8 - The Twistor Transform

Summary

Introduction and Motivation

There are two ways of describing a free Maxwell field on Minkowski space:

1. As a 2-form, F, subject to dF = 0 and d * F = 0 where * is the Hodge *-operator.

2. As an equivalence class of I-forms, Φ subject to d * dΦ = 0 where two 1-forms are said to be equivalent if their difference is of the form df for some function f.

Locally, these two descriptions are the same, the relationship between them being given by F = dΦ. The 1-form Φ is called a potential for the field F. Globally, this mapping Φ → F is neither injective nor surjective. There is a topological obstruction to finding a Φ for a given F and a similar topological obstruction to finding an f such that df = Φ when dΦ = O. This global inequivalence is of great interest since it requires one to determine experimentally which, if any, is the physically relevant description. The rather unexpected answer is ‘none of the above.’ Nowadays, it is well known that some combination of these two is required. The equivalence class of potentials Φ may be precisely re-interpreted as a connection on a trivial line bundle and allowing non-trivial line bundles with connection allows arbitrary fields F (as the curvature of this connection). The Aharonov-Bohm effect [2] provides a physical justification for this reformulation.