Momentum relation and classical limit in the future-not-included complex action theory

TL;DR

This paper investigates the momentum relation and classical limit in the future-not-included complex action theory, revealing that the standard relation does not hold and providing a corrected formulation with implications for classical dynamics.

Contribution

It introduces the correct momentum relation for the future-not-included complex action theory and links it to a real action-based classical theory, extending previous work.

Findings

The standard momentum relation is invalid in the future-not-included theory.

A corrected momentum relation is derived using a formal Lagrangian approach.

The classical limit is described by a real action, connecting to classical mechanics.

Abstract

Studying the time development of the expectation value in the future-not-included complex action theory, we point out that the momentum relation (the relation analogous to $p=\frac{\partial L}{\partial \dot{q}}$), which was derived via the Feynman path integral and was shown to be correct in the future-included theory in our previous papers, is not valid in the future-not-included theory. We provide the correct momentum relation in the future-not-included theory, and argue that the future-not-included classical theory is described by a certain real action. In addition, we provide another way to understand the time development of the future-not-included theory by utilizing the future-included theory. Furthermore, properly applying the method used in our previous paper to the future-not-included theory by introducing a formal Lagrangian, we derive the correct momentum relation in the future-not-included theory.