A particle-field Hamiltonian in relativistic quantum electrodynamics

Abstract

We mathematically analyze a Hamiltonian Hτ(V,g) of a Dirac particle—a relativistic charged particle with spin 1/2—minimally coupled to the quantized radiation field, acting in the Hilbert space F≔[⊕4L2(R3)]⊗Frad, where Frad is the Fock space of the quantized radiation field in the Coulomb gauge, V is an external potential in which the Dirac particle moves, g is a photon-momentum cutoff function in the interaction between the Dirac particle and the quantized radiation field, and τ∈R is a deformation parameter connecting the Hamiltonian with the “dipole approximation” (τ=0) and the original Hamiltonian (τ=1). We first discuss the self-adjointness problem of Hτ(V,g). Then we consider Hτ≔Hτ(0,g), the Hamiltonian without the external potential. It is shown that, under a general condition on g, the closure of Hτ is unitarily equivalent to a direct integral ∫R3⊕Hτ(p)¯dp with a fiber Hamiltonian Hτ(p) acting in the four direct sum ⊕4Frad of Frad, physically the polaron Hamiltonian of the Dirac particle with total momentum p∈R3.