In Section III, we go on to discuss complications that ensue due to a finite truncation of the infinite Hilbert space Hamiltonians in Section II and their corresponding resolutions. When projecting the dipole gauge Hamiltonian (Sect. II B) and minimal coupling Hamiltonian (Sect. II A) to a finite matter eigenbasis, the polariton energies do not match, indicating that performing a simple projection breaks the gauge invariance (Sect. III A). These gauge ambiguities can be understood since projecting the PZW operator looses its unitary property. By properly truncating all operators, these ambiguities can be resolved a so-called properly truncated Coulomb gauge Hamiltonian is formed (Sect. III B). Building off of this, by performing a change of basis and a phase rotation, the polarized Fock state representation can be formed from this properly truncated Coulomb gauge Hamiltonian, which for molecular systems can have its eigenspectrum converge for a very small basis set (Sect. III C). It has also been shown that truncating the number of photonic modes also leads to gauge ambiguities that can be resolved using a method similar to the resolution for the matter DOFs (Sect. III D). Additionally, these gauge ambiguities have recently been resolved for Hamiltonians beyond the long-wavelength approximation (Sect. III E).