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Physical Review

Self-Interaction Corrections of a Charge in a Nonrelativistic Particle Formalism

John E. Krizan

Published Jan 25, 1968• 3 citations
Publisher

Abstract

A nonlinear equation for the description of self-interaction corrections is developed by means of stochastic mechanics. It is derived within a nonrelativistic framework in which fields do not appear. The linearized approximation of this equation is also examined, and an alternative, more conventional, interpretation of the resulting effective interaction is given. Calculations involving effects which have a possible interpretation as the Lamb shift and anomalous magnetic moment of the electron, and which are free of cutoffs, are made; these give semiquantitative agreement, offering some encouragement for a possible attempt at a relativistic generalization.

DOI
10.1103/physrev.165.1725

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A nonlinear equation for the description of self-interaction corrections is developed by means of stochastic mechanics. It is derived within a nonrelativistic framework in which fields do not appear. The linearized approximation of this equation is also examined, and an alternative, more conventional, interpretation of the resulting effective interaction is given. Calculations involving effects which have a possible interpretation as the Lamb shift and anomalous magnetic moment of the electron, and which are free of cutoffs, are made; these give semiquantitative agreement, offering some encouragement for a possible attempt at a relativistic generalization.
The radiation damping terms of classical electrodynamics are introduced into a set of stochastic equations of nonrelativistic quantum mechanics to study their effects on the energy levels of a stationary system to first order in perturbation theory. In the potential case there appear correction terms to Schr\"odinger's equation, one of which is calculated by solving a subsidiary equation. In particular, we obtain for the Lamb shift an expression of essentially stochastic origin, whose structure is in complete agreement with known results. According to this calculation, which has the advantage of being devoid of divergence problems, the non-relativistic spinless contribution to the Lamb shift of the 2s2s2s level amounts of 60% of the total effect.
The interaction of an atomic system with an unquantized electromagnetic field is studied by means of the Heisenberg-operator equations of motion. The electromagnetic fields created by the atom are taken into account by assuming that the charge and probability current densities are the actual charge and current source terms in Maxwell's equation. When Ehrenfest's theorem is written to order (1/c)2{)}^{2})2, the equations of motion are found to have a constant of motion that can be interpreted as stating that the sum of the atomic energy, energy of interaction, and energy stored in the electromagnetic field is constant. This constant of integration can be expressed as an expectation of a new semiclassical Hamiltonian that now includes the effects of the atomic self-fields to order 1/c2{\mathit{c}}^{2}c2. This Hamiltonian is related to the classical Darwin Lagrangian. Since the new approximate Hamiltonian is the sum of an atomic and a field Hamiltonian, it provides a formulation of semiclassical radiation theory that is formally close to the usual formulation of quantum electrodynamics. A Schr\"odinger equation can be derived by applying the variational principle to the expectation of the new Hamiltonian. The result is a nonlinear integro-differential equation, in \ensuremath{\Psi}, which is somewhat similar to Hartree's self-consistent equation for a multielectron atom.When the Heisenberg equations are written to include the next-higher-order terms, 1/c3{\mathit{c}}^{3}c3, it is found that the total energy of the atom and the electromagnetic field in the vicinity of the atom decreases at a rate that is given by the Larmor power formula (2e2{\mathit{e}}^{2}e2/3c3{\mathit{c}}^{3}c3)(d〈v〉/dt)2{)}^{2})2, where 〈v〉 is the quantum-mechanical expectation of the electron's velocity operator. This approximate formulation of semiclassical radiation theory is applied to semiclassical calculations of atomic radiative shifts. It is then shown that, if some retardation effects are included to all orders of 1/c in the vector potential, the semiclassical analysis provides a formula for radiative energy-level shifts that are time dependent and that contain coefficients similar to the starting point of the Bethe calculation of the Lamb shift. The incorporation of mass renormalization in semiclassical theory is then discussed.
We describe here the coherent formulation of electromagnetism in the nonrelativistic quantummechanical many-body theory. We use the mathematical frame of the field theory and its quantization in the spirit of the QED. This is necessary because of the manifold of misinterpretations emerging from the hystorical development of quantum mechanics, starting from the Schrödinger equation of a single particle in the presence of given electromagnetic fields, followed by the many-body theories of many charged identical particles having just Coulomb interactions inspired from the classical electromagnetic theory of point-like charges. However, this later is known to be inconsistent due to the self-interaction. This way could not be continued further to include properly the magnetic forces between the charged particles and lead to a lot of confusion about the interpretation of the magnetic field in the Hamiltonian, as well as about the gauge invariance. We emphasize the importance of the distinction between the applied (external fields) and the field in the matter. All these problems are length properly solved within the non-relativistic QED, nevertheless the confusion dominates in all the problems related to the magnetic properties of the solid state.
In a previous paper by the author a new approach in quantum electrodynamics was proposed in which the electronic and electromagnetic fields are ordinary c-numbers in contradistinction to noncommuting q-numbers used in the standard formulation of quantum electrodynamics. The factual development of standard quantum electrodynamics began when the anomalous correction to the electron magnetic moment and the Lamb shift were calculated and the calculations were confirmed by experiment. It is demonstrated in the present paper how the anomalous electron magnetic moment and Lamb shift can be obtained in the new approach in quantum electrodynamics.
A systematic method of handling a system composed of a number of particles and an intermediary Bose field has been developed. The main idea consists in linearizing the interaction by taking certain average values so that it may be amalgamated to the free Hamiltonian. Only the remaining fluctuational interaction is treated by perturbation theory. The effect of the averaged interaction appears in the form of two generalized Hartree fields associated with the particle and the Bose quantum, respectively. The self-consistent equations for these Hartree fields are derived from an analysis of the SSS matrix, keeping close analogy to the renormalization procedure in quantum electrodynamics. The formalism includes such approximations as the Bohm-Pines, Hartree-Fock, and Thomas-Fermi theories as special cases. General criteria about the validity and plausibility of the picture involved in the present formulation are discussed from various points of view.
A relativistically invariant representation of the generalized momentum of a particle in an external field is proposed. In this representation, the dependence of the potentials of the interaction of the particle with the field on the particle velocity is taken into account. The exact correspondence of the expressions of energy and potential energy for the classical Hamiltonian is established, which makes identical the solutions to the problems of mechanics with relativistic and nonrelativistic approaches. The invariance of the proposed representation of the generalized momentum makes it possible to equivalently describe a physical system in geometrically conjugate spaces of kinematic and dynamic variables. Relativistic invariant equations are proposed for the action function and the wave function based on the invariance of the representation of the generalized momentum. The equations have solutions for any values of the constant interaction of the particle with the field, for example, in the problem of a hydrogen-like atom, when the atomic number of the nucleus is Z > 137. Based on the parametric representation of the action, the expression for the canonical Lagrangian, the equations of motion, and the expression for the force acting on the charge are derived when moving in an external electromagnetic field. The Dirac equation with the correct inclusion of the interaction for a particle in an external field is presented. In this form, the solutions of the equations are not limited by the value of the interaction constant. The solutions of the problem of charge motion in a constant electric field, the problems for a particle in a potential well and the passage of a particle through a potential barrier, the problems of motion in an exponential field (Morse), and also the problems of a hydrogen atom are given.