Descriptions of Relativistic Dynamics with World Line Condition

TL;DR

This paper introduces a generalized relativistic dynamics framework using Poincaré algebra representations on tangent bundles, clarifying the algebra's role and analyzing the 'eleventh generator' formalism to address the no-interaction theorem.

Contribution

It presents a new realization of relativistic dynamics via vector fields on tangent bundles, elucidates the algebraic structure, and analyzes alternative formalisms that bypass the no-interaction theorem.

Findings

Explicit construction of Poincaré algebra realization on tangent bundles

Clarification of algebraic versus Poisson bracket descriptions

Analysis of the 'eleventh generator' formalism as a no-interaction theorem bypass

Abstract

In this paper a generalized form of relativistic dynamics is presented: A realization of the Poincar\'e algebra is provided in terms of vector fields on the tangent bundle of a simultaneity surface in $\mathbb{R}^4$. The construction of this realization is explicitly shown to clarify the role of the commutation relations of the Poincar\'e algebra versus their description in terms of Poisson brackets in the no-interaction theorem. Moreover, a geometrical analysis of the ``eleventh generator'' formalism introduced by Sudarshan and Mukunda is outlined, this formalism being at the basis of many proposals which evaded the no-interaction theorem.