``Lorentz Basis'' of the Poincaré Group

Abstract

An explicit derivation is given for the matrix elements of the translation generators Pμ 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 Gμ = i[(N2 − M2), Pμ] are considered and several interesting aspects of the algebras generated by N, M′, and Pμ′=(ε1Pμ+ε2Gμ) are discussed for the cases of positive as well as zero rest mass.