Interaction of an electron with a multimode quantized radiation field

Photon operators that obey commutation relations form Lie algebras. By identifying the Hamiltonian of an interacting system of one electron and a multimode many-photon field in a momentum representation as an element of a Cartan subalgebra, we are able to find the automorphism that maps the Hamiltonian on to that of the noninteracting system. By this method we find the exact solutions as new solutions of the Schrödinger equation for the case of one electron in a multimode many-photon field. The solutions show the existence of both ponderomotive energy and momentum, which obey the rules of linear superposition among the photon modes.