Zero Mass as a Borel Structure

The Lorentz group Lor1,3=SO0(1,3) has two point fixgroups, namely SO(3) for time-like translations and SO0(1,1)×IR2 for light-like translations. However, for light-like translations, it is reasonable to consider a line fixgroup that leads to the Borel structure of the Lorentz group and provides appropriate helicities for massless particles. Therefore, whether a particle is massless or massive is not so much a physical question but rather a question of the underlying Lie group symmetry.