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.
To describe physics properly, we need to take into account quantum effects. Thus, for every non-quantum physical theory, we must come up with an appropriate quantum theory. A traditional approach is to replace all the scalars in the classical description of this theory by the corresponding operators. The problem with the above approach is that due to non-commutativity of the quantum operators, two mathematically equivalent formulations of the classical theory can lead to different (non-equivalent) quantum theories. An alternative quantization approach that directly transforms the non-quantum action functional into the appropriate quantum theory, was indeed proposed by the Nobelist Richard Feynman, under the name of path integration. Feynman path integration is not just a foundational idea, it is actually an efficient computing tool (Feynman diagrams).\nFrom the pragmatic viewpoint, Feynman path integral is a great success. However, from the foundational viewpoint, we still face an important question: why the Feynman's path integration formula? In this paper, we provide a natural explanation for Feynman's path integration formula.