Gauge transformations in relativistic two-particle constraint theory

TL;DR

This paper explores gauge transformations in relativistic two-particle constraint theory, establishing their laws, invariance properties, and implications for gauge-dependent potentials and critical coupling constants.

Contribution

It introduces explicit gauge transformation laws for wave functions and potentials, linking them to quantum field theory and simplifying for local potentials with multiphoton exchange effects.

Findings

Weak invariance of wave equations under gauge transformations

Explicit representation of finite gauge transformations as local dilatation operators

Identification of a gauge invariant critical coupling constant  = 1/2

Abstract

Using connection with quantum field theory, the infinitesimal covariant abelian gauge transformation laws of relativistic two-particle constraint theory wave functions and potentials are established and weak invariance of the corresponding wave equations shown. Because of the three-dimensional projection operation, these transformation laws are interaction dependent. Simplifications occur for local potentials, which result, in each formal order of perturbation theory, from the infra-red leading effects of multiphoton exchange diagrams. In this case, the finite gauge transformation can explicitly be represented, with a suitable approximation and up to a multiplicative factor, by a momentum dependent unitary operator that acts in $x$-space as a local dilatation operator. The latter is utilized to reconstruct from the Feynman gauge the potentials in other linear covariant gauges. The resulting effective potential of the final Pauli-Schr\"odinger type eigenvalue equation has the gauge invariant attractive singularity $\alpha^2/r^2$, leading to a gauge invariant critical coupling constant $\alpha_c =1/2$.