Tutorial: Macroscopic QED and vacuum forces

This tutorial introduces the theory of macroscopic QED, where a Hamiltonian is found that represents the electromagnetic field interacting with a dispersive, dissipative material. Using a one dimensional theory as motivation, we then build up the more cumbersome three dimensional theory. Then considering the extension of this theory to moving materials, where the material response changes due to both the Doppler effect and the mixing of electric and magnetic responses, it is shown that one gets the theory of quantum electromagnetic forces for free. We finish by applying macroscopic QED to reproduce Pendry’s expression for the quantum friction force between sliding plates. The universe was a language with a perfectly ambiguous grammar. Every physical event was an utterance that could be parsed in two entirely different ways, one causal and the other teleological. Ted Chiang, Story of your life and other stories, 1998 I. PRELIMINARY REMARKS This tutorial describes the basics of a fully quantum mechanical approach to the theory of vacuum forces, one based upon the principle of least action. As well as reclaiming and justifying some of the key equations of Lifshitz theory [1] (the workhorse of vacuum force calculations) here we shall also find some new ones, and all within a framework familiar from basic quantum mechanics: we shall work with a Hamiltonian operator and a wavefunction to describe the field, the body, and its motion. The advantage of this approach is that it contains no assumptions about the state of the medium or the field, beyond the fact that the macroscopic Maxwell equations are valid. In the domain of Casimir physics we are in an interesting regime where we wish to calculate tiny forces on objects that are too large for us to use a microscopic theory. Yet the force stems from an electromagnetic field with a very low amplitude, so that the description must also be quantum mechanical. As previous chapters have indicated, in this situation we might expect the electromagnetic field to obey quantised versions of the macroscopic