graviton couples to stress-energy.
Focusing particularly on the last part-- it seems as though if GR is what you get when you gauge one part of the Poincaré group (translations), it would be natural to gauge the other part (boosts and rotations). Following this chain of logic, this would yield a gauge boson that would need to couple to (relativistic) angular momentum. To the best of my (limited) knowledge there exists no theory of such a boson. My question is-- what (if any) obstruction is there that prevents us from "gauging" the Lorentz group in this way?
