Spectral Analysis of the Dirac Polaron

Abstract

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, $\mathbf{A}(\hat{\mathbf{x}})$ the quantized vector potential and $H_f$ 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 momentums.