Carrier cones of analytic functionals

TL;DR

This paper establishes the existence and uniqueness of minimal carrier cones for continuous linear functionals on a space of entire analytic functions, with implications for nonlocal quantum field theory.

Contribution

It proves the existence and uniqueness of minimal carrier cones for functionals on the space $S^0(R^d)$, extending previous results to more complex topological spaces.

Findings

Every continuous linear functional has a unique minimal carrier cone.

The results are applicable to nonlocal quantum field theory.

The proof involves a decomposition theorem for spaces associated with cones.

Abstract

We prove that every continuous linear functional on the space $S^0(R^d)$ consisting of the entire analytic functions whose Fourier transforms belong to the Schwartz space $\mathcal D$ has a unique minimal carrier cone in $R^d$, which substitutes for the support. The proof is based on a relevant decomposition theorem for elements of the spaces $S^0(K)$ associated naturally with closed cones $K\subset R^d$. These results, essential for applications to nonlocal quantum field theory, are similar to those obtained previously for functionals on the Gelfand-Shilov spaces $S^0_\alpha$, but their derivation is more sophisticated because $S^0(K)$ are not DFS spaces and have more complicated topological structure.