When deriving QED from first principles, one often uses the minimal coupling Hamiltonian in the Coulomb gauge . From there, the electric-dipole Hamiltonian can be found via a gauge transformation. The commonly used Pauli-Fierz (PF) QED Hamiltonian in recent studies of polariton chemistry can be obtained by applying another gauge transformation on the electric-dipole Hamiltonian. We will further discuss the consequence of matter state truncation on gauge invariance, the connection with the commonly used quantum optics model Hamiltonians, and when they will break down in molecular QED.