Self-adjointness of the semi-relativistic Pauli-Fierz Hamiltonian

The spinless semi-relativistic Pauli-Fierz Hamiltonian H= (p o×1-A)²+M²+Vo× 1+1o×H_f, in quantum electrodynamics is considered. Here p denotes a momentum operator, A a quantized radiation field, M ≥ 0, H_f the free Hamiltonian of a Boson Fock space and V an external potential. The self-adjointness and essential self-adjointness of H are shown. It is emphasized that it includes the case of M = 0. Furthermore, the self-adjointness and the essential self-adjointness of the semi-relativistic Pauli-Fierz model with a fixed total momentum P ε^d: H(P) = s(P P _f - A(0))² + M² + H_f, M ≥ 0 is also proven for arbitrary P.