The propagator field theory revisited: A Lorentz symmetry breaking approach

It is well known that the propagator for a massive scalar field is ill-defined in the coordinate space for $d\ge 2$, 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 infinitesimal 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, eventually dropping 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 $2<s\le d+1$ will guarantee the positivity for the amplitudes near the light cone.