Relativistic Particle on Light-Front

We construct one-particle states as unitary, irreducible representations of Poincare group in front form, characterized by a special null vector, dubbed reference vector. We demonstrate that this construction has massive-massless continuation. The state is defined by the reference vector. The little group transformation, defined at a general moving momentum, is equivalent to a change of reference vector. The resulting Wigner D-matrix is parameterized by the rapidities, in addition to the two reference vectors before and after transformation. Boosting the rapidities to infinity, it obtain the massless limit smoothly. We then apply those results to massive spin-1 particle and compute the corresponding Wigner D-matrices. The resulting polarization vectors are equivalent to those in spinor-helicity formalism. In the massless limit, it is shown that longitudinal polarization decouples from the spectrum. The $\epsilon^\mu_\pm \rightarrow \epsilon^\mu_\pm +\xi k^\mu$ shift turns out to be remnant of this decoupling, with $\xi$ determined by the angle between the reference vectors. Our results thus give us a deeper understanding of gauge symmetry: massless spin-1 particle is defined as the infinite boost limit of massive spin-1 particle, gauge symmetry can be understood to origin from obtaining the massless limit for polarization vectors through different reference vectors.