A new approach to finding the exact Green's function to the time-dependent Schr\"odinger equation is described. The approach is conceptually simple: one seeks an operator which when commuted through the Schr\"odinger operator transforms it into the Schr\"odinger operator of a soluble problem. This approach provides an alternative to both the eigenfunction expansion of the propagator and the path integral. The first example of an exact propagator represented as a ``sum over classical paths'' which is not equal to the Gaussian approximation of the path integral is found.
In the complex action theory (CAT) we explicitly examine how the momentum and Hamiltonian are defined from the Feynman path integral (FPI) point of view based on the complex coordinate formalism of our foregoing paper. After reviewing the formalism briefly, we describe in FPI with a Lagrangian the time development of a
ξ-parametrized wave function, which is a solution to an eigenvalue problem of a momentum operator. Solving this eigenvalue problem, we derive the momentum, Hamiltonian, and Schrodinger equation. Oppositely, starting from the Hamiltonian we derive the Lagrangian in FPI, and we are led to the momentum relation again via the saddle point for
p. This study confirms that the momentum and Hamiltonian in the CAT have the same forms as those in the real action theory. We also show the third derivation of the momentum relation via the saddle point for
q.