Semiclassical trace formulas in the presence of continuous symmetries

We derive generalizations of the semiclassical trace formula of Gutzwiller [J. Math. Phys. 12, 343 (1971)] and Balian and Bloch [Ann. Phys. 69, 76 (1972)] that are valid for systems exhibiting continuous symmetries. In particular, we consider symmetries for which the associated set of conserved quantities Poisson-commute. For these systems, the periodic orbits of a given energy occur in continuous families and the usual trace formula, which is valid only when the periodic orbits of a given energy are isolated, does not apply. In the trace formulas we derive, the density of states is determined by a sum over continuous families of periodic orbits rather than a sum over individual periodic orbits. Like Gutzwiller's formula for isolated orbits, the sum involves intrinsic, canonically invariant properties of the periodic orbits. We illustrate the theory with two important special cases: axial symmetry and integrable systems.