  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.
  The scope 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.
  The field theoretic setup permits to work explicitly at every step of the
derivation of the trace formula with invariant quantities of classical periodic
orbits. R. Forman's theory of functional determinants of linear, non singular
elliptic operators yields the expression of quantum quadratic fluctuations
around classical periodic orbits directly in terms of the monodromy matrix of
the periodic orbits.
  The phase 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.
