A Non-Perturbative Construction of the Fermionic Projector on Globally Hyperbolic Manifolds I - Space-Times of Finite Lifetime

TL;DR

This paper constructs the fermionic projector on finite-lifetime globally hyperbolic manifolds using functional analysis, linking it to quantum field theory concepts and analyzing specific cosmological models.

Contribution

It provides a non-perturbative, functional analytic method to construct the fermionic projector on such manifolds, connecting it to causal fermion systems and cosmological solutions.

Findings

Representation of the fermionic projector as a two-point distribution.

Ultraviolet regularization connects to causal fermion systems.

Quantitative analysis of the fermionic projector in Friedmann-Robertson-Walker universes.

Abstract

We give a functional analytic construction of the fermionic projector on a globally hyperbolic Lorentzian manifold of finite lifetime. The integral kernel of the fermionic projector is represented by a two-point distribution on the manifold. By introducing an ultraviolet regularization, we get to the framework of causal fermion systems. The connection to the "negative-energy solutions" of the Dirac equation and to the WKB approximation is explained and quantified by a detailed analysis of closed Friedmann-Robertson-Walker universes.