As discussed in the previous chapter, we adopt the Coulomb gauge for all our further investigations starting from the many-body Hamiltonian (8.86) and the mode expansion (8.87). Before we proceed to quantize the Hamiltonian, we want to make sure that our analysis is focused on the nontrivial quantum phenomena. Thus, we first have to identify and efficiently deal with the trivial parts of the problem.

Often, the experimental conditions are chosen such that only a subset of all the electrons in a solid interacts strongly with the transversal electromagnetic fields while the remaining electrons and the ions are mostly passive. To describe theoretically such a situation in an efficient way, it is desirable to separate the dynamics of reactive electrons from the almost inert particles that merely produce a background contribution. This background can often be modeled as an optically passive response that is frequency independent and does not lead to light absorption.

In this chapter, we show how the passive background contributions can be systematically identified and included in the description. As the first step, we introduce the generalized Coulomb gauge to eliminate the scalar potential and to express the mode functions and the canonical variables. This leads us to a new Hamiltonian with altered Coulomb potential and mode functions. This generalized Hamiltonian allows us to efficiently describe optically active many-body systems in the presence of an optically passive background.