Algebraic Structure of Classical Field Theory: Kinematics and Linearized Dynamics for Real Scalar Fields

Abstract

We describe the elements of a novel structural approach to classical field theory, inspired by recent developments in perturbative algebraic quantum field theory. This approach is local and focuses mainly on the observables over field configurations, given by certain spaces of functionals which are studied here in depth. The analysis of such functionals is characterized by a combination of geometric, analytic and algebraic elements which (1) make our approach closer to quantum field theory, (2) allow for a rigorous analytic refinement of many computational formulae from the functional formulation of classical field theory and (3) provide a new pathway towards understanding dynamics. Particular attention will be paid to aspects related to nonlinear hyperbolic partial differential equations and their linearizations.

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Notes

  1. 1.

    A causal (resp. timelike, null) curve \(\gamma :I\rightarrow {\mathscr {M}}\) is said to be inextendible if there is no causal (resp. timelike, null) curve \({\tilde{\gamma }}:{\tilde{I}}\rightarrow {\mathscr {M}}\) such that \({\tilde{I}}\supsetneqq I\) and \({\tilde{\gamma }} \vert _{I}=\gamma \).

  2. 2.

    Some physics texts, such as [30], call \({\mathscr {C}}^\infty ({\mathscr {M}})\) the space of field histories on \({\mathscr {M}}\).

  3. 3.

    In the context of field theory, such connections were formally introduced in [30]. They allow one to extend to higher orders the notion of fiber derivative employed in the calculus of variations [11]. For a precise, general concept of ultralocal lifts of connections on target spaces, see for instance Example 4.5.3, pp. 94 of [45].

  4. 4.

    The restriction on \({\mathscr {U}}\) can be weakened in a certain sense. See Lemma 3.3.

  5. 5.

    For simplicity, here we allow ourselves a slight abuse of notation—strictly speaking, the smooth density supported in \(\mathrm {supp}\ F\) representing \(F^{(1)}[\varphi ]\) for each \(\varphi \in {\mathscr {U}}\) is only defined up to an exact d-form, so when we write \(*_{\!g}F^{(1)}\) we apply \(*_{\!g}\) simultaneously to all representatives of \(F^{(1)}[\varphi ]\) for each \(\varphi \in {\mathscr {U}}\). In other words, we are dealing with all d-forms representing \(F^{(1)}[\varphi ]\) simultaneously. We shall be more precise with this from the proof of Proposition 2.2 onwards.

  6. 6.

    This property is assumed a priori in Definition 6.1 of [15].

  7. 7.

    Here “continuous” means sequentially continuous with respect to the (weak) Hörmander topology on the extended domain (see Sect. 4.1 below), as shown e.g. by Theorem 8.2.13, pp. 268–269 of [50] and, more precisely, by Theorems 8.2.9. (iii) and 8.2.10, pp. 515–520 of [23]. One can see indirectly from the arguments in [13] that one cannot hope to upgrade this result to full continuity, unless one uses instead the strong Hörmander topology (see also Remark 4.3 below).

  8. 8.

    Recall that a completely regular topological space X is said to be compactly generated or a k-space if the topology of X coincides with the final topology induced by the inclusions of compact subsets of X. This is equivalent to the space of continuous real-valued functions on X being complete with respect to the topology of uniform convergence on compact subsets of X (see e.g. Theorem 3.6.4, pp. 70 of [54]).

  9. 9.

    Nonetheless, in this case the \(c^\infty \)-topology coincides with the so-called Kelleyfication of \({\mathscr {F}}\), which is the final topology induced by all compact subsets of \({\mathscr {F}}\) through their respective inclusions (see e.g. Theorem 4.11 (3), pp. 39–40 of [63]). It is clear that the Kelleyfication of \({\mathscr {F}}\) coinciding with the original topology of \({\mathscr {F}}\) amounts to \({\mathscr {F}}\) being compactly generated (see footnote 8 above). This happens if e.g. \({\mathscr {F}}\) is metrizable.

  10. 10.

    MB differentiability and MB smoothness are respectively listed in Keller’s treatise [55] as “\({\mathscr {C}}^k_c\)- and \({\mathscr {C}}^\infty _c\)-differentiability”. Here we avoid his nomenclature, for it clashes with the usual notation for differentiable and smooth functions with compact support.

  11. 11.

    However, as argued e.g. in Proposition 2.7, pp. 17 of [63], if \(\gamma \) is Lipschitz (i.e. the subset \(\{(t-s)^{-1}(\gamma (t)-\gamma (s))\ |\ t\ne s,\,a\le t,s\le b\}\) is bounded) then it suffices to assume that \({\mathscr {F}}\) is convenient to get the Riemann integral of \(\gamma \) along [ab] with all the properties discussed in this Appendix.

