``Lorentz Basis'' of the Poincaré Group

Abstract

An explicit derivation is given for the matrix elements of the translation generators P_μ of the Poincaré algebra with respect to the ``Lorentz basis,'' namely, in terms of states which diagonalize the two Casimir operators of the homogeneous Lorentz group (HLG). The results are given for the cases mass μ > 0 and μ = 0 and, for the latter, for discrete and continuous spin. The transforms connecting the momentum and Lorentz bases are discussed, a detailed derivation being given for the zero‐mass discrete‐spin case. The matrix elements of G_μ = i[(N² − M²), P_μ] are considered and several interesting aspects of the algebras generated by N, M′, and $P_{\mu }{\mathrm{\prime =(\epsilon }}_1{P}_{\mu }{\mathrm{+\epsilon }}_2{G}_{\mu })$ are discussed for the cases of positive as well as zero rest mass.