Section II discusses three different representations to express the cavity-matter hybrid system in full space. It begins with the minimal coupling Hamiltonian in the Coulomb gauge (Sect. II A), where the light-matter interaction is mediated through the matter momentum and the photonic field's vector potential. Then, the Power-Zienau-Woolly (PZW) transformation is introduced under the long-wavelength approximation (or equivalently dipole approximation), and the dipole gauge and Pauli-Fierz Hamiltonians (Sect. II B) are derived from the minimal coupling Hamiltonian. Finally, a new representation, called the Asymptotically Decoupled Hamiltonian, is presented from the minimal coupling Hamiltonian and its advantageous convergence properties are discussed (Sect. II C).