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Acknowledgements

We would like to thank Prof. Frank Michael Forger for a critical reading of an early version of the Introduction, as well as for invaluable advice on the mathematical literature on classical field theory and general presentation details. We are specially grateful to him for discussions on physically relevant functionals, which led to most of the examples presented in Sect. 2.3. We would also like to thank Prof. Stefan Waldmann for pointing out a mistake in the proof of Corollary 4.1 in a previous version of the present paper, and also for his clarifying comments. Finally, we are much grateful to Prof. Christian Brouder for his several inquires about our work, particularly for pointing out a substantial gap in the previous proofs of of Proposition 3.4 and Corollary 3.3, and for numerous discussions, as well as to Prof. Peter Michor for the enlightening observations on MathOverflow, which led us to the crucial Lemma 2.6, and the anonymous referees for the valuable comments. The junior author (P.L.R.) would like to thank the hospitality of the II. Institut für theoretische Physik, Universität Hamburg, the Dipartimento di Matematica, Facoltà di Scienze della Università di Trento, and the Instituto de Matemática e Estatística, University of São Paulo, where most of the work presented in this paper was developed, and also the support from the Research Training Group 1670—“Mathematics Inspired by String Theory and Quantum Field Theory”, Universität Hamburg as well as from the Centro Italiano di Ricerca Matematica (CIRM) and the Bruno Kessler Foundation in the final stages of the writing.

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Communicated by Y. Kawahigashi

A Short Review of Differential Calculus on Locally Convex Topological Vector Spaces

A Short Review of Differential Calculus on Locally Convex Topological Vector Spaces

In this Appendix we list the basic definitions and results of differential calculus we need. Our basic references are [45] and [63], to whom we refer for more details and proofs. The first reference works only with Fréchet spaces, but the proofs of the results quoted below work in the general case with little or no change.

The notion of differentiability of curves in locally convex topological vector spaces is straightforward.

Definition A.1

Let \(\gamma :(a,b)\rightarrow {\mathscr {F}}\), \(a<b\in {\mathbb {R}}\cup \{\pm \infty \}\) be a continuous curve into a locally convex topological vector space \({\mathscr {F}}\). We say that \(\gamma \) is a \({\mathscr {C}}^1\)curve if for all \(t\in (a,b)\) the limit

$$\begin{aligned} \gamma '(t)\doteq \lim _{s\rightarrow 0}\frac{1}{s}(\gamma (t+s)-\gamma (t)) \end{aligned}$$

exists and defines a continuous curve \(\gamma ':(a,b)\rightarrow {\mathscr {F}}\) (continuity of \(\gamma \) actually follows from these conditions alone, hence it does not hurt to assume it from the start). We also say that \(\gamma \) is a \({\mathscr {C}}^m\) curve, \(m\ge 1\), if \(\gamma ^{(k)}\doteq (\gamma ^{(k-1)})'\) exists and is continuous for all \(1\le k\le m\), where \(\gamma ^{(0)}\doteq \gamma \). If \(\gamma \) is a \({\mathscr {C}}^m\) curve for all m, we say that \(\gamma \) is a smooth curve.

We stress that there would be no loss of generality if we required the domain of smooth curves to be the whole real line: by the chain rule (A.3), \(\gamma :(a,b)\rightarrow {\mathscr {F}}\) is smooth if and only if \(\gamma \circ f:{\mathbb {R}}\rightarrow {\mathscr {F}}\) is smooth for any diffeomorphism \(f:{\mathbb {R}}\rightarrow (a,b)\) (e.g. \(f(\lambda )=\frac{b+a}{2} +\frac{b-a}{2}\tanh (\lambda )\)). Once this is said, let us see how Definition A.1 is realized in the concrete cases that interest us.

  • \({\mathscr {F}}={\mathscr {C}}^\infty ({\mathscr {M}})\) (endowed with the compact-open topology): \(\gamma :{\mathbb {R}}\rightarrow {\mathscr {F}}\) is smooth if and only if \(\gamma (\lambda )(p)={\varPhi }(\lambda ,p)\) for all \((\lambda ,p)\in {\mathbb {R}}\times {\mathscr {M}}\), where \({\varPhi }\in {\mathscr {C}}^\infty ({\mathbb {R}}\times {\mathscr {M}})\);

  • \({\mathscr {F}}={\mathscr {C}}^\infty _c({\mathscr {M}})\) (endowed with the usual inductive limit topology): \(\gamma :{\mathbb {R}}\rightarrow {\mathscr {F}}\) is smooth if and only if \(\gamma (\lambda )(p)={\varPhi }(\lambda ,p)\) for all \((\lambda ,p)\in {\mathbb {R}}\times {\mathscr {M}}\), where \({\varPhi }\in {\mathscr {C}}^\infty ({\mathbb {R}}\times {\mathscr {M}})\) is such that for any \(a<b\in {\mathbb {R}}\) there is a compact subset \(K\subset {\mathscr {M}}\) such that \({\varPhi }(\lambda ,p)={\varPhi }(a,p)\) for all \(p\not \in K\), \(\lambda \in [a,b]\).

