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Properties of field functionals and characterization of local functionals. (English) Zbl 1456.81296
Summary: 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 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 Poincaré lemma and defining multi-vector fields and graded functionals within our framework.{
©2018 American Institute of Physics}

MSC:
81T05 Axiomatic quantum field theory; operator algebras
82D20 Statistical mechanics of solids
46F10 Operations with distributions and generalized functions
46F05 Topological linear spaces of test functions, distributions and ultradistributions
26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
46S60 Functional analysis on superspaces (supermanifolds) or graded spaces
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