DIFFERENT FORMULATIONS OF GRAVITY AND THEIR EQUIVALENCE AT CLASSICAL AND QUANTUM LEVEL

General Relativity is usually formulated as a theory with gauge invariance under the diffeomorphism group, but there is a ``dilaton'' formulation where it is in addition invariant under Weyl transformations, and a ``unimodular'' formulation where it is only invariant under the smaller group of special diffeomorphisms. Other formulations with the same number of gauge generators, but a different gauge algebra, also exist. These different formulations provide examples of what we call ``inessential gauge invariance'', ``symmetry trading'' and ``linking theories''; they are locally equivalent, but may differ when global properties of the solutions are considered. We discuss these notions in the Lagrangian and Hamiltonian formalism. The discussion is then extended to the quantum level. By making suitable choices of parametrization and gauge we show that the alternative formulations are equivalent to quantum EG, in the sense that the effective actions are the same. In particular, in the dilaton formulation Weyl invariance can be maintained also in the quantum theory.