  A Duistermaat-Guillemin-Gutzwiller trace formula for Dirac-type operators on
a globally hyperbolic spatially compact stationary spacetime is achieved by
generalising the recent construction by Strohmaier and Zelditch [Adv. Math.
\textbf{376}, 107434 (2021)] to a vector bundle setting. We have analysed the
spectrum of the Lie derivative with respect to a global timelike Killing vector
field on the solution space of the Dirac equation and found that it consists of
discrete real eigenvalues. The distributional trace of the time evolution
operator has singularities at the periods of induced Killing flow on the
manifold of lightlike geodesics. This gives rise to the Weyl law asymptotic at
the vanishing period. A pivotal technical ingredient to prove these results is
the Feynman propagator. In order to obtain a Fourier integral description of
this propagator, we have generalised the classic work of Duistermaat and
H\"{o}rmander [Acta Math. \textbf{128}, 183 (1972)] on distinguished
parametrices for normally hyperbolic operators on a globally hyperbolic
spacetime by propounding their microlocalisation theorem to a bundle setting.
As a by-product of these analyses, another proof of the existence of Hadamard
bisolutions for a normally hyperbolic operator (resp. Dirac-type operator) is
reported.
