Momentum and Hamiltonian in Complex Action Theory

TL;DR

This paper investigates how momentum and Hamiltonian are defined within complex action theory using Feynman path integrals, confirming their forms are consistent with real action theory.

Contribution

It provides a detailed derivation of momentum and Hamiltonian in complex action theory from FPI, connecting Lagrangian and Hamiltonian formalisms.

Findings

Momentum and Hamiltonian in CAT match those in real action theory.

Derived the Schrödinger equation within the complex formalism.

Confirmed the consistency of formalism via multiple derivations.

Abstract

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 $q$.