Matter, Fields, and Reparametrization-Invariant Systems

TL;DR

This paper investigates reparametrization-invariant systems such as relativistic particles and d-branes, analyzing their matter Lagrangians with background interactions and deriving universal equations like the Klein-Gordon and Dirac equations.

Contribution

It introduces a novel approach to formulating the Dirac equation without relying on the Klein-Gordon equation, within the context of reparametrization-invariant systems.

Findings

Universal form of the mass-shell constraint and Klein-Gordon equation with gravity-like interactions.

Non-relativistic equations derived assuming integral sub-manifold embedding of d-branes.

A new technique for the Dirac equation based on Rund's algebra of gamma matrices.

Abstract

We study reparametrization-invariant systems, mainly the relativistic particle and its D-dimensional extended object generalization--d-brane. The corresponding matter Lagrangians naturally contain background interactions, like electromagnetism and gravity. For a d-brane that doesn't alter the background fields, we define non-relativistic equations assuming integral sub-manifold embedding of the d-brane. The mass-shell constraint and the Klein-Gordon equation are shown to be universal when gravity-like interaction is present. Our approach to the Dirac equation follows Rund's technique for the algebra of the $\gamma $-matrices that doesn't rely on the Klein-Gordon equation.