Vector parametrization, partial angular momenta and unusual commutation relations in physics

When studying an N-particle system by means of N-1 vectors i.e., by means of a vector parametrization, one unavoidably comes across several angular momenta: not only the total angular momentum of the system but also the various partial angular momenta corresponding to the motion of the various vectors. All these momenta can, in addition, be referred to a variety of reference frames. The use of vector parametrizations and partial angular momenta in physics greatly simplifies the classical as well as quantum expressions of the kinetic energy. The present paper is devoted to a detailed and rigorous study of the partial angular momenta and the various commutation relations they satisfy, in particular the unusual commutation relations whose origin is traced back to the very structure of the coordinate changes used to define the Body-Fixed (BF) frames. The direct quantization of the classical expressions of the kinetic energy obtained in the context of various vector parametrizations is also given in detail. It turns out to be an efficient extension of well-known quantization procedures to the case where supernumerary quasi-momenta are used. As an illustration, the case of a four-particle system is treated in detail for a particular choice of the BF frames. Finally, the analogies between the classical and quantum approaches are emphasized