Conformal Invariance and Electrodynamics: Applications and General Formalism

TL;DR

This paper explores the application of conformal invariance to electrodynamics, demonstrating how conformal transformations can be used to analyze electromagnetic fields and reformulate Maxwell's equations on a conformal compactification of Minkowski space.

Contribution

It introduces a new reformulation of Maxwell's equations on the projective cone that respects additional gauge invariances and provides insights into electromagnetic fields of accelerating charges.

Findings

Derived electromagnetic fields for uniformly accelerating charges using conformal transformations.

Reformulated Maxwell's equations on the projective cone with enhanced gauge invariances.

Demonstrated the linear realization of conformal transformations in electrodynamics.

Abstract

The role of the conformal group in electrodynamics in four space-time dimensions is re-examined. As a pedagogic example we use the application of conformal transformations to find the electromagnetic field for a charged particle moving with a constant relativistic acceleration from the Coulomb electric field for the particle at rest. We also re-consider the reformulation of Maxwell's equations on the projective cone, which is isomorphic to a conformal compactification on Minkowski space, so that conformal transformations, belonging to the group O(4,2), are realised linearly. The resulting equations are different from those postulated previously and respect additional gauge invariances which play an essential role in ensuring consistency with conventional electrodynamics on Minkowski space. The solution on the projective cone corresponding to a constantly accelerating charged particle is discussed.