Relativistic particle on light-front

Abstract We construct one-particle states as unitary, irreducible representations of Poincare group in front form, characterized by a special null vector, dubbed the 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 proved to be 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 reaches the massless limit smoothly. We then apply those results to massive spin-1 particle and compute the corresponding Wigner D-matrix. 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 $$\epsilon ^\mu _\pm \rightarrow \epsilon ^\mu _\pm +\xi k^\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>ϵ</mml:mi> <mml:mo>±</mml:mo> <mml:mi>μ</mml:mi> </mml:msubsup> <mml:mo>→</mml:mo> <mml:msubsup> <mml:mi>ϵ</mml:mi> <mml:mo>±</mml:mo> <mml:mi>μ</mml:mi> </mml:msubsup> <mml:mo>+</mml:mo> <mml:mi>ξ</mml:mi> <mml:msup> <mml:mi>k</mml:mi> <mml:mi>μ</mml:mi> </mml:msup> </mml:mrow> </mml:math> shift turns out to be remnant of this decoupling, with $$\xi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ξ</mml:mi> </mml:math> determined by the angle between the reference vectors. Our results thus give us a new perspective on gauge symmetry: it can be understood as the equivalence between different massless spin-1 polarization vectors from reaching the massless limit through different reference vectors.