Renormalization and Scattering in Relativistic Particle Theories

William Klink, Professor emeritus of Mathematics, University of Iowa

While there seems to be general agreement that a relativistic quantum theory is a field theory, I investigate whether it is possible to formulate a relativistic particle theory, in which the fundamental mathematical ingredients are creation and annihilation operators, rather than quantum fields. Starting with the irreps of the Poincare group, a many-particle theory is given in terms of bare creation and annihilation operators rather than field operators. Equations that renormalized creation and annihilation operators must satisfy are specified by eigenvalue commutators, as well as automorphisms connecting to the bare creation and annihilation operators. Because particles are thought of as localized bundles of energy and momentum, a relativistic position operator is introduced to give meaning to the localization of particles. Scattering is formulated using ideas from a paper of Eckstein and applied to nonrelativistic scattering. Two applications of the formalism are presented, one involving the renormalization of a charged scalar meson, the other the interaction of a charged meson with an external electromagnetic field.