On the generalized function calculus for infrared and ultraviolet singular quantum fields

Abstract

New theorems on properties of the generalized functions defined on Gelfand–Shilov’s spaces Sα0 are established. These functional classes are universal for the operator realization of quantum field theories whose infrared or/and ultraviolet behavior is more singular than that of the standard Wightman quantum field theories (QFT’s). The leading role in these applications is played by the notion of a carrier cone of analytic functional which generalizes and replaces the notion of support of distribution. An explicit representation for the generalized functions with a given carrier cone is obtained. It is proved that the restrictions of functionals defined on Sα0 to the spaces with smaller subscripts have the same carrier cones. The precise characterization of the relation between the carrier cones of multilinear forms with respect to their arguments and the carrier cones of their associated generalized functions is given. Applications of the obtained results to indefinite metric QFT and to nonlocal models are discussed.