Behavior of Commutator Matrix Elements at Small Distances. I. Existence and Structure of Equal-Time Limits

The behavior of commutator matrix elements at small distances is investigated in the framework of general quantum field theory. The equivalence between current density commutators and the commutators of a finite number of charge moments and a current density at equal times is proved by means of microcausality but without using the spectrum condition. If $N$ is the (finite) order of the commutator, then the equal-time limits of the commutators between all charge moments of degree higher than $2N+m-1$ ($m$ fixed between zero and $N$) and a current density vanish. The equal-time commutators between the first $2N+m-1$ charge moments and a current density exist if and only if the equal-time limit of the corresponding current densities exists and is a sum of $2N+m-1$ derivatives of the $\delta$ function in the space variables x. The coefficients of the individual $\delta$ functions are identical to the equal-time limits of the charge moments and one density. If the spectrum condition holds in addition, then $m$ is equal to zero.