Divergent Self-energy Research Articles

Dirac's “Fine-Tuning Problem”: A Constructive Use of Anachronism?

In order to shed light on contemporary arguments about “fine-tuning” in cosmology, I examine a possible historical precedent for fine-tuning from the early years of Quantum Electrodynamics: the divergent self-energy of the electron in Dirac's theory. I argue that viewing this problem as a fine-tuning problem involves an anachronistic reconstruction, but that such reconstructions can be philosophically useful so long as they are not confused with real historical understanding. I relate how, historically, this problem really was conceived, and show how one important step toward its solution drew upon an interpretation of Dirac's formalism in terms of “hole theory.” In light of the subsequent demise of hole theory, I argue that my anachronistic reconstruction can serve as a cautionary tale that should considerably weaken the conclusions that might otherwise be drawn from attempts to give theistic or multi-verse solutions to cosmological fine-tuning problems.

Stability and mass of point particles

In this paper we consider classical point particles in full interaction with an arbitrary number of dynamical scalar and (abelian) vector fields. It is shown that the requirement of stability — vanishing self-force — is sufficient to remove the well-known inconsistencies of the classical theory: the divergent self-energy, as well as the failure of Lorentz covariance of the energy-momentum when including the contributions of the fields. As a result, in these models the mass of a point particle becomes finitely computable. We discuss how these models are connected to quantum field theory via the path-integral representation of the propagator.

Virtual-pair effects in atomic structure theory.

A modified perturbation procedure for computing relativistic atomic bound-state energies is described, in which the instantaneous Coulomb interaction is included in the unperturbed QED Hamiltonian and the effect of virtual electron-positron pairs and of virtual transverse photons is treated as the perturbation. To illustrate the method in the context of a simple model, a one-electron atom is considered and a ``one-pair'' approximation is analyzed in detail. The approach is based on the resolvent-operator formulation of many-body time-independent perturbation theory. The removal of divergent vacuum-to-vacuum matrix elements is easily accomplished in this formulation, and it leads rather directly to the derivation of a modified Dirac equation in which all perturbative effects are contained in an effective one-electron potential. A set of integral equations for the evaluation of the effective potential in the one-pair model is derived. These equations have the same structure as the Dyson-Schwinger equations of covariant QED and serve a similar purpose. That is, they allow for the isolation of the divergent self-energy and vertex parts, leaving a finite kernel (the K matrix) which contains the effects of multiparticle intermediate states. In the one-pair model considered here the intermediate states consist of an electron and a virtual pair interacting Coulombically. It is shown that the K matrix can be constructed with the aid of a rigorous minimum principle of the Rayleigh-Ritz type. That such a minimum principle exists appears to be an advantage of the noncovariant Hamiltonian procedure. The inapplicability of standard renormalization methods is, however, a distinct disadvantage. While some progress (the removal of vacuum divergences and the isolation of divergent parts in the integral equation) is reported, a full resolution of the renormalization problem in the Hamiltonian formulation remains to be developed.

Implications of scaling for the proton-neutron mass difference

We assume that the structure functions V 1, V 2 scale and do not contain fixed poles at integer values of α = 0, 1, 2, …. This leads to a logarithmically divergent self-energy. We include a tadpole term in the Lagrangian. The divergent part of the self-energy is absorbed in a renormalization of the tadpole coupling constant. Using the available data we then carefully evaluate low- and high-energy contributions to the electromagnetic mass shift of the nucleons. The conclusions agree with the tadpole model of Coleman and Glashow. In the quark-model language our result corresponds to the values m u ≅ 4 MeV, m d ≅ 6 MeV and m s ≅ 135 MeV for the masses of up, down and strange quarks, respectively.

Electromagnetic Mass Shift of Hadrons

Based on the indicated possibility of the divergence of the electromagnetic mass shift of hadrons, we propose a simple method of computing a finite value out of the divergent self­ energy integral. The divergent selfenergy integral, here, is assumed to be given by bare-like contributions of one-particle intermediate states in the lowest-order selfenergy diagrams.

Particle-like solutions in field theory

A number of quantum field theories have the property that the field equations, studied as classical equations of motion, have particle-like solutions. These are spatially localized exact solutions, with a definite value of angular momentum. The role of these solutions in a fully quantum theory is the subject of investigation. We examine, in particular, the theory of a nonrelativistic scalar nucleon, coupled by a local interaction to a scalar meson field. The classical solution (whose spatial extent is a nucleon Bohr radius) is identified with a self-consistent field approximation to the one-nucleon state. However, quantum corrections to this approximation yield the same divergent self-energy as perturbation theory. In this example, the particle-like solutions have nothing to do with the ultraviolet divergences, but are relevant to the treatment of the finite part of the theory for strong coupling. The separation of the infinite part is trivial and is accomplished by a canonical transformation. The particle solutions still exist for the finite part for strong coupling.