  We show that a notion of one-particle state and the corresponding vacuum
state exists in general curved backgrounds for spin $\frac{1}{2}$ fields. A
curved spacetime can be equipped with a coordinate system in which the metric
component $g_{--}=0$. We separate the component of the left-handed massless
Dirac field which is annihilated by the null vector $\partial_-$ and compute
the corresponding Feynman propagator. We find that the propagating modes are
localized on two dimensional subspaces and the Feynman propagator is similar to
the Feynman propagator of chiral fermions in two dimensional Minkowski
spacetime. Therefore, it can be interpreted in terms of one-particle states and
the corresponding vacuum state similarly to the second quantization in
Minkowski spacetime.
