Singularity-Free Breit Equation From Constraint Two-Body Dirac Equations

Abstract

We examine the relation between two approaches to the quantum relativistic two-body problem: (1) the Breit equation, and (2) the two-body Dirac equations derived from constraint dynamics. In applications to quantum electrodynamics, the former equation becomes pathological if certain interaction terms are not treated as perturbations. The diculty comes from singularities which appear at nite separations r in the reduced set of coupled equations for attractive potentials even when the potentials themselves are not singular there. They are known to give rise to unphysical bound states and resonances.In contrast, the two-body Dirac equations of constraint dynamics do not have these pathologies in many nonperturbative treatments. To understand these 1 marked dierences we rst express these contraint equations, which h a v e a n \external potential" form similar to coupled one-body Dirac equations, in a hyperbolic form. These coupled equations are then re-cast into two equivalent equations: (1) a covariant Breit-like equation with potentials that are exponential functions of certain \generator" functions, and (2) a covariant orthogonality constraint on the relative momentum. This reduction enables us to show in a transparent w a y that nite-r singularities do not appear as long as the the exponential structure is not tampered with and the exponential generators of the interaction are themselves nonsingular for nite r. These Dirac or Breit equations, free of the structural singularities which plague the usual Breit equation, can then be used safely under all circumstances, encompassing numerous applications in the elds of particle, nuclear, and atomic physics which i n v olve highly relativistic and strong binding congurations.2

Tests of two-body Dirac equation wave functions in the decays of quarkonium and positronium into two photons

“…The conversion from sixteen component spinor wave functions to four by four matrix wave functions now can be carried out in a two-step process. First, as in [34,35], the "energy" or q space column vector direct products are converted to 4x4 matrices as follows (recall the factor of iα y plus the transpose operation changes particle spinor into antiparticle spinor)…”

RelativisticN-body problem in a separable two-body basis

“…Much progress has been made in the study of relativistic two-body bound state problems [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. In the 1970s, several authors used Dirac's constraint mechanics [1] to attack the relativistic two-body problem at its classical roots [2], successfully evading the so-call "no interaction theorem" [3].…”

“…The quantum version of the constraint approach was extended to pairs of spin one-half particles. The results were two-body quantum bound state equations that correct the defects in the Breit equation, correct the defects in the ladder approximation to the Bethe-Salpeter equation, and control covariantly the relative time and energy variables [4]. Those bound state equations for fermions are the two-body Dirac equations of constraint dynamics, which we shall also call the constraint equations [5].…”

Variational wave equations of two fermions interacting via scalar, pseudoscalar, vector, pseudovector and tensor fields

“…A nonperturbative treatment of a Breit-type equation, appropriate to this case, may meet with difficulties, even without accounting for retardation. It is known that, depending on the choice of interaction, the radially reduced set of Breit equations may possess energy-dependent poles at finite r even if the original potential (the function G(r) in our case) is regular at these points (see [25,26] and references therein). Correspondingly, an exact boundary-value problem becomes improper and therefore needs to be modified.…”

“…For example, considering the relationship of the Breit equation with the constrained two-body Dirac equations, one can approximate singular terms by means of singularity-free energy-dependent exponential potentials, as is done in [26,21]. In so doing, the perturbative spectrum does not change but the exact formulation of the boundary-value problem becomes correct.…”