Introduction to the Fock Quantization of the Maxwell Field

In this article we give an introduction to the Fock quantization of the Maxwell field. At the classical level, we treat the theory in both the covariant and canonical phase space formalisms. The approach is general since we consider arbitrary (globally-hyperbolic) space-times. The Fock quantization is shown to be equivalent to the definition of a complex structure on the classical phase space. As examples, we consider stationary space-times as well as ordinary Minkowski space-time. The account is pedagogical in spirit and is tailored to beginning graduate students. The paper is selfcontained and is intended to fill an existing gap in the literature.