A system of a Dirac particle interacting with the radiation field is considered. The Hamiltonian of the system is defined by $H = \alpha\cdot(\hat\mathbf{ p}-q\mathbf{ A}(\hat\mathbf{ x}))+m\beta + H_f$ where $q\in\mathbb{ R}$ is a coupling constant, $\mathbf{ A}(\hat\mathbf{ x})$ denotes the quantized vector potential and $H_f$ denotes the free photon Hamiltonian. Since the total momentum is conserved, $H$ is decomposed with respect to the total momentum with fiber Hamiltonian $H(\mathbf{ p}), (\mathbf{ p}\in\mathbb{ R}^3)$. Since the self-adjoint operator $H(\mathbf{ p})$ is bounded from below, one can define the lowest energy $E(\mathbf{ p},m):=\inf\sigma(H(\mathbf{ p}))$. We prove that $E(\mathbf{ p},m)$ is an eigenvalue of $H(\mathbf{ p})$ under the following conditions: (i) infrared regularization and (ii) $E(\mathbf{ p},m)<E(\mathbf{ p},0)$. We also discuss the polarization vectors and the angular momenta.
Categories: math-ph math.MP