A non-perturbative construction of the fermionic projector on globally hyperbolic manifolds I: Space-times of finite lifetime

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.

The chiral index of the fermionic signature operator

“…But both conditions are satisfied if we assume that the space-time (M, g) has finite lifetime in the sense that it admits a foliation (N t ) t∈(t 0 ,t 1 ) by Cauchy surfaces with t 0 , t 1 ∈ R such that the function ν, ∂ t is bounded on M (see [6,Definition 3.4] …”

“…We recall a few basic constructions from [6]. Let (M, g) be a smooth, globally hyperbolic Lorentzian spin manifold of even dimension k ≥ 2.…”

“…For the construction of the fermionic signature operator, we need to extend the bilinear form (2.4) to the solution space of the Dirac equation. In order to ensure that the integral in (2.4) exists, we need to make the following assumption (for more details see [6,Section 3.2]). Under this assumption, the space-time inner product is well-defined as a bounded bilinear form on H m , <.|.> :…”

“…Starting from a Lorentzian spin manifold, one can construct a corresponding causal fermion system by choosing H as a suitable subspace of the solution space of the Dirac equation, forming the local correlation operators (possibly introducing an ultraviolet regularization) and defining ρ as the push-forward of the volume measure on M (see [6,Section 4] or the examples in [4]). The advantage of working with a causal fermion system is that the underlying space-time does not need to be a Lorentzian manifold, but it can be a more general "quantum space-time" (for more details see [2]).…”

“…In the recent papers [6,7] the fermionic signature operator was introduced on globally hyperbolic Lorentzian spin manifolds. It is a bounded symmetric operator on the Hilbert space of solutions of the Dirac equation which depends on the global geometry of space-time.…”