Functional integral approach to semi-relativistic Pauli–Fierz models

By means of functional integrations spectral properties of semi-relativistic Pauli–Fierz HamiltoniansH=(p−αA)2+m2−m+V+Hrad in quantum electrodynamics are considered. Here p is the momentum operator, A a quantized radiation field on which an ultraviolet cutoff is imposed, V   an external potential, HradHrad the free field Hamiltonian and m⩾0m⩾0 describes the mass of electron. Two self-adjoint extensions of a semi-relativistic Pauli–Fierz Hamiltonian are defined. The Feynman–Kac type formula of e−tHe−tH is given. A self-adjointness, a spatial decay of bound states, a Gaussian domination of the ground state and the existence of a measure associated with the ground state are shown. All the results are independent of values of coupling constant α  , and it is emphasized that m=0m=0 is included.