  Classical field theory is insensitive to the split of the field into a
background configuration and a dynamical perturbation. In gauge theories, the
situation is complicated by the fact that a covariant (w.r.t. the background
field) gauge fixing breaks this split independence of the action. Nevertheless,
background independence is preserved on the observables, as defined via the
BRST formalism, since the violation term is BRST exact. In quantized gauge
theories, however, BRST exactness of the violation term is not sufficient to
guarantee background independence, due to potential anomalies. We define
background independent observables in a geometrical formulation as flat
sections of the observable algebra bundle over the manifold of background
configurations, with respect to a flat connection which implements background
variations. A theory is then called background independent if such a flat
(Fedosov) connection exists. We analyze the obstructions to preserve background
independence at the quantum level for pure Yang-Mills theory and for
perturbative gravity. We find that in the former case all potential
obstructions can be removed by finite renormalization. In the latter case, as a
consequence of power-counting non-renormalizability, there are infinitely many
non-trivial potential obstructions to background independence. We leave open
the question whether these obstructions actually occur.