The notion of smooth curves allows one to introduce another topology on \({\mathscr {F}}\), given by the final topology induced by \({\mathbb {R}}\) through all smooth curves \(\gamma : {\mathbb {R}}\rightarrow {\mathscr {F}}\). We call this topology the \(c^\infty \)-topology on \({\mathscr {F}}\). This topology is necessarily finer than the original one, but it is not in general a vector space topology—the finest locally convex vector space topology on \({\mathscr {F}}\) that is coarser then the \(c^\infty \)-topology is the bornologification of \({\mathscr {F}}\)’s original topology. The \(c^\infty \)- and the original locally convex vector space topologies coincide if \({\mathscr {F}}\) is e.g. metrizable (such as \({\mathscr {C}}^\infty ({\mathscr {M}})\)), but are distinct for \({\mathscr {F}}={\mathscr {C}}^\infty _c({\mathscr {M}})\) if \({\mathscr {M}}\) is non-compact since then the \(c^\infty \)-topology is not a vector space topology (see e.g. Proposition 4.26 (ii), pp. 45 of [63]).Footnote 9

Given two locally convex vector spaces \({\mathscr {F}}_1\), \({\mathscr {F}}_2\), \({\mathscr {U}}\subset {\mathscr {F}}_1\)\(c^\infty \)-open, we say that a map \({\varPhi }:{\mathscr {U}}\rightarrow {\mathscr {F}}_2\) is conveniently smooth if \({\varPhi }\circ \gamma \) is a smooth curve on \({\mathscr {F}}_2\) for every smooth curve \(\gamma : {\mathbb {R}}\rightarrow {\mathscr {U}}\). We stress that conveniently smooth maps need not even be continuous (see [41] for a counterexample). A simple non-trivial example of a conveniently smooth map \({\varPhi }:{\mathscr {F}}\rightarrow {\mathscr {F}}\) is, of course, the translation \(\varphi \mapsto {\varPhi }(\varphi )=\varphi +\varphi _0\) by a fixed element \(\varphi _0\in {\mathscr {F}}\). In particular, the coordinate change maps \(\kappa _{\varphi _2}\circ \kappa _{\varphi _1}^{-1}:{\mathscr {C}}^\infty _c({\mathscr {M}}) \rightarrow {\mathscr {C}}^\infty _c({\mathscr {M}})\) in the affine flat manifold \({\mathscr {C}}^\infty ({\mathscr {M}})\) (endowed with the Whitney topology) are conveniently smooth for all \(\varphi _1,\varphi _2\in {\mathscr {C}}^\infty ({\mathscr {M}})\) such that \(\varphi _1-\varphi _2\in {\mathscr {C}}^\infty _c({\mathscr {M}})\). This shows that the atlas \({\mathfrak {U}}\) defined in (2.7) induces a smooth structure on \({\mathscr {C}}^\infty ({\mathscr {M}})\); the corresponding smooth manifold topology is, of course, the manifold topology generated by the \(c^\infty \)-open subsets of the modelling vector space \({\mathscr {C}}^\infty _c({\mathscr {M}})\), which is even finer than the Whitney topology. The connected components of this topology are, however, also of the form \({\mathscr {C}}^\infty _c({\mathscr {M}})+\varphi _0\), \(\varphi _0\in {\mathscr {C}}^\infty ({\mathscr {M}})\); therefore, the smooth curves in \({\mathscr {C}}^\infty ({\mathscr {M}})\) with respect to the smooth structure induced by the atlas \({\mathfrak {U}}\) must be of the form \({\mathbb {R}}\ni \lambda \mapsto \gamma (\lambda )=\varphi _0+\gamma _0(\lambda )\), where \(\gamma _0:{\mathbb {R}}\rightarrow {\mathscr {C}}^\infty _c({\mathscr {M}})\) is smooth. Hence, it is just fair to say that such \(\gamma \) is a smooth curve with respect to the Whitney topology, and the smooth structure induced by the atlas \({\mathfrak {U}}\), the smooth structure on\({\mathscr {C}}^\infty ({\mathscr {M}})\)induced by the Whitney topology.

Remark A.1

It can be shown [63] that, for \({\mathscr {C}}^\infty ({\mathscr {M}})\) endowed with the smooth structure induced by the Whitney topology, the bundles

$$\begin{aligned} T^{r,s}{\mathscr {C}}^\infty ({\mathscr {M}})=\left( \otimes ^sT^*{\mathscr {C}}^\infty ({\mathscr {M}})\right) \otimes \left( \otimes ^rT{\mathscr {C}}^\infty ({\mathscr {M}})\right) \end{aligned}$$

