Relativistic Particle on Light-Front

TL;DR

This work constructs one-particle states as unitary irreducible representations of the Poincaré group in the front form, labeled by a null reference vector $n$, and demonstrates a smooth massive–massless continuation through infinite rapidity. The little group is shown to act simply as a change of the reference vector, yielding a Wigner $D$-matrix $D_{'}(,',n|n')$ that depends on two rapidities and the two reference vectors. Applied to a massive spin-1 particle, the authors derive explicit polarization vectors and Wigner $D$-matrix elements, recovering the standard rest-frame results and, in the massless limit, the gauge-shift $oldsymbol\u03b5_ o oldsymbol_ + ext{shift}$ with coefficients determined by angular parameters. A parallel contraction of the little-group algebra confirms the equivalence of the front-form construction with the conventional Lie-algebra approach, clarifying the origin of gauge symmetry as the massless limit across different reference vectors. The framework connects to spinor-helicity formalisms and suggests extensions to spin-$ frac{1}{2}$ and higher spins, offering a deeper, mathematically controlled bridge between massive and massless relativistic states and their amplitudes.

Abstract

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 $ε^μ_\pm \rightarrow ε^μ_\pm +ξk^μ$ shift turns out to be remnant of this decoupling, with $ξ$ 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.