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.

65 pages, LaTeX, 4 figures, many small improvements (published version)

References in corpus (5)

• A Lorentzian Quantum Geometry
• Causal Fermion Systems: A Quantum Space-Time Emerging from an Action Principle
• A Non-Perturbative Construction of the Fermionic Projector on Globally Hyperbolic Manifolds I - Space-Times of Finite Lifetime
• From Discrete Space-Time to Minkowski Space: Basic Mechanisms, Methods and Perspectives
• An Action Principle for an Interacting Fermion System and its Analysis in the Continuum Limit

Cited by in corpus (33)

• A Lorentzian Quantum Geometry
• A Non-Perturbative Construction of the Fermionic Projector on Globally Hyperbolic Manifolds I - Space-Times of Finite Lifetime
• Causal Fermion Systems: A Quantum Space-Time Emerging from an Action Principle
• Causal Fermion Systems as a Candidate for a Unified Physical Theory
• A Hamiltonian Formulation of Causal Variational Principles
• Causal Fermion Systems: An Elementary Introduction to Physical Ideas and Mathematical Concepts
• On the Support of Minimizers of Causal Variational Principles
• A Non-Perturbative Construction of the Fermionic Projector on Globally Hyperbolic Manifolds II -- Space-Times of Infinite Lifetime
• On the Structure of Minimizers of Causal Variational Principles in the Non-Compact and Equivariant Settings
• Noether-Like Theorems for Causal Variational Principles
• Causal Fermion Systems: A Primer for Lorentzian Geometers
• Entanglement and Second Quantization in the Framework of the Fermionic Projector
• Spinors on Singular Spaces and the Topology of Causal Fermion Systems
• The Regularized Hadamard Expansion
• Two-Dimensional Area and Matter Flux in the Theory of Causal Fermion Systems
• The Linear Dynamics of Wave Functions in Causal Fermion Systems
• A Mechanism of Baryogenesis for Causal Fermion Systems
• A Gauge Fixing Procedure for Causal Fermion Systems
• The Fermionic Projector, Entanglement, and the Collapse of the Wave Function
• A holographic map of action onto entropy
• Causal Fermion Systems: Classical Gravity and Beyond
• Causal Fermion Systems and the ETH Approach to Quantum Theory
• Modified Measures as an Effective Theory for Causal Fermion Systems
• On the Mathematical Foundations of Causal Fermion Systems in Minkowski Space
• Causal Fermion Systems and Octonions
• On the Initial Value Problem for Causal Variational Principles
• Dynamical Gravitational Coupling as a Modified Theory of General Relativity
• Baryogenesis in Minkowski Spacetime
• The Chiral Index of the Fermionic Signature Operator
• A Mechanism for Dark Matter and Dark Energy in the Theory of Causal Fermion Systems
• A Positive Quasilocal Mass for Causal Variational Principles
• Holographic Mixing and Fock Space Dynamics of Causal Fermion Systems
• Introduction to Causal Fermion Systems