of tensors of contravariant rank r and covariant rank s are given at each \(\varphi \in {\mathscr {C}}^\infty ({\mathscr {M}})\) by the space of bounded linear mappings from \(\otimes ^s_\beta {\mathscr {C}}^\infty _c({\mathscr {M}})\) to \(\otimes ^r_\beta {\mathscr {C}}^\infty _c({\mathscr {M}})\). Here \(\otimes _\beta \) denotes the bornological tensor product, whose topology is the finest locally convex topology on the algebraic tensor product such that the canonical quotient map is bounded; this topology is finer than the projective tensor product topology. Nonetheless, \(T{\mathscr {C}}^\infty ({\mathscr {M}})\) and \(T^*{\mathscr {C}}^\infty ({\mathscr {M}})\) do assume the form given in Sect. 2.2 (see the proof of Theorem 42.17, pp. 447–448 of [63]). It also turns out that the particular structure of \({\mathscr {C}}^\infty _c({\mathscr {M}})\), together with Theorems 6.14, pp. 72–73 and 28.7, pp. 280–281 of [63], imply that every kinematical tangent vector on \({\mathscr {C}}^\infty ({\mathscr {M}})\) is also an operational one, i.e. it defines a point derivation on (conveniently) smooth maps \(F:{\mathscr {C}}^\infty ({\mathscr {M}})\rightarrow {\mathbb {R}}\).

In principle, we could develop essentially all tools of differential calculus by using convenient smoothness. However, for the purposes of this paper, it is often preferrable to use a stronger concept of smoothness. Such a notion is provided, for instance, by Michal [70] and Bastiani [5]. This is also the notion employed in the accounts of infinite dimensional differential calculus done by Milnor [71] and Hamilton [45], and all the basic results of Calculus we present in the remainder of this Appendix are formulated in this context (see, however, Remark A.2 below). The basic definition is as follows (See also Definition 2.3 for the special case of real-valued maps):

Definition A.2

Let \({\mathscr {F}}_1,{\mathscr {F}}_2\) be locally convex topological vector spaces, \({\mathscr {U}}\subset {\mathscr {F}}_1\) open, and \(F:{\mathscr {U}}\rightarrow {\mathscr {F}}_2\) a continuous map. We say that F is (MB-)differentiable of order\(m(>0)\) (“MB” stands for the names of Michal and Bastiani) if for all \(k=1, \ldots ,m\) the k-th order directional (Gâteaux) derivatives

$$\begin{aligned} F^{(k)}[\varphi ](\mathbf {\varphi }_1,\ldots ,\mathbf {\varphi }_k)\doteq \frac{\partial ^k}{\partial \lambda _1 \cdots \partial \lambda _k}\left. \phantom {\frac{}{}}\!\right| _{\lambda _1=\cdots =\lambda _k=0}F \left( \varphi +\sum ^k_{j=1}\lambda _j \mathbf {\varphi }_j\right) \end{aligned}$$
(A.1)

exist as jointly continuous maps from \({\mathscr {U}}\times {\mathscr {F}}^k_1\ni (\varphi ,\mathbf {\varphi }_1,\ldots , \mathbf {\varphi }_k)\) to \({\mathscr {F}}_2\). If F is differentiable of order m for all \(m\in {\mathbb {N}}\), we say that F is (MB-)smooth.Footnote 10

The right-hand side of formula (A.1) should be understood as the differentiation of a k-parameter curve taking values in \({\mathscr {F}}_2\), for fixed \(\varphi ,\mathbf {\varphi }_1,\ldots , \mathbf {\varphi }_k\). The argument of F inside the limit is guaranteed to lie inside \({\mathscr {U}}\) for sufficiently small \(\lambda _1,\ldots ,\lambda _k\).

It follows from Definition A.2 that if \(F:{\mathscr {U}}\subset {\mathscr {F}}_1\rightarrow {\mathscr {F}}_2\) is MB-differentiable of order \(m>0\) then the maps \({\mathscr {U}}\ni \varphi \mapsto F^{(k)}[\varphi ]\in {\mathscr {L}}^k({\mathscr {F}}_1,{\mathscr {F}}_2)\) are continuous for all \(1\le k\le m\), where \({\mathscr {L}}^k({\mathscr {F}}_1,{\mathscr {F}}_2)\) is the locally convex topological vector space of all k-linear maps from \({\mathscr {F}}_1^k\) to \({\mathscr {F}}_2\) endowed with the compact-open topology. If \({\mathscr {F}}_1\) is semi-Montel (i.e. closed and bounded subsets of \({\mathscr {F}}_1\) are compact), such topology amounts to uniform convergence in bounded subsets of \({\mathscr {F}}_1^k\). If \({\mathscr {F}}_1^k\) is compactly generated (e.g. when \({\mathscr {F}}_1\) is metrizable, see e.g. Proposition 3.3.20, pp. 152 of [36] and footnote 9 above) and \({\mathscr {F}}_2\) is complete, then by Proposition 16.6.2, pp. 361 of [54] \({\mathscr {L}}^k({\mathscr {F}}_1,{\mathscr {F}}_2)\) is also complete.

