Spectrum of the semi-relativistic Pauli–Fierz model I

Abstract

A HVZ type theorem for the semi-relativistic Pauli–Fierz Hamiltonian,H=(p⊗1−A)2+M2⊗1+V⊗1+1⊗Hf,M≥0, in quantum electrodynamics is studied. Here H is a self-adjoint operator in Hilbert space L2(Rd)⊗F≅∫Rd⊕Fdx, A=∫Rd⊕A(x)dx is a quantized radiation field and Hf is the free field Hamiltonian defined by the second quantization of a dispersion relation ω:Rd→R. It is emphasized that massless case, M=0, is included. Let E=inf⁡σ(H) be the bottom of the spectrum of H. Suppose that the infimum of ω is m>0. Then it is shown that σess(H)=[E+m,∞). In particular the existence of the ground state of H can be proven.