Canonical Proper Time Formulation for Physical Systems

TL;DR

This paper introduces a canonical proper time framework for relativistic dynamics that describes classical and quantum systems using their own local time, providing new insights into system evolution and interactions.

Contribution

It presents a novel formalism that employs a canonical transformation of the time variable, applicable to both classical and quantum systems, without altering other dynamical variables.

Findings

Eigenvalues of the hydrogen atom calculated within the framework

Application to near horizon scalar field dynamics near a Schwarzschild black hole

Formalism summarized with example calculations

Abstract

The canonical proper time formulation of relativistic dynamics provides a framework from which one can describe the dynamics of classical and quantum systems using the clock of those very systems. The framework utilizes a canonical transformation on the time variable that is used to describe the dynamics, and does not transform other dynamical variables such as momenta or positions. This means that the time scales of the dynamics are described in terms of the natural local time coordinates, which is the most meaningful parameterization of phenomena such as the approach to equilibrium, or the back reaction of interacting systems. We summarize the formalism of the canonical proper time framework, and provide example calculations of the eigenvalues of the hydrogen atom and near horizon description of a scalar field near a Schwarzschild black hole.