  Possible (algebraic) commutation relations in the Lagrangian quantum theory
of free (scalar, spinor and vector) fields are considered from mathematical
view-point. As sources of these relations are employed the Heisenberg
equations/relations for the dynamical variables and a specific condition for
uniqueness of the operators of the dynamical variables (with respect to some
class of Lagrangians). The paracommutation relations or some their
generalizations are pointed as the most general ones that entail the validity
of all Heisenberg equations. The simultaneous fulfillment of the Heisenberg
equations and the uniqueness requirement turn to be impossible. This problem is
solved via a redefinition of the dynamical variables, similar to the normal
ordering procedure and containing it as a special case. That implies
corresponding changes in the admissible commutation relations. The introduction
of the concept of the vacuum makes narrow the class of the possible commutation
relations; in particular, the mentioned redefinition of the dynamical variables
is reduced to normal ordering. As a last restriction on that class is imposed
the requirement for existing of an effective procedure for calculating vacuum
mean values. The standard bilinear commutation relations are pointed as the
only known ones that satisfy all of the mentioned conditions and do not
contradict to the existing data.
