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Pré-Publication, Document De Travail Année : 2017

Properties of field functionals and characterization of local functionals

Résumé

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. By leaning 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 it 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 within our framework.
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Dates et versions

hal-01654120 , version 1 (03-12-2017)
hal-01654120 , version 2 (05-12-2017)
hal-01654120 , version 3 (16-05-2019)

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  • HAL Id : hal-01654120 , version 1

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Christian Brouder, Nguyen Viet Dang, Camille Laurent-Gengoux, Kasia Rejzner. Properties of field functionals and characterization of local functionals. 2017. ⟨hal-01654120v1⟩
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