Quantum Field Theory and Gravity: Conceptual and Mathematical ...

Causal Fermion Systems 175
In order to compare with the metric connection ∇ of Definition 3.11, we
subdivide γ (for simplicity with equal spacing, although a non-uniform spacing
would work just as well). Thus for any given N, we define the points
x 0 ,...,x N by
x n = γ(t n ) with t n = nT N .
We define the parallel transport ∇ N x,y by successively composing the parallel
transport between neighboring points,
∇ N x,y := ∇ xN ,x N−1
∇ xN−1 ,x N−2 ···∇ x1 ,x 0
: T y → T x .
Our first theorem gives a connection to the Minkowski vacuum. For any
ε>0 we regularize on the scale ε>0 by inserting a convergence-generating
factor into the integrand in (5),
∫
P ε d 4 k
(x, y) =
(2π) (k/ + m) 4 δ(k2 − m 2 )Θ(−k 0 ) e εk0 e −ik(x−y) . (41)
This function can indeed be realized as the kernel of the fermionic operator
(15) corresponding to a causal fermion system (H, F,ρ ε ). Here the measure
ρ ε is the push-forward of the volume measure in Minkowski space by an
operator F ε , being an ultraviolet regularization of the operator F in (2)-(4)
(for details see [14, Section 4]).
Theorem 3.15. For given γ, we consider the family of regularized fermionic
projectors of the vacuum (P ε ) ε>0 as given by (41). Then for a generic curve
γ and for every N ∈ N, thereisε 0 such that for all ε ∈ (0,ε 0 ] and all
n =1,...,N, the points x n and x n−1 are spin-connectable, and x n+1 lies in
the future of x n (according to Definition 3.10). Moreover,
∇ LC
x,y = lim lim
N→∞ ε↘0 ∇N x,y .
By a generic curve we mean that the admissible curves are dense in the
C ∞ -topology (i.e., for any smooth γ and every K ∈ N, there is a sequence γ l
of admissible curves such that D k γ l → D k γ uniformly for all k =0,...,K).
The restriction to generic curves is needed in order to ensure that the Euclidean
and directional sign operators are generically separated (see Definition
3.8(b)). The proof of the above theorem is given in [14, Section 4].
Clearly, in this theorem the connection ∇ LC
x,y is trivial. In order to show
that our connection also coincides with the Levi-Civita connection in the case
with curvature, in [14, Section 5] a globally hyperbolic Lorentzian manifold
is considered. For technical simplicity, we assume that the manifold is flat
Minkowski space in the past of a given Cauchy hypersurface.
Theorem 3.16. Let (M,g) be a globally hyperbolic manifold which is isometric
to Minkowski space in the past of a given Cauchy hypersurface N . For given
γ, we consider the family of regularized fermionic projectors (P ε ) ε>0 such
that P ε (x, y) coincides with the distribution (41) if x and y lie in the past of
N . Then for a generic curve γ and for every sufficiently large N, thereisε 0
such that for all ε ∈ (0,ε 0 ] and all n =1,...,N, the points x n and x n−1 are