a New Angular Momentum Sum Rule

Sum rules, relating the total angular momentum of a nucleon to the spin and orbital angular momentum carried by its constituents, are interesting and important in understanding the internal structure of the nucleon. In a much cited paper, Jaffe and Manohar stressed the subtleties involved in deriving general angular momentum sum rules. As they point out, too naive an approach leads immediately to highly ambiguous divergent integrals, and a careful limiting procedure has to be introduced in order to obtain physically meaningful results. In this it is essential to work with non-diagonal matrix elements < {p',σ |J|p,σ } ; >, and this can have some unexpected consequences. Jaffe and Manohar comment that to justify rigorously the steps in such a procedure requires the use of normalizable wave packets, though they do not do this explicitly in their paper. We show that the results in the literature are incorrect. Surprisingly it turns out that the results are very sensitive to the type of relativistic spin state used to describe the motion of the particle i.e. whether a standard canonical (i.e. boost, as in e.g. Bjorken-Drell) state or a helicity state is utilized. We present results for the matrix elements of the angular momentum operators, valid in an arbitrary Lorentz frame, both for helicity states and canonical states. We present a new sum rule for transversely polarized nucleons.