SEMICLASSICAL PROPAGATION - PHASE INDEXES AND THE INITIAL-VALUE FORMALISM

Initial-value phase-space integral representations for a time-dependen t propagator are obtained in the coordinate and momentum representatio ns. To do so we first derive nonuniform semiclassical propagators for the various representations, obtaining the global-time semiclassical p hase indices (Maslov indices) due to caustics. Results include readily implementable general phase index formulas for any type of caustic, i ncluding cases where the Morse index theorem is inapplicable. The meth od of obtaining the indices is general and based simply on concatenati ng uniform short-time propagators which also gives rise to alternative path-integral forms. Initial-value integral representations are then derived by introducing a method of extending short-time initial-value propagator formulas to global times via a simple stationary-phase asym ptotic-equivalence approach. The integrals reduce to the nonuniform se miclassical propagators within the stationary-phase approximation, are uniform about caustics, and have integrand phases which properly acco unt for the global-time phases in terms of appropriate Maslov indices. The initial-value integrals are also consistently derived via a canon ical mapping procedure on the coordinate-space path integral. Initial- value integrals for time-dependent wave-function propagation are also given. Evaluation of the initial-value integral expressions do not req uire trajectory root searches for propagation.