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Gutzwiller's famous semiclassical trace formula plays an important role intheoretical and experimental quantum mechanics with tremendous success. Wereview the physical derivation of this deep periodic orbit theory in terms ofthe phase space formulation with an view towards the Hamiltonian dynamicalsystems. The Maslov phase appearing in the trace formula is clarified byMeinrenken as Conley-Zhender index for periodic orbits of Hamiltonian systems.We also survey and compare various versions of Maslov indices to establish thisfact. A refinement and improvement to Conley-Zehnder's index theory which wewill recall all essential ingredients is the Maslov-type index theory forsymplectic paths developed by Long and his collaborators which would shed newlight on the computations and understandings on the semiclassical traceformula. The insights in Gutzwiller's work also seems plausible to the studieson Hamiltonian systems.

Gutzwiller's Semiclassical Trace Formula and Maslov-Type Index Theory for Symplectic Paths