Even with the numerous recent advances in the field, there are still many mysteries to be solved in polariton chemistry. Typically, the first step in approaching these challenges is to formulate the Hamiltonian to describe the system. With a summary of the different representations, truncation schemes, and levels of approximation in various cQED Hamiltonians, we hope that this review will provide theorists with a full toolbox such that they can fit the best method to their own application and start unraveling the mysteries of the field. is defined as being perpendicular to the propagation direction and parallel to the interface. The TM mode's polarization, êk,TM , is defined as being perpendicular to both the propagation direction and the TE polarization. For a given polarization, êk , the transverse electric field is along êk and the magnetic field is along the k×ê k direction. For example, for the TM mode, the transverse electric field polarization is along êk,TE and the transverse magnetic field polarization is along -ê k,TM .