Inhomogeneities, loop quantum gravity corrections, constraint algebra and general covariance

Loop quantum gravity corrections, in the presence of inhomogeneities, can lead to a deformed constraint algebra. Such a deformation implies that the effective theory is no longer generally covariant. As a consequence, the geometrical concepts used in the classical theory lose their meaning. In the present paper we propose a method, based on canonical transformation on the phase space of the spherically symmetric effective theory, to systematically recover the classical constraint algebra in the presence of the inverse triad corrections as well as in the presence of the holonomy corrections. We show, by way of explicit example, that this also leads to the recovery of general covariance of the theory in the presence of inverse triad corrections, implying that one can once again use the geometrical concepts to analyze the solutions in the presence of these quantum gravity corrections.