Given \({\mathscr {U}}\) an arbitrary (i.e. not necessarily open) subset of \({\mathscr {F}}_1\), we say that a continuous map \(F:{\mathscr {U}}\rightarrow {\mathscr {F}}_2\) is differentiable of order m (resp. smooth) if there is \({\mathscr {V}}\supset {\mathscr {U}}\) open in the compact-open topology and a functional \({\tilde{F}}:{\mathscr {V}}\rightarrow {\mathscr {F}}_2\) extending F (i.e. \({\tilde{F}}\vert _{{\mathscr {U}}}=F\)) such that \({\tilde{F}}\) is differentiable of order m (resp. smooth). For completely arbitrary \({\mathscr {U}}\), the derivatives of F on \({\mathscr {U}}\) depend on the choice of extension \({\tilde{F}}\) (take for instance \({\mathscr {U}}=\{\varphi \}\) for some \(\varphi \in {\mathscr {F}}_1\)). However, if \({\mathscr {U}}\) happens to have a nonvoid interior, then it is easily shown that the derivatives of F on \({\mathscr {U}}\) do not depend on the choice of extension. Under certain conditions on F, one can weaken this condition (see, for instance, Remark 2.4).

Remark A.2

For Mackey-complete locally convex topological vector spaces (also called \(c^\infty \)-complete or convenient topological vector spaces), convenient smoothness enjoys essentially all the rules of Calculus presented in the remainder of this Appendix assuming MB differentiability (see e.g. footnote 11 below). Moreover, for Fréchet spaces (which are convenient and whose topology coincides with the corresponding \(c^\infty \)-topology) convenient and MB smoothness coincide (see e.g. Theorem 1, pp. 77 of [39] together with Theorem 2.14, pp. 20–21 of [63]).

Let \(\gamma :[a,b]\rightarrow {\mathscr {F}}\), \(a<b\in {\mathbb {R}}\), be a continuous curve segment in the complete locally convex topological vector space \({\mathscr {F}}\). We can define the (Riemann) integral of \(\gamma \) along [ab]

$$\begin{aligned} \int ^b_{a}\gamma (\lambda )\mathrm {d}\lambda \in {\mathscr {F}}\end{aligned}$$

as the unique linear map from the space \({\mathscr {C}}([a,b],{\mathscr {F}})\) of continuous curves from [ab] to \({\mathscr {F}}\) into the space \({\mathscr {F}}\) such that:Footnote 11

  1. 1.

    For any continuous linear functional \(u:{\mathscr {F}}\rightarrow {\mathbb {R}}\), we have that \(u\left( \int ^b_{a} \gamma (\lambda )\mathrm {d}\lambda \right) =\int ^b_{a}u(\gamma (\lambda ))\mathrm {d}\lambda \);

  2. 2.

    For any continuous seminorm \(\Vert \cdot \Vert \) on \({\mathscr {F}}\), we have that \(\left\| \int ^b_{a} \gamma (\lambda )\mathrm {d}\lambda \right\| \le \int ^b_{a}\Vert \gamma (\lambda )\Vert \mathrm {d}\lambda \);

  3. 3.

    If \(a<c<b\in {\mathbb {R}}\), then \(\int ^b_{a}\gamma (\lambda )\mathrm {d}\lambda =\int ^c_{a}\gamma (\lambda ) \mathrm {d}\lambda +\int ^b_{c}\gamma (\lambda )\mathrm {d}\lambda \).

The Fundamental Theorem of Calculus holds for the Riemann integral of curves taking values in \({\mathscr {F}}\):

Theorem A.1

([45], Theorems 2.2.3 and 2.2.2). Let \(\gamma _0:[a,b]\rightarrow {\mathscr {F}}\) be a continuous curve, \(a\le t\le b\), and define \(\gamma _1(t)\doteq \int ^t_{a}\gamma _0(\lambda )\mathrm {d}\lambda \). Then \(\gamma _1:[a,b]\rightarrow {\mathscr {F}}\) is a \({\mathscr {C}}^1\) curve, and \(\gamma '_1(t)=\gamma _0(t)\). Conversely, if \(\gamma _1:[a,b]\rightarrow {\mathscr {F}}\) is a \({\mathscr {C}}^1\) curve, then \(\gamma _1(b)-\gamma _1(a)=\int ^b_{a}\gamma '_1(\lambda )\mathrm {d}\lambda \)\(\quad \square \)

Corollary A.1

([45], Theorem 3.2.2). Let \(F:{\mathscr {U}}\subset {\mathscr {F}}_1\rightarrow {\mathscr {F}}_2\) be a continuous map with \({\mathscr {F}}_2\) complete, \(\varphi _0 \in {\mathscr {U}}\), and \(\mathbf {\varphi }\in {\mathscr {U}}-\varphi _0\doteq \{\varphi -\varphi _0\in {\mathscr {F}}_1\ |\ \varphi \in {\mathscr {U}}\}\). Assume that \({\mathscr {U}}\) is convex for simplicity. If F is differentiable of order one in the sense of Definition A.2, then

$$\begin{aligned} F(\varphi _0+\mathbf {\varphi })-F(\varphi _0)=\int ^1_0F^{(1)} [\varphi _0+\lambda \mathbf {\varphi }] (\mathbf {\varphi })\mathrm {d}\lambda . \end{aligned}$$
(A.2)

