On the existence of the Deser–Gilbert–Sudarshan representation

The Jost–Lehmann representation for the single-particle matrix element $\langle p|[j(x),j(0)]|p\rangle=\varepsilon(x_0)\widehat C(x^2,x_0)$ of the current commutator is used to study the existence of the Deser–Gilbert–Sudarshan representation. Using the analytic and functional properties of this matrix element, one can show that the spectral function for the Deser–Gilbert–Sudarshan representation exists in the ordinary sense if and only if $\widehat C(x^2,x_0)\in S'(\overline R_+\otimes R)$. In the general case, the spectral function is an element of a more complicated space.