# Properties of field functionals and characterization of local functionals

## TL;DR

This paper establishes a rigorous mathematical framework for functionals in quantum field theory, characterizing local functionals and linking their derivatives to distributions, with implications for physical applications and cohomological approaches.

## Contribution

It introduces a new rigorous framework for functionals, characterizes local functionals via additivity and Peetre's theorem, and advances cohomological methods in quantum field theory.

## Key findings

- Most formal manipulations are rigorously justified.
- Local functionals are characterized by additivity and Peetre's theorem.
- A cohomological approach to quantum field theory is initiated.

## Abstract

Functionals (i.e. functions of functions) are widely used in quantum field theory and solid-state physics. In this paper, functionals are given a rigorous mathematical framework and their main properties are described. The choice of the proper space of test functions (smooth functions) and of the relevant concept of differential (Bastiani differential) are discussed. The relation between the multiple derivatives of a functional and the corresponding distributions is described in detail. It is proved that, in a neighborhood of every test function, the support of a smooth functional is uniformly compactly supported and the order of the corresponding distribution is uniformly bounded. Relying on a recent work by Yoann Dabrowski, several spaces of functionals are furnished with a complete and nuclear topology. In view of physical applications, it is shown that most formal manipulations can be given a rigorous meaning. A new concept of local functionals is proposed and two characterizations of them are given: the first one uses the additivity (or Hammerstein) property, the second one is a variant of Peetre's theorem. Finally, the first step of a cohomological approach to quantum field theory is carried out by proving a global Poincar\'e lemma and defining multi-vector fields and graded functionals within our framework.