Renormalization and dressing in quantum field theory

We illustrate the mass and charge renormalization procedures in quantum field theory using, as an example, a simple model of interacting electrons and photons. It is shown how addition of infinite renormalization counterterms to the Hamiltonian helps to obtain finite and accurate results for the S-matrix. In order to remove the ultraviolet divergences from the Hamiltonian, we apply the Greenberg-Schweber dressing transformation'' and the Glazek-Wilson similarity renormalization’‘. The resulting ``dressed particle’’ Hamiltonian is finite in all orders of the perturbation theory and yields accurate S-matrix and bound state energies. The bare and virtual particles are removed from the theory, and physical dressed particles interact via direct action-at-a-distance.