A Note on Angular Momentum Commutators in Light-Cone Formulation of Open Bosonic String Theory

We recalculate in a systematic and pedagogical way one of the most important results of Bosonic open string theory in the light-cone formulation, namely the [J^{-i},J^{-j}] commutators, which together with Lorentz covariance, famously yield the critical dimension D=26 and the normal order constant a=1. We use traditional transverse oscillator mode expansions (avoiding the elegant but more advanced language of operator product expansions). We streamline the proof by introducing a novel bookkeeping/regularization parameter \kappa to avoid splitting into creation and annihilation parts, and to avoid sandwiching between bras and kets.