Derivation of semiclassical asymptotic initial value representations of the quantum propagator

We present a formalism based on the Bargmann (coherent state) representation of states and operators to derive asymptotic semiclassical initial value representations of the quantum propagator for general multidimensional systems. We first derive a semiclassical WKB-like approximation to the general solution of the multidimensional time dependent Schrödinger equation in the Bargmann representation. From here, we readily obtain the semiclassical asymptotic form of the coherent-state matrix elements of the propagator. This form includes terms depending on the quantization scheme chosen to quantize a classical Hamilton function or the classical symbol chosen for a given quantum Hamiltonian. From this expression and its analytic properties we derive through asymptotic saddle point approximations a whole family of semiclassical initial value representations of the quantum propagator, all of which belong to the Herman-Kluk (HK) class. A parameter in this family determines either the quantization scheme or the Hamiltonian classical symbol. The Wigner-Weyl choice for it leads to the HK propagator. Potential applications for other choices are discussed.