  The emergence of the concept of a causal fermion system is revisited and
further investigated for the vacuum Dirac equation in Minkowski space. After a
brief recap of the Dirac equation and its solution space, in order to allow for
the effects of a possibly nonstandard structure of spacetime at the Planck
scale, a regularization by a smooth cutoff in momentum space is introduced, and
its properties are discussed. Given an ensemble of solutions, we recall the
construction of a local correlation function, which realizes spacetime in terms
of operators. It is shown in various situations that the local correlation
function maps spacetime points to operators of maximal rank and that it is
closed and homeomorphic onto its image. It is inferred that the corresponding
causal fermion systems are regular and have a smooth manifold structure. The
cases considered include a Dirac sea vacuum and systems involving a finite
number of particles and antiparticles.
