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Possible (algebraic) commutation relations in the Lagrangian quantum theoryof free (scalar, spinor and vector) fields are considered from mathematicalview-point. As sources of these relations are employed the Heisenbergequations/relations for the dynamical variables and a specific condition foruniqueness of the operators of the dynamical variables (with respect to someclass of Lagrangians). The paracommutation relations or some theirgeneralizations are pointed as the most general ones that entail the validityof all Heisenberg equations. The simultaneous fulfillment of the Heisenbergequations and the uniqueness requirement turn to be impossible. This problem issolved via a redefinition of the dynamical variables, similar to the normalordering procedure and containing it as a special case. That impliescorresponding changes in the admissible commutation relations. The introductionof the concept of the vacuum makes narrow the class of the possible commutationrelations; in particular, the mentioned redefinition of the dynamical variablesis reduced to normal ordering. As a last restriction on that class is imposedthe requirement for existing of an effective procedure for calculating vacuummean values. The standard bilinear commutation relations are pointed as theonly known ones that satisfy all of the mentioned conditions and do notcontradict to the existing data.

Lagrangian quantum field theory in momentum picture. IV. Commutation relations for free fields