A system of a Dirac particle interacting with the radiation field is considered. The Hamiltonian of the system is defined by \( H = \balpha\cdot(\hat\bp-q\bA(\hat\bx))+m\beta + H_f \), where \( q\in\BR \) is a coupling constant, \( \bA(\hat\bx) \) the quantized vector potential and the free photon Hamiltonian. Since the total momentum is conserved, is decomposed with respect to the total momentum with fiber Hamiltonian \( H(\bp), (\bp\in\BR^3) \). Since the self-adjoint operator \( H(\bp) \) is bounded from below, one can define the lowest energy \( E(\bp,m):=\inf\sigma(H(\bp)) \). We prove that \( E(\bp,m) \) is an eigenvalue of \( H(\bp) \) under the following conditions: (i) infrared regularization and (ii) \( E(\bp,m)<E(\bp,0) \). We also discuss the polarization vectors and the angular momentums.
Cite this article
Itaru Sasaki, Spectral Analysis of the Dirac Polaron. Publ. Res. Inst. Math. Sci. 50 (2014), no. 2, pp. 307–339DOI 10.4171/PRIMS/135