With the aid of the fundamental theorem of Calculus A.1, the following key results can be proven. First, the usual linearity property for first-order derivatives holds:

Lemma A.1

([45], Lemma 3.2.3 and Theorem 3.2.5). Let \(F:{\mathscr {U}}\subset {\mathscr {F}}_1\rightarrow {\mathscr {F}}_2\) be a continuous map with \({\mathscr {F}}_2\) complete, \(\varphi \in {\mathscr {U}}\). If F is differentiable of order one in the sense of Definition A.2, then for all scalars \(\lambda ,\mu \) and all \(\mathbf {\varphi },\mathbf {\varphi }'\in {\mathscr {F}}_1\) we have that

$$\begin{aligned} F^{(1)}[\varphi ](\lambda \mathbf {\varphi }+\mu \mathbf {\varphi }')=\lambda F^{(1)}[\varphi ](\mathbf {\varphi }) +\mu F^{(1)}[\varphi ](\mathbf {\varphi }'). \end{aligned}$$

\(\square \)

Next, the chain rule holds:

Theorem A.2

([45], Theorem 3.3.4). Let \(F:{\mathscr {U}}\subset {\mathscr {F}}_1\rightarrow {\mathscr {F}}_2\), \(G:{\mathscr {V}}\subset {\mathscr {F}}_2\rightarrow {\mathscr {F}}_3\) be respectively continuous maps from open subsets \({\mathscr {U}},{\mathscr {V}}\) of locally convex topological vector spaces \({\mathscr {F}}_1,{\mathscr {F}}_2\) into \({\mathscr {F}}_2\) and the locally convex topological vector space \({\mathscr {F}}_3\), such that \(F({\mathscr {U}})\subset {\mathscr {V}}\). Suppose that \({\mathscr {F}}_2\) and \({\mathscr {F}}_3\) are complete. If F (resp. G) is once differentiable on \({\mathscr {U}}\) (resp. \({\mathscr {V}}\)) in the sense of Definition A.2, then for all \(\varphi \in {\mathscr {U}}\), \(\mathbf {\varphi }\in {\mathscr {F}}_1\) we have that

$$\begin{aligned} (G\circ F)^{(1)}(\varphi )(\mathbf {\varphi })= G^{(1)}[F(\varphi )](F^{(1)}[\varphi ](\mathbf {\varphi })). \end{aligned}$$
(A.3)

\(\square \)

The chain rule (A.3) yields, after taking direct sums, the Leibniz’s rule for derivatives of composition of n-tuples of maps \(F_1,\ldots ,F_n\) with a continuous n-linear map \(\psi \)

$$\begin{aligned} (\psi (F_1,\ldots ,F_n))^{(1)}[\varphi ](\mathbf {\varphi })= \sum ^n_{j=1}\psi (F_1[\varphi ], \ldots ,F^{(1)}_j[\varphi ](\mathbf {\varphi }),\ldots ,F_n[\varphi ]). \end{aligned}$$
(A.4)

This, together with the fundamental theorem of Calculus (A.2), yields the integration by parts formula and, even more importantly, Taylor’s formula with (integral) remainder

$$\begin{aligned} F(\varphi _0+\mathbf {\varphi })= & {} \sum ^k_{j=0}\frac{1}{j!}F^{(j)}[\varphi _0] (\mathbf {\varphi },\ldots ,\mathbf {\varphi }) \nonumber \\&+\int ^1_0\frac{(1-\lambda )^k}{k!} F^{(k+1)} [\varphi _0+\lambda \mathbf {\varphi }](\mathbf {\varphi },\ldots ,\mathbf {\varphi }) \mathrm {d}\lambda . \end{aligned}$$
(A.5)

To see this, note that Leibniz’s rule implies the following key formula:

$$\begin{aligned} \begin{aligned}&\frac{(1-\lambda )^{k-1}}{(k-1)!}F^{(k)}[\varphi _0+\lambda \mathbf {\varphi }](\mathbf {\varphi }, \ldots ,\mathbf {\varphi }) \\&\quad =\frac{(1-\lambda )^k}{k!}F^{(k+1)}[\varphi _0+\lambda \mathbf {\varphi }] (\mathbf {\varphi },\ldots ,\mathbf {\varphi })\\&\qquad -\frac{\mathrm {d}}{\mathrm {d}\lambda } \left[ \frac{(1-\lambda )^k}{k!}F^{(k)} [\varphi _0+\lambda \mathbf {\varphi }](\mathbf {\varphi },\ldots , \mathbf {\varphi })\right] . \end{aligned} \end{aligned}$$
(A.6)

Integrating both sides of formula (A.6) from \(\lambda =0\) to \(\lambda =1\) by means of the fundamental theorem of Calculus (A.2) yields the fundamental induction step from \(k-1\) to k. Since the case \(k=0\) of (A.5) is settled by the fundamental theorem of Calculus itself, we are done.

