A generally accepted set of axioms of quantization is introduced and applied to the quantization of the multilinear momentum observables so as to delimit the possible forms of the corresponding quantum operators. This analysis leads to a canonical decomposition of the quantum observables into a series of symmetric operators each of which is determined by an unknown auxilliary tensor generated by the multilinear momentum. Methods of removing the residual indeterminates in the differential operators are then critically reviewed, and a particular choice illustrated by means of examples defined on the real line
