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A Lorentzian quantum geometry. (English) Zbl 1346.82002

Summary: 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.

MSC:

82B10 Quantum equilibrium statistical mechanics (general)
83C65 Methods of noncommutative geometry in general relativity
83C60 Spinor and twistor methods in general relativity and gravitational theory; Newman-Penrose formalism
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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