1 - The classical electromagnetic field in the absence of sources

Summary

Introduction. The principal aim of this chapter is to familiarize the reader with the notation adopted in the text, as well as to introduce some concepts, such as the energy-momentum tensor of the electromagnetic field, the partition of its total angular momentum into an orbital and a spin contribution and its expansion in vector spherical harmonics, which are not usually included in an undergraduate course on electrodynamics. The chapter is entirely dedicated to the classical electromagnetic field in the absence of charges and currents. In the first two sections we present Maxwell's equations, the vector potential and different forms of the Lagrangian density of the free field from which Maxwell's equations can be obtained as Euler-Lagrange equations. In Section 1.3 we discuss briefly the properties of the field under pure Lorentz transformation and tensor notation. Then we introduce the concept of local gauge invariance and of gauge transformation, and we define the constraints leading to the Lorentz and to the Coulomb gauge. Using a canonical formalism, in Section 1.5 we obtain the Hamiltonian density of the field in the Coulomb gauge. The energy-momentum tensor of the field, the momentum and the angular momentum, along with their important conservation properties, are discussed in Section 1.6.