For the convenience of the reader, we prove the generalization of the chain rule (A.3) for higher derivatives, since this proof is not easy to find in the literature at the present level of generality. We follow the argument employed in [56].

Corollary A.2

(Faà di Bruno’s formula). Let \(F:{\mathscr {U}}\subset {\mathscr {F}}_1\rightarrow {\mathscr {F}}_2\), \(G:{\mathscr {V}}\subset {\mathscr {F}}_2\rightarrow {\mathscr {F}}_3\) satisfy the hypotheses of Theorem A.2. If F (resp. G) is m-times differentiable on \({\mathscr {U}}\) (resp. \({\mathscr {V}}\)), then \(G\circ F\) is also m-times differentiable on \({\mathscr {U}}\), and for all \(1\le k\le m\),

$$\begin{aligned} (G\circ F)^{(k)}[\varphi ](\mathbf {\varphi }_1,\ldots , \mathbf {\varphi }_k)=\sum _{\pi \in P_k} G^{(|\pi |)}[F(\varphi )]\left( \bigotimes _{I\in \pi } F^{(|I|)}[\varphi ](\otimes _{j\in I} \mathbf {\varphi }_j)\right) ,\qquad \end{aligned}$$
(A.7)

where \(P_k\) is the set of all partitions \(\pi =\{I_1,\ldots ,I_l\}\) of \(\{1,\ldots ,k\}\), that is, \(I_j\ne \varnothing \), \(I_j\cap I_{j'}=\varnothing \) for \(j\ne j'\) and \(\cup ^l_{j=1}I_j=\{1,\ldots ,k\}\).

Proof

We proceed by induction on k. The case \(k=1\) is just the usual chain rule (A.3). Assume that the formula is valid up to order \(k-1\) along \(\mathbf {\varphi }_1,\ldots , \mathbf {\varphi }_{k-1}\). Then for each partition \(\pi \) of \(\{1,\ldots ,k-1\}\) in the above sum we have, by Leibniz’s rule (A.4),

$$\begin{aligned} \begin{aligned}&\Bigg [G^{(|\pi |)}\left. \circ \,F\left( \bigotimes _{I\in \pi } F^{(|I|)}(\otimes _{j\in I}\mathbf {\varphi }_j) \right) \right] ^{(1)}[\varphi ](\mathbf {\varphi }_k) \\&\quad =G^{(|\pi |+1)}[F(\varphi )]\left( F^{(1)}[\varphi ] (\mathbf {\varphi }_k)\otimes \bigotimes _{I\in \pi } F^{(|I|)}[\varphi ](\otimes _{j\in I}\mathbf {\varphi }_j)\right) \\&\qquad +\sum _{I'\in \pi }G^{(|\pi |)}[F(\varphi )]\Bigg (F^{(|I'|+1)} [\varphi ] \bigg (\mathbf {\varphi }_k\otimes \bigotimes _{j\in I'}\mathbf {\varphi }_j\bigg )\\&\qquad \otimes \bigotimes _{I\in \pi {\smallsetminus }\{I'\}} F^{(|I|)}[\varphi ](\otimes _{l\in I}\mathbf {\varphi }_l)\Bigg ). \end{aligned} \end{aligned}$$

However, any partition \(\pi '\) of \(\{1,\ldots ,k\}\) is either of the form \(\pi '=\{\{k\}\}\cup \pi \) or \(\pi '=(\pi {\smallsetminus }\{I'\})\cup \{I'\cup \{k\}\}\) for some \(I'\in \pi \), \(\pi \in P_{k-1}\). Hence, summing the above identities over all such \(\pi \) gives the desired result. \(\quad \square \)

A consequence of Faà di Bruno’s formula (A.7) is the generalization of Leibniz’s rule (A.4) for higher order derivatives of composition of l-tuples of maps \(F_1,\ldots ,F_l\) with a continuous l-linear map \(\psi \)

$$\begin{aligned} \begin{aligned}&(\psi (F_1,\ldots ,F_l))^{(k)}[\varphi ]\left( \mathbf {\varphi }_1, \ldots ,\mathbf {\varphi }_k\right) \\&\quad =\sum _{\{I_1,\ldots ,I_l\}\in {\tilde{P}}_{k,l}}\psi \left( F_1^{(|I_1|)}[\varphi ](\otimes _{j\in I_1} \mathbf {\varphi }_j),\ldots ,F_l^{(|I_l|)}[\varphi ] (\otimes _{j\in I_l}\mathbf {\varphi }_j)\right) , \end{aligned} \end{aligned}$$
(A.8)

