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

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.