  We construct the propagator of the massless Dirac operator $W$ on a closed
Riemannian 3-manifold as the sum of two invariantly defined oscillatory
integrals, global in space and in time, with distinguished complex-valued phase
functions. The two oscillatory integrals -- the positive and the negative
propagators -- correspond to positive and negative eigenvalues of $W$,
respectively. This enables us to provide a global invariant definition of the
full symbols of the propagators (scalar matrix-functions on the cotangent
bundle), a closed formula for the principal symbols and an algorithm for the
explicit calculation of all their homogeneous components. Furthermore, we
obtain small time expansions for principal and subprincipal symbols of the
propagators in terms of geometric invariants. Lastly, we use our results to
compute the third local Weyl coefficients in the asymptotic expansion of the
eigenvalue counting functions of $W$.
