Path integration over closed loops and Gutzwillerʹs trace formula

In 1967 M.C. Gutzwiller succeeded to derive the semiclassical expression of the quantum energy density of systems exhibiting a chaotic Hamiltonian dynamics in the classical limit. The result is known as the Gutzwiller trace formula. ope of this review is to present in a self-contained way recent developments in functional determinant theory allowing to revisit the Gutzwiller trace formula in the spirit of field theory. eld theoretic setup permits to work explicitly at every step of the derivation of the trace formula with invariant quantities of classical periodic orbits. R. Formanʹs theory of functional determinants of linear, nonsingular elliptic operators yields the expression of quantum quadratic fluctuations around classical periodic orbits directly in terms of the monodromy matrix of the periodic orbits. ase factor associated to quadratic fluctuations, the Maslov phase, is shown to be specified by the Morse index for closed extremals, also known as Conley and Zehnder index.