The Gauge Group in the Ashtekar-barbero Formulation of Canonical Gravity

It still not universally recognized that all the gauge symmetries present in the Lagrangian formalism for a physical theory must also manifest themselves in the canonical formalism. This issue has been particularly obscure and controversial in the case of generally covariant theories, where the full realization of the di eomorphisms-induced gauge group in phase space has met with the di culty of the lack of projectability of some generators from the con guration-velocity space to phase space. In a collaboration that started some time ago, we have tried to clarify the different issues involved, particularly that of projectability. 1;2;3;4 We have obtained the general result that, in generally covariant theories with a metric , the arbitrary functions x that describe an in nitesimal di eomorphism in spacetime must contain speci c and compulsory dependences on the lapse and shift gauge variables in order for the transformations of the elds be projectable to phase space. If a Yang-Mills type eld is included and the gauge symmetry is larger that that of the spacetime di eomorphisms, other dependences with respect to other gauge variables may become necessary in order to ensure projectability. In turn, the fact that arbitrary functions of the gauge transformations explicitly depend on some elds makes the structure \constants" associated with the algebra of generators no longer constants, but functions of the elds. Since the gauge group, acting on the space of eld con gurations, contains the transformations induced by di eomorphisms acting on spacetime, the fact that the di eomorphism group by itself is not realizable in phase space is compatible with the fact that a wise selection of the generators of the gauge group can make it fully realizable in phase space. The application of this program to the Ashtekar formulation of complex canonical relativity has, besides the requirements of projectability the additional subtlety that the reality conditions need to be preserved. In fact, we nd that both aspects are interrelated. 4 Our presentation here concerns the extension of our previous work to the case of the Ashtekar-Barbero real connection or, even more generally, to connection-based models described by an arbitrary value of the Immirzi-Barbero parameter. Our results, showing how the di eomorphisms-induced gauge group is fully implemented in phase space, suggest the possibility of applying new techniques to the loop quantum gravity program; it is possible in principle to retain all the variables