  Gutzwiller's famous semiclassical trace formula plays an important role in
theoretical and experimental quantum mechanics with tremendous success. We
review the physical derivation of this deep periodic orbit theory in terms of
the phase space formulation with an view towards the Hamiltonian dynamical
systems. The Maslov phase appearing in the trace formula is clarified by
Meinrenken as Conley-Zhender index for periodic orbits of Hamiltonian systems.
We also survey and compare various versions of Maslov indices to establish this
fact. A refinement and improvement to Conley-Zehnder's index theory which we
will recall all essential ingredients is the Maslov-type index theory for
symplectic paths developed by Long and his collaborators which would shed new
light on the computations and understandings on the semiclassical trace
formula. The insights in Gutzwiller's work also seems plausible to the studies
on Hamiltonian systems.
