Smooth 1-dimensional algebraic quantum field theories

This paper proposes a refinement of the usual concept of algebraic quantum field theories (AQFTs) to theories that are smooth in the sense that they assign to every smooth family of spacetimes a smooth family of observable algebras. Using stacks of categories, this proposal is realized concretely for the simplest case of 1-dimensional spacetimes, leading to a stack of smooth 1-dimensional AQFTs. Concrete examples of smooth AQFTs, of smooth families of smooth AQFTs and of equivariant smooth AQFTs are constructed. The main open problems that arise in upgrading this approach to higher dimensions and gauge theories are identified and discussed.

An m-dimensional algebraic quantum field theory (AQFT) is a functor A : Loc m → * Alg C from a suitable category of m-dimensional Lorentzian spacetimes to the category of associative and unital * -algebras over C. The algebra A(M ) that is assigned by this functor to a spacetime M is interpreted as the algebra of quantum observables of the theory A that can be measured in M . Such functors are also required to satisfy a list of physically motivated axioms, see e.g. [BFV03,FV15], which includes most notably the Einstein causality axiom.
Even though this axiomatic definition of AQFTs is by now widely used in the relevant research community and has led to interesting model-independent results, we would like to point out the following issue that is usually not discussed: Suppose that we consider a family of spacetimes {M s ∈ Loc m } s∈R that depends smoothly (in some appropriate sense as explained in this paper) on a parameter s ∈ R. This s-dependence could, for example, be due to changing smoothly the coefficients of the metric tensor. Evaluating an AQFT A : Loc m → * Alg C on this smooth family results in a family of algebras {A(M s ) ∈ * Alg C } s∈R which, however, will in general not be smooth in any appropriate sense because smoothness is not covered by the usual AQFT axioms. Similarly, given a smooth family {f s : M s → N s } s∈R of spacetime morphisms, the associated family {A(f s ) : A(M s ) → A(N s )} s∈R of * Alg C -morphisms will in general not be smooth. In our opinion, encoding a suitable concept of smoothness for these families as part of the axioms of AQFT is desirable for several reasons: 1.) Physically speaking, smoothness of these families means that a small variation at the level spacetimes does not have a too drastic effect on the observable content of the theory, hence it excludes models with unpleasant discontinuous behavior. 2.) Certain standard constructions in AQFT, such as the computation of the stressenergy tensor as a derivative of the relative Cauchy evolution [BFV03], only exist for models that react sufficiently smoothly to metric perturbations. 3.) In the context of AQFT on spacetimes with background fields, see e.g. [Zah14], such a smooth dependence may be used to describe an adiabatic switching of the interaction with the background fields. Similarly, it enables us to introduce AQFTs that are smoothly equivariant with respect to an action of a Lie group.
The main goal of this paper is to make some first steps towards developing a refinement of the axiomatic foundations of AQFT that encodes the preservation of smooth families as part of its structure. The key idea behind our approach is to refine the ordinary categories Loc m and * Alg C that enter the definition of AQFTs to stacks of categories, which will encode precise concepts of smooth families of spacetimes and algebras, and to introduce a concept of smooth AQFTs in terms of morphisms between these stacks. A similar program of smooth refinements of QFTs has been developed successfully within other approaches, in particular for functorial QFTs in the sense of Atiyah and Segal [ST11,BEP15,BW21,LS21], however as of now this idea seems to be unexplored in the context of AQFT. In order to outline our proposal in the simplest possible setting and to circumvent in this first paper certain technical challenges (of both analytical and algebraic nature, see Section 6), we consider only the simplest case of dimension m = 1, which physically represents AQFTs on time intervals (i.e. quantum mechanics). We believe that, due to its simplicity, the case of 1-dimensional AQFTs is perfectly suited to explain the main ideas and features of our proposed framework for smooth AQFTs and to illustrate this formalism through the simplest possible examples.
