A Lorentzian Quantum Geometry

TL;DR

This paper introduces a Lorentzian quantum geometry framework based on causal fermion systems, deriving classical geometric objects from quantum structures and demonstrating their correspondence to traditional Lorentzian geometry in the continuum limit.

Contribution

It formulates a novel Lorentzian quantum geometry using causal fermion systems, connecting quantum structures to classical Lorentzian geometry objects.

Findings

Derived Lorentzian metric, connection, and curvature from causal fermion systems.

Constructed examples from regularized Dirac sea configurations in Minkowski space.

Showed that quantum geometric objects reduce to classical Lorentzian geometry objects upon removing regularization.

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.