where \({\tilde{P}}_{k,l}\) is the set of all partitions \(\pi =\{I_1,\ldots ,I_l\}\) of \(\{1,\ldots ,k\}\) in lpossibly (but not all) empty subsets, i.e. \(I_j\cap I_{j'}=\varnothing \) for \(j\ne j'\) and \(\cup ^l_{j=1}I_j=\{1,\ldots ,k\}\). As another application, we obtain the so-called k-th order resolvent formula (A.12) below which shall often be useful. Consider two MB-differentiable maps \(F:{\mathscr {U}}\times {\mathscr {F}}_1\rightarrow {\mathscr {F}}_2\), \(G:{\mathscr {U}}\times {\mathscr {F}}_2\rightarrow {\mathscr {F}}_1\) of order one, where \({\mathscr {F}}_1,{\mathscr {F}}_2\) are locally convex topological vector spaces and \({\mathscr {U}}\subset {\mathscr {F}}\) is a nonvoid open subset of the locally convex topological vector space \({\mathscr {F}}\). For notational convenience, we also occasionally write \(F(\varphi ,\mathbf {\varphi })\doteq F[\varphi ] \mathbf {\varphi }\), \(G(\varphi ,\mathbf {\psi })\doteq G[\varphi ]\mathbf {\psi }\). Suppose that both F and G are linear in their second arguments and satisfy

$$\begin{aligned} \begin{aligned} F[\varphi ]G[\varphi ]\mathbf {\psi }&=\mathbf {\psi }\ ,\quad \forall \varphi \in {\mathscr {U}},\, \mathbf {\psi }\in {\mathscr {F}}_2,\\ G[\varphi ]F[\varphi ]\mathbf {\varphi }&=\mathbf {\varphi }\ ,\quad \forall \varphi \in {\mathscr {U}},\, \mathbf {\varphi }\in {\mathscr {F}}_1. \end{aligned} \end{aligned}$$
(A.9)

If we define

$$\begin{aligned}&D^k_1F[\varphi ](\mathbf {\varphi }_1,\ldots ,\mathbf {\varphi }_k) \mathbf {\varphi }=F^{(k)}[\varphi , \mathbf {\varphi }] ((\mathbf {\varphi }_1,0),\ldots ,(\mathbf {\varphi }_k,0))\ , \nonumber \\&D^1_1\doteq D_1,\, D^0_1=\mathbb {1}, \end{aligned}$$
(A.10)

then by the chain rule (A.3) applied to the pair of maps \(F,(\mathbb {1},G)\) and (A.9) we have the (first-order) resolvent formula

$$\begin{aligned} D_1G[\varphi ](\mathbf {\varphi }_1)\mathbf {\psi }= -G[\varphi ]D_1F[\varphi ](\mathbf {\varphi }_1)G[\varphi ] \mathbf {\psi }. \end{aligned}$$
(A.11)

It follows from the above formula that if in addition F is MB-smooth, then so is G. More precisely, in this case we obtain the following (not so pleasant) higher-order generalization of (A.11), obtained by induction on \(k\ge 1\) from (A.11) and an argument analogous to the one used in the proof of Corollary A.2:

$$\begin{aligned} \begin{aligned}&D^k_1G[\varphi ]\left( \mathbf {\varphi }_1,\ldots , \mathbf {\varphi }_k\right) \mathbf {\psi }\\&\quad =\sum ^k_{l=1}(-1)^l\sum _{\{I_1,\ldots ,I_l\} \in P_k}\sum _{\sigma \in S_l}\left( \prod ^l_{j=1}G[\varphi ] D^{|I_{\sigma (j)}|}F[\varphi ](\otimes _{i\in I_{\sigma (j)}}\mathbf {\varphi }_i)\right) G[\varphi ] \mathbf {\psi }.\qquad \end{aligned} \end{aligned}$$
(A.12)

Here, \(P_k\) is again the set of all partitions of \(\{1,\ldots ,k\}\) as in the statement of Corollary A.2, whereas \(S_l\) is the set of all permutations of \(\{1,\ldots ,l\}\).

Finally, one can show that the order of differentiation for higher order derivatives is irrelevant:

Theorem A.3

([45], Theorem 3.6.2). Let \(F:{\mathscr {U}}\subset {\mathscr {F}}_1\rightarrow {\mathscr {F}}_2\) be a continuous map with \({\mathscr {F}}_2\) complete. If F is differentiable of order \(m>1\) in the sense of Definition A.2, then \(F^{(k)} [\varphi ]:{\mathscr {F}}^k_1\ni (\mathbf {\varphi }_1,\ldots ,\mathbf {\varphi }_k)\mapsto F^{(k)}[\varphi ] (\mathbf {\varphi }_1,\ldots ,\mathbf {\varphi }_k)\in {\mathscr {F}}_2\) is a symmetric, k-linear map for all fixed \(\varphi \in {\mathscr {U}}\), \(2\le k\le m\)\(\quad \square \)

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Brunetti, R., Fredenhagen, K. & Ribeiro, P.L. Algebraic Structure of Classical Field Theory: Kinematics and Linearized Dynamics for Real Scalar Fields. Commun. Math. Phys. 368, 519–584 (2019). https://doi.org/10.1007/s00220-019-03454-z

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