Two classes of generalized functions used in nonlocal field theory

TL;DR

This paper explores two formulations of causality in nonlocal quantum field theory using different classes of test functions, establishing their equivalence in terms of carrier cones and extending classical theorems to singular generalized functions.

Contribution

It proves the equivalence of carrier cones for functionals on different Gelfand-Shilov spaces and extends the Paley-Wiener-Schwartz theorem to singular generalized functions.

Findings

Carrier cones are the same for functionals on $S^0$ and $S^0_eta$ spaces.

Extended Paley-Wiener-Schwartz theorem applies to arbitrarily singular generalized functions.

Derived an extension of Vladimirov's algebra of holomorphic functions.

Abstract

We elucidate the relation between the two ways of formulating causality in nonlocal quantum field theory: using analytic test functions belonging to the space $S^0$ (which is the Fourier transform of the Schwartz space $\mathcal D$) and using test functions in the Gelfand-Shilov spaces $S^0_\alpha$. We prove that every functional defined on $S^0$ has the same carrier cones as its restrictions to the smaller spaces $S^0_\alpha$. As an application of this result, we derive a Paley-Wiener-Schwartz-type theorem for arbitrarily singular generalized functions of tempered growth and obtain the corresponding extension of Vladimirov's algebra of functions holomorphic on a tubular domain.