Our framework for smooth AQFTs introduces naturally a second layer of smoothness. Because we realize smooth 1-dimensional AQFTs as the points of a stack AQFT ∞ 1 , we can also make precise sense of questions like what are "smooth families of smooth AQFTs" and in particular what are "smooth curves of smooth AQFTs". We shall illustrate through simple examples that smooth variations (e.g. an adiabatic switching) of the external parameters of a theory, such as the mass parameter, define such smooth families of smooth AQFTs. Furthermore, we show that each smooth AQFT has an associated smooth automorphism group, refining the discrete automorphism groups of ordinary AQFTs [Few13], and explain how these are related to smooth AQFTs that are equivariant with respect to a smooth action of a Lie group. We construct a concrete example of the latter that captures the global U (1)-symmetry of the 1-dimensional massless Dirac field.
The outline of the remainder of this paper is as follows: Section 2 contains a brief review of some relevant aspects of the theory of stacks of categories that we shall need in this work. In Section 3 we introduce the stacks of categories Loc ∞ 1 and * Alg ∞ C that provide smooth refinements of the category Loc 1 of 1-dimensional spacetimes and of the category * Alg C of associative and unital * -algebras. The stack of smooth 1-dimensional AQFTs is then defined as the mapping stack AQFT ∞ 1 := Map(Loc ∞ 1 , * Alg ∞ C ) and we shall explore some interesting consequences of this definition, including a natural notion of smooth automorphism group of a smooth AQFT and its relation to G-equivariant smooth AQFTs, for G a Lie group. Section 4 develops two stack morphisms CCR and CAR that are smooth refinements of the usual canonical (anti-)commutation relation quantization functors for Bosonic (resp., Fermionic) theories. These are later used for constructing explicit examples in Section 5, which illustrate our proposed approach to smooth AQFT. In Subsection 5.1, we introduce smooth refinements of retarded/advanced Green operators and prove their existence in simple cases through explicit formulas. We then construct in Subsection 5.2 a concrete example of a smooth family of smooth 1-dimensional AQFTs, which can be interpreted physically as (a smooth analog of) the 1-dimensional massive scalar field (quantum harmonic oscillator) in the presence of a smooth variation of the mass (frequency) parameter. In Subsection 5.3, we construct a concrete example of a U (1)-equivariant smooth 1-dimensional AQFT, which can be interpreted physically as (a smooth analog of) the 1-dimensional massless Dirac field, together with its global U (1)-symmetry. Section 6 provides a concise list of open problems that have to be solved to upgrade our approach to encompass higher dimensions m ≥ 2 and gauge theories. Most pressingly, Open Problem 6.1 poses the question of existence of smoothly parametrized retarded/advanced Green operators for vertical normally hyperbolic operators on smooth families of Lorentzian spacetimes, which goes beyond the standard results developed e.g. in [BGP07] and might be of interest to researchers in hyperbolic PDE theory.

Preliminaries on stacks of categories
We shall briefly review some basic concepts from the theory of stacks of categories that we need to describe a smooth refinement of algebraic quantum field theories (AQFTs). Our perspective on smoothness is through the functor of points approach, see e.g. [BS19, Section 3.2] and [BS17] for introductions in the context of AQFT and also [Sch13] for a more detailed overview. We also refer to [Lei04,Lac10] for the relevant 2-categorical background and to [Vis05] for a detailed introduction to the theory of stacks.
Let Man denote the category of (finite-dimensional) smooth manifolds and smooth maps. The usual open cover Grothendieck topology endows Man with the structure of a site. We choose the site Man because our aim is to formalize smooth families. The framework of stacks is however very flexible and can be adapted to model other types of families, such as continuous or algebraic, by an appropriate choice of site.
Definition 2.1. A prestack (of categories) is a pseudo-functor X : Man op → Cat to the 2category Cat of categories, functors and natural transformations. Explicitly, this consists of the following data: (1) For each object U ∈ Man, a category X(U ).
(3) For each pair of composable morphisms h : U → U ′ and h ′ : (4) For each object U ∈ Man, a natural isomorphism X U : id X(U ) ⇒ X(id U ) of functors from X(U ) to X(U ).
These data have to satisfy the following axioms: (i) For all triples of composable morphisms h : of natural transformations commute.
