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``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*[(N^{2} − M^{2}), *P* _{μ}] are considered and several interesting aspects of the algebras generated by N, M′, and are discussed for the cases of positive as well as zero rest mass.

© 1968 The American Institute of Physics

Received 21 June 1967
Published online 28 October 2003