Quantum optics is the study of light interacting with matter on a small scale. Many-body interactions in quantum optics are of interest to the study of cold atom systems and quantum semiconductors, and have application in quantum information. Frequently, these interactions are studied within the framework of quantum field theory, with photons described by their frequency and atoms described by their positions. I will describe a model which places both the electromagnetic field and matter on equal footing described spatially. This model produces a pair of nonlocal partial differential equations describing the energy eigenstates that have at most one photon present in the field.

Next, I will consider the case that the density of atoms is periodic with respect to a lattice and describe a structure theorem for the spectrum (allowable energies) of the Hamiltonian. Finally I will discuss the existence of linear crossings of the spectral bands, also known as Dirac points, in two cases:

1. Low contrast: Atomic densities which are weak perturbations of a constant.

2. High contrast: Atomic densities which are sums of shifted scaled atomic inclusions."

**Note**:

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