Maxwell's equations and quantization of the electromagnetic field

Abstract

In 1861, J. C. Maxwell published a theoretical and important work on electromagnetism. Indeed, this work contains essentially the equations that today are known as Maxwell's equations . These equations, together with the Lorentz force law F = q ( E + v × B ), constitute the complete set of laws for classical electromagnetism. An important property of Maxwell's equations is that they are Lorentz covariant. However, classical electromagnetism is, of course, non-quantized, as the prefix ‘classical’ suggests. Later on, in 1900, M. Planck introduced the concept of quantization of radiation that can be studied by using emission of radiation from heated bodies, which has become known as black-body radiation . Thus, the idea of quantization was born by Planck's discovery. Then, in 1905, Einstein introduced the concept of ‘quanta of light’, since the quantum nature of light was revealed by the photoelectric effect that was claimed by himself. In addition, the quanta of light were dubbed photons . In this chapter, we will first present Maxwell's equations, we will then discuss different gauges and quantization of the electromagnetic field, we will next consider the Casimir effect that arises when quantizing the electromagnetic field, and finally, we will try to obtain a covariant quantization of the electromagnetic field.