A self-adjoint decomposition of the radial momentum operator

With acceptance of the Dirac's observation that the canonical quantization entails using Cartesian coordinates, we examine the operator e(r)P(r) rather than P-r itself and demonstrate that there is a decomposition of e(r)P(r) into a difference of two self-adjoint but noncommutative operators, 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, offering an explanation of why the mean value of P-r over a quantum mechanical state makes sense and supporting Dirac's claim that P-r "is real and is the true momentum conjugate to r".