A self-adjoint decomposition of radial momentum implies that Dirac's   introduction of the operator is insightful
  With acceptance of the Dirac's observation that the \textit{canonical quantization entails using Cartesian coordinates, }we examine the\textit{\ }% operator $\mathbf{e}_{r}P_{r}$ rather than $P_{r}$ itself and demonstrate that there is a decomposition of $\mathbf{e}_{r}P_{r}$ into two self-adjoint but non-commutative parts, in which one is the total momentum and another is the transverse one. This study renders the operator $P_{r}$ indirectly measurable and physically meaningful. 
Q. H. Liu, S. F. Xiao
quant-ph