Remark 2.2. From now on we shall often follow the usual convention of suppressing the symbols X h ′ ,h and X U denoting the coherence isomorphisms of a prestack X. Definition 2.1 should help readers unfamiliar with this convention to extrapolate from the context which coherence isomorphism is relevant at any given point. △ Given any prestack X : Man op → Cat, one can define, for every manifold U ∈ Man and every open cover {U α ⊆ U }, the associated descent category X({U α ⊆ U }) ∈ Cat, see e.g. [Vis05,Definition 4.2]. An object in this category is a tuple of families of objects x α ∈ X(U α ) and isomorphisms ϕ αβ in X(U αβ ) satisfying for all α, β, γ. Here we denoted by U α 1 α 2 ···αn := U α 1 ∩ U α 2 ∩ · · · ∩ U αn the intersection of open subsets and by |Ũ := X(ι Ũ U ) : X(U ) → X(Ũ ) the functor associated with a subset inclusion morphism ι Ũ U :Ũ → U in Man. The unlabeled isomorphisms in (2.3b) are given by the coherence isomorphisms associated with the pseudo-functor X : for all α, β. There exists a canonical functor (1) For each U ∈ Man, a functor F U : X(U ) → Y (U ).
(2) For each morphism h : U → U ′ in Man, a natural isomorphism These data have to satisfy the following axioms: (i) For all pairs of composable morphisms h : U → U ′ and h ′ : of natural transformations commutes.
(ii) For all U ∈ Man, the diagram of natural transformations commutes.
Definition 2.6. A 2-morphism ζ : F ⇒ G between two stack morphisms F, G : X → Y is a modification between the underlying pseudo-natural transformations. Explicitly, this consists of the following data: (1) For each U ∈ Man, a natural transformation ζ U : These data have to satisfy the following axioms: of natural transformations commutes.
It is well-known that pseudo-functors, pseudo-natural transformations and modifications form a 2-category, see e.g. [SP09, Appendix A.1] and [Fio04,Chapter 3]. Selecting only those pseudofunctors that satisfy descent leads to the 2-category defined below.
Definition 2.7. We denote by St the 2-category of stacks (of categories). Its objects are stacks (see Definition 2.3), 1-morphisms are stack morphisms (see Definition 2.5) and 2-morphisms are given in Definition 2.6. We conclude this section by recalling briefly some important constructions involving stacks that will be needed in the bulk of our paper. morphisms. This 2-functor is fully faithful, i.e. manifolds can be equivalently regarded as stacks. Even more, for every U ∈ Man and X ∈ St, there exists a natural equivalence between the category of morphisms from U to X and the category obtained by evaluating X on U . As a consequence, the category X(U ) admits a useful interpretation as the category of "smooth maps" U → X from the manifold U to the stack X. In particular, for U = { * } the point, we can interpret X({ * }) as the category of "global points" { * } → X and similarly, for U = R the line, we can interpret X(R) as the category of "smooth curves" R → X in the stack X.
Products of stacks: Given any two stacks X, Y ∈ St, one defines the product stack X × Y ∈ St in terms of the pseudo-functor together with the obvious coherence isomorphisms induced from X and Y . For the particular case of two manifolds M, N ∈ Man, one finds that the product stack M × N ≃ M × N is equivalent to the stack associated with the product manifold.
Mapping stacks: Given any two stacks X, Y ∈ St, one defines the mapping stack Map(X, Y ) ∈ St in terms of the (strict) 2-functor

Smooth 1-dimensional AQFTs
An m-dimensional algebraic quantum field theory (AQFT) [BFV03,FV15] is a functor A : Loc m → * Alg C from the category Loc m of m-dimensional (globally hyperbolic) Lorentzian spacetimes to the category * Alg C of associative and unital * -algebras over C. This functor is required to satisfy certain physically motivated axioms, most notably the Einstein causality axiom expressing that every two causally disjoint observables must commute with each other. Such structures can be described most effectively in terms of operad theory and one observes that the category AQFT m of m-dimensional AQFTs is the category of algebras over a suitable colored operad, see [BSW21,BSW19a] for the details. The case of m = 1 dimensions, which physically represents AQFTs on time intervals (i.e. quantum mechanics), is structurally much simpler because causal disjointness, and hence the associated Einstein causality axiom, is a phenomenon arising only in dimension m ≥ 2. As a consequence, the category AQFT 1 = Fun(Loc 1 , * Alg C ) of 1-dimensional AQFTs is simply a functor category.
