Aspects of Diffeomorphism Invariant Theory of Extended Objects

TL;DR

This paper develops a diffeomorphism invariant Lagrangian framework for extended objects like branes, unifying various models and exploring their covariant quantization, including the Dirac equation formulation.

Contribution

It introduces a universal covariant Lagrangian approach for extended objects, connecting point particles, strings, and branes with a novel Dirac equation formulation.

Findings

Unified Lagrangian structure for extended objects.

Universal mass-shell and Klein-Gordon equations with gravity.

A Dirac equation for branes using non-commuting gamma variables.

Abstract

The structure of a diffeomorphism invariant Lagrangians for an extended object W embedded in a bulk space M is discussed by following a close analogy with the relativistic particle in electromagnetic field as a system that is reparametrization-invariant. The current construction naturally contains, relativistic point particle, string theory, and Dirac--Nambu--Goto Lagrangians with Wess--Zumino terms. For Lorentzian metric field, the non-relativistic theory of an integrally submerged W-brane is well defined provided that the brane does not alter the background interaction fields. A natural time gauge is fixed by the integral submergence (sub-manifold structure) within a Lorentzian signature structure. A generally covariant relativistic theory for the discussed brane Lagrangians is also discussed. The mass-shell constraint and the Klein--Gordon equation are shown to be universal when gravity-like interaction is present. A construction of the Dirac equation for the W-brane that circumvents some of the problems associated with diffeomorphism invariance of such Lagrangians by promoting the velocity coordinates into a non-commuting gamma variables is presented.