Electromagnetism, Elliptic Systems and Tractor Calculus
by
A. Rod Gover
University of Auckland
Coauthors: Tom Branson (University of Iowa)

In the classical theory of electromagnetism and in quantised versions thereof the ``Lorentz gauge'' has often been used to restrict the gauge freedom of the field potential. However this gauge equation is not conformally invariant even though the field equations themselves are. In 1984 M.G. Eastwood and M. Singer produced an alternative gauge equation that is conformally invariant on solutions of the field equations. Their system can be very elegantly described in terms of a new tool in differential geometry called tractor calculus. Investigating why this is the case has revealed a rather beautiful picture involving some elementary Lie algebra representation theory. Here via purely algebraic tools we can see not only how to predict the Eastwood-Singer construction but how to manufacture similar gauge systems for other field equations. Switching to the positive signature setting we see the gauge operators naturally complete the field operators to elliptic differential operators.

This is collaborative work with Tom Branson (University of Iowa).

Date received: September 29, 2000