Maitra and coworkers 138 , as well as Tokatly and coworkers 137 , laid the basis for the conception of the exact-factorization formalism to treat the PEN wavefunction. Specifically, the factored form of the wavefunction Ψ(x, X,t) = χ(X,t)Φ(x,t; X) has been used to analyze various kinds of problems, like photon-electron dynamics, by interpreting X = q as the photon displacement and x = r as the electronic coordinate, or the full PEN dynamics, by interpreting X = R as the nuclear coordinates and x = r, q as the polaritonic coordinates. Maitra's preliminary work demonstrates that different factorizations yield complementary information on the fundamental interpretation of the PEN problem , as well as, perhaps most importantly, on the directions to follow to introduce approximations for an efficient computational treatment. For instance, employing the factorization Ψ(r, q,t) = χ(q,t)Φ(r,t; q), it was observed that even though the free-photon Hamiltonian is harmonic in the photon displacement coordinate and the light-matter coupling is bilinear in the nonrelativistic dipole approximation, the q-TDPES is far from harmonic. Therefore, a quasi-classical trajectory-based representation of the photon displacements and conjugated momenta should be carefully assessed and does not guarantee reproducing correctly the quantum dynamics. Specifically, the limitations of the multi-trajectory Ehrenfest approach have been discussed in the light of the different forces guiding the photon-displacement dynamics within such a mean-field theory and the exact factorization 141 . Fur-thermore, the factorization Ψ(r, R, q,t) = χ(R,t)Φ(r, q,t; R) was used to introduced the TDPES as an alternative tool to the polaritonic surfaces to interpret how photochemical reactivity is affected by the cavity in a model proton coupledelectron transfer . Such an exact factorization naturally inspires simulations based on the coupled-trajectory schemes presented in previous sections, similarly to what the community already proposed using, for instance, surface hopping 287 or ab initio multiple spawning 291 and in the spirit of the Floquet-based CTMQC 147 method discussed briefly in Section III A. Nonetheless, this possibility has not been yet investigated.