  We construct, in classical two-time physics, the necessary structure for the
most general configuration space formulation of quantum mechanics containing
gravity in d+2 dimensions. This structure is composed of a symmetric Riemannian
metric tensor and of a vector field that defines a section of a flat U(1)
bundle over space-time. This construction is possible because of the existence
of a finite local scale invariance of the Hamiltonian and because two-time
physics contains, at the classical level, a local generalization of the
discrete duality symmetry between position and momentum that underlies the
structure of quantum mechanics.
