Gauge transformations in relativistic two-particle constraint theory

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–Schrödinger type eigenvalue equation has the gauge invariant attractive singularity α2/r2, leading to a gauge invariant critical coupling constant αc=1/2.