DIFFEOMORPHISMS VERSUS NON-ABELIAN GAUGE TRANSFORMATIONS - AN EXAMPLEOF 1+1 DIMENSIONAL GRAVITY

We investigate the phase space of a typical model of (1 + 1) dimension al gravity (Jackiw-Teitelboim model with cylindrical topology) using i ts reformulation as a non abelian gauge theory based on the sl(2, R) a lgebra. Modifying the conventional approach we argue that one should t ake the universal covering of SL(2, R) rather than PSL (2, R) as the g auge group of the theory. We discuss the consequences for the quantiza tion of the model and find that the spectrum of the Dirac observables is sensitive to this modification. Our analysis further provides an ex ample for a gravity theory where the standard Hamiltonian formulation identifies gravitationally inequivalent solutions.