  We propose a formulation of a Lorentzian quantum geometry based on the
framework of causal fermion systems. After giving the general definition of
causal fermion systems, we deduce space-time as a topological space with an
underlying causal structure. Restricting attention to systems of spin dimension
two, we derive the objects of our quantum geometry: the spin space, the tangent
space endowed with a Lorentzian metric, connection and curvature. In order to
get the correspondence to differential geometry, we construct examples of
causal fermion systems by regularizing Dirac sea configurations in Minkowski
space and on a globally hyperbolic Lorentzian manifold. When removing the
regularization, the objects of our quantum geometry reduce precisely to the
common objects of Lorentzian spin geometry, up to higher order curvature
corrections.
