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

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