  Spherically symmetric models of loop quantum gravity have been studied
recently by different methods that aim to deal with structure functions in the
usual constraint algebra of gravitational systems. As noticed by Gambini and
Pullin, a linear redefinition of the constraints (with phase-space dependent
coefficients) can be used to eliminate structure functions, even Abelianizing
the more-difficult part of the constraint algebra. The Abelianized constraints
can then easily be quantized or modified by putative quantum effects. As
pointed out here, however, the method does not automatically provide a
covariant quantization, defined as an anomaly-free quantum theory with a
classical limit in which the usual (off-shell) gauge structure of hypersurface
deformations in space-time appears. The holonomy-modified vacuum theory based
on Abelianization is covariant in this sense, but matter theories with local
degrees of freedom are not. Detailed demonstrations of these statements show
complete agreement with results of canonical effective methods applied earlier
to the same systems (including signature change).
