A Lorentzian quantum geometry

Abstract

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.