Study of a Model of Quantum Electrodynamics

This paper studies the model of the quantum electrodynamics (QED) of a singlenonrelativistic electron due to W. Pauli and M. Fierz and studied further by P.Blanchard. This model exhibits infrared divergence in a very simple context.The infrared divergence is associated with the inequivalence of the Hilbertspaces associated with the free Hamiltonian and with the complete Hamiltonian.Infrared divergences that are visible in the perturbative description disappearin the space of the clothed electrons. In this model when the Hamiltonian isexpressed in terms of the ``physical'' fields that create the electron togetherwith its cloud of soft photons the variational principle suggested earlier canbe applied. At finite time the Heisenberg field of the model acts in the spaceof the perturbative electron together with a finite number of perturbativephotons, while the ``physical'' field can be chosen to act in the space of theexact (``physical'') electron eigenstates together with a finite number ofphysical photons. The space of the physical (or clothed) electron states can bechosen to be a Fock space