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Properties of Field Functionals and Characterization of Local Functionals This is a repository copy of Properties of field functionals and characterization of local functionals. White Rose Research Online URL for this paper: https://eprints.whiterose.ac.uk/126454/ Version: Accepted Version Article: Brouder, Christian, Dang, Nguyen Viet, Laurent-Gengoux, Camille et al. (1 more author) (2018) Properties of field functionals and characterization of local functionals. Journal of Mathematical Physics. ISSN 0022-2488 Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ Properties of field functionals and characterization of local functionals Christian Brouder,1, a) Nguyen Viet Dang,2 Camille Laurent-Gengoux,3 and Kasia Rejzner4 1)Sorbonne Universit´es, UPMC Univ. Paris 06, UMR CNRS 7590, Institut de Min´eralogie, de Physique des Mat´eriaux et de Cosmochimie, Mus´eum National d’Histoire Naturelle, IRD UMR 206, 4 place Jussieu, F-75005 Paris, France. 2)Institut Camille Jordan (UMR CNRS 5208) Universit´eClaude Bernard Lyon 1, Bˆat. Braconnier, 43 bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France. 3)Institut Elie Cartan de Lorraine (IECL) Universit´ede Lorraine, Bˆat. A, Ile du Saulcy F-57045 Metz cedex 1, France. 4)Department of Mathematics, University of York, Heslington, York YO10 5DD, United Kingdom. (Dated: 19 January 2018; Revised 19 January 2018) 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´elemma and defining multi-vector fields and graded functionals within our framework. CONTENTS C. The regular functionals 13 D. The microcausal functionals 14 I. Motivation 2 E. Local functionals 14 II. Functionals and their derivatives 3 V. Additivity 15 A. The space of classical fields 3 A. Partial additivity 15 B. Locally convex spaces 3 B. A non-local partially additive functional 15 C. Functional derivatives 4 C. Additive functionals 16 1. Examples 5 VI. Characterization of smooth local 2. Historical remarks 5 functionals 17 D. Properties of the differential 6 A. The manifold of jets of functions on a 1. Continuity 6 manifold 17 2. The fundamental theorem of calculus 6 (2) B. Properties of F 19 3. Additional properties 6 ϕ C. Proof of Theorem VI.3 22 E. Smooth functionals 7 D. Representation theory of local III. Properties of functionals 8 functionals. 24 1. Explicit forms 26 A. Support of a functional 8 B. A multilinear kernel theorem with VII. Peetre theorem for local and multilocal parameters. 9 functionals 27 1. Proof of the main result. 10 A. Peetre theorem for local functionals 27 C. Order of distributions 12 B. Multilocal functionals and first Peetre D. Derivatives as smooth functionals 12 theorem 28 C. The second Peetre Theorem 29 IV. Topologies on spaces of functionals 12 A. Bastiani’s topology 13 VIII. Multi-vector fields and graded B. Nuclear and complete topologies 13 functionals 29 1. Locality of functionals on graded space. 29 2. The contraction operation. 30 3. Some conjectures on local graded a)Electronic mail: [email protected] functionals. 30 2 IX. Acknowledgements 31 topological framework for local functionals as understood by16–18. The present paper puts forth the following formulation I. MOTIVATION of the concept of locality: Definition I.1. Let M be a manifold19. Let U be an Functionals (i.e. functions of functions) are mathe- open subset of C∞(M). A smooth functional F : U matical objects successfully applied in many areas of K is said to be local if, for every ϕ U, there is→ a physics. Since Schwinger’s ground-breaking papers1,2, neighborhood V of ϕ, an integer k, an∈ open subset k V ⊂ Green functions of quantum field theory are obtained as J M and a smooth function f C∞( ) such that x functional derivatives of the generating functional Z(j) k ∈ V ∈ M f(jx ψ) is supported in a compact subset K M with respect to the functions j (external sources). In and7→ ⊂ solid-state and molecular physics, the exchange and cor- relation potential of density functional theory is com- k F (ϕ + ψ)= F (ϕ)+ f(jx ψ)dx, puted from the functional derivative of the total energy ZM E(ρ) with respect to the electron density ρ3,4. In per- whenever ϕ + ψ V and where jkψ denotes the k-jet of turbative algebraic quantum field theory (pAQFT), the x ψ at x. ∈ observables are functionals F (ϕ) of the classical field ϕ5. 6 This formulation was possible due to the crucial result In other words, we require F to be local in the sense of that allowed to realize abstract quantum fields as con- Eq. (1), but only around each ϕ U because the integer crete functionals on the space of classical configurations. k and the function f can depend∈ on the neighborhood This viewpoint is not only simplifying computations, but V . In short, our local functionals are local in the “tradi- also allows to construct new perturbative and exact mod- tional sense”, but only locally in the configuration space 7,8 els of QFT’s . It is, therefore, crucial to understand (i.e. in a neighborhood of each ϕ). We do not need functional analytic properties of classical functionals to global locality to apply variational methods and derive be able to use these in quantization and obtain even more Euler-Lagrange equations. We will show by exhibiting models. The importance of this endeavour is justified by an example that this concept of locality is strictly more the fact that presently we do not know any exact inter- general than the traditional one. Our first main result is acting QFT models in 4 spacetime dimensions. a simple characterization of local functionals in the sense Functionals are also used in pure mathematics, for ex- of Def. 1.1: ample loop space cohomology9 and infinite dimensional integrable systems: the hierarchy of commuting Hamil- Theorem I.2. Let U be an open subset of C∞(M). A tonians for the Korteweg de Vries equation is for instance smooth functional F : U K (where K = R or C) is → all made of functionals10. local if and only if In all these fields, the concept of locality is crucial: the Lagrangian of quantum field theory is local and the coun- 1. F is additive (i.e. it satisfies F (ϕ1 + ϕ2 + ϕ3) = terterms of the renormalization process have to be local, F (ϕ1+ϕ2)+F (ϕ2+ϕ3) F (ϕ2) whenever supp ϕ1 supp ϕ = ) − ∩ the approximations of E(ρ) used in practice are local and 3 ∅ it is an open question whether the true density functional 2. For every ϕ U, the differential DFϕ of F at ϕ is a E(ρ) is local or not. Therefore, it is crucial to determine distribution∈ with empty wave front set. Thus, it can precisely what is meant by a local functional. According 11–14 be represented by a function Fϕ (M) (with to the standard definition , if ϕ is a classical field (i.e. (M) the space of compactly∇ supported∈ D smooth a smooth section of a vector bundle over M and we mo- functionsD on M, i.e. “test functions”). mentarily consider M = Rd for notational convenience), then a functional F (ϕ) is local if it is of the form 3. The map U (M) defined by ϕ Fϕ is smooth (in the→ sense D of Bastiani). 7→ ∇ F (ϕ)= dxf x,ϕ(x),∂µϕ(x),...,∂µ1...µk ϕ(x) .(1) Rd Our characterization of locality is inspired by the mi- Z crolocal functionals proposed by Brunetti, Fredenhagen where f is a smooth compactly supported function with and Ribeiro20. However, the proof of their Proposi- a finite number of arguments. tion 2.3.12 is not complete because the application of However, this definition of local functionals is not very the Fubini theorem and the second use of the fundamen- handy in practice because it is global and sometimes too tal theorem of calculus are not justified. Our condition 3 restrictive.
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