# Momentum and Hamiltonian in Complex Action Theory. (arXiv:1105.1294v5 [quant-ph] UPDATED)

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 $\xi$-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 Schr\"{o}dinger 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 $q$.