The semiclassical initial value series representation of the quantum propagator

One of the central open challenges of the 21st century is the computation of real time quantum dynamics for systems with "many" degrees of freedom. A promising approach for obtaining approximate real time quantum dynamics is through the use of the semiclassical initial value approximation for the exact quantum propagator. The main drawback of this class of approximations was its ad hoc nature, it was in many senses an uncontrolled approximation scheme. This drawback has been recently remedied by showing that the semiclassical initial value representation (SCIVR) propagator is just a leading order term in a formally exact series representation of the true quantum propagator. In this review we present the SCIVR series representation, its successes and future challenges in applications to "large" systems. In addition, a new interaction representation initial value series representation for the exact quantum propagator is formulated.