The aim of this section is to introduce a smooth refinement of 1-dimensional AQFTs. This means that we will upgrade the categories Loc 1 and * Alg C to stacks of categories, which encode suitable concepts of smoothly U -parametrized families of spacetimes and algebras, for all manifolds U ∈ Man. A smooth 1-dimensional AQFT will then be defined as a stack morphism between these two stacks, which in particular means that smooth AQFTs map smooth U -families of spacetimes to smooth U -families of algebras. Loosely speaking, one may say that "smooth AQFTs respond smoothly to smooth variations of spacetimes".
Our approach to smooth AQFTs introduces also a further layer of smoothness, namely we can define a stack AQFT ∞ 1 ∈ St of smooth 1-dimensional AQFTs. Through this stack we obtain a natural concept of smoothly U -parametrized families of smooth AQFTs, which for the special case U = R leads to a notion of smooth curves of smooth AQFTs. We shall illustrate later in Section 5 that smooth variations of the external parameters of a theory, such as the mass parameter, gives rise to such smooth families.
Throughout the whole paper we restrict our attention to the simplest case given by 1dimensional AQFTs. We expect that a generalization to higher-dimensional AQFTs is possible by using similar techniques, however there are certain additional technical difficulties and challenges that we explain in more detail in Section 6. Given any manifold U ∈ Man, the category Loc ∞ 1 (U ) is supposed to describe smooth Ufamilies of 1-dimensional spacetimes. A suitable way to formalize those is through fiber bundles and their vertical geometry. Remark 3.2. The interpretation of this definition is as follows: Given any pair (π : M → U, E) as in Definition 3.1, one obtains, for every point x ∈ U , a 1-dimensional spacetime (M | x , E| x ) := (π −1 ({x}), E| π −1 ({x}) ) ∈ Loc 1 by restricting to the fiber over x ∈ U . Due to the smooth fiber bundle structure, it makes sense to interpret this pair as depending smoothly on x ∈ U . △ A natural concept of morphisms f : (π : M → U, E) → (π ′ : M ′ → U, E ′ ) between smooth U -families of 1-dimensional spacetimes is given by fiber bundle maps

The stack Loc
that preserve the 1-forms, i.e. f * (E ′ ) = E, and that are in a suitable sense "open embeddings of fiber bundles". Indeed, from the AQFT point of view, it is quite natural to consider open embeddings as they allow to push forward compactly supported sections of vector bundles, which is crucial to construct examples. We would like to emphasize that there exist a priori different concepts of what open embeddings of fiber bundles could be. For example, we could demand the point-wise condition that the restriction f | x : M | x → M ′ | x to the fiber over every point x ∈ U is an open embedding of manifolds. Unfortunately, this simple point-wise condition is incompatible with pushing forward vertically compactly supported functions on the total spaces. Hence, the correct concept of "open embeddings of fiber bundles" should be in some sense more uniform on U . There exist a priori different options to formalize this, but fortunately the three main candidates are equivalent.
In order to define a (pre)stack Loc ∞ 1 : Man op → Cat (see Definition 2.1), we have to assign to every morphism h : U → U ′ in Man a functor Let us recall that, given any fiber bundle π : M → U ′ , one may form the pullback bundle which is a locally trivializable fiber bundle with the same typical fiber as π : M → U ′ . We then define the functor (3.2) on objects as and on morphisms f : (π : where the fiber bundle map h * f is defined uniquely through the universal property of pullback bundles by the commutative diagram Proof. This is a direct consequence of descent for fiber bundles and differential forms and of the fact that the first condition on the fiber bundle morphisms stated in Lemma 3.3 is a local condition on U ∈ Man. In more detail, spelling out descent for objects (π : M → U, E) ∈ Loc ∞ 1 (U ), one observes that it involves descent for the underlying fiber bundles π : M → U and also for the underlying 1-forms E, which are straightforward consequences of descent for fiber bundles and differential forms. Similarly, descent for and the verification that the 1-forms are preserved and that any one of the equivalent conditions from Lemma 3.3 holds, which are again both consequences of descent for fiber bundles and differential forms and the fact that the descent data fulfill these properties. The aim of this subsection is to develop a stack * Alg ∞ C that provides a smooth refinement of the usual category * Alg C of associative and unital * -algebras over C. Let us recall that the latter category may be defined as the category * Mon rev (Vec C ) of order-reversing * -monoids in the involutive symmetric monoidal category Vec C of complex vector spaces, see e.g. [Jac12,BSW19a] for the relevant background on involutive category theory. Our strategy is to introduce first a stack (of involutive symmetric monoidal categories) that refines the category Vec C of vector spaces over C and then discuss how to form order-reversing * -monoids at the level of stacks.
