Properties of Field Functionals and Characterization of Local Functionals

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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

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[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 Universités, UPMC Univ. Paris 06, UMR CNRS 7590, Institut de Minéralogie, de Physique des Matériaux et de Cosmochimie, Muséum National d’Histoire Naturelle, IRD UMR 206, 4 place Jussieu, F-75005 Paris, France. 2)Institut Camille Jordan (UMR CNRS 5208) UniversitéClaude Bernard Lyon 1, Bât. Braconnier, 43 bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France. 3)Institut Elie Cartan de Lorraine (IECL) Universitéde Lorraine, Bât. 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 Poincarélemma 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- Dfunctions 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. For example, general relativity has no lo- solves that problem. On the other hand, we do not need cal gauge-invariant observables in the sense of Eq. (1), their assumption that F is compactly supported. whereas it has local gauge-invariant observables when the Let us stress that the notion of locality is quite sub- concept of locality is slightly generalized, as discussed tle and depends strongly on the functional analytic set- in7 (see also the parallel work15). Note that the concept ting. A functional characterization of a notion of lo- of locality presented in the present paper gives a proper cal functionals on measurable functions might not be 3 valid anymore when applied to smooth functions as because the rapid decrease of test functions at infinity is is shown by the simple counterexample of Section VB. controlled by some Euclidian distance which is not canon- We also make a conjecture as to how to generalize our ically defined on general spacetime manifolds [25, p. 339]. main result to multi-vector fields and graded functionals, The most natural spaces of test functions on a general which is crucial for a rigorous version of the Batalin- spacetime M are the space C∞(M) of real valued smooth Vilkovisky approach to gauge field theory and quantum functions on M and its subspace (M) of compactly sup- gravity. The second main result is a proof of the global ported functions. These two spacesD are identical when Poincarélemma (in our context), which is crucial to set M is compact, but physically relevant spacetimes are up the BRST and variational complexes. The last one is not compact because they are globally hyperbolic, and another characterization of local (and multilocal) func- a choice must be made. tionals in the form of a Peetre’s theorem. In this paper, we choose C∞(M) (or the set Γ(M,B) Along the way to these results, we prove interesting of smooth sections of a vector bundle B). There is a properties of general functionals that we briefly describe strong physical reason for this26: in the quantization pro- now. In section 2, we explain why we choose test func- cess we must be able to deal with on-shell fields ϕ, that tions that are only smooth instead of smooth and com- are smooth solutions to normally hyperbolic equations pactly supported, we describe the topology of the space and as such cannot be compactly supported. Therefore, of test functions and we present the concept of Bastiani the domain of the functionals can be C∞(M) but not differentiability and its main properties. In section 3, (M). There are also good mathematical arguments D we show that a smooth functional is locally compactly for this choice: In particular, C∞(M) is a Fréchet space supported (i.e. in a neighborhood of every test func- and its pointwise multiplication is continuous [27, p. 119]. tion), we prove that the kth derivative of a functional Moreover, the Fréchet property of C∞(M) saves us the defines a continuous family of distributions whose order trouble of distinguishing Bastiani from convenient differ- is locally bounded. Section 4, which relies heavily on entiability which is treated in Ref. 28. 21,22 Dabrowski’s work , describes in detail a nuclear and The choice of C∞(M) has, however, several drawbacks: complete topology on several spaces of functionals used (i) Since smooth functions are generally not integrable in quantum field theory. Section 5 discusses the con- over M, the Lagrangian density (ϕ) must be multi- cept of additivity which characterizes local functionals. plied by a smooth compactly supportedL function g so that Sections 6 and 7 prove the main results discussed above. (ϕ)g is integrable over M 29. As a result, long-range in- Note that the present paper has a somewhat foundational Lteractions are suppressed and infrared convergence is en- character, in as much as the choice of test-functions, ad- forced. This simplifies the problem but makes it difficult ditivity property and differential are carefully justified to deal with the physics of infrared divergence. (ii) The from the physical and mathematical points of view. It function g breaks the diffeomorphism invariance of the contributes to the formulation of a mathematically rig- Einstein-Hilbert action. (iii) The effect of a perturbation orous basis on which the quantum field theory of gauge ϕ + ǫψ is easier to deal with when ψ is compactly sup- fields and gravitation can be built. ported because it avoids the presence of boundary terms. Note also that this paper aims at both functional an- This problem can be solved by considering C∞(M) as a alysts and theoretical physicists. Because of this dual manifold modeled on (M)5,20, but this is an additional readership, the proofs are often more detailed than what complication. D would be required for experts in functional analysis.

B. Locally convex spaces II. FUNCTIONALS AND THEIR DERIVATIVES The spaces of test functions and functionals consid- To set up a mathematical definition of functionals, we ered in the paper are all locally convex. The most peda- need to determine precisely which space of test functions gogical introduction to locally convex spaces is probably (i.e. classical fields and sources) we consider and what Horvath’s book30, so we refer the reader to it for more we mean by a functional derivative. details. We describe now the topology of the spaces of test functions that we use. For the space of smooth test func- d A. The space of classical fields tions C∞(R ), the topology is defined by the seminorms

α πm,K (f)= sup sup ∂ f(x) , (2) Propagators and Green functions of quantum ﬁelds x K α m | | in ﬂat spacetimes are tempered distributions23,24 and ∈ | |≤ d the corresponding test functions are rapidly decreasing. where f C∞(R ), m is an integer, K is a compact ∈d Tempered distributions are computationaly convenient subset of R , α =(α1,...,αd) is a d-tuple of nonnegative α α1 αd because they have Fourier transforms. However, tem- integers, with α = α1 + + αd and ∂ = ∂1 ...∂d , | i| ··· pered distributions cannot be canonically extended to with ∂i = ∂/∂x the derivative with respect to the i-th curved spacetimes (i.e. Lorentzian smooth manifolds) coordinate of x [27, p. 88]. 4

d If U is open in R , we denote by C∞(U) the space Locally convex spaces are very versatile and they are of all functions defined on U which possess continuous the proper framework to define spaces of smooth func- partial derivatives of all orders. We equip C∞(U) with tionals, i.e. smooth functions on a space of functions (or the topology defined by the seminorms πm,K where K sections of a bundle). The first step towards this goal is runs now over the compact subsets of U [30, p. 89]. For to provide a rigorous definition of functional derivatives. d every open set U R , the space C∞(U) is Fréchet, reflexive, Montel, barrelled⊂ [30, p. 239], bornological [30, p. 222] and nuclear [31, p. 530]. We define now C (M), where M is a d-dimensional ∞ C. Functional derivatives manifold (tacitly smooth, Hausdorff, paracompact and orientable) described by charts (Uα,ψα). If for every Uα M we are given a smooth function gα To define the space of functionals, we consider the main ⊂ 1 ∈ C∞(ψα(Uα)) such that gβ = gα ψα ψ− on ψβ(Uα Uβ), examples Z(j) and F (ϕ). These two functionals send ◦ ◦ β ∩ we call the system gα a smooth function g on M. smooth classical fields to K, where K = R or K = C. The space of smooth functions on M is denoted by Moreover, functional derivatives of Z and F of all orders C∞(M) [32, p. 143]. This definition is simple but to de- are required to obtain the Green functions from Z(j) and scribe the topological properties of C∞(M) the following to quantize the product F (ϕ)G(ϕ). Therefore, we must more conceptual definition is useful. define the derivative of a function f : E K, where E Let M be a manifold and B M a smooth vector is the space of classical fields. → → bundle of rank r over M with projection π. Let E = It will be useful to generalize the problem to functions Γ(M,B) be the space of smooth sections of B equipped 28 f between arbitrary locally convex spaces E and F . To with the following topology : define such a derivative we start from Definition II.1. The topology on Γ(M,B) is defined as follows. Choose a chart (Uα,ψα)α and a trivialization Definition II.2. Let U be an open subset of a Hausdorff 1 r map Φα : π− (Uα) Ω R , where Ω is a fixed open locally convex space E and let f be a map from U to a d → × set in R . Then the map Φα allows to identify Γ(Uα,B) Hausdorff locally convex space F . Then f is said to have r 1 d with C∞(Ω, R ) by Φα : π− Uα Ω R such that a derivative at x U in the direction of v E if the → × following limit exists∈ 33 ∈ Φα(x,s(x))=(ψα(x),Kα(s)(ψα(x))), where f(x + tv) f(x) Dfx(v) := lim − . r t 0 t Kα : s Γ(Uα,B) Kα(s) C∞(Ω, R ). → ∈ 7→ ∈ The topology on Γ(M,B) is the weakest topology making One can also consider the same definition restricted all the maps Kα continuous. to t > 034. A function f is said to have a Gâteaux dif- 35,36 37 This topology does not depend on the choice of charts ferential (or a Gâteaux variation ) at x if Dfx(v) exists for every v E. However, this definition is far or trivialization maps [28, p. 294]. To interpret this topol- ∈ too weak for our purpose because Dfx(v) is generally ogy, denote by ρα : s Γ(M,B) s Uα Γ(Uα,B) the restriction map of sections∈ on open7→ sets| of∈ our open cover neither linear nor continuous in v and it can be linear without being continuous and continuous without being (Uα)α I of M. The space Γ(M,B) fits into the following complex∈ of vector spaces linear [38, p. 7]. Therefore, we will use a stronger definition, namely Bastiani differentiability39, which is the (ρα)α∈I r fundamental concept of differentiability used throughout 0 Γ(M,B) Γ(Uα,B) C∞(Ω, R ) → −→ ≃ the paper: α I α I Y∈ Y∈ (ρα ρβ )α6=β − Γ(Uα Uβ,B). (3) −→ ∩ Definition II.3. Let U be an open subset of a Hausdorff (α=β) I2 6 Y∈ locally convex space E and let f be a map from U to a Hausdorff locally convex space F . Then f is Bastiani The topology on Γ(Uα,B) is given by the isomorphism 1 r differentiable on U (denoted by f C (U)) if f has a Γ(Uα,B) C∞(Ω, R ) hence it is nuclear Fréchet. The ≃ Gâteaux differential at every x U∈ and the map Df : countable products α I Γ(Uα,B) and (α,β) I2 Γ(Uα ∈ ∩ U E F defined by Df(x,v)= Dfx(v) is continuous U ,B) are therefore nuclear∈ Fréchet. For every∈ pair × → β Q Q on U E. (α,β) of distinct elements of I, the difference of re- × striction maps ρα ρβ is continuous and the topology on Γ(M,B) is the− weakest topology which makes the With this definition, most of the properties used in above complex topological, which implies that it is nu- physics textbooks (e.g. chain rule, Leibniz rule, linearity) clear Fréchet as the kernel of α=β ρα ρβ. are mathematically valid. 6 − Q 5

1. Examples not normable, no topology on L(E,F ) provides the nice properties of Bastiani’s definition [44, p. 6] (Hamilton [40, We shall consider several examples of functions from p. 70] gives a simple example of a map which is contin- d C∞(M) to R or C, where M = R : uous U E E but such that the corresponding map U L×(E,E→) is not continuous). Thus, L(E,F ) was → F (ϕ)= f(x)ϕn(x)dx, equipped with various non-topological convergence struc- M tures [44, p. 23]. The result is an impressive zoology of Z differentiabilities. Twenty-five of them were reviewed and G(ϕ)= g(x1,...,xn)ϕ(x1) ...ϕ(xn)dx1 ...dxn, classified by Averbukh and Smolyanov45. Still more can n ZM be found in the extensive lists given by Gähler46 and Ver µν 47 38,44,48 H(ϕ)= g (x)h(x)∂µϕ(x)∂ν ϕ(x)dx, Eecke covering the period up to 1983 (see also ). µν M Nowadays, essentially two concepts of differentiabil- X Z ity survive, Bastiani’s and the so-called convenient ap- ϕ(x) I(ϕ)= f(x)e dx, proach developed by Kriegl–Michor in the reference ZM monograph28, which is weaker than Bastiani’s for gen- R f(x)ϕ(x)dx J(ϕ)= e M , eral Hausdorff locally convex spaces. In particular, on any locally convex space which is not bornological, there K(ϕ)= f(x)sin ϕ(x) dx, is a conveniently smooth map which is not continuous [41, ZM p. 19]. However, a nice feature of both approaches is that where f, g, and h are smooth compactly supported func- for a Fréchet space E, a function f : E K is smooth in tions and where g is a symmetric function of its argu- → 17,40 the sense of Bastiani iff it is smooth in the sense of the ments. Further examples can be found in . It is im- convenient calculus49. Bastiani differentiability became mediate to check that widespread after it was used by Michor50, Hamilton40 51 n 1 (for Fréchet spaces) and Milnor and it is now vigor- DFϕ(v)= n g(x)ϕ − (x)v(x)dx, ously developed by Glöckner and Neeb (see also52). M Z To complete this section, we would like to mention that DGϕ(v)= n g(x1,...,xn)ϕ(x1) ...ϕ(xn 1) the Bastiani differential is sometimes called the Michal- n − ZM Bastiani differential53–55 (or even Michel-Bastiani differ- v(xn)dx1 ...dxn, ential54). This is not correct. The confusion comes from the fact that Bastiani defines her differentiability in sev- DH (v) = 2 gµν (x)h(x)∂ ϕ(x)∂ v(x)dx, ϕ µ ν eral steps. She starts from the Gâteaux derivability, then µν M X Z she says that a map f : U F is differentiable at x ϕ(x) → DIϕ(v)= f(x)e v(x)dx, (see [41, p. 18] and [39, p. 18]) if: i) Dfx is linear and M continuous from E to F and ii) the map mx : R E F Z × → R f(x)ϕ(x)dx defined by DJϕ(v)= e M f(x)v(x)dx, ZM f(x + tv) f(x) mx(t,v)= − Dfx(v), DKϕ(v)= f(x) cos ϕ(x) v(x)dx. t − ZM for x + tv U, is continuous at (0,v) for all v E. This differentiability∈ at x is indeed equivalent to the∈ dif- 2. Historical remarks ferentiability defined by Michal56 in 1938, as proved in Refs. 45 and 57, [44, p. 72] and [47, p. 202]. What we Definition II.3 is due to Bastiani39,41 and looks quite call Bastiani differentiability is called differentiability on natural. In fact, it is not so. For a long time, many differ- an open set by Bastiani (see [41, p. 25] and [39, p. 44]) ent approaches were tried. For any reasonable definition and is strictly stronger than Michal differentiability. of differentiability, the map Dfx : E F is linear and The same distinction between Michal-Bastiani dif- → continuous, so that Dfx L(E,F ). If E and F are Ba- ferentiability and Bastiani differentiability is made by nach spaces, then a map ∈f : E F is defined to be con- Keller [44, p. 72] in his thorough review. Bastiani’s dif- → 1 44 tinuously (Fréchet) differentiable if the map x Dfx is ferentiability is denoted by Cc by Keller , who also → 1 continuous from U to Lc(E,F ), where Lc(E,F ) is the set attributes the definition equivalent to Cc to Bastiani of continuous maps from E to F equipped with the opera- alone [44, p. 11]. tor norm topology. But Fréchet differentiability is strictly In her PhD thesis, Andrée Bastiani developed her con- stronger than Bastiani’s differentiability specialized to cept of differentiability to define distributions on a locally Banach spaces42. This is why Bastiani’s definition was convex space E with values in a locally convex space F . often dismissed in the literature43 and, for locally convex She started from Schwartz’ remark that a distribution is, spaces that are not Banach, the map Df was generally locally, the derivative of a continuous function. She used required to be continuous from U to L(E,F ) equipped her differential D to define F -valued distributions over with some well-chosen topology. However, when E is E58. A drawback of Bastiani’s framework with respect 6 to the convenient framework is that her category is not Theorem II.7. Let f be a Bastiani differentiable map Cartesian closed for locally convex spaces that are not between two Hausdorff locally convex spaces E and F . Fréchet. Let U be an open set in E, x in U and v in E such that (x + tv) U for every t in an open neighborhood I of [0, 1], so∈ that g : t f(x + tv) is a map from I to F . D. Properties of the differential Then, 7→

1 We review now some of the basic properties of func- f(x + v)= f(x)+ g′(t)dt tional derivatives which will be used in the sequel. We Z0 strongly recommend Hamilton’s paper40, adapted to lo- 1 cally convex spaces by Neeb42. = f(x)+ Dfx+tv(v)dt. (4) Z0 To give a meaning to Eq. (4), we need to define an 1. Continuity integral of a function taking its values in a locally convex space. To cut a long story short64,65: We characterize continuous (nonlinear) maps between two locally convex spaces. Definition II.8. Let X be a locally compact space (for example Rn or some finite dimensional manifold), µ a Lemma II.4. Let E and F be locally convex spaces measure on X and F a Hausdorff locally convex space. whose topology is defined by the families of seminorms Let f be a compactly supported continuous function from (pi)i I and (qj)j J , respectively. Then f is continu- X to F . Let F be the topological dual of F (i.e. the ∈ ∈ ′ ous at x iff, for every seminorm qj of F and every space of continuous linear maps from F to K). If there

ǫ > 0, there is a ﬁnite number pi1 ,...,pik of semi- is an element y F such that { } norms of E and k strictly positive numbers η1,...,ηk ∈ such that pi (x y) < η ,...,pi (x y) < ηk imply 1 − 1 k − α,y = α,f dµ, qj f(y) f(x) <ǫ. h i h i − ZX Proof. This is just the translation in terms of seminorms for every α F ′, where , denotes the duality pairing, of the fact that f is continuous at x if, for every open set then we say∈ that f has ah· weak·i integral and we denote y

V containing f(x) , there is an open set U containing x by X fdµ. such that f(U) V [59, p. 86]. ⊂ TheR uniqueness of the weak integral follows from the When the seminorms of E are saturated [30, p. 96], as fact that F is Hausdorff. In general, the existence of d the seminorms πm,K of C∞(R ), the condition becomes a weak integral requires some completeness property for d simpler: a map f : C∞(R ) K is continuous at x if and F [64, p. 79]. However, this is not the case for the fun- → only if, for every ǫ> 0, there is a seminorm πm,K and an damental theorem of calculus [41, p. 27]. This point was η > 0 such that πm,K (x y) <η implies f(y) f(x) <ǫ. stressed by Glöckner66. Since Fréchet spaces are− metrizable, we| can also− use| the following characterization of continuity [60, p. 154]: 3. Additional properties Proposition II.5. Let E be a metrizable topological space and F a topological space. Then, a map f : E F is continuous at a point x iff, whenever a sequence→ For maps between locally convex spaces, the linearity 40 (xn)n N converges to x in E, the sequence f(xn)n N con- of the differential is not completely trivial . verges∈ to f(x) in F . ∈ Proposition II.9. Let E and F be locally convex spaces Another useful theorem is [61, p. III.30]: and f be a Bastiani differentiable map from an open subset U of E to F . Then, for every x U, the differential Proposition II.6. Let E and F be two Fréchet spaces ∈ Dfx : E F is a linear map. and G a locally convex space. Every separately continu- → ous bilinear mapping from E F to G is continuous. The chain rule for Bastiani-differentiable functions was × 4139 53 This result extends to multilinear mappings from a first proved by Bastiani herself (see also ). product E1 En of Fréchet spaces to a locally convex Proposition II.10. 62,63 ×···× Assume that E,F,G are locally con- space . vex spaces, U E and V F are open subsets and f : V G and ⊂g : U V are⊂ two Bastiani-differentiable maps.→ Then, the composite→ map f g : U G is Bastiani 2. The fundamental theorem of calculus ◦ → differentiable and D(f g)x = Df Dgx. ◦ g(x) ◦ The fundamental theorem of calculus for functionals By using these properties, the reader can prove that reads our examples are all Bastiani differentiable. 7

E. Smooth functionals 5. If E is metrizable, then f Ck(U, F ) iff f belongs k 1 k 1 ∈ k 1 to C − (U, F ) and D − f : U L(E − ,F ) is → To define smooth functionals we first define multiple Bastiani differentiable [41, p. 43]. Here the metriz- derivatives. ability hypothesis is used to obtain a canonical in- k 1 k jection from C(U E,L(E − ,F )) to C(U E ,F ). Definition II.11. Let U be an open subset of a locally × × convex space E and f a map from U to a locally convex We refer the reader to40,42 and Bastiani’s cited works space F . We say that f is k-times Bastiani differentiable for the proofs. Other results on Ck(U) functions can be on U if: found in Keller’s book44. All the statements of Propo- sition II.13 are valid for k = , i.e. smooth functions. The kth Gâteaux differential ∞ • Bastiani also defines jets of smooth functions between k locally convex spaces [41, p. 52] and [39, p. 75]. k ∂ f(x + t1v1 + + tkvk) 67 68 D fx(v1,...,vk)= ··· , Note that Neeb and Glöckner agree with Bastiani ∂t1 ...∂tk t=0 for the definition of the first derivative but they use an ap-

parently simpler definition of higher derivatives by saying where t = (t1,...,tk), exists for every x U and k k 1 k 1 1 ∈ that f is C iff df is C − iff d − f is C . However, this every v ,...,vk E. 1 ∈ definition is less natural because, for example, f C2 if The map Dkf : U Ek F is continuous. df : U E F is C1. In the definition of the first∈ deriva- • × → tive, U×is now→ replaced by U E and E by E E. In other Notice that for a function f assumed to be k-times words, d2 is a continuous map× from U E3×to F . More Bastiani differentiable, the restriction to any finite di- × k generally dk is a continuous map from U E2 1 to F [68, mensional affine subspace is not only k-times differen- − p. 20]. Moreover, according to Proposition× 1.3.13 [68, tiable (in the usual sense) but indeed of class Ck. The p. 23], a map f belongs to Ck if and only if it belongs set of k-times Bastiani differentiable functions on U is to Ck(U) is the sense of Bastiani, and Bastiani’s Dkf denoted by Ck(U), or Ck(U, F ) when the target space is denoted by d(k)f by Glöckner [68, p. 23] and called F has to be specified. Bastiani gives an equivalent def- the k-th differential of f. The k-th derivatives dkf and inition, called k-times differentiability on U [41, p. 40], (k) k k 44 d f = D f are not trivially related. For example [68, which is denoted by Cc by Keller . 2 2 p. 24]: d f(x,h1,h2,h3)= D f(x,h1,h2)+ Df(x,h3). Definition II.12. Let U be an open subset of a locally The Taylor formula with remainder for a function in convex space E and f a map from U to a locally convex Cn+1(U) reads [41, p. 44]: k space F . We say that f is smooth on U if f C (U, F ) n ∈ tk for every integer k. f(x + th)= f(x)+ Dkf (hk) k! x We now list a number of useful properties of the k-th k=1 t X n Bastiani differential: (t τ) n+1 n+1 + − D fx+τh(h )dτ, (5) Proposition II.13. Let U be an open subset of a locally 0 n! Z n convex space E and f Ck(U, F ), where F is a locally tk ∈ = f(x)+ Dkf (hk) convex space, then k! x k=1 k X 1. D fx(v1,...,vk) is a k-linear symmetric function t n 1 (t τ) − n n n n of v ,...,v [40, p. 84]. + − D fx τh(h ) D fx(h ) dτ. 1 k (n 1)! + − Z0 − 2. The function f is of class Cm for all 0 m k [41, Taylor’s formula with remainder is a very important tool p. 40]. In particular, f is continuous.≤ ≤ to deal with smooth functions on locally convex spaces. 3. The compositions of two functions in Ck is in Ck The reader can check that all our examples are smooth and the chain rule holds [41, p. 51] and [40, p. 84]. functionals in the sense of Bastiani.

m k m m k n! n k 4. The map D f is in C − (U, L(E ,F )) [41, p. 40] D Fϕ(v1,...,vk)= f(x)ϕ − (x) m (n k)! where L(E ,F ) is the space of jointly continuous − ZM m-linear maps from E to F , equipped with the lo- v1(x) ...vk(x)dx, cally convex topology of uniform convergence on the k for k n and D Fϕ = 0 for k>n. compact sets of E: i.e. the topology generated by ≤ the seminorms k n! D Gϕ(v1,...,vk)= g(x1,...,xn) (n k)! M n pC,j(α)= sup qj α(h1,...,hm) , − Z (h1,...,hm) C v1(x1) ...vk(xk)ϕ(xk+1) ...ϕ(xn) ∈ dx1 ...dxn, where C = C1 Cm, Ci runs over the com- ×···× k pact sets of E and (qj)j J is a family of seminorms for k n and D Gϕ = 0 for k>n. Recall that g is defining the topology of∈F . a symmetric,≤ smooth compactly supported function of 8 its arguments. The functional H has only two non-zero from Ek to K, and the quantum field amplitudes (e.g. derivatives and represented pictorially by Feynman diagrams) that are ˆ k k ⊠k continuous linear maps from E⊗π = Γ(M ,B ) to K, 2 µν k ⊠k D Hϕ(v1,v2) = 2 g (x)h(x)∂µv1(x)∂ν v2(x)dx. i.e. elements of the space Γ′(M , (B∗) ) of compactly µν M X Z supported distributions with values in the k-th external tensor power of the dual bundle B∗. It is easy to see that The example I has an infinite number of nonzero deriva- k tives: there is a canonical correspondence between D Fϕ and ˆ k its associated distribution on E⊗π , that we denote by k ϕ(x) (k) D Iϕ(v1,...,vk)= f(x)e v1(x) ...vk(x)dx. Fϕ . However, the equivalence between the continuity of M k k (k) ˆ k Z D F on U E and the continuity of F on U E⊗π Finally requires a proof.× × Finally, we show that the order of F (k) is locally k R f(x)ϕ(x)dx D Jϕ(v1,...,vk)= e M f(x1) ...f(xk) bounded. k ZM v1(x1) ...vk(xk)dx1 ...dxk. The functionals F , G and H are polynomials in the A. Support of a functional sense of Bastiani [41, p. 53]: Definition II.14. Let E and F be locally convex spaces. Brunetti, Dütsch and Fredenhagen proposed to define A polynomial of degree n on E is a smooth function f : the support of a functional F by the property that, if E F such that Dkf = 0 for all k>n. the support of the smooth function ψ does not meet the → support of F , then F (ϕ + ψ) = F (ϕ) for all ϕ. More k 17 Let u be a distribution in ′(M ), then the functional precisely : D k f : (M) K defined by f(ϕ) = u(ϕ⊗ ) is polynomial in theD sense→ of Bastiani and its k-derivative is: Definition III.1. Let F : U K be a Bastiani smooth → function, with U a subset of C∞(M). The support of F k D fϕ(v1,...,vk)= u(vσ(1) vσ(k)), is the set of points x M such that, for every open set x ⊗···⊗ ∈ U σ containing x, there is a ϕ U and a ψ in C∞(M) with X ∈ ϕ + ψ U such that supp ψ x and F (ϕ + ψ) = F (ϕ). where σ runs over the permutations of 1,...,k and the ∈ ⊂U 6 canonical inclusion (M) k (M k){ was used.} ⊗ We want to relate this definition of the support of F If F and G are smoothD maps⊂D from E to K, we can with the support of D F , which is compactly supported compose the smooth map ϕ F (ϕ),G(ϕ) and the ϕ as every distribution over C (M)32. To do so, we need a multiplication in K to show that:7→ ∞ technical lemma about connected open subsets in locally Proposition II.15. Let E be a locally convex space, and convex spaces. U an open set in E. Then the space of smooth functionals from U to K is a sub-algebra of the algebra of real valued Lemma III.2. Let U be a connected open set in a lo- 2 functions. cally convex space E then any pair (x,y) U can be connected by a piecewise affine path. ∈

III. PROPERTIES OF FUNCTIONALS Proof. Define the equivalence relation in E as follows, two elements (x,y) are equivalent iff they∼ are connected by a piecewise affine path. Let us prove that this equiv- We now prove important properties of smooth func- alence relation is both open and closed hence any non tionals. We first investigate the support of a functional. empty equivalence class for is both open and closed in The fact that DF is continuous from C (M) to K ex- ϕ ∞ U hence equal to U. ∼ actly means that DFϕ is a compactly supported distribution for every ϕ. The support of F is then essentially Let x U then there exists a convex neighborhood V of x in∈U which means that every element in V lies the union over ϕ of the supports of DFϕ. We prove that, in the class of x, the relation is open. Conversely let y for any smooth functional and any ϕ C∞(M), there ∈ be in the closure of the equivalence class of x, then any is a neighborhood V of ϕ such that F V is compactly supported. | neighborhood V of y contains an element equivalent to x. The second property that we investigate is required to Choose some convex neighborhood V then we find z V s.t. z x, but z y hence x z y and we just proved∈ establish a link with quantum field theory. In this paper, ∼ ∼ ∼ ∼ we deal with functionals that are smooth functions F on that the equivalence class of x was closed. an open subset U of E = Γ(M,B), where Γ(M,B) is the space of smooth sections of some finite rank vector We can now prove an alternative formula, due to bundle B on the manifold M. There is a discrepancy Brunetti, Fredenhagen and Ribeiro20. k between D Fϕ, which is a continuous multilinear map 9

Lemma III.3. For every Bastiani smooth function F : Of course, this inequality implies that DFϕ(ψ) = 0 when U K, with U a connected open subset of C∞(M): ψ is identically zero on the compact subset K. → Let us show that this implies that the support of DFϕ supp(F )= supp DFϕ. (6) is contained in K. To avoid possible problems at the ϕ U boundary, take any compact neighborhood K of K. [∈ ′ Now, take an open set Ω in M such that Ω K′ = . Proof. Using the result of Lemma III.2, we may reduce ∩ ∅ to the case where U is an open convex set. We prove Then, for every smooth function ψ supported in Ω, we have πm,K (ψ) = 0 because the seminorm πm,K takes that both sets ϕ supp DFϕ and supp F as defined in Def. III.1 have identical complements. Indeed, for every only into account the points of K. As a consequence, point x M, xS / supp F means by definition of the the restriction of DFϕ to Ω is zero, which means that ∈ ∈ Ω is outside the support of DFϕ for every ϕ V . support that there exists an open neighborhood Ω of x ∈ Thus, supp F V K because, for every ϕ V , the such that (ψ,ϕ) (Ω) C∞(M), F (ϕ + ψ) = F (ϕ). | ⊂ ∈ It follows that∀ for∈ all Dψ × (Ω), there exists ε > 0 such support of DFϕ is included in every compact neighbor- that t 6 ε = ϕ + tψ∈ DU and t [ ε,ε] F (ϕ + hood of K. Finally we show that, if F is compactly n | | ⇒ ∈ ∈ −dF (ϕ+7→tψ) supported, then all D Fϕ are compactly supported with tψ) is a constant function of t therefore dt t=0 = n n | supp D Fϕ (supp F )× . This is easily seen by the DFϕ(ψ) = 0. This means that for all ϕ U, the support ⊂ ∈ following cutoff function argument: if F is compactly of DFϕ ′(M) does not meet Ω hence Ω lies in the ∈ E supported then for every cutoff function χ equal to 1 on complement of ϕ U supp (DFϕ) and therefore x Ω ∪ ∈ ∈ an arbitrary compact neighborhood of supp F , we have: does not meet the closure ϕ U supp (DFϕ). ∪ ∈ F (ϕ)= F (χϕ), ϕ E. Then it is immediate by defini- Conversely if x does not meet the closure n ∀ ∈ tion of D Fϕ that ϕ U supp (DFϕ), then there is some neighborhood ∪ ∈ n Ω of x which does not meet ϕ U supp (DFϕ) therefore n d F (χ(ϕ + t1ψ1 + + tnψn)) ∪ ∈ D Fϕ(ψ1,...,ψn)= ··· t=0 for all (ϕ,ψ) U (Ω) s.t. ϕ + ψ U, the whole dt1 ...dtn | straight path∈ [ϕ,ϕ×D+ ψ] lies in U (by convexity∈ of U) n = D Fχϕ(χψ1,...,χψn) hence n n 1 thus supp D Fϕ supp χ× for all test function χ s.t. ⊂ n n t [0, 1],DFϕ tψ(ψ)=0 = dtDFϕ tψ(ψ) = 0, χ supp F = 1 and therefore supp D Fϕ (supp F )× ∀ ∈ + ⇒ + | ⊂ Z0 since supp χ = supp F . χ supp F =1 and by the fundamental theorem of calculus F (ϕ + ψ)= | T F (ϕ). Now we show that any smooth functional is locally B. A multilinear kernel theorem with parameters. compactly supported: We work with M a smooth manifold and B M a Proposition III.4. Let F : U K be a Bastiani smooth smooth vector bundle of finite rank over M. Let→ E = function, where U is an open7→ connected subset of E = Γ(M,B) be its space of smooth sections and U an open C (M). For every ϕ U, there is a neighborhood V ∞ subset of E. We consider smooth maps F : E K of ϕ in U and a compact∈ subset K M such that the where K is the field R or C. In this section we relate7→ the support of F restricted to V is contained⊂ in K. Moreover Bastiani derivatives DkF , which are k-linear on Γ(M,B) for all integers n and all ϕ U, the distributional support n ∈ n n to the distributions used in quantum field theory, which supp (D Fϕ) is contained in K M . ⊂ are linear on Γ(M k,B⊠k). Since the k-th derivative DkF Proof. By definition of the support of a functional, it is of a smooth map is multilinear and continuous in the last enough to show that, for every ϕ V , supp DFϕ K. k variables, we can use the following result [69, p. 471] If F is smooth, then DF : U E∈ K is continuous.⊂ and [70, p 259]: × → Thus, it is continuous in the neighborhood of (ϕ0, 0) for Lemma III.5. Let E be a Hausdorff locally convex every ϕ0 U. In other words, for every ǫ > 0, there ∈ space. There is a canonical isomorphism between any is a neighborhood V of ϕ0, a seminorm πm,K and an k-linear map f : Ek K and the map f : E k K, η > 0 such that DFϕ(χ) <ǫ for every ϕ V and every ⊗ | | ∈ where is the algebraic→ tensor product, which is→ linear χ E such that πm,K (χ) < η. Now, for every ψ E ∈ ∈ and defined⊗ as follows: if χ = χj χj is a finite such that πm,K (ψ) = 0, we see that χ = ψη/(2πm,K (ψ)) j 1 k 6 sum of tensor products, then ⊗···⊗ satisfies πm,K (χ) < η. Thus, DFϕ(χ) < ǫ for every | | P ϕ V and, by linearity, DFϕ(ψ) < (2ǫ/η)πm,K (ψ). On ∈ | | f¯(χ)= f(χj ,...,χj ). (7) the other hand, if πm,K (ψ) = 0, then for any µ > 0 1 k j ψm,K (µψ) = 0 <η, so that DFϕ(µψ) <ǫ. By linearity, X | | DFϕ(ψ) < ǫ/µ for any µ > 0 and we conclude that | | Let us give a topological version of this lemma, us- DFϕ(ψ) = 0. Thus, for every ϕ V and every ψ E, | | ∈ ∈ ing the projective topology. We recall the definition of ǫ a family of seminorms defining the projective topology DFϕ(ψ) 2 πm,K (ψ). | |≤ η on tensor powers of locally convex spaces following [71, 10 p. 23]. For arbitrary seminorms p1,...,pk on E there U1,...,Uk of zero such that ϕ V and ei Ui for i = k ∈ ∈ exists a seminorm p1 pk on E⊗ defined for every 1,...,k imply F (ϕ,e1,...,ek) ǫ. Since E is locally k ⊗···⊗ | | ≤ ψ E⊗ by convex, there are continuous seminorms p ,...,pk on E ∈ 1 and strictly positive numbers η ,...,ηk such that ei Ui 1 ∈ p1 pk(ψ) = inf p1(e1,n) ...pk(ek,n), if pi(ei) ηi. Consider now arbitrary elements e ,...,ek ⊗···⊗ ≤ 1 n of E such that pi(ei) = 0. Then, if fi = eiηi/pi(ei) X 6 we have pi(fi) ηi. Thus, F (ϕ,f ,...,fk) < ǫ and, where the infimum is taken over the representations of ≤ | 1 | by multilinearity, F (ϕ,e1,...,ek) < Mp1(e1) ...pk(ek) ψ as finite sums: ψ = n e1,n ek,n. Following | | ⊗···⊗ where M = ǫ/(η1,...,ηk). The argument in the proof Köthe [72, p. 178], one can prove that p1 pk is the Pk ⊗···⊗ of Proposition III.4 shows that F (ϕ,e1,...,ek) = 0 largest seminorm on E⊗ such that | | if pi(ei) = 0. Therefore, for every ϕ V and every ∈ p(x1 xk)= p1(x1) ...pk(xk), (8) e1,..., ek in E, F (ϕ,e1,...,ek) Mp1(e1) ...pk(ek). ⊗···⊗ By defining C =| 2M we obtain| for ≤ every ϕ V and for all x1,...,xk in E. More precisely, if p is a seminorm k ∈ k every (e1,...,ek) E on E⊗ satisfying Eq. (8), then p(X) 6 (p1 pk)(X) ∈ k ⊗···⊗ for every X E⊗ . 6 ∈ k F (ϕ,e1,...,ek) F (ϕ0,e1,...,ek) Cp1(e1) ...pk(ek). The projective topology on E⊗ is defined by the fam- | − | (9) ily of seminorms p pk where each pi runs over k 1 By definition of F , for all (ϕ,ψ) V E⊗ : a family of seminorms⊗···⊗ defining the topology of E [71, ∈ × p. 24]. F (ϕ,ψ) F (ϕ0,ψ) 6 F (ϕ,e1,n,...,ek,n) When E is Fréchet, its topology is defined by a count- | − | n | X able family of seminorms and it follows that the family of F (ϕ0,e1,n,...,ek,n) k seminorms p1 pk on E⊗ is countable. Hence they − | ⊗···⊗ k 6 C p (e ) ...p (e ), can be used to construct a metric on E⊗ which defines 1 1,n k k,n ˆ k n the same topology as the projective topology and E⊗π X k is defined as the completion of E⊗ relative to this metric for all representations of ψ as finite sum ψ = e ,n n 1 ⊗ or equivalently with respect to the projective topology. A ek,n. Taking the infimum over such representations fundamental property of the projective topology is that yields···⊗ the estimate: P k k f : E K is (jointly) continuous iff f¯ : E⊗ K, still → → k defined by Eq. (7) is continuous with respect to the pro- (ϕ,ψ) V E⊗ , F (ϕ,ψ) F (ϕ0,ψ) 6 Cq(ψ), (10) jective topology [73, p. I-50]. Then f¯ extends uniquely to ∀ ∈ × | − | π k a continuous map (still denoted by f¯) on the completed for the seminorm q = p1 pk on E⊗ and the above ˆ k ⊗···⊗ ˆ π k tensor product E⊗π [61, p. III.15]. inequality extends to any ψ of E⊗ since, in a Fréchet ˆ π k k space, ψ can be approximated by a convergent sequence If E = C∞(M), then E⊗ = C∞(M ) [31, p. 530], k k ˆ k and f¯ becomes a compactly supported distribution on of elements in E⊗ by density of E⊗ in E⊗π and by M k. More generally, if E = Γ(M,B), then E is Fréchet continuity of the seminorm p1 pk for the topology ˆ ⊗···⊗ ˆ k k ⊠k π k nuclear and E⊗π = Γ(M , ) [74, p. 72]. Thus, f¯ of E⊗ . becomes a compactly supportedE distributional section on Another way to state the previous result is to say that k k ¯ (k) M . If f = D Fϕ we denote f by Fϕ . the family of linear maps F (ϕ, ); ϕ V is equicontin- Recall that, if U is an open subset of a Hausdorff locally uous [61, p. II.6]. { · ∈ } convex space E, a map F : U K is smooth iff every DkF is continuous from U E→k to K. According to × k the previous discussion, continuity on E is equivalent to 1. Proof of the main result. ˆ k continuity on E⊗π . Therefore, it is natural to wonder when joint continuity on U Ek is equivalent to joint ˆ k × We are now ready to prove continuity on U E⊗π . This is the subject of the next paragraphs. × Proposition III.7. Let E be a Fréchet space and U E We now prove an equicontinuity lemma. an open subset. Then F : U Ek K, multilinear in⊂ the last k variables, is jointly continuous× 7→ iff the corresponding Lemma III.6. Let E be a Fréchet space, U open in E, ˆ π k K and F : U Ek K a continuous map, multilinear in map F : U E⊗ is jointly continuous. × 7→ × → the last k variables. Then for every ϕ0 U, there exist Proof. One direction of this theorem is straightforward ∈ ˆ k a neighborhood V of ϕ0, a seminorm q of E⊗π and a and holds if E is any locally convex space. Indeed, by constant C > 0 such that definition of the projective tensor product, the canonical k ˆ k ˆ k multilinear mapping E E⊗ is continuous [73, p. I- ϕ V, ψ E⊗π , F (ϕ,ψ) F (ϕ ,ψ) 6 Cq(ψ). 0 → ˆ k ∀ ∈ ∀ ∈ | − | 50]. Therefore, if F is continuous on U E⊗ then, by Proof. By continuity of F : U Ek K, for every ǫ> 0, composition with the canonical multilinear× mapping, F there exist a neighborhood V×of ϕ7→and neighborhoods is continuous on U Ek. 0 × 11

Let us prove that the continuity of F implies the con- some neighborhood V3 of ϕ0 such that for all ϕ V3 the tinuity of F . According to Lemma II.4, we have to show following bound ∈ that, for every ϕ0 U and every ε > 0, there exist a ∈ ˆ k p ﬁnite number of continuous seminorms qi on E⊗π , a ε F (ϕ,e1,j,...,ek,j) F (ϕ0,e1,j,...,ek,j) 6 , neighborhood V of ϕ0 and ηi > 0 such that, if ϕ be- | − | 3 ˆ k j=1 longs to V and ψ E⊗π satisﬁes qi(ψ ψ ) 6 ηi, then X ∈ − 0 F (ϕ,ψ) F (ϕ0,ψ0) 6 ε. holds true. | In order− to bound |F (ϕ,ψ) F (ϕ ,ψ ), we cut it into − 0 0 Finally we found some neighborhood V = V2 V3 of three parts ˆ π k ∩ ϕ0, two seminorms q1 and q2 of E⊗ , and two numbers η1 > 0 and η2 = ǫ/6C such that q1(ψ ψ0) < η1 and F (ϕ,ψ) F (ϕ0,ψ0) = F (ϕ,ψk) F (ϕ0,ψk) − − − q2(ψ ψ0) <η2 imply + F (ϕ,ψ ψk) F (ϕ ,ψ ψk) − − − 0 − + F (ϕ0,ψ) F (ϕ0,ψ0), F (ϕ,ψ) F (ϕ0,ψ0) 6 ε. − (11) | − | where ψk is some element of the algebraic tensor prod- The proposition is proved. k uct E⊗ close enough to ψ0 that we choose now. The equicontinuity Lemma III.6 yields a neighborhood V2 of Now we can specialize our result to the space of smooth ϕ0, a constant C > 0 and a continuous seminorm q2 on sections of vector bundles. We recall a fundamental result ˆ k on the projective tensor product of sections [74, p. 72]: E⊗π so that

ˆ k Proposition III.8. Let Γ(M,B) be the space of smooth ϕ V , ψ E⊗π , F (ϕ,ψ) F (ϕ ,ψ) 6 Cq (ψ). ∀ ∈ 2 ∀ ∈ | − 0 | 2 sections of some smooth finite rank vector bundle B M ˆ π k k ⊠k→ Now we use the fact that the algebraic tensor product on a manifold M. Then Γ(M,B)⊗ = Γ(M ,B ). k ˆ k E⊗ is everywhere dense in E⊗π hence there is some Note that we could remove the index π in ˆ because k π element ψk in the algebraic tensor product E⊗ such that we saw that Γ(M,B) is nuclear. If we specialize⊗ Proposi- ε q (ψ ψk) 6 η with η := . 2 0 − 2 2 6C tion III.7 to sections of vector bundles (which is a Fréchet Now that ψk is chosen, we can bound the second term space) we obtain of the sum (11), namely F (ϕ,ψ ψk) F (ϕ0,ψ ψk). From the previous relation, for every− ϕ− V and− every Theorem III.9. Let E = Γ(M,B) be the space of ∈ 2 ψ such that q2(ψ ψ0) η2, the triangle inequality for smooth sections of some smooth finite rank vector bundle − ≤ k q2(ψ ψk) gives us B M. Then F : U E K multilinear in the last k − variables→ is jointly continuous× → iff the corresponding map F (ϕ,ψ ψk) F (ϕ0,ψ ψk) 6 C(q2(ψ ψ0) F : U Γ(M k,B⊠k) K is jointly continuous. | − − − | − ε × → +q (ψ ψk)) 6 . 2 0 − 3 Proof. The proof is an immediate consequence of the fact ˆ k k ⊠k that E⊗π = Γ(M ,B ) and Proposition III.7. We continue by bounding the last term F (ϕ ,ψ) 0 − F (ϕ0,ψ0) in the sum (11). Since ϕ0 is fixed, the map The definition of a Bastiani smooth functional implies ψ F (ϕ0,ψ) is continuous in ψ because, since F (ϕ0, ) the following corollary: 7→ π k · is continuous on E⊗ , its extension to the completion ˆ π k Theorem III.10. Let E = Γ(M,B) be the space of E⊗ , also denoted by F (ϕ0, ), is continuous. It follows · ˆ π k smooth sections of some smooth finite rank vector bundle that there is some seminorm q1 of E⊗ and a number K η > 0 such that if ψ U satisfies q (ψ ψ ) 6 η then B M. A map F : U , where U is open in E, is 1 ∈ 1 − 0 1 Bastiani→ smooth iff the maps→ F (k) : U Γ(M k,B⊠k) K ε are (jointly) continuous for every k ×1. → F (ϕ ,ψ) F (ϕ ,ψ ) 6 . ≥ | 0 − 0 0 | 3 To interpret Theorem III.7 in terms of distributional kernels, let B M denote a smooth vector bundle of To bound the first term F (ϕ,ψk) F (ϕ0,ψk) in the → − k finite rank over a manifold M equipped with a fixed sum (11), we use the fact that ψk E⊗ . Thus, ψk = p (e e ) for some (e ∈,...,e ) Ek. By density dx and B∗ M the corresponding dual bun- j=1 1,j k,j 1,j k,j | | → 75 ⊗···⊗ ∈ dle. Recall that Γ(M,B)′ Γ(M,B∗) ∞ ′(M) , definition of F , ≃ ⊗C (M) E P where Γ(M,B∗) ∞ ′(M) denotes the compactly ⊗C (M) E p supported distributional sections of the dual bundle B∗. F (ϕ,ψk) F (ϕ ,ψk)= F (ϕ,e ,j,...,ek,j) In global analysis, to every continuous linear map L : − 0 1 j=1 Γ(M,B) Γ(M,B)′, we associate the continuous bilin- X → 2 K F (ϕ0,e1,j,...,ek,j). ear map B :(ϕ1,ϕ2) Γ(M,B) ϕ1,Lϕ2 where − the pairing is understood∈ as a pairing7→ h betweeni∈ a smooth By continuity of F in the first factor, the finite sum function and a distribution once the smooth density on p j=1 F (ϕ,e1,j,...,ek,j) is continuous in ϕ and there is M is fixed. P 12

The usual kernel theorem of the theory of distributions Proof. The functional F is smooth because, for every states that a bilinear map can be represented by a dis- ϕ0 C∞(R), we can define a neighborhood of ϕ0 by 2 ∈ tribution: KL ′(M M) ∞ 2 Γ(M ,B∗ ⊠ B∗) V = ϕ ; π ,K (ϕ ϕ ) < ǫ , where K is a compact ∈ E × ⊗C (M ) { 0 − 0 } living on configuration space M 2 such that, for every neighborhood of the support of g. Let N be the small- 2 (ϕ1,ϕ2) Γ(M,B) est integer strictly greater than π0,K (ϕ0) + ǫ. Then, ∈ N <ϕ(x) < N for every ϕ V and every x K − ∈ ∈ KL,ϕ1 ⊠ ϕ2 Γ′ ,Γ = ϕ1,Lϕ2 Γ(M,B),Γ(M,B)′ , and h i 2 2 h i N+1 2 ( n ) where Γ2 = Γ(M ,B ⊠ B). Theorem III.7 generalizes F (ϕ)= χn(ϕ(x))ϕ | | (x)g(x)dx, the kernel theorem by using multilinear maps instead of n= N 1 Z bilinear ones and by considering that these multilinear X− − maps depend continuously and non-linearly on a param- is a finite sum of smooth functionals. eter ϕ. However, the order of

(1) ∞ ( n ) Fϕ (ψ)= χn(ϕ(x))ψ | | (x) C. Order of distributions n= X−∞ Z (1) ( n ) +χn′ (ϕ(x))ψ (x)ϕ | | (x) g(x)dx, If F is a Bastiani smooth map from an open subset U k R of E = C∞(M) to K, then, for every ϕ U, D Fϕ is a is not bounded on C∞( ). Indeed, for any positive compactly supported distribution. Therefore,∈ the order integer n, we can ﬁnd a smooth function ϕ such that (k) (1) of F is ﬁnite [27, p. 88]. For some applications, for χn ϕ(x) g(x) = 0 for some x supp g. Since Fϕ (ψ) ϕ 6 ∈ example to local functionals, it is important to require contains a factor ψ(n)(x) it is at least of order n. (k) the order of Fϕ to be locally bounded:

Proposition III.11. Let E = C∞(M) and F : E D. Derivatives as smooth functionals K be a smooth functional on an open subset U of E→. Then, for every ϕ U and every integer k, there is 0 ∈ In the next section we equip several spaces of function- a neighborhood V of ϕ0, an integer m and a compact als with a topology. As a warm-up exercise, we show here k (k) (k) K M such that, for every ϕ V , the order of Fϕ is that the maps F are smooth functionals from C∞(M) ⊂ (k) ∈ k smaller than m and Fϕ is supported in K. to ′(M ). WeE adapt to the case of functionals the general re- Proof. According to Lemma III.6, for every ϕ0 in U, there sult given in item 4 of Prop. II.13 stating that, if F k is a neighborhood V of ϕ0, a constant C and a seminorm is a smooth functional on U, then D F is a Bastiani πn,K of C∞(M) such that smooth map from U to L(Ek, K). We need to identify k K (k) (k) the topology of L(E , ) used by Bastiani. Let us start Fϕ (ψ) Fϕ (ψ) 6 C πn,K (ψ). K | − 0 | with L(E, ). Bastiani furnishes E with the topology of (k) (k) convergence on all compact sets of E. In other words, This means that the order of Fϕ Fϕ0 is bounded by the seminorms that define the topology of L(E, K) are −(k) n [27, p. 64], and the order of Fϕ is bounded by n pC (u) = supf C u,f , where C runs over the compact (k) ∈ |h i| plus the order of Fϕ0 . Moreover, if supp ψ K = , subsets of C∞(M). Since C∞(M) is a Montel space [30, (k) (k) ∩ ∅ p. 239], the topology of uniform convergence on compact then πn,K (ψ) = 0 and Fϕ (ψ) Fϕ0 (ψ) = 0. This (k) − (k) sets is the same as the strong topology [30, p. 235]. This means that the support of Fϕ Fϕ0 is contained in K R (k) − means that L(E, ) is the space ′(M) of compactly sup- and the support of Fϕ (ψ) is contained in the compact ported distributions with its usualE topology. Similarly, (k) k R k K supp Fϕ0 . L(E , ) can be identified to a subset of ′(M ) with its ∪ usual topology. We just obtained the followingE result: Note also that, in general, the order of the distributions is not bounded on U: Proposition III.13. Let U be an open subset of C∞(M) and F : U K a Bastiani smooth functional. Then, for R → (k) k Lemma III.12. Let g ( ) and (χn)n Z a sequence of every integer k, the map F : U ′(M ) is smooth ∈D ∈ → E functions such that χn ([n 1,n+1]) and n Z χn = in the sense of Bastiani. 1. Then, the functional∈D − ∈ P n ∞ d| |ϕ IV. TOPOLOGIES ON SPACES OF FUNCTIONALS F (ϕ)= χn(ϕ(x)) n (x)g(x)dx, R dx n= Z | | X−∞ We need to define a topology on the various spaces of (k) is Bastiani smooth on C∞(R) but the order of F is not functionals used in quantum field theory. The generally (k) bounded on C∞(R). idea is to define seminorms on F and its derivatives F . 13

k The topology proposed by Brunetti, Dütsch and Freden- front set) which enables him to equip Γ′ k (M ) with a 17 (k) E hagen is the initial topology of all the maps F Fϕ , Montel, complete, ultrabornological and nuclear topol- (k) → ogy. He also considers support conditions which are dif- where each Fϕ belongs to a nuclear space determined by a wavefront set condition. This topology is nuclear, but ferent from compact. the absence of a control of the dependence on ϕ makes To describe the second difference, recall that Bastiani’s it generally not complete. We then describe Bastiani’s topology gives a locally convex space which is complete. However it is generally not nuclear. This is due to a topology, which is complete but has two drawbacks: it 80 does not take wavefront set conditions into account and theorem by Colombeau and Meise which says, broadly it is generally not nuclear. Finally we shall describe the speaking, that a function space over a Fréchet space family of topologies proposed by Dabrowski21 which are cannot be nuclear for the topology of convergence over both nuclear and complete. some balanced, convex, compact sets of infinite dimension. To avoid that problem, the variable ϕ is made to run over finite dimensional compact sets. More precisely, Rm A. Bastiani’s topology Dabrowski considers compact sets in for any finite value of m and smooth maps f from Rm to an open subset of C (M). He defines two families of seminorms: Bastiani defines several topologies on the space of Bas- ∞ tiani smooth maps between two locally convex spaces [41, pf,K (F )= sup F (ϕ) , (12) ϕ f(K) | | p. 65]. For the case of functionals, we consider the topol- ∈ (n) ogy defined by the following seminorms: pn,f,K,C (F )= sup sup Fϕ ,v , (13) ϕ f(K) v C |h i| ∈ ∈ pC0 (F )= sup F (ϕ) , m ϕ C | | where K is a compact subset of R for some m and C is ∈ 0 k an equicontinuous subset of the dual of the space of dis- pC0,C (F )= sup D Fϕ(h1,...,hk) , (n) (ϕ,h1,...,hk) C0 C | | tributions to which Fϕ belong. Dabrowski proved that, ∈ × with this family of seminorms, the space of functionals F where C = C Ck and Ci runs over the compact 22 1 ×···× is a complete locally convex nuclear space . sets of Γ(M,B). By using Bastiani’s results [41, pp. 66] We describe now several types of functionals that have we obtain been used in the literature and we specify more precisely π Proposition IV.1. Let B M be a finite rank vector their topologies. bundle over the manifold M →and Γ(M,B) be the space of smooth sections of B. Then, with the seminorms defined C. The regular functionals above, the space of smooth functionals on Γ(M,B) is a complete locally convex space. A polynomial functional of the form A similar topology was used by Glöckner [76, p. 367] and Wockel [77, p. 29] and [78, p. 12]. F (ϕ)= dx1 ...dxnfn(x1,...,xn)ϕ(x1) ...ϕ(xn), n n M X Z n where the sum over n is finite and fn (M ), is B. Nuclear and complete topologies called a regular functional 81, because all its∈D derivatives 82 (k) are smooth functions , i. e. the wave front set of Fϕ Quantum field theory uses different spaces of function- is empty. More generally, we define the space Freg(M) (k) als defined by conditions on the wave front set of Fϕ . of regular functionals to be the set of Bastiani smooth Recall that the wave front set describes the points and functionals F such that WF(F (n)) = for every n > 0. 79 (n) n n ∅ the directions of singularity of a distribution . Yoann Thus, F ′ (M )= (M ) and the sets C in Equa- 21 ∈E∅ D n Dabrowski recently described nuclear and complete tion (13) are the equicontinuous sets of ′(M ). By a topologies for spaces of functionals with wave front set general theorem [30, p. 200], the topologyD of uniform con- n conditions. We present some of his topologies for several vergence on the equicontinuous sets of ′(M ) is equiv- common spaces of functionals. alent to the topology given by the seminormsD of its dual n Dabrowski’s definition differs from Bastiani in two re- (M ). In other words, the topology of Freg(M) is de- spects. To describe the first difference, recall that, ac- Dfined by the seminorms22: cording to Proposition III.13, if F : U R is a smooth functional, then the derivatives F (k) : →U (M k) are pf,K (F )= sup F (ϕ) , (14) ′ ϕ f(K) | | smooth functionals. To add the wave front→ E set condi- ∈ k (n) tions, Dabrowski requires F ( ) to be smooth from U to pn,f,K,α(F )= sup pα,n Fϕ , (15) k ϕ f(K) ′ (M ), which is the space of compactly supported dis- ∈ EΓk tributions whose wave front sets are included in Γk, a where pα,n runs over a defining family of seminorms of k n 83 cone in T ∗M . In fact, Dabrowski supplements this def- (M ) . With this topology, Freg(M) is nuclear and inition with a more refined wave front set (the dual wave Dcomplete. 14

Note that the tensor product of elements of E. Local functionals (M m) with elements of (M n) is not continuous in D(M m+n)84. Thus, the productD in F (M) is hypocon- D reg As discussed in the introduction, local functionals are tinuous but not continuous. the basic building block (Lagrangian) of quantum field theory. We shall see that local functionals are a closed (1) subset of the set of smooth functionals such that Fϕ can be identified with an element of (M) that we de- D (k) D. The microcausal functionals note by Fϕ and the wave front set of Fϕ is included ∇ k in the conormal Ck of Dk = (x ,...,xk) M ; x = { 1 ∈ 1 = xk . Recall that the conormal of Dk is the set of It is possible to describe quantum field theory (up to ··· } k (x ,...,xk; ξ ,...,ξk) T ∗M such that x = = xk renormalization) as the deformation quantization of clas- 1 1 ∈ 1 ··· and ξ + + ξk = 0. sical field theory26. For the deformation quantization of 1 ··· the product FG of two functionals to first order in ~, we Since the additivity property (defined in Section VC) of local functionals complicates the matter, we follow need to evaluate DFϕ DGϕ, ∆+ , where ∆+ is a singu- h ⊗ i Dabrowski21 and, for any open set Ω M, we first define lar distribution (the Wightman propagator) and , is ⊂ h· ·i FC (Ω) to be the set of smooth maps such that ϕ Fϕ an extension of the duality pairing between distributions 7→ ∇ and test functions85. For this pairing to be meaningful is Bastiani smooth from C∞(Ω) to (M) and, for ev- (k) D to all orders in ~, the wave front set of ∆+ imposes that ery integer k, ϕ Fϕ is Bastiani smooth from C∞(Ω) k 7→ k (k) to ′ (M ) (we do not need to index ′(M ) with two the wave front set of Dϕ must not meet the cone Γk Ck E 21 E defined as follows26. cones because Ck is closed ). The set Floc(Ω) of local functionals is then the subset of F (Ω) satisfying the ad- Let M be a Lorentzian manifold with pseudo-metric C + divity condition. g. Let Vx (resp. Vx−) be the set of (x; ξ) Tx∗M such µν ∈ The topology of FC is induced by the family of that g (x)ξµξν 0 and ξ0 0 (resp. ξ0 0), where we 00 ≥ ≥ ≤ seminorms given by Eq. (15) that depend on the assume that g > 0. We define the closed cone k k equicontinuous sets of the dual ′ (M ) of C′ (M ), DΛk,Λk D k where Λ = T˙ M k C . They were determined by ˙ n k ∗ k Γn = (x1,...,xn; ξ1,...,ξn) T ∗M ;(ξ1,...,ξn) Dabrowski [21, Lemma\ 28]: { + + ∈ ∈ (V V ) (V − V − ) , x1 ×···× xn ∪ x1 ×···× xn } Proposition IV.2. k A subset B of Λ′ k (M ) is equicontinuous if and only if there is a closedD cone Γ Λ such ˙ n n k where T ∗M is the cotangent bundle T ∗M without its that WF(u) Γ for every u B and B is bounded⊂ in zero section. The space Fmc of microcausal functionals k ⊂ ∈ ′ (M ). was originally defined as the set of Bastiani smooth func- DΓ (n) n tionals such that Fϕ Ξ′ (M ) for every ϕ, where k ∈ E n The bounded sets of Γ′ (M ) are characterized in de- ˙ n 16,17,20,26,81,86,87 D k Ξn = T ∗M Γn is an open cone . tail in Ref. 85. The topology of ′ (M ), where Λk is \ Λk n D However, the space ′ (M ) being not even sequen- open, can be described as a non-countable inductive limit Ξn c 85 E as follows. Write the complement Λ = Γn, where each tially complete , it is not suitable to define a complete k ∪ Γn is a compactly supported closed set. We write the space of functionals. Therefore, Dabrowski defines the c c open set Γn as a countable union of closed sets Γn = space Fmc of microcausal functionals to be the set of Bas- c c c (n) n mΛn,m, so that Γn = mΛn,m and Λk = n m Λn,m is tiani smooth functionals such that Fϕ ′ (M ), ∪ ∩ ∪ ∩ ∈ EΞn,Ξn a countable union of countable intersections of open sets. which is the completion of (M n). Dabrowski proved Ξ′ n We obtain Λk = n m Λn,m. We define for a sequence α n E that ′ (M ) is the set of compactly supported dis- ∩ ∪ Ξn,Ξn the closed set Πα = nΛn,α(n), such that α β implies E n ∩ ≤ tributions u ′(M ) such that the dual wavefront set Πα Πβ. Then Λ = αΠα is a non-countable induc- ∈ E ⊂ ∪ of u is in Ξn and the wavefront set of u is in its closure tive limit of closed cones from which we can define the 21 k Ξn (see for a precise definition of these concepts and topology of Λ′ k (M ) as a non-countable inductive limit of the topology). This completion is not only complete, k D of Π′ (M ). but even Montel and nuclear21. According to the general D α The space FC furnished with the topology induced by results of Ref. 21, the sets C are now equicontinuous sets the seminorms defined by Eqs. (12) and (13) is complete of the bornologification of the normal topology of ′ . Γn and nuclear. The space Floc of local functionals is the 21 D However, it was shown that these equicontinuous sets closed subset of FC defined by the additivity condition are the same as the bounded sets of Γ′ with its nor- defined in the next section. As a closed subspace of a D n mal topology. Therefore, the sets C are the well-known nuclear complete space, the space of local functionals is 85 bounded sets of Γ′ . nuclear and complete. D n With this topology, the space Fmc is a complete nuclear Further examples of spaces of functionals are given by algebra with hypocontinuous product. Dabrowski22. 15

V. ADDITIVITY whose supports are disjoint. It is not a vector subspace 1 of C∞(S ). The characterization of local functionals is a long- The idea of the construction is the following. In the 1 88 metric space C∞(S ), we will show that the subset is standing mathematical problem. According to Rao , I the first criterium was proposed by Pinsker in 1938 and bounded away from the constant function f = 1. This called partial additivity89. This criterium is also used in means that the functional equation (16) only concerns physics, but we shall see that it is not what we need by the restriction F to a subset which is bounded away |I exhibiting a partially additive functional which is not lo- from 1. Therefore there is some open neighborhood of f = 1 which does not meet . Then we use Sobolev cal. Then, we shall discuss a more stringent criterium I which is exactly what we need. norms to build some cut–off function χ to glue a local functional near with a nonlocal functional near f = 1. I Lemma V.1. The constant function f = 1 is bounded A. Partial additivity 1 away from in C∞(S ): if f , then f 1 0 = I ∈ I || − ||C supx S1 f(x) 1 1. When looking for an equation to characterize function- ∈ | − |≥ als having the form of Eq. (1), one can make the follow- Proof. Let us denote by . 0 the norm f 0 = k kC k kC ing observation. Let ϕ1 and ϕ2 be two smooth func- supx S1 f(x) . It is a continuous norm for the Fréchet ∈ | | 1 tions with disjoint support K1 and K2 and assume that topology of C∞(S ) because f C0 = π0,S1 (f). Then, 89 || || f(x,ϕ(x),... ) = 0if ϕ = 0 on a neighborhood of x , if supp ϕ supp ϕ = we have ϕ + ϕ 1 0 > 1. 1 ∩ 2 ∅ k 1 2 − kC so that F (0) = 0. Then, since the support of ϕ1 + ϕ2 is Indeed, the supports of ϕ1 and ϕ2 being compact, the included in K1 K2, fact that they do not meet implies that they are at a ∪ finite distance. Thus, there is a point x S1 such that ∈ F (ϕ1 + ϕ2)= dxf(x,ϕ1(x)+ ϕ2(x),... ) ϕ1(x) = ϕ2(x) = 0. Hence, ϕ1(x)+ ϕ2(x) 1 = 1 and K1 | − | Z supx S1 ϕ1(x)+ ϕ2(x) 1 1. ∈ | − |≥ + dxf(x,ϕ1(x)+ ϕ2(x),... ) K The second step is to build a smooth function χ such Z 2 that χ(1) = 1 and χ = 0. = dxf(x,ϕ (x),... ) |I 1 1 K Lemma V.2. There is a smooth function χ : C (S ) Z 1 ∞ R such that χ = 1 on a neighborhood of f = 1 and χ(f)=→ + dxf(x,ϕ2(x),... )= F (ϕ1)+ F (ϕ2). 0 if f 1 C0 1. In particular, χ = 0. ZK2 || − || ≥ |I Therefore, it is tempting to use the condition of locality: Proof. First recall that the Sobolev norm H2k on S1 is defined as F (ϕ1 + ϕ2)= F (ϕ1)+ F (ϕ2), (16) k 2 f H2k = ((1 ∆) f(x)) dx for ϕ1 and ϕ2 with disjoint support and functionals F k k S1 − sZ such that F (0) = 0. And indeed, many authors since 1 1938, including Gelfand and Vilenkin [90, p. 275], used 2 = 2π (1 + n2)2k f(n) 2 , (17) condition (16), but with disjoint support replaced by | | n Z ! ϕ1ϕ2 = 0 and smooth functions by measurable functions X∈ 88 (see for a review). In perturbative quantum field the- where the last representation uses theb Fourier series ory, partial additivity in our sense is also used when the f(x) = fˆ(n)einx. By the Sobolev injections, H2(S1) 17,91,92 n function f in Eq. (1) is polynomial because, in injects continuously in C0(S1). In other words, there is that case, partial additivity is equivalent to locality in P a constant C > 0 such that f C0 6 C f H2 for every the sense of Eq. (1)17. 1 k k k k f C∞(S ). However, this definition of locality does not suit our ∈ Now we take a function g C∞(R) such that g(t) = 1 purpose, essentially because the set of functions ϕ that when t 1/3C2 and g(t) =∈ 0 when t 1/2C2 and we can be written as ϕ = ϕ1+ϕ2 (with supp ϕ supp ϕ2 = ) ≤ C S1 R ≥ ∩ ∅ define χ : ∞( ) by composing g with the square is not dense in the space of smooth functions. We show of the Sobolev norm.→ this now and we construct a partially additive functional 2 which is not local. χ(f)= g 1 f 2 . || − ||H If 1 f 0 > 1 (in particular, if f by Lemma V.1) k − kC ∈I B. A non-local partially additive functional the Sobolev injection leads to:

2 1 S1 1 6 1 f C0 6 C 1 f H2 = 1 f 2 > We work in the space C∞( ) of smooth functions on k − k k − k ⇒ k − kH C2 the unit circle. We denote by the subset of functions I S1 2 f = ϕ + ϕ which are sums of two elements of C∞( ) hence g 1 f 2 = 0 by deﬁnition of g. 1 2 k − kH 16

1 On the other hand 1 f 2 6 means that f be- Proposition V.5. A smooth functional F on C (M) H ) √3C ∞ k − k (2) longs to the neighborhood of the constant function f = 1 is additive iff supp Fϕ D2 for every ϕ C∞(M), 2⊂ ∈ defined by V = f ; 1 f H2 6 1/√3C . On this neigh- where D2 = (x,y) M ; x = y . If F is an additive { k2 − k } { ∈ (n) } borhood, g 1 f 2 = 1. The smoothness of χ is an k − kH functional, then supp Fϕ Dn for every ϕ C∞(M), immediate consequence of the chain rule, the smoothness ⊂ n ∈ where Dn = (x ,...,xn) M ; x = = xn . 2 { 1 ∈ 1 ··· } of g and of the squared Sobolev norm . H2(S1). k k Proof. We first prove that the second derivative of an 17 We are now ready to define our counterexample: additive functional is localized on the diagonal . If we use the additivity property with ϕ1 = λψ, ϕ3 = µχ and Theorem V.3. S1 supp ψ supp χ = , then The functional Fnl on C∞( ) defined ∩ ∅ for any integer N > 1 by F (λψ + ϕ + µχ)= F (λψ + ϕ )+ F (ϕ + µχ) F (ϕ ). 2 2 2 − 2 N Since no term on the right hand side of this equation Fnl(f)= 1 χ(f) f(x)dx + χ(f) f(x)dx , depends on both λ and µ, we have − S1 S1 Z Z (18) ∂2F (λψ + ϕ + µχ) 2 = D2F (ψ,χ) is partially additive but not local. ∂λ∂µ λψ+ϕ2+µχ (2) 1 2 Proof. For every (ϕ ,ϕ ) C∞(S ) whose supports are = Fλψ+ϕ +µχ(ψ χ) = 0. 1 2 ∈ 2 ⊗ disjoint, f = ϕ1 + ϕ2 hence χ(f) = 0 by Lemma ∈ I This equation, being true for every ϕ2, can be written V.2. Moreover, we saw that, if supp ϕ1 supp ϕ2 = , (2) S1 ∩ ∅ Fϕ (ψ χ) = 0 for every ϕ and every pair (ψ,χ) with then there is a point x such that ϕ1(x) = ϕ2(x) = ⊗ ∈ disjoint supports. Now for every point (x,y) M 2 such 0. Thus, 1 ϕ1 C0 1 and 1 ϕ2 C0 1. As ∈ || − || ≥ || − || ≥ that x = y, there are two open sets Ux containing x and a consequence, χ(ϕ1) = χ(ϕ2) = 0 by Lemma V.2 and 6 Uy containing y such that Ux Uy = . Then, any pair Fnl(ϕ1 +ϕ2)= S1 (ϕ1(x)+ϕ2(x))dx = Fnl(ϕ1)+Fnl(ϕ2). ∩ ∅ of functions ψ and χ supported in Ux and Uy satisfies On the other hand, in the neighborhood V of f = (2) 1 given by LemmaR V.2, χ(f) = 1 hence F (f) = Fϕ (ψ χ) = 0. Since the functions ψ χ are dense in nl 2 ⊗ ⊗ 2 N (M ), this implies that every test functions f (M ) S1 f(x)dx which is not local. It is even a typical D (2) ∈D example of a multilocal functional16. supported in Ux Uy satisfies Fϕ (f) = 0. Thus (x,y) / R (2) × (2) ∈ supp Fϕ and supp Fϕ D2. To determine the support (n) ⊂ Since partial additivity is equivalent to locality for of Fϕ , consider a point (x1,...,xn) which is not in Dn. polynomial functions, the non-locality of F can be con- Then, there are two indices i and j such that xi = xj. nl 6 sidered to be non perturbative. Moreover, the fact that Denote by Ux an open neighborhood of xi and by Uy an n the derivatives D Fnl calculated at f = 0 are supported open neighborhood of xj and repeat the previous proof S1 n (2) in the thin diagonal of ( ) , although Fnl is not local, to obtain Fϕ (ψ χ) = 0 for every ϕ and every pair ⊗ means that locality cannot be controlled by the support (ψ,χ) with supports in Ux and Uy. Now, rewrite ϕ = of differentials taken at a single function f. We come now ϕ0 + λkψk, where ψk(xk) = 0 and the sum is over all to the property that is relevant for quantum field theory. integers from 1 to n except i and6 j. Then, the derivatives P with respect to λk are all zero and we find again with the same argument that (x1,...,xn) is not in the support of (n) C. Additive functionals Fϕ for every ϕ. 20,92 (2) Conversely , assume that supp Fϕ D2 for ev- In 1965, Chacon and Friedman93 introduced a more ery ϕ. As we have seen in the first part⊂ of the proof, stringent concept of additivity which meets our needs: this means that, if ψ and χ have disjoint support, then 2 (2) D Fϕ(ψ,χ) = Fϕ (ψ χ) = 0. By the fundamental Definition V.4. We say that a Bastiani smooth map theorem of calculus, ⊗ F : C∞(M) K is additive if, for every triple 1 (ϕ ,ϕ ,ϕ ) of→ smooth functions on M, the property d 1 2 2 F (ϕ + ψ + χ)= F (ϕ + ψ)+ dµ F (ϕ + ψ + µχ), supp ϕ supp ϕ = implies the property dµ 1 ∩ 3 ∅ Z0 1 d F (ϕ1 + ϕ2 + ϕ3)= F (ϕ1 + ϕ2)+ F (ϕ2 + ϕ3) F (ϕ2). F (ϕ + ψ + µχ)= F (ϕ + µχ)+ dλ F (ϕ + λψ + µχ). − dλ (19) Z0 Thus, In the literature, the additivity equation (19) is also 1 d called the Hammerstein property94–99. The additivity F (ϕ + ψ + χ)= F (ϕ + ψ)+ dµ F (ϕ + µχ) dµ property is equivalent to the fact that the functional Z0 1 1 2 derivatives are supported on the thin diagonal Dn = ∂ n 17,92 + dλ dµ F (ϕ + λψ + µχ). (x1,...,xn) M ; x1 = = xn . ∂λ∂µ { ∈ ··· } Z0 Z0 17

2 The last term is zero because D Fϕ(ψ,χ) = 0 and the The main theorem of this section is second term is F (ϕ+χ) F (ϕ). We recover the additivity − condition. Theorem VI.3. Let U be an open subset of C∞(M) and F : U K be Bastiani smooth. Then, F is local if and Finally, additivity is stronger than partial additivity only if→ the following two conditions are satisfied: because the latter corresponds to the case ϕ2 = 0 and F (0) = 0. It is strictly stronger because Fnl is not addi- 1. F is additive, tive. (1) 2. for every ϕ U, the differential DFϕ = Fϕ of F at ϕ has an∈ empty wave front set and the map VI. CHARACTERIZATION OF SMOOTH LOCAL ϕ Fϕ is Bastiani smooth from U to (M). FUNCTIONALS 7→ ∇ D Note that our definition of locality is strictly more gen- In this section, we give a characterization of local eral than the usual one because the counterexample de- functionals inspired by the topology described in Sec- scribed in Lemma III.12 is local in our sense but not in tion IVE. In the sequel, we shall deal with compactly the sense of Eq. (1) since its order is infinite. supported distributions u with empty wavefront sets. We The proof is delayed to Section VI C. Since this the- repeat the definition of local functionals in terms of jets: orem deals with jets, we start with a short presentation of the jet bundle. Our point of view on jets is based Definition VI.1. Let U be an open subset of C∞(M). A Bastiani smooth functional F : U K is said to be local on the concept of infinitesimal neighborhoods due to → Grothendieck and is closely related to several expositions if, for every ϕ U, there is a neighborhood V of ϕ, an 100–102 integer k, an open∈ subset J kM and a smooth func- in the literature . V ⊂ k tion f C∞( ) such that x M f(jx ψ) is supported in a compact∈ V subset K M ∈and 7→ ⊂ A. The manifold of jets of functions on a manifold k F (ϕ + ψ)= F (ϕ)+ f(jx ψ)dx, ZM Let M be a manifold. For every smooth real-valued k whenever ϕ + ψ V and where jx ψ denotes the k-jet of function ϕ on M, we call k-jet of ϕ at a point x M ∈ k k+1 ∈ ψ at x. the class jx (ϕ) of ϕ in the quotient C∞(M)/Ix , with k+1 We invite the reader not familiarized with jet bundles the understanding that Ix stands for the (k + 1)-th to have a look at Section VI A, where these objects are power of the ideal Ix of smooth functions on M vanishing at x M. Recall that Ik+1 coincides with the ideal carefully defined. Note that the representation of F by ∈ x f is not unique: adding the total derivative of a function of smooth functions on M whose k + 1 first derivatives does not change the result. We shall see that f belongs to vanish at the point x. k a unique cohomology class for some specific cohomology For all x M, the space Jx (M) of all k-jets of func- ∈ k+1 theory on the space of local functionals. tions on M at x coincides with C∞(M)/Ix and is called Before we state the main Theorem of this section, let the space of k-jets at x. It is clearly a vector space. The k k us start by some useful definition-lemma : disjoint union J (M) := x M Jx (M) is a smooth vector bundle over M called the∈ bundle of k-th jets. Con- Lemma VI.2. Let U be an open subset of C∞(M) and sider the map: ` F : U K be Bastiani smooth. For every ϕ such that → the distribution DFϕ ′(M) has empty wave front set, k ∈E J∆ : C∞(M M) Γ(J (M)) there exists a unique function F (M) such that × → k ϕ ψ x j (i∗ ψ), ∇ ∈D 7→ 7→ x x

DFϕ[h]= Fϕ(x)h(x)dx. (20) where ix : M M M is the map y (x,y). It is M ∇ → × 7→ Z known that J∆ is surjective onto the space of smooth Proof. Once a density dx is fixed on M, functions in sections of J k(M) and its kernel is the (k + 1)-th power 1 Lloc(M) (in particular in C∞(M)) can be identified with of the ideal ∆ of functions on M M vanishing on the distributions by the map : diagonal. I × Last, the projection p1 : M M M onto the first 1 × → f Lloc(M) φ fφdx component dualizes in an algebra morphism ϕ p1∗ϕ ∈ 7→ 7→ M 7→ Z from C∞(M) to C∞(M M) which endows C∞(M M) × × and [32, Theorem 1.2.4] shows that the distribution is with a C∞(M)-module structure. The space of sections k uniquely defined when f is continuous hence when f is of J (M) is also a C∞(M)-module, and it is routine to smooth. check that J∆ is a morphism of C∞(M)-modules. There- k Since WF (DFϕ) = , there exists a unique C∞ func- fore, the space of sections of J (M) is, as a C∞(M)- ∅ k+1 tion Fϕ which represents the distribution DFϕ ′(M) module, isomorphic to the quotient C∞(M M)/ ∆ by integration∇ on M against dx. ∈E and Γ(J k(M)) fits into the following exact× sequenceI of 18

C∞(M)-modules: Before establishing these results, we shall need several lemmas. k+1 k 0 C∞(M M) Γ(J (M)) →I∆ → × → ≃ k+1 Lemma VI.6. jk C∞(M M)/ 0. The vector bundle morphism described × I∆ → in (21) is surjective and admits a smooth section sk. And the map that to f associates its k-jet reads: Proof. The section sk, when it exists, being by construc- k+1 f C∞(M) [(p∗f)] C∞(M M)/ . k ∈ 7→ 1 ∈ × I∆ tion a right inverse of j , the latter is surjective. It suffices sk The purpose of the rest of this section is to prove the therefore to prove the existence of . technical Proposition VI.4, the statement of which we We first prove that the lemma holds true for M an N open subset V of Rn. In that case, the bundle of k-jets now explain. For all integer k , there is a natu- k ral vector bundle morphism jk from∈ the trivial bundle jx (V ) is isomorphic to the trivial bundle over V with typical fiber the space of polynomials of degree less or over M with typical fiber E = C∞(M) to the bundle k equal to k. There is an obvious candidate for the section J (M) M of k-jets. This morphism simply consists k k in assigning→ to a pair (f,x) in E M M the k-jet of of j : it consists in mapping αx jx (V ) to the unique × → polynomial of degree k whose k-jet∈ at x V is α. The f at x. In equation: ∈ k henceforth obtained assignment, that we denote by sV , k jk : E M jk(M) is a smooth vector bundle morphism from j (V ) V to (21) → × 7→ k the trivial bundle C∞(V ) V V . It is by construction (f,x) jx (f). → a section of jk. × → The result goes as follows. We now go back to the general case of an arbitrary manifold M. For every point x M, choose Vx a coordi- Proposition VI.4. Let E = C∞(M) and V E be an nate neighbourhood and let χ be∈ a smooth function with open subset and k N an integer. ⊂ ∈ compact support on Vx which is identically equal to 1 in 1. The subset jk(V M) is an open subset of jk(M). a neighbourhood Vx′ Vx of x. Since Vx is a coordinate × neighbourhood, it can⊂ be identified with an open subset 2. Let c be a smooth K-valued function on V M, with of Rn, which allows to consider V E an open subset. Assume that c(f,x×) depends ⊂ k k only on the k-jet of the function f at the point x. s : j (Vx) C∞(Vx) Vx Vx 7→ × Then there exists a unique smooth K-valued function c˜ on the open subset jk(V M) jk(M) that as in the previous paragraph. We can then consider the × ⊂ makes the following diagram commutative: composition of vector bundle morphisms over Vx:

c k V M / K. (22) sV mχ id : k k x × × j (M) Vx j (Vx) / C∞(Vx) Vx / E Vx c˜ | ≃ × × jk k k k where j (M) j (Vx) is the obvious identiﬁcation of j (V M) Vx ≃ × the k-jet bundle of V to the restriction to V of the k-jet x x bundle on M , and where m is the smooth linear map i.e, such that the relation c(f,x) =c ˜(jk(f)) holds χ x from C (V ) to E = C (M) deﬁned by m (f) = fχ. for all f V and x M. ∞ x ∞ χ ∈ ∈ Since χ = 1 identically equal to 1 on Vx′, the restriction When V = E, Proposition VI.4 specializes to the fol- to Vx′ of this vector bundle morphism is by construction k lowing easier statement: a section of the restriction of j to Vx′. Since the manifold M is paracompact103, the latter Corollary VI.5. Let c be a smooth function from E M point implies that the manifold M can be covered by K × k to . Assume that there exists an integer k such that open subsets (Ui)i I such that the restriction of j to c(f,x) depends only on the k-jet of the function f at ∈ k Ui admits a section si . Without any loss of generality, the point x. Then there exists a unique smooth K-valued we can assume the existence of a smooth partition of k function c˜ on j (M) such that the following diagram com- unity (χi)i I relative to this open cover. A global smooth mutes: section of ∈jk is then given by the explicit formula sk = k c i I χi si , as follows from the obvious computation: E M / K. (23) ∈ × ; c˜ P k k k k k j s = χi j s = χi id k = id k j ◦ ◦ i j (M) j (M) i I i I jk(M) X∈ X∈ where we used the fact that jk commutes with multipli- k k k i.e. such that the relation c(f,x) =c ˜(jx (f)) holds for all cations by χi since j : E M J M is a vector bundle f V and x M. morphism. This completes× the7→ proof. ∈ ∈ 19

Since c : V M K is only defined on the open subset simply a set-theoretic property: the difficulty is to show V M of E× M→, we need the following refinement of that this functionc ˜ is smooth. × × k k Lemma VI.6 where the local sections tx of j are valued When V = E (i.e. under the assumptions of Corollary in V M: VI.5), the smoothness ofc ˜ follows directly from Lemma × VI.6, which implies that the commutative diagram (23) Lemma VI.7. For every (f,x) E M, the vector can be completed to bundle morphism jk described in∈ (21)× admits a smooth k 104 section t through (f,x). c E M / K. ×I ; Proof. Notice that Lemma VI.7 can be derived from c˜ k k Lemma VI.6 for any vector bundle morphism over the s j identity of M. A careful check shows that the arguments jk(M) below are absolutely general and indeed show that for any two vector bundles E1,E2 over M, any vector bun- which amounts to say that the following relation holds dle morphism E E over M that admits a section is 1 7→ 2 a submersion, and admits a section through every point c˜ = c sk. (24) of E1. ◦ We prefer to do it, however, in our particular setting The latter formula and the smoothness of sk implies that, – since one of the bundles is infinite dimensional and when c is assumed to be a smooth function, so is the requires careful attention. functionc ˜ by composition. This proves Corollary VI.5. k k Let s be a section of j as in Lemma VI.6. Consider For the general case, we have to choose, for all α the smooth map defined at all point y M by jk(V M) a section tk of jk such that tk(α) V M∈. ∈ × k ∈ × k k Such a section t always exists by Lemma VI.7. Since t : j (M) (E M)y E, k y y → × ≃ t is smooth, there exists a neighbourhood Wα of α in β sk(β)+(f sk jk(f)) k k 7→ y − y ◦ y j (V M) on which t takes values in the domain of definition× V M of c, which implies that the commutative This map is smooth by construction. It is again a section diagram (22×) can be completed to jk, as follows from the following computation, valid for all y M,β jk(M): y k 1 c K ∈ ∈ (V M) (j )− (Wα) 6/ . k k k k k k × ∩I jy ty(β)= jy sy(β)+(f sy jy (f)) c˜ ◦ − ◦ tk jk k k k k k k = jy sy(β)+ jy(f) jy sy jy (f) Ù ◦ − ◦ ◦ W = β + jk(f) jk(f)= β. α y − y Then the section tk above satisfies by construction: In turn, the commutativity of this diagram gives the explicit description ofc ˜ through the following formula, valid k k k k k k t j (f)= s j (f)+ f s j (f)= f. on Wα: x ◦ x x ◦ x − x ◦ x This completes the proof. c˜ = c tk. (25) ◦ Lemma VI.7 has the following immediate consequence. Formula (25) and the smoothness of tk imply that, when c is assumed to be a smooth function, so is, by com- Lemma VI.8. The vector bundle morphism jk described position, the restriction to W of the functionc ˜. Since in (21) is a submersion. α every α jk(V M) admits a neighbourhood on which ∈ × Proof. For every (f,x) in E M, let us choose tk to be the restriction ofc ˜ is smooth, the functionc ˜ is a smooth a section through (f,x) as in× Lemma VI.7. By construc- function. This completes the proof. tion, the differential of jk at (f,x) admits the differential of tk at jk(f,x) as right inverse, so it is surjective. (2) B. Properties of Fϕ We can now prove Proposition VI.4. Proof. Since the vector bundle morphism jk described in We first show that the two assumptions of our theorem (21) is a submersion by Lemma VI.8 and since V M is are equivalent to some strong assumptions on the second open in E M, the subset jk(V M) is an open× subset derivative of F : of jk(M).× This proves the first item× in Proposition VI.4. Lemma VI.9. Let us now prove the second item. Assume that we Let U be an open subset of C∞(M) and K F : U K be Bastiani smooth. Assume that for every are given a function c : V M such that the value → (1) c(f,x) at an arbitrary f ×E and7→ x M depends only ϕ U, the differential DFϕ = Fϕ of F at ϕ has no wave ∈ ∈ ∈ (1) the k-jet of f at x. The existence of an unique function front set, i.e. WF(Fϕ ) = . Then the two following c˜ from jk(M) to K making the diagram (22) commute is properties are equivalent: ∅ 20

1 k k 1. F is additive and the map ϕ Fϕ is Bastiani (ϕ,ψ1,...,ψk) 0 dss D Fsϕ(ψ1,...,ψk) is continu- smooth from U to (M). 7→ ∇ ous and the proof7→ is complete because continuity holds D true for every k. R 2. For every ϕ0 U, there is a neighborhood V of ϕ , a compact ∈K M and a finite family of Bas- 0 ⊂ Let us now prove Lemma VI.9. tiani smooth maps fα : V (K) with α 6 k, such that in any system of→ local D coordinates| |(x,y) 2 Proof. First of all, by Proposition V.5, F is additive iff on M : its second derivative is represented by a distribution sup- (2) α ported in the diagonal. We start by proving the direct Fϕ (x,y)= fα(ϕ)(x)∂y δ(x y), (26) − sense assuming that ϕ U Fϕ C∞(M) is Bastiani α 6k ∈ 7→ ∇ ∈ |X| smooth. for every ϕ V . We first show that item 1 implies item 2 in ∈ Lemma VI.9. Since F is Bastiani smooth for any ϕ0 U, 2 ∈ In particular, both conditions imply that D Fϕ is rep- we already know by Proposition III.4 that there is some (2) resented by a distribution F whose wave front set is neighborhood V of ϕ0 on which F V has fixed compact ϕ | the conormal bundle of the diagonal in M 2 [105, p. 32]. support that we denote by K. Therefore, Fϕ belongs to (2) ∇ In the sequel, we shall often use the following simple (K) for every ϕ V and Fϕ is supported in K K. lemma : D (2) ∈ ×2 Since Fϕ is also supported in the diagonal of M by (2) Lemma VI.10. Let E, F and G be locally convex spaces. Proposition V.5, the support of Fϕ is contained in the If f : E F is Bastiani smooth and ℓ : F G is linear diagonal of K K which can be identified with K itself. → → × and continuous, then ℓ f : E G is Bastiani smooth Since DFϕ has an empty wavefront set by assumption, and Dk(ℓ f)= ℓ Dkf◦. → its singular support is empty and it can be represented by ◦ ◦ a unique smooth compactly supported function Fϕ [32, Proof. This is a consequence of three facts: the map ℓ is p. 37] such that ∇ Bastiani smooth because it is linear and continuous, ℓ f ◦ is Bastiani smooth because it is the composition of two d Bastiani smooth maps and the chain rule. F (ϕ + th) t=0 = DFϕ(h)= Fϕ(x)h(x)dx. (27) dt | ∇ ZM We also need the following lemma in the proof of (2) Lemma VI.9: The main step is to represent Fϕ as the Bastiani differential of Fϕ by calculating the second derivatives in Lemma VI.11. Let U be a convex open subset of E = two different∇ ways. The Bastiani smoothness of F yields: C∞(M) containing the origin and F : U E a Bastiani → 2 smooth map. Then, G : U E defined by G(ϕ) = 2 d 1 → D Fϕ(g,h)= F (ϕ + t1h + t2g) t1=t2=0 0 F (sϕ)ds is Bastiani smooth. dt1dt2 | d d Proof.R The first step is to define a candidate for the Bas- = F (ϕ + t h + t g) k k 1 2 t1=0 t2=0 tiani differential D G by determining D G(x) pointwise dt2 dt1 | | k in x M. For every (t1,...,tk,x) [0, 1] M and d ∈ k ∈ × = Fϕ t g(x)h(x)dx t , (ϕ,ψ1,...,ψk) U E , the function (t1,...,tk,x) dt ∇ + 2 | 2=0 1 ∈ × 7→ 2 ZM dsF (s(ϕ + t ψ + + tkψk))(x) is smooth in 0 1 1 ··· (t1,...,tk,x) by dominated convergence theorem since where we used the Schwarz lemma and Equation (27). xR can always be restricted to some compact subset K To justify switching d and integration over M, observe ⊂ dt2 M to obtain uniform bounds. We can differentiate in that the map ϕ U Fϕ (M) is Bastiani smooth (t ,...,t ) outside and inside the integral and both dif- 1 ∈ 7→ ∇ ∈D d 1 k hence C . It follows by the chain rule that t dt Fϕ+tg 0 7→ ∇ ferentials coincide. Therefore, for every x M, the Bas- is a C map valued in (M). Since Fϕ is actually in k ∈1 D ∇ tiani kth–differential D G(x) of G(x) = 0 dsF (sϕ(x)) (K) for every ϕ V and the topology induced by (M) exists and satisfies the relation DkG (ψ ,...,ψ )(x) = onD (K) is the usual∈ Fréchet topology of (K), theD map ϕ 1R k D D 1 k k k F is smooth from V to the Fréchet space (K). dss D Fsϕ(ψ1,...,ψk)(x). Let us show that D G : 0 ∇ 0 D k Since (K) injects continuously in (C (K), π0,K ), this U E E is jointly continuous in (ϕ,ψ1,...,ψk). D R × 7→ implies that (t,x) [ 1, 1] K d F (x) We know that the map χ : (s,ϕ,ψ1,...,ψk) [0, 1] dt ϕ+tg k k k ∈ × 0 ∈ − × 7→d ∇ ∈ C ([ 1, 1] K). Hence the integrand Fϕ t g(x)h(x) U E s D Fsϕ(ψ1,...,ψk) E is continuous by dt2 + 2 × 7→ k k ∈ − × ∇ joint continuity of D F : U E E and composition is in C0([ 1, 1] K) and is bounded on the integration × 7→ of the continuous maps domain.− A continuous× map u : t [ 1, 1] u(t,.) 0 ∈ − 7→ ∈ k k (C (K), π0,K ) corresponds to a map also denoted by (s,ϕ,ψ) (sϕ,ψ) s D Fsϕ(ψ), u C0([ 1, 1] K). Indeed for every convergent se- 7→ 7→ ∈ − × quence (tn,xn) (t,x)in[ 1, 1] K, the simple esti- where ψ = (ψ1,...,ψk). Then by [40, Thm 2.1.5 n →∞→ − × p. 72] applied to the function χ, the integrated map mate u(t,x) u(tn,xn) 6 u(t,x) u(t,xn) + u(t,xn) | − | | − | | − 21

6 (2) α u(tn,xn) u(t,x) u(t,xn) + π0,K (u(tn,.) u(t,.)) Fϕ = α 6k fα(ϕ,x)∂y δ(x y). We denote the dis- | | − | − | | − shows that u(tn,xn) u(t,x). n tributions fα(ϕ) by fα(ϕ,x) because we shall show that →∞→ P By the dominated convergence theorem, we can differ- ϕ fα(ϕ) is Bastiani smooth from V to (K). 7→ D entiate under the integral sign: By Equation (30), we know that for every (ϕ,g) ∈ V C∞(M) the map from V C∞(M) to (K) defined × × D 2 d by D Fϕ(g,h)= Fϕ+t2g(x)h(x)dx t2=0 dt2 M ∇ | Z (2) α α d (ϕ,g) Fϕ (x, )g(x)dx = ( 1)| |fα(ϕ,.)∂ g(.) 7→ M · − = Fϕ+t2g(x) t2=0 h(x)dx. Z α 6k dt ∇ | |X| ZM 2 is smooth. Choosing g to be equal to the Fourier oscil- d i ξ.x By definition Fϕ+t2g t2=0 is only the Bastiani deriva- latory function e , we obtain by the chain rule the dt2 ∇ | − h i tive maps that sends (ϕ,ξ) to

D F :(ϕ,g) V C∞(M) D F [g] (K)(28), (2) i ξ.x α ϕ F (x,y)e− h idy = ( 1)| |fα(ϕ,y) ∇ ∈ × 7→ ∇ ∈D ϕ − ZM ZM α 6k where D F is a Bastiani smooth map since F is Bas- |X| ∇ ∇ α i ξ.x tiani smooth. δ(x y)∂y e− h idy − Note also that by Theorem III.9, the second deriva- α α 2 = ( 1)| |fα(ϕ,x)( iξ) , tive D Fϕ(g,h) can be represented by a map ϕ − − α 6k (2) 2 ∈ |X| U Fϕ ′(M ) such that (ϕ,g,h) U 7→ ∈ E ∀ ∈ × 2 2 (2) is Bastiani smooth. Moreover, since the image of the map C∞(M) , D Fϕ(g,h)= Fϕ ,g h which means that ⊗ in Eq. (30) is in (K) for every smooth g, we obtain that (2) α D α Fϕ is the distributionalD kernel ofE the second deriva- α 6k( 1)| |fα(ϕ, )( iξ) is in (K) for every ξ. This 2 | | − · − D tive D Fϕ. We now arrive at the following equality is only possible if fα(ϕ) (K) for every α 6 k. There- P ∈Dd α (2) | i| ξ.y which identifies two different representations of the sec- fore ϕ fα(ϕ)=(i ) Fϕ (.,y)e− h idy ξ=0 is 2 7→ dξ M | ond derivative. For every (ϕ,g,h) U C∞(M) Bastiani smooth from V to (K) and the proof of the ∈ × R direct sense is complete. D (2) Fϕ ,g h = D Fϕ[g](y)h(y)dy. (29) Conversely, we want to prove that if there is a neigh- ⊗ M ∇ borhood V of ϕ , a compact K M and a finite family D E Z 0 of smooth maps ϕ f (ϕ) (⊂K), α 6 k such that in By the same Theorem III.9 and the chain rule, α any system of local7→ coordinates∈D (x,y)| on| M 2: D Fϕ[g](y) = evyD Fϕ[g] is linear continuous in ∇ ∇ g C∞(M), hence there is a distribution, denoted (2) α ∈ Fϕ (x,y)= fα(ϕ)(x)∂y δ(x y), by D Fϕ(x,y), such that D Fϕ(x,y)g(x)dx = − ∇ M ∇ α 6k D Fϕ[g](y) and D Fϕ(x,y)g(x)dx is in (K) by |X| ∇ M ∇ R D Eq. (28). Since the above identity holds for all (g,h) then ϕ Fϕ (M) is Bastiani smooth. Without 2 R ∈ 7→ ∇ ∈ D C∞(M) , we have in the sense of distributions that loss of generality, we assume that V is convex. By the (2) Fϕ (x,y)= D Fϕ(x,y) where the map Taylor formula with remainder for Bastiani smooth func- 2 ∇ tions, for every (ϕ,ψ ,ψ ) V C∞(M) : 1 2 ∈ × (2) 2 (ϕ,g) V C∞(M) F (x, )g(x)dx (K), D Fϕ+s ψ +s ψ (ψ1,ψ2)= ∂s ∂s F (ϕ + s1ψ1 + s2ψ2) ∈ × 7→ ϕ · ∈D 1 1 2 2 1 2 ZM = ∂ DF (ψ ), (30) s2 ϕ+s1ψ1+s2ψ2 1 for s1 and s2 small enough. It follows by the fundamen- is Bastiani smooth. tal theorem of calculus and by evaluating at s1 = 0 the It suffices to do the last part of the proof, which previous relation that d is local in nature, on M = R . We now represent t (2) Fϕ (x,y) as a C∞(M)-linear combination of deriva- DFϕ+tψ2 (ψ1)= DFϕ(ψ1)+ ∂s2 DFϕ+s2ψ2 (ψ1)ds2 tives of Dirac distributions concentrated on the diago- 0 Z t nal. By Proposition V.5, the additive property satis- 2 = DFϕ(ψ )+ D Fϕ sψ (ψ ,ψ )ds, fied by F implies that the distribution F (2) associated 1 + 2 1 2 Z0 to the second derivative D2F is supported in the diag- where by assumption DFϕ(ψ1) is represented by integra- onal D2 M M. By Proposition III.11, the kernel (2) ⊂ × tion against a smooth function Fϕ (x,y) ′(M M) has bounded distributional or- ∈ E × der uniformly in ϕ V . Schwartz’ theorem on distri- DFϕ(ψ )= Fϕ(x)ψ (x)dx, ∈ 1 ∇ 1 butions supported on a submanifold [27, p. 101] states ZM that in local coordinates, there exists a finite sequence of D2F (ψ ,ψ )= F (2) (x,y)ψ (x)ψ (y)dxdy, ϕ+sψ2 1 2 ϕ+sψ2 1 2 distributions (ϕ V fα(ϕ,.) (K)) α 6k such that M M ∈ 7→ ∈D | | Z × 22

(2) and Fϕ+sψ is supported on a subset of the diagonal D2 Proof. The first Bastiani differential DΦϕ(h) can be iden- M M that can be identified with K. Hence, for ψ ⊂ tified with the smooth function x (0,h(x)), which is × ∈ 7→ C∞(M) such that ϕ + ψ V : linear continuous in h and does not depend on ϕ. It is ∈ 1 thus smooth and so is Φ. (2) Fϕ ψ(x)= Fϕ(x)+ F (x,y)ψ(y)dy ds ∇ + ∇ ϕ+sψ Therefore, the composition Z0 ZM α α d (d+k)! = Fϕ(x)+ ( 1)| |∂ ψ(x) R R d! ∇ − Φ Ψ: C∞(Ω) C∞(Ω, ) α 6k ◦ → × |X| 1 defined by fα(ϕ + sψ)(x)ds. α Φ Ψ(ψ): x x,χ∂ ψ(x) α 6k , Z0 ◦ 7→ | | To show that the map χ : V (K) defined by χ(ψ)= is Bastiani smooth and finally →D Fϕ+ψ is smooth, we notice that, according to the last ∇ k k equation, Fϕ+ψ is the sum of the constant Fϕ and a ψ C∞(M) f(.,jx ψ(.)) (K) f(x,jx ψ)dx, ∇ ∇α finite linear combination of products of ψ ∂ ψ by an ∈ 7→ ∈D 7→ Ω 7→ Z integral over s. The integrand fα(ϕ + sψ) is smooth by is Bastiani smooth by the chain rule and since the last in- assumption. Therefore, the map tegration map is linear continuous thus Bastiani smooth. 1 Now let us prove the direct sense of Thm. VI.3, where ψ (V ϕ) fα(ϕ + sψ)(x)ds (K) we start from a functional characterization of F and ∈ − 7→ ∈D Z0 end up with a representation as a function F (ϕ + ψ) = k is smooth by Lemma VI.11 and the fact that the topology M f(x,jx ψ)dx on jet space, for ϕ+ψ in a neighborhood induced on (K) by the topology of C∞(M) is the stan- V of ϕ, that we assume convex. We start by deriving dard topologyD of (K). The map ψ ∂αψ is smooth Ra candidate for the function f. According to the funda- because it is linearD and continuous. Finally,7→ the product mental theorem of calculus, α of the integral by ∂ ψ is smooth by a trivial extension 1 of Lemma VI.15. This completes the proof of Lemma F (ϕ + ψ)= F (ϕ)+ dtDFϕ+tψ(ψ). (31) VI.9. Z0 As discussed at the beginning of this section, since we We are now ready to prove Theorem VI.3 characteriz- (1) ing local functionals. assume that WF(Fϕ ) = for every ϕ U, there exists a unique smooth compactly∅ supported∈ function x Fϕ(x) such that: 7→ ∇ C. Proof of Theorem VI.3 (1) Fϕ (ψ)= dx Fϕ(x)ψ(x). (32) M ∇ Let us start by proving the converse sense where we Z assume that F is the integral of some local function on Therefore equation (31) reads: jet space. Let ϕ U and V some neighborhood of ϕ 1 ∈ k F (ϕ + ψ)= F (ϕ)+ dt Fϕ+tψ(x)ψ(x)dx.(33) such that F (ϕ + ψ) = M f(x,jx ψ)dx for every ψ V ∈ 0 M ∇ where jkψ is the k-jet of ψ at x and where f is smooth Z Z x R and compactly supported in the variable x in some fixed We show that Fubini’s theorem can be applied to the compact K M. Without loss of generality, we can function χ : (x,t) ϕ+tψF (x)ψ(x). By Prop. III.4, (1) 7→ ∇ restrict the support⊂ K of f by a smooth partition of unity F is locally compactly supported, so that there is a and assuming that K is contained in some open chart of convex neighborhood V of ϕ and a compact subset K of (1) M, we may reduce to the same problem for f C∞(Ω) Ω such that Fϕ+ψ is supported in K for every ϕ+ψ V . d ∈ ∈ where Ω is some open set in R and K Ω. The function χ is defined on [0, 1] K and supported ⊂ We choose a smooth compactly supported function χ on K for fixed t [0, 1]. Moreover,× by imposing the ad- ∈ (Ω) such that χ = 1 on a compact neighborhood of K ditional assumption∈ carried by item 2 in Theorem VI.3, D with supp χ Ω and we observe that namely that ϕ Fϕ be Bastiani smooth from U to ⊂ 7→ ∇ α (d+k)! (M), the support property of F implies that the im- Ψ: ψ C∞(Ω) (∂ ψ) χ (supp(χ)) d! , D α 6k age of Fϕ tψ is actually in (K) and F is smooth ∈ 7−→ | | ∈D ∇ + D ∇ is linear continuous hence Bastiani smooth. We need a from V to (K) because the topology induced on (K) by (M) isD the Fréchet topology of (K) determinedD simple D D by the seminorms πm,K [30, p. 172]. Since (K) in- Rd 0 D Lemma VI.12. Let Ω be an open set in then the jects continuously in (C (K), π0,K ), ϕ DϕF is a con- 0 7→ map tinuous (C (K), π0,K )-valued map. This implies that r d r (t,x) Fϕ+tψ(x) is continuous as a K-valued func- Φ: ϕ C∞(Ω, R ) x (x,ϕ(x)) C∞(Ω, R R ), ∈ 7→ { 7→ }∈ × tion on7→ [0 ∇, 1] K. Hence so is the integrand of (33), is Bastiani smooth. Fubini theorem× holds and we obtain: 23

Lemma VI.13. Let U be an open subset of E = C∞(M) u DFϕ+tψi (u) on ′(M) by duality pairing. By conti- K 7→ D and F : E be Bastiani smooth. Assume that for nuity, DFϕ+tψi (hn) DFϕ+tψi (δx0 ) and Eq. (32) yields → (1) (1) → every ϕ U, WF(Fϕ ) = and F : U (M) is h = DFϕ+tψ (δx ) DFϕ+tψ (δx ) Bastiani∈ smooth, then, for every∅ ϕ U, there→ is D a convex 2 0 − 1 0 ∈ (1) neighborhood V of ϕ such that, if ϕ + ψ V , then = dx Fϕ+tψ (x) F (x) δ(x x0) ∈ ∇ 2 −∇ ϕ+tψ1 − ZM 1 = Fϕ+tψ2 (x0) Fϕ+tψ1 (x0). F (ϕ + ψ)= F (ϕ)+ dx Fϕ+tψ(x)ψ(x)dt.(34) ∇ −∇ ∇ ZM Z0 For the right hand side, we know by Lemma VI.9 that From now on, we consider ϕ U to be fixed. Our for every s [0, 1], the wave front set of the distribution k ∈ (2) ∈ candidate for f(jx ψ) is Fϕ+tψ +st(ψ ψ ) is in the conormal C2 and the sequence 1 2− 1 1 (ψ2 ψ1) hn converges to (ψ2 ψ1) δx0 in N′ ∗(M x ), − ⊗ − ⊗ D ×{ 0} cψ(x)= Fϕ tψ(x)dtψ(x). (35) where N ∗ (M x0 ) is the conormal of the submanifold ∇ + × { } Z0 M x0 M M in T ∗ (M M). Therefore, by transversality× { } ⊂ of the× wave front sets× and hypocontinuity By definition and LemmaVI.13, for all ψ such that ϕ + 106 ψ V , of the duality pairings , the following limit exists ∈ (2) lim Fϕ+tψ +st(ψ ψ ), (ψ2 ψ1) hn . n h 1 2− 1 − ⊗ i F (ϕ + ψ)= F (ϕ)+ cψ(x)dx. M Moreover, still by Lemma VI.9, we have for ϕ+ψ V : Z ∈ To show that cψ(x) is the right candidate we first need (2) α α α F ,g h = ( 1)| | dxθ (x)g(x)∂ h(x), h ϕ+ψ ⊗ i − ψ α k ZM Proposition VI.14. The function cψ depends only on |X|≤ a finite jet of ψ. More precisely, for every ϕ U, there 2 α for every (g,h) C∞(M) and all θψ belong to (K). is a convex neighborhood V of ϕ and an integer∈ k 0 ∈ D ≥ An integration by parts yields such that, for all x M, for every ψ1 and ψ2 such that ∈ k k (2) α α ϕ + ψ1 and ϕ + ψ2 are in V and jx ψ1 = jx ψ2, then Fϕ+ψ,g h = dxfψ (x)h(x)∂ g(x), h ⊗ i M cψ1 (x)= cψ2 (x). α k Z |X|≤ α β β α β The beginning of the proof is inspired by Ref. 20. For where fψ = β α ∂ − θψ where the sum is over the fixed ϕ U, by Proposition III.11, there exists an integer multi-indices such that β α and β 6 k. ∈ P ≥ | | k, a compact K and a convex neighborhood V of ϕ such As a consequence, for ϕ + ψ1 + ψ2 in the convex neigh- (2) that the order of Fϕ+ψ is smaller than k and the support borhood V : of DFϕ+ψ is in K if ϕ + ψ V . (2) ∈ X = Fϕ+tψ +st(ψ ψ ), (ψ2 ψ1) δx0 Let us choose some point x0 M. Consider a pair h 1 2− 1 − ⊗ i ψ ,ψ of smooth functions such∈ that ψ (x ) = ψ (x ). α α 1 2 1 0 2 0 = ftψ +st(ψ ψ )(x0)∂ (ψ2 ψ1)(x0). 1 2− 1 − Then, α k |X|≤ 1 k k If, at the point x0, jx0 ψ1 = jx0 ψ2, then cψ1 (x0) cψ2 (x0)= dt Fϕ+tψ2 (x0) ψ2(x0) − ∇ 1 1 Z0 F (x ) ψ (x ) cψ1 (x0) cψ2 (x0)= ψ1(x0) tdt ds ϕ+tψ1 0 1 0 − −∇ α k Z0 Z0 1 |X|≤ α α α = ψ1(x0) dt Fϕ+tψ2 (x0) ftψ +st(ψ ψ )(x0) ∂ ψ2(x0) ∂ ψ1(x0) = 0. ∇ 1 2− 1 − Z0 We showed that cψ depends only on the k-jet of ψ at x0. Fϕ+tψ (x0) . −∇ 1 Moreover, the number k depends only on V and not on We use the fundamental theorem of analysis again x0, so that cψ depends on the k-jet for every x M. In ∈ for DFϕ(h) = dx Fϕ(x)h(x) for an arbitrary h other words there is an integer k and a function f such M ∇ ∈ α C∞(M) to get that cψ(x) = f(x,ψ(x),...,∂ ψ(x)) for every x M, R where 1 α k. ∈ 1 ≤| |≤ (2) We want to show that f is smooth in its arguments DFϕ tψ (h) DFϕ tψ (h)= t ds F , + 2 + 1 ϕ+tψ1+st(ψ2 ψ1) − 0 h − hence we now investigate in which manner cψ depends Z on ψ. This suggests to study the regularity of the (K)- (ψ2 ψ1) h . D − ⊗ i valued function ψ Fϕ tψψ. More precisely, we need 7→ ∇ + Now we take a sequence of smooth functions (hn)n N to show that the map ψ Fϕ+tψψ is Bastiani smooth ∈ 7→ ∇ which converges to δx in ′(M) when n goes to in- from U to (K). This is not completely trivial because 0 D D finity and show that both the left and right hand side the map x Fϕ+tψ(x) is in (K) and ψ in C∞(M) have limits. For the left hand side, the distribution and we must7→ check ∇ that the productD of a function in (K) D DFϕ+tψi being smooth, it defines the continuous form by a function in C∞(M) is continuous [27, p. 119]. 24

∂ α (α) (α) ∂ Lemma VI.15. If U is an open set in C∞(M) and EL = u ∂u + α( 1)| |∂ u ∂u(α) . Let us discuss F : U (K) is a compactly supported Bastiani-smooth the nature of the objects− involved, ρ is a vertical vector →D P k map, then the function G : U (M) defined by field and acts on C∞(J M) as a C∞(M) linear map, for → D d k d G(ϕ) = F (ϕ)ϕ is compactly supported Bastiani-smooth χ C∞(R ),f C∞(J R ), ρ(χf) = χ(ρf). For every with the same support as F . i ∈ 1,...,d , ∂∈i is a vector field on J kRd but it has a ∈ { } d horizontal component, therefore it is not C∞(R ) linear d Proof. Yoann Dabrowski pointed out to us the follow- and the Euler–Lagrange operator is not C∞(R )–linear ing fact. For any compact subset K of Ω, both (Ω) D either. and C∞(Ω) induce on (K) the usual Fréchet topology D What follows is a definition–proposition where we give of (K) [30, p. 172]. Thus to establish the smoothness an intrinsic and global definition of the Euler–Lagrange of GD, it suffices to show that the multiplication (u,v) ∈ operator in terms of the operator F associated to a (K) C∞(M) (K) is continuous then it would functional. ∇ Dbe Bastiani× smooth7→ and D by the chain rule it follows that ϕ (F (ϕ),ϕ) F (ϕ)ϕ is smooth. Since both (K) Proposition VI.16 (Euler-Lagrange operator is intrin- 7→ 7→ D and C∞(M) are Fréchet, the product (K) C∞(M) sic). Let U be an open subset of C∞(M) and F : U K endowed with the product topology is metrizableD × and it a Bastiani smooth local functional. For ϕ U, if there→ is is enough to prove that the product is sequentially con- an integer k, a neighborhood V of ϕ, an open∈ subset of k k V tinuous. Indeed, let (un,vn) (u,v) in (K) C∞(M), J M and f C∞( ) such that x f(jx ψ) is compactly we can find some cut–off function→ χ D(M)× such that supported and∈ V 7→ ∈ D χ = 1 on the support of all un, and for all m, by [85, p. 1351] k F (ϕ + ψ)= F (ϕ)+ f(jx ψ)dx ZM πm,K (uv unvn) 6 πm,K ((u un)vχ) − − whenever ϕ + ψ V then in every local chart +πm,K (unχ(v vn)) ∈ m − k 6 2 πm,K (u un)πm,K (χv) Fϕ = EL(f)(j ψ) (36) − ∇ +πm,K (un)πm,K ((v vn)χ) 0. − → α (α) ∂f k where EL(f)(ψ) = α 6k( 1)| | ∂ ∂u(α) (jx ψ) Hence G is smooth. | | − is the Euler–LagrangeP operator and EL (f)(ψ) is uniquely 1 determined by F . This implies that ψ cψ = 0 Fϕ+tψψdt is smooth since the above Lemma7→ shows the∇ smoothness R The above proposition means that EL(f) does not de- of a(t,ψ) Fϕ tψψ and integration over t conserves 7→ ∇ + pend on the choice of representative f and is intrinsic smoothness by Lemma VI.11. At this point, Theo- (i.e. it does not depend on the choice of a local chart). rem VI.3 follows directly from Proposition VI.4. Proof. Indeed, assume that we make a small perturbation ϕ + ψ of the background field ϕ by ψ which is compactly D. Representation theory of local functionals. supported in some open chart U of M. Then a local calculation yields In this section, we discuss the issue of representation of ∂f (α) our local functionals and the relations between the func- DFϕ(ψ)= (α) ψ (x)dx k M ∂u (x) tionals (cψ, F,f(jx ψ)) which are defined or constructed α Z ∇ X in the course of our proof of Theorem VI.3. In the sequel, α (α) ∂f = ( 1)| | ψ(x) ∂ dx, we assume that our manifold M is connected, oriented − ∂u(α) α k ZM without boundary, hence we can fix a density dx on M |X|≤ which is also a differential form on M of top degree. where we used an integration by parts to recover the In the sequel, we shall work out all explicit formu- Euler-Lagrange operator and all boundary terms van- las in local charts which means without loss of gen- ish since f is compactly supported in x and ψ (U). Rd erality that we work on and the reference den- We have just proved that for all open chart U∈ D M, sity dx is chosen to be the standard Lebesgue mea- ⊂ α Fϕ U = EL(f) U . But Fϕ is intrinsically defined sure. We will denote by (x,u,u ) α 6k where α are ∇ | | ∇ | | on M therefore so is EL(f) and we have the equality multi–indices, some local coordinates on the jet bun- F = EL(f). The unique determination of F follows k Rd ϕ dle J ( ). Introduce the vertical Euler vector field from∇ Lemma VI.2. ∇ (α) ∂ k Rd ρ = u ∂u(α) on the bundle J ( ). In the manifold case if we work on J k(M), this vector field is in- Theorem VI.17. [Global Poincaré]Assume that M is a P trinsic since it generates scaling in the fibers of J k(M). smooth, connected, oriented manifold without boundary. For all multiindex (α)=(α1 ...αp),αi 1,...,d , in- Let U be an open subset of C (M) and F : U K ∈ { } ∞ troduce the operators ∂(α) = ∂α1 ...∂αp where ∂i = a Bastiani smooth local functional. Then the following→ ∂ (αi) ∂ ∂xi + α u ∂u(α) and the Euler–Lagrange operator statements are equivalent: P 25

k d d d two functions (f1,f2) C∞( ) for an open sub- jet spaces. For all fdx C∞(J R ) Ω (R ) : • set of the jet space J k∈M, areV two representationsV ∈ ⊗ ∂f ∂f ∂f of F in a neighborhood V of ϕ U : (ρf) dx = u(α) dx = u + u(α) dx ∈ ∂u(α) ∂u ∂u(α) α >1 X |X| k d F (ϕ + ψ)= F (ϕ)+ f1(j ψ)dx x = uEL(f)dx + ∂µj (j2kψ)dx ZM µ µ=1 = F (ϕ)+ f (jkψ)dx X 2 x d M Z 2k ∂ = uEL(f)dx + d jµ(j ψ) ydx ∂xµ whenever ϕ + ψ V µ=1 ! ∈ X 2k d where jµ C∞(J R ) is a local functional. for all ψ V ϕ : ∈ • ∈ − To prove the second identity, we shall use the funda- k k 2k mental Theorem of calculus and the first identity : f1(jx ψ)dx f2(jx ψ)dx = dβ(jx ψ), (37) − k k f(j (ψ1 + ψ2))dx = f(j ψ1)dx 2k d 1 where β(jx ψ) Ωc− (M) is a differential form of 1 ∈ dt k degree d 1 whose value at a point x depends only + (ρf)(j (ψ1 + tψ2))dx on the 2k−-jet of ψ at x. 0 t Zk = f(j ψ1)dx Let us stress that we do not need to constraint the 1 k topology of M in the above Theorem only the compact- + dtψ2EL(f)(j (ψ1 + tψ2))dx k 0 ness of the support of fi(jx ψ)dx,i 1, 2 really matters. Z ∈ { } 1 dt 2k + d jµ(j (ψ1 + tψ2))∂xµ ydx Proof. One sense of the equivalence is trivial since the in- 0 t tegral of a compactly supported exact form on M always Z vanishes. By Proposition VI.4, we know that the map To prove the claim of the Lemma is equivalent to show ψ V ϕ jkψ has an open image in J k(M) denoted the following statement: if a local functional F is locally ∈ − 7→ constant i.e. F (ϕ+ψ)= F (ϕ) whenever ϕ+ψ V , then and we only need (f1,f2) to be defined on . We denote 2k 2k ∈ n 1 V α V F (ϕ+ψ)= F (ϕ)+ dβ(jx ψ) and β(jx ψ) Ωc − (M) by (x,u,u ) α 6k where α are multi–indices, some local M ∈ | | is a compactly supported n 1 form. For all ψ in V ϕ, coordinates on the jet bundle J k(Rd). We use the vertical R − − (α) ∂ k Rd 1 Euler vector field ρ = u ∂u(α) on the bundle J ( ). dt k For every multiindex (α)=(α ...α ),α 1,...,d , F (ϕ + tψ)= F (ϕ) = (ρf)(jx (tψ))dx = 0, 1 p i ⇒ Rd 0 t P (α) α αp ∈ { i } Z Z introduce the operators ∂ = ∂ 1 ...∂ where ∂ = 1 ∂ (αi) ∂ k ∂xi + α u ∂u(α) and the Euler–Lagrange operator = dt ψEL(f)(jx (tψ))dx = 0. ⇒ Rd reads EL = u ∂ + ( 1) α ∂(α)u(α) ∂ . Z Z0 P ∂u α | | ∂u(α) We shall prove two related− identities, in local chart This means that EL(f) = 0 therefore on any open chart P U (U is contractible), Eq. (39) yields (ρf)(jkψ)dx =(uEL(f))(jkψ)dx 1 d p 2p 2k ∂ f(jx(ψ)) = f(0) + d dt jµ(jx (tψ))∂xµ ydx . +d jµ(j ψ) ydx (38) ∂xµ 0 µ ! µ=1 ! Z X X k k k f(j (ψ1 + ψ2))dx = f(j ψ1)dx We want to prove that EL(f)=0 = f(jx (ψ))dx 2k ⇒ 2p − 1 f(0)dx M = dβ(j ψ) where β C∞(J (M)) know- k | p+1 ∈ + dtψ2EL(f)(j (ψ1 + tψ2))dx ing that this holds true on any local chart, and that Z0 EL(f) = 0 is equivalent to assuming that F (ϕ + ψ) := 1 k dt 2k f(j ψ)dx is locally constant. We cover M by some +d jµ(j (ψ1 + tψ2))∂xµ ydx .(39) M x 0 t countable union i NUi of contractible open charts such Z ∈ thatR every element∪ x M belongs to a finite number kRd 2 ∈ For all (f,g) C∞(J ) and all multiindices α, the of charts Ui, set Mp = (U Up) and we arrange ∈ 1 ∪···∪ generalized Leibniz-like identity holds true: the cover in such a way that Mp Up+1 = for all p which is always possible. Assume by∩ induction6 ∅ on p that (∂α1 ...∂αp f)g =( 1)pf(∂αp ...∂α1 g) EL(f)=0 and supp (f) Mp implies p − ⊂ i+1 αi αi αp αi α k 2k + ( 1) ∂ (∂ +1 ...∂ f)∂ −1 ...∂ 1 g f(j (ψ))dx f(0)dx M = dβ(j ψ), − x − | p i=1 2p d 1 X where β C∞(J (M)) Ωc− (Mp). where the second term is a sum of total derivatives. Using We want∈ to prove that⊗ EL(f) = 0, supp (f) k 2k ⊂ this we derive the following key identity which is valid on Mp = f(j (ψ))dx f(0)dx M = dβ(j ψ) where +1 ⇒ x − | p+1 26

2p d 1 α (α) ∂f k β C∞(J (M)) Ω − (Mp+1). Choose a partition of c where EL(f)(ψ) = α 6k( 1)| | ∂ ∂u(α) (jx ψ) unity∈ (χ, 1 χ) subordinated⊗ to M U , the key idea | | − p p+1 is the Euler–Lagrange operator and EL (f)(ψ) is uniquely is to decompose− the variation ψ of the∪ background field P determined by F . ϕ as the sum of two components χψ +(1 χ)ψ where χψ − Furthermore, we find that : (resp (1 χ)ψ) vanishes outside Up (resp Up ) which − +1 yields : k F (ϕ + ψ)= F (ϕ)+ f(jx ψ)dx f(jkψ)= f(jk(χψ + (1 χ)ψ)) f(jk((1 χ)ψ)) ZM − − − 1 +f(jk((1 χ)ψ)) f(0) + f(0). = F (ϕ)+ dtψEL(f)(tψ)ψ dx − − M 0 The second idea is to note that for every fixed ψ, the new Z Z 1 functional where f(jkψ) dtψEL(f)(tψ)ψ dx = dβ(j2kψ) and x − 0 x 2k d 1 β(j ψ) Ω − (M) is a compactly supported d 1 form. φ f˜(jkφ)= f(jk(χφ + (1 χ)ψ)) f(jk((1 χ)ψ)) x ∈ c R − 7→ − − − ˜ k has trivial Euler–Lagrange equation EL(f)(j φ) = 1. Explicit forms EL(f)(jk(χφ + (1 χ)ψ)) = 0 since EL(f) = 0 and − its support is contained in Mp. Therefore : In this section we derive the explicit expression of Fϕ ∇ f(jkψ)= f˜(jkψ)+ f(jk((1 χ)ψ)) f(0) + f(0) and F (α)(ϕ) in terms of f when M = Rd. Since the − − = dβ˜(j2kψ)+ f(jk((1 χ)ψ)) f(0) +f(0) general expression is not very illuminating, let us start − − with the following simple example: by the inductive assumption. To treat the term under F (ϕ)= h(x)ϕ4(x)+ gµν (x)∂ ϕ(x)∂ ϕ(x)dx, brace, define a new functional| {z } µ ν ZM µν ψ g(jkψ)= f(jk((1 χ)ψ)) f(0) where h and g are smooth and compactly supported 7→ − − and gµν is symmetric. We compute whose support is contained in Up+1 and whose Euler- 3 µν Lagrange equation vanishes, EL(g) = 0 again by the DFϕ(u) = 2 dx2h(x)ϕ (x)u(x)+ g (x)∂µϕ(x)∂ν u(x) M fact that EL(f) = 0. Since Up+1 is contractible we Z k 2k 3 µν know that f(j ((1 χ)ψ)) f(0) = dα(j ψ) where = 2 dx 2h(x)ϕ (x) ∂ν g (x)∂µϕ(x) u(x), 2k −d 1 − M − α C∞(J M) Ωc− (Up+1) and therefore we found Z that∈ ⊗ where we used integration by parts. Thus, 3 µν k 2k Fϕ(x) = 4h(x)ϕ (x) 2∂ν g (x)∂µϕ(x) . f(j ψ)dx = dβ(j ψ)+ f(0)dx ∇ − Moreover, 2k d 1 β C∞(J M) Ωc− (Mp+1). Therefore for all ψ V ∈ ϕ, f(jk(ψ))dx⊗ = dβ(j2kψ)+ f(0)dx. Now we con-∈ 2 2 − x D Fϕ(u,v) = 2 dxu(x) 6h(x)ϕ (x)v(x) clude by using the fact that F is a constant functional M Z thus 0 = F (ϕ + ψ) F (ϕ)= f(0)dx + dβ(j2kψ) = µν − M ∂ν g (x)∂µv(x) . f(0)dx. But f(0)dx is a top form in Ωd(M) which − M R c does not depend on ψ and whose integral over M van- To write this as a distribution, we need to integrate over R d 1 ishes hence f(0)dx = dk for some k Ωc− (M) since two variables: Hd(M, R) R for the top de Rham∈ cohomology with c 2 2 compact support≃ when M is connected [107, Theorem D Fϕ(u,v) = 2 dxdyu(x)δ(x y) 6h(y)ϕ (y)v(y) 2 − 17.30 p. 454]. ZM µν ∂ν g (y)∂µv(y) . The next Theorem summarizes the above results : − Now we can use integration by parts over y to recover Theorem VI.18. Let U be an open subset of C∞(M) v(y): and F : U K a Bastiani smooth local functional. For ϕ U, if→ there is an integer k, a neighborhood V of 2 α α D Fϕ(u,v)= dxdyu(x)v(y)f (ϕ)(y)∂y δ(x y), ∈ k 2 ϕ, an open subset of J M and f C∞( ) such that α M − k V ∈ V Z x f(j ψx) compactly supported and X 7→ where the non-zero f α(ϕ) are k 0 2 F (ϕ + ψ)= F (ϕ)+ f(jx ψ)dx f (ϕ)(y) = 12ϕ (y), M µ µν Z f (ϕ)(y)= ∂ν g (y), − whenever ϕ + ψ V then in every local chart f µν (ϕ)(y)= gµν (y). ∈ − k Fϕ = EL(f)(j ψ) (40) More generally ∇ 27

Proposition VI.19. If VII. PEETRE THEOREM FOR LOCAL AND MULTILOCAL FUNCTIONALS F (ϕ)= f(ϕ(α)(x))dx, ZM In this section, we propose an alternative characterization of local functionals in terms of a nonlinear Peetre then theorem. We do not characterize the locality of the ac- 2 tion F but the locality of the Lagrangian density, that α β β ∂ f F (ϕ)(x)= ( 1)| | ∂ β−γ . we denoted F in the previous section. We first state − γ y ∂ϕ(α γ)(x)∂ϕ(β)(x) ∇ β γ − our theorems for local functionals, and then we prove X≤ them for the case of multilocal functionals, which are Proof. The proof is a straightforward generalization of a natural generalization of local functionals in quantum the example. Indeed, field theory. Our proof is inspired by recent works on the Peetre theorem108,109, however it is formulated in ∂f (α) the language of Bastiani smoothness and uses simpler DFϕ(u)= u (x)dx 110 ∂ϕ(α)(x) assumptions than Slovák’s paper . α ZM X α α ∂| | ∂f = ( 1)| | u(x) dx, ∂xα (α) A. Peetre theorem for local functionals α − M ∂ϕ (x) X Z where we used an integration by parts to recover the Let Ω be some open set in a manifold M. We first Euler-Lagrange operator. The second derivative is begin with an alternative definition of a local map from C∞(Ω) to itself, that we call Peetre local. 2 α D Fϕ(u,v)= ( 1)| | u(x) Definition VII.1. A map F : C∞(Ω) C∞(Ω) is Pee- − M → αβ Z tre local if for every x Ω, if ϕ1 = ϕ2 on some neigh- X ∈ α 2 borhood of x then F (ϕ1)(x)= F (ϕ2)(x). ∂| | ∂f (β) α (α) (β) v (x) dx. ∂x ∂ϕ (x)∂ϕ (x) The relation with the additivity condition is given by

We write this as a double integral Proposition VII.2. Let F : C∞(Ω) C∞(Ω) be Pee- →2 tre local. For every (ϕ ,ϕ ) C∞(Ω) if supp ϕ and 1 2 ∈ 1 2 α supp ϕ2 do not meet then for every x Ω and for all ϕ, D Fϕ(u,v)= ( 1)| | u(x)δ(x y) ∈ − 2 − αβ M X Z F (ϕ1 + ϕ2 + ϕ)(x)= F (ϕ1 + ϕ)(x) 2 ∂f (β) +F (ϕ2 + ϕ)(x) F (ϕ)(x).(41) ∂yα v (y) dxdy. − ∂ϕ(α)(y)∂ϕ(β)(y) Proof. If x / (supp ϕ1 supp ϕ2) then ϕ1 = ϕ2 = 0 ∈ ∪ A first integration by parts gives us in some neighborhood of x, it follows that F (ϕ1 + ϕ2 + ϕ)(x)= F (0+0+ ϕ)(x)= F (ϕ)(x) and F (ϕ1 + ϕ)(x)+ F (ϕ +ϕ)(x) F (ϕ)(x) = 2F (ϕ)(x) F (ϕ)(x)= F (ϕ)(x) ∂f 2 2 2 (β) hence Eq. (41−) holds true. − D Fϕ(u,v)= u(x) (α) (β) v (y) 2 ∂ϕ (y)∂ϕ (y) αβ M If x supp ϕ then necessarily there is some neighbor- X Z ∈ 1 hood U of x on which ϕ2 U = 0 hence ϕ1 + ϕ2 + ϕ U = ∂yα δ(x y)dxdy. | | − ϕ1 + ϕ U and F (ϕ1 + ϕ2 + ϕ)(x) = F (ϕ1 + ϕ)(x). Also F (ϕ +|ϕ)(x)+F (ϕ +ϕ)(x) F (ϕ)(x)= F (ϕ +ϕ)(x)+ A second integration by parts isolates v(y): 1 2 1 F (ϕ)(x) F (ϕ)(x)= F (ϕ +−ϕ)(x) hence again Eq. (41) − 1 holds true. The case where x supp ϕ2 can be treated 2 β β ∈ D Fϕ(u,v)= ( 1)| | dxdyu(x)v(y) by similar methods which yields the final result. − γ 2 αβ γ β ZM X X≤ The Peetre theorem for local functionals is ∂f 2 ∂yβ−γ ∂yα+γ δ(x y). ∂ϕ(α)(y)∂ϕ(β)(y) − Theorem VII.3. Let F : C∞(Ω) C∞(Ω) be a Bas- tiani smooth Peetre local map. Then,→ for every ϕ ∈ C∞(Ω) there is a neighborhood V of ϕ in C∞(Ω) and an integer k such that for all ψ such that ϕ + ψ V , k ∈ k F (ϕ + ψ)(x) = c(j ψx) for some smooth function c on If we calculate higher differentials D Fϕ(u1,...,uk) we J kΩ. see that we always obtain products of smooth functions by derivatives of products of delta functions. This shows In other words, if F is a Bastiani smooth Peetre local (k) that the wavefront set of Fϕ is in the conormal Ck. map, then for every g (M), F (ϕ)g is a Bastiani ∈ D M R 28 smooth local map in the sense of the rest of the paper. Therefore it suffices to find some sequence λn 0 such → This relation between a priori different concepts of lo- that πm,K′ (ϕ ,λ ϕ ,λ ) 6 ηn. Since ϕ ϕ vanishes 1 n − 2 n 1 − 2 cality strongly supports the idea that our definition is a at order m+1 on the set X = x1,...,xk , Lemma VII.6 natural one. yields the estimate { } If F is only assumed to be a continuous local map, πm,K′ (ϕ ,λ ϕ ,λ) 6 Cλπ˜ m ,K (ϕ ϕ ) , then a similar theorem exists for which the function c is | 1 − 2 | +1 1 − 2 not necessarily smooth. These theorems are proved in which implies that the next section for the more general case of multilocal lim πm,K′ (ϕ1,λ ϕ2,λ)= lim πm,K′ ((ϕ1 ϕ2)χλ) = 0. functionals. λ 0 − λ 0 − → → Finally, we obtain that if ϕ1,ϕ2 have same (m + 1)-jet at X = x ,...,xk then for all n> 0: B. Multilocal functionals and first Peetre theorem { 1 } F (ϕ )(x ,...,xk) F (ϕ )(x ,...,xk) = | 1 1 − 2 1 | By generalizing Definition VII.1 of local maps, we can 1 6 define multilocal maps. These maps appear naturally F (ϕ1,λn )(x1,...,xk) F (ϕ2,λn )(x1,...,xk) | − | 2n in quantum field theory as the product of several La- which implies F (ϕ )(x ,...,x ) = F (ϕ )(x ,...,x ). grangian densities (x ) ... (xk). 1 1 k 2 1 k L 1 L Definition VII.4. Let k be an integer. A map F : k d C∞(Ω) C∞(Ω ) is k-local if for every (x1,...,xk) Lemma VII.6. Let X be any closed subset of R . Let k → ∈ m+1 d Ω , if ϕ1 = ϕ2 on some neighborhood of x1,...,xk Ω (X, R ) denote the closed ideal of functions of reg- { }⊂ I m+1 then F (ϕ1)(x1,...,xk)= F (ϕ2)(x1,...,xk). ularity C which vanish at order m + 1 on X. Then there is a function χ C (Rd) parametrized by λ The multilocal maps are the maps that are k-local for λ ∞ (0, 1] s.t. χ = 1 (resp χ∈ = 0) when d(x,X) 6 λ (resp∈ some k. We emphasize that Peetre local maps in the λ λ 8 Rd sense of definition VII.1 correspond with 1-local maps d(x,X) > λ) such that for all compact subset K , ˜ ⊂ in the above sense. For M a smooth manifold, we denote there is a constant C such that, for every λ (0, 1] and p ⊠k k every ϕ m+1(X, Rd) ∈ by J M the bundle over M whose fiber over a k-tuple ∈I of points (x ,...,x ) M k is J pM J pM . 1 k x1 xk ˜ ∈ ×···× πm,K (χλϕ) 6 Cλπm+1,K d(x,X)6λ (ϕ) . (42) k ∩{ } Theorem VII.5. Let F : C∞(Ω) C∞(Ω ) be a con- → d tinuous k-local map. Then, for every ϕ C∞(Ω) there Proof. Choose φ > 0 s.t. Rd φ(x)d x = 1 and φ = 0 if ∈ 3 d 1 is a neighborhood V of ϕ in C∞(Ω), p N such that for x > . Then set φ = λ φ(λ .) and set α to be ∈ 8 λ R − − λ all ψ such that ϕ + ψ V , the| | characteristic function of the set x s.t. d(x,X) 6 ∈ λ { p p 2 then the convolution product χλ = φλ αλ satisfies F (ϕ + ψ)(x1,...,xk)= c(j ψx1 ,...,j ψxk ) } λ ∗ χλ(x)=1if d(x,X) 6 and χλ(x)=0if d(x,X) > λ. p ⊠k k 8 k for some function c : J M (M Dk) M , where Since by Leibniz rule one has k | \ → k M Dk denotes the configuration space M minus all \ diagonals. α α k α k ∂ (χλϕ)(x)= ∂ χλ∂ − ϕ(x), k k Proof. Fix a k–tuple of points (x1,...,xk) Ω and k 6 α ∈ k | X| | | some compact neighborhood K of (x1,...,xk) in Ω . k α k Continuity of F implies that for all ε > 0, there ex- it suffices to estimate each term ∂ χλ∂ − ϕ(x) of the ists η > 0 and a seminorm πm,K′ of C∞(Ω) such that above sum. For every multi-index k, there is some con- Rd k Ck π ′ (ϕ ϕ ) 6 η implies stant Ck such that x X, ∂ χλ 6 |k| and m,K 1 2 ∀ ∈ \ | x | λ − k 6 6 supp ∂x χλ d(x,X) λ . Therefore for all ϕ sup F (ϕ1)(y1,...,yk) F (ϕ2)(y1,...,yk) ε. m+1 Rd⊂ { } k α k ∈ (y1,...,yk) K | − | (X, ), for all x supp ∂x χλ∂ − ϕ, for y X ∈ I ∈ α k ∈ such that d(x,X)= x y , we find that ∂ − ϕ vanishes Assume that (ϕ1,ϕ2) have same (m + 1)-jets at at y at order k + 1.| Indeed− | ϕ vanishes at order m + 1 x ,...,xk . Let (χλ)λ be the family of compactly sup- α k | | { 1 } hence ∂ − ϕ vanishes at order m + 1 α + k > k + 1 ported cut-off functions equal to 1 in some neighbor- since α 6 m. Therefore: −| | hood of X = x ,...,xk defined in lemma VII.6. It | | { 1 } follows that ϕ1,λ = ϕ1χλ (resp. ϕ2,λ = ϕ2χλ) coin- α k β ∂x − ϕ(x)= (x y) Rβ(x), cides with ϕ1 (resp. ϕ2) near x1,...,xk . Hence, for − { } β = k +1 all λ > 0, F (ϕ1,λ)(x1,...,xk) = F (ϕ1)(x1,...,xk) and | |X| | 1 F (ϕ2,λ)(x1,...,xk) = F (ϕ2)(x1,...,xk). Set εn = 2n where the right hand side is just the integral remainder then there exists η such that π ′ (ψ ψ ) 6 η im- α k n m,K 1 2 n in Taylor’s expansion of ∂ − ϕ around y. Hence: plies − 1 k α k Ck β ∂ χλ∂ − ϕ(x) 6 (x y) Rβ(x) . sup F (ψ1)(y1,...,yk) F (ψ2)(y1,...,yk) 6 . k n | | λ| | | − | (y ,...,yk) K | − | 2 β = k +1 1 ∈ | |X| | 29

It is easy to see that Rβ only depends on the jets of ϕ of “odd” space E∗[1] through the space of functions on it, order 6 m + 1. Hence understood as multilinear smooth, totally antisymmetric, functionals. Then we shall make a conjectural claim k α k ∂ χλ∂ − ϕ(x) 6 Ckλ sup Rβ(x) on the meaning of locality in that context. | | x K,d(x,X)6λ | | ∈ β = k +1 | |X| | and the conclusion follows easily. 1. Locality of functionals on graded space.

We consider a graded space E0 E1[1], where E0 = C. The second Peetre Theorem ⊕ Γ(M,B0) and E1 = Γ(M,B1) are spaces of smooth sections of finite rank vector bundles B0 and B1 over M k Theorem VII.7. Let F : C∞(Ω) C∞(Ω ) be a Bas- respectively. Before giving formal definitions, let us ex- → tiani smooth k-local map. Then, for every ϕ C∞(Ω) plain the idea of our construction. We will first define ∈ there is a neighborhood V of ϕ in C∞(Ω), p N such the space (E0 E1[1]) to be space of maps from E0 to that for all ψ such that ϕ + ψ V , ∈ , where O ⊕ ∈ A p p ∞ ∞ F (ϕ + ψ)(x1,...,xk)= c(j ψx1 ,...,j ψxk ) . k . k ⊠k = = Γ′ (M ,B ) , A A a 1 p ⊠k k k k =0 =0 for some smooth function c on J M (M Dk) where Y Y k | k\ M Dk denotes the configuration space M minus all satisfying an appropriate smoothness condition. Let us \ diagonals. clarify the notation Γa′ . We first define the iterated wedge product of k elements u ,...,uk of the space of distribu- Proof. Without loss of generality, we may assume 1 tional sections Γ′(M,B ) by that M = Rd and to go back to arbitrary man- 1 σ ifolds, we use partitions of unity as in the proof u1 uk,h1 hk = ( 1) u1,hσ(1) of Lemma VI.6. The coordinates on the jet space h ∧···∧ ⊗···⊗ i σ − h i p d α X J (R ) are denoted by (x,p ) 6 . Let (U ,...,Uk) α p 1 ... uk,hσ(k) , | | d h i be two by two disjoint open subsets of R , then U1 d k × where h ,...,h are sections in Γ(M,B ) and σ runs over Uk is an open subset of (R ) Dk. We de- 1 k 1 ··· × \ the permutations of 1,...,k . Then, the k-th exterior fine the smooth map: Φ : (x1,...,xk; p1,...,pk) p k { } p Rd ⊠k i,α α ∈ power Λ Γ′(M,B1) is the vector space of finite sums of J ( ) U1 Uk ( 6i6k (. xi) χi(. xi)) | ×···× 7→ 1 α! − − ∈ k ⊠k Rd Rd such iterated wedge products and Γa′ (M ,B1 ) is the C∞( ) where the functions χi Cc∞( ) are cut– k P ∈ completion of Λ Γ′(M,B1) with respect to the topology off functions equal to 1 near 0 and such that for ˆ k k ⊠k of Γ′(M,B1)⊗π = Γ′(M ,B ) where all the duals are all (x1,...,xk) U1 Uk, the support of ∼ 1 ∈ × ··· × d strong. The subscript “a” stands for antisymmetry. the functions χi(. xi) are disjoint on R . Then − In the case of multilinear symmetric functions, we can the map sending (x1,...,xk; p1,...,pk), (y1,...,yk) to identify a k-linear map f(h1,...,hk) of k variables with a F (ϕ +Φ(x1,...,xk; p1,...,pk))(y1,...,yk) is smooth by smoothness of F and Φ. Hence, its pull–back on the polynomial map of one variable f(h,...,h) by using the polarization identity. There is no polarization identity in diagonal x1 = y1,...,xk = yk is also smooth and reads the antisymmetric case and we must consider a function k F (ϕ + Φ(x1,...,xk; p1,...,pk))(x1,...,xk)= F : E0 as a function of one variable ϕ0 in E0 and k →A ˆ π k c(x1,...,xk; p1,...,pk) variables (h1,...,hk) in E1 (or a variable in H E1⊗ ). k∈ Then, we can identify a function F : E0 and the as the composition of smooth functions and it follows ˆ → A ˜ π k K p ⊠k function F : E0 E1⊗ defined by that c is smooth on J M U Uk . | 1×···× × → F˜(ϕ ; h hk)= F (ϕ )(h hk). 0 1 ⊗···⊗ 0 1 ⊗···⊗ VIII. MULTI-VECTOR FIELDS AND GRADED This motivates the following FUNCTIONALS Definition VIII.1. Let M be a smooth manifold, (B ,B ) are smooth vector bundles on M and E = In the quantum theory of gauge fields, especially in the 0 1 0 Γ(M,B ), E = Γ(M,B ) are spaces of smooth sections Batalin-Vilkovisky approach, it is necessary to deal, not 0 1 1 of the respective bundles. We say that a function F from only with functionals as discussed above, but also with k k E0 to is an element of (E0 E1[1]) if there exists a multi-vector fields on the configuration space E (assumed A O ˆ⊕ k 16 Bastiani smooth map F˜ : E E⊗π K which is linear to be the space of sections of some vector bundle B) . 0 × 1 → ˆ π k Such multi-vector fields can be seen as functionals on the in E1⊗ and antisymmetric w.r.t. the natural action . . ˆ k graded space T ∗[1]E = E E∗[1], where E∗ = Γ(M,B∗) of permutations on E⊗π such that : is the space of smooth sections.⊕ To make this notion pre- 1 111 cise, we use the ideas presented in and characterize the F˜(ϕ ; h hk)= F (ϕ )(h hk). (43) 0 1 ⊗···⊗ 0 1 ⊗···⊗ 30

We denote by (E0 E1[1]) the direct product of all Let us now discuss the notion of support which is the k O ⊕ 0 (E0 E1[1]), over k N0 and set (E0 E1[1]) K. appropriate generalization of the notion of support for O ⊕ ∈ O ⊕ ≡ graded functionals, generalizing the definitions in Sec- Let us now discuss the notion of derivative for the type tion IIIA. of functionals introduced above. Clearly, if F belongs to (E0 E1[1]), there are two natural ways to differentiate Definition VIII.5. Let F k(U E [1]) be a graded O ⊕ ˜ ∈O ⊕ 1 it. In the first instance we can differentiate F in the sense functional, with U an open subset of E0. The support of of Bastiani in the first variable (ϕ E0) and we denote F is defined by supp F = A B, where this derivative as ∈ ∪

. ˜ A = supp (ϕ (ιh1 ...ιhk F )(ϕ)) D0F(ϕ;u)(g) = DF(ϕ,u)(g, 0) , k 7→ (h ,...,hk) E 1 [ ∈ 1 ˆ π k K δ where u E⊗ , g E0 or F . B = supp h ιh ...ιh F (ϕ,h) . ∈ 1 → ∈ δϕ0 1 k−1 k−1 7→ ϕ U,(h ,...,hk ) E ∈ 1 [ −1 ∈ 1 2. The contraction operation. 3. Some conjectures on local graded functionals. Let us now consider contraction of the graded part with some h E1, sometimes referred to as derivations k with respect to∈ odd variables. This concept is needed in Let F (E0 E1[1]) be such that the WF set of ∈ O (1)⊕ order to define the Koszul complex and the Chevalley- both (ιh1 ...ιhk F )ϕ and ιh1 ...ιhk−1 F (ϕ,.) is empty for k Eilenberg complex in the Batalin–Vilkovisky formalism all ϕ U and (h1,...,hk) E1 . We conjecture that in infinite dimension. The definition is spelled out below. some∈ version of Lemmas VI.2∈ and VI.9 should hold in the graded case. The “standard” characterization of lo- Definition VIII.2. k Let F (E0 E1[1]), h E1. cality for a functional F k(E E [1]) is the re- The contraction of F by h is∈ defined,O ⊕ for every integer∈ ∈ O 0 ⊕ 1 k 1 quirement that F is compactly supported and for each k > 0 and u E1⊗ − , by k N ∈ (ϕ; u1,...,uk) E0 E1 there exists i0,...,ik such that ∈ × ∈ ιhF,u = F˜(h u), h i ⊗ 0 and ιhF = 0 if F . i0 i1 ik ∈A F (ϕ; u1,...,uk)= α(jx (ϕ),jx (u1),...,jx (uk)) , 1 M In particular, ιhF = F,h˜ if F . We extend this Z ∈ A (44) definition to by linearity.D E where α is a density-valued function on the jet bundle. A To conclude, we conjecture some graded analogue of In view of (43) and the definition of k(E E [1]), it is 0 1 Theorem VI.3 whose formulation would be as follows : clear that ι F k 1(E E [1]) forO all F⊕ k(E h − 0 1 0 Let U be an open subset of E and F k(U E [1]) E [1]). Equation∈ O (43) allows⊕ also to make∈ sense O of⊕ a 0 1 1 be a graded functional. Assume that ∈ O ⊕ second important operation on (E E [1]): O 0 ⊕ 1 k Definition VIII.3. The wedge product : (E0 1. F is additive in some suitable sense, still to be writ- k′ k+k′ ∧ O ⊕ ten with care (conceivably this would be additivity E1[1]) (E0 E1[1]) (E0 E1[1]) is defined by ×O ⊕ →O ⊕ of F˜ as a function of several variables).

^ ˜ (1) F G (u1,...,uk+k′ )= sgn(σ)F (uσ(1),...,uσ(k)) 2. (ι ...ι F ) and ι ...ι F (ϕ,.) have ∧ h1 hk ϕ h1 hk−1 σ empty wave front set for all ϕ U X k ∈ G˜(u ,...,u ′ and (h1,...,hk) E and the maps σ(k+1) σ(k+k ) ∈ 1 (ϕ,u) (ι ...ι F )(1) ,ι ...ι F (ϕ,.) (where the sum runs over k k shuffles) and extended h1 hk ϕ h1 hk−1 ′ 7→ ˆ − π k by linearity on (E0 E1[1]) (E0 E1[1]). are Bastiani smooth from U k N E1⊗ to O ⊕ ×O ⊕ × ∈ Γc(M,B0∗) and Γc(M,B1∗), respectively. Here B0∗ Again, in view of (43) and the definition of (E0 L O ⊕ and B1∗ denote dual bundles. E1[1]), it is clear that the wedge product of an element ′ k k ˆ k in (E0 E1[1]) with an element in (E0 E1[1]) is an ⊗π O ⊕ ′ O ⊕ Then, for every ϕ U, u k N E1 , there is a neigh- k+k ∈ ∈ ∈ element in (E0 E1[1]). The contraction and wedge borhood V of the origin in E0, an integer N and a smooth product satisfyO the following⊕ relation on (E E [1]): K L O 0 ⊕ 1 -valued function f on the N-jet bundle such that Lemma VIII.4. The contraction satisfies the graded k i0 i1 ik Leibniz rule: if F (E0 E1[1]), G (E0 E1[1]) F (ϕ+ψ; v1 vk)= α(jx (ψ),jx (v1),...,jx (vk)) , and h E , then ∈O ⊕ ∈O ⊕ ⊗···⊗ M ∈ 1 Z (45) k ιh(F G)=(ιhF ) G +( 1) F ιhG. for every ψ V and some i ,...,ik

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