Resolving the Weinberg Paradox with Topology

TL;DR

This paper resolves Weinberg's paradox by demonstrating that soft-photon resummation in a toy model with electric and magnetic charges restores Lorentz invariance through a topological phase, aligning with topological monopoles.

Contribution

It shows how Lorentz invariance is preserved in monopole theories via soft-photon resummation and topological phases, reconciling topological monopoles with quantum field theory.

Findings

Soft-photon resummation exponentiates Lorentz-violating terms into a topological phase.

The amplitude modulus remains Lorentz invariant, and the full amplitude is invariant when Dirac quantization is imposed.

A topological component of the phase relates to 4D topological quantum field theory.

Abstract

Long ago Weinberg showed, from first principles, that the amplitude for a single photon exchange between an electric current and a magnetic current violates Lorentz invariance. The obvious conclusion at the time was that monopoles were not allowed in quantum field theory. Since the discovery of topological monopoles there has thus been a paradox. On the one hand, topological monopoles are constructed in Lorentz invariant quantum field theories, while on the other hand, the low-energy effective theory for such monopoles will reproduce Weinberg's result. We examine a toy model where both electric and magnetic charges are perturbatively coupled and show how soft-photon resummation for hard scattering exponentiates the Lorentz violating pieces to a phase that is the covariant form of the Aharonov-Bohm phase due to the Dirac string. The modulus of the scattering amplitudes (and hence observables) are Lorentz invariant, and when Dirac charge quantization is imposed the amplitude itself is also Lorentz invariant. For closed paths there is a topological component of the phase that relates to aspects of 4D topological quantum field theory.