Extended Particles and the Exterior Calculus

TL;DR

This paper introduces an intrinsic geometric calculus for analyzing classical extended relativistic systems like particles, strings, and membranes, simplifying their equations of motion and boundary conditions in a gauge-invariant manner across various spacetimes.

Contribution

It presents a unified geometric framework for classical extended systems and derives their equations of motion and boundary conditions using a gauge-invariant approach.

Findings

Unified treatment of particles, strings, and membranes

Simplified derivation of equations of motion

Gauge-invariant boundary conditions

Abstract

These notes were delivered as a series of NIMROD lectures at the Rutherford Appleton Laboratory by the author in February 1976 (RL-76-022). The purpose of these lectures was primarily two-fold: to discuss the classical theory of free point particles, free strings and free membranes from a unified viewpoint; and to present in the process of doing this the rudiments of an intrinsic geometrical calculus that the author has found of immense value in investigating these systems. It is shown how the equations of motion for such classically extended relativistic systems arise in a very simple manner from a principle of stationary action and furthermore how the boundary conditions for finite systems may be derived in a gauge invariant way. Momenta are naturally introduced and the primary constraints that exist in a Hamiltonian description follow simply. Calculations may proceed in an index-free manner until components are required. It is at this stage that one can, if one desires, impose gauge conditions and remove non-independent degrees of freedom. Such methods can be applied in any spacetime of any dimension and metric, and examples are given throughout.