A nonperturbative approach to solving bound state problems of quantum electrodynamics

Bound state calculations in quantum electrodynamics (QED) become unnecessarily complicated in high orders of perturbation theory when conventional relativistic formalisms are used. The complication arises from the presence of different energy scales in the bound state system. A separation of scales is accomplished by using a nonrelativistic effective field theory, Nonrelativistic QED (NRQED). NRQED is much simpler to work with, and has become the primary tool in modern bound state analyses.; The present work builds on the ideas of NRQED and develops a nonperturbative approach to solving for bound state properties. A fixed and finite UV cutoff is incorporated into the effective Hamiltonian to eliminate UV divergences, making the effective theory well-defined beyond perturbation theory; a series of local operators compensates for the large-momentum/short distance modes excluded by the cutoff. The state space is truncated to the valence Fock state, replacing the original field-theoretical system by a system of fixed particle number whose properties are straightforwardly calculated by standard methods of fixed-particle-number quantum mechanics. Effects of higher Fock states are included as perturbations of the effective Hamiltonian acting only on the valence Fock state.; The replacement of the field theory Hamiltonian by a fixed-particle-number Hamiltonian and the incorporation of a fixed UV cutoff is illustrated with the calculation of the energy levels of the hydrogen atom accurate through O(m_eα5), in the approximation of an infinite-mass, structureless nucleus, but including all relativistic and radiative corrections; an appendix extends this analysis for hydrogen to O(m_eα 6). Following this illustration in the hydrogen system, state-of-the-art calculations are presented for three problems: the O( mα6) pure-recoil contribution to the muonium and positronium hyperfine splitting; the O(α 2) and O(α3 In α) non-relativistic corrections to the orthopositronium decay rate; and the O( mα7 In α) contribution to the muonium and positronium hyperfine splittings.