The propagator field theory revisited: a Lorentz symmetry breaking approach

TL;DR

This paper introduces a Lorentz symmetry breaking approach to construct a family of well-defined propagators for massive scalar fields near the light-cone, addressing divergence issues in coordinate space.

Contribution

It presents a novel method using weighted measures that break Lorentz symmetry to regularize propagators and ensure finite quantum amplitudes near the light-cone.

Findings

Constructed an infinite family of propagators near the light-cone.

Demonstrated positivity of amplitudes under certain measure restrictions.

Showed propagators decrease beyond the Compton wavelength.

Abstract

It is well known that the propagator for a massive scalar field is ill-defined in the coordinate space for $d\geq2$, in particular it diverges at the light-cone; we show that by using Lorentz symmetry breaking weighted measures, an infinite family of propagators can be constructed in an in\-finite\-simal strip near the light-cone, which are labeled by the weight of the measure; hence, the results will provide a finite quantum amplitude for a massive particle for propagating on the light-cone. The propagators regarded as smooth two-points functions, increase within a region smaller than the Compton wavelength, and decrease beyond that wavelength, and eventually drop off for large arguments. Although the time ordered propagators retain negative values regions for arbitrary values of the weight $s$ for the measures, the restriction $2<s\leq d+1$ will guarantee the positivity for the amplitudes near the light cone.