Let us consider for now the case where K is either R or C. As a first attempt to introduce a smooth refinement of the category Vec K , we could consider the stack VecBun K : Man op → Cat of K-vector bundles introduced in Example 2.4. To a manifold U ∈ Man, this stack assigns the category VecBun K (U ) of (locally trivializable and finite rank) K-vector bundles over U . Considering as in Remark 3.2 the fibers over points x ∈ U , every vector bundle can be interpreted as a smooth U -family of K-vector spaces. The problem with this first attempt is that the fibers of vector bundles are (by definition) finite-dimensional vector spaces, while examples of AQFTs, even in dimension 1, require infinite-dimensional vector spaces, such as the vector spaces underlying the canonical commutation relation algebras. A natural way to enlarge the category VecBun K (U ) in order to capture such infinite-dimensional aspects is to pass (via the sheaf of sections functor) to the category Sh C ∞ K (U ) of sheaves of C ∞ K,U -modules over U ∈ Man.
denotes the sheaf of K-valued smooth functions on U . Indeed, VecBun K (U ) embeds fully faithfully in Sh C ∞ K (U ) and the essential image consists of locally free C ∞ K,U -modules of finite rank, see e.g. [Ram05, Chapter 2].
Remark 3.7. We would like to note that there are also alternative candidates to enlarge the category VecBun K (U ) to include such infinite-dimensional aspects. For example, one could imagine to work with bundles over U whose fibers are e.g. locally convex, bornological or diffeological vector spaces. However, to make this a valid choice, one would have to confirm that such categories assemble into a stack, as it is the case for the sheaf categories U → Sh C ∞ K (U ), see Proposition 3.8 below. As another alternative, one could search directly for a stack providing a smooth refinement of the category of C * -algebras. To the best of our knowledge, such a stack has not yet been studied, but we believe that this may be related to the concept of continuous bundles/fields of C * -algebras, see e.g [KW95] and [Dix77, Section 10.3]. △ where h −1 is the inverse image sheaf functor and ⊗ h −1 (C ∞ K,U ′ ) denotes the relative tensor product of sheaves of modules. Together with the canonical coherence isomorphisms associated with relative tensor products and inverse image functors, this defines a prestack Sh C ∞ K : Man op → Cat in the sense of Definition 2.1. The following result is well-known, see e.g. [KS06,Proposition 19.4.7].
Proposition 3.8. For K being either R or C, the prestack Sh C ∞ K : Man op → Cat defined above is a stack, i.e. it satisfies the descent condition from Definition 2.3.
Remark 3.9. Observe that the category K is the ordinary category Vec K of vector spaces over K.
△ As explained at the beginning of this subsection, we interpret the stack Sh C ∞ K as a smooth refinement of the category Vec K of vector spaces over K. In order to introduce a smooth refinement of the category * Alg C = * Mon rev (Vec C ) of associative and unital * -algebras over C, we have to define an involutive symmetric monoidal structure on Sh C ∞ C . To achieve this goal, let us first observe that, for both K = R or C, the category Sh C ∞ K (U ) of sheaves of C ∞ K,U -modules over each U ∈ Man is symmetric monoidal with respect to the relative tensor product (3.7b) One can check that the canonical coherence isomorphisms of the stack Sh C ∞ K are monoidal natural transformations.
Corollary 3.10. The stack Sh C ∞ K in Proposition 3.8 is canonically a stack Sh C ∞ K : Man op → SMCat with values in the 2-category SMCat of symmetric monoidal categories, strong symmetric monoidal functors and monoidