An Extension of Distribution Theory Related to Gauge Field Theory

TL;DR

This paper extends distribution and hyperfunction theory to more singular generalized functions, facilitating the mathematical treatment of gauge quantum field theories with indefinite metrics in covariant gauges.

Contribution

It introduces an extension of distribution theory using Gelfand-Shilov spaces, generalizes the Paley--Wiener--Schwartz theorem for singularities, and develops tools for gauge field models.

Findings

Extended distribution theory to more singular functions.

Generalized Paley--Wiener--Schwartz theorem for singularities.

Established nuclearity and tensor product formulas for Gelfand-Shilov spaces.

Abstract

We show that a considerable part of the theory of (ultra)distributions and hyperfunctions can be extended to more singular generalized functions, starting from an angular localizability notion introduced previously. Such an extension is needed to treat gauge quantum field theories with indefinite metric in a generic covariant gauge. Prime attention is paid to the generalized functions defined on the Gelfand-Shilov spaces $S_\alpha^0$ which gives the widest framework for construction of gauge-like models. We associate a similar test function space with every open and every closed cone, show that these spaces are nuclear and obtain the required formulas for their tensor products. The main results include the generalization of the Paley--Wiener--Schwartz theorem to the case of arbitrary singularity and the derivation of the relevant theorem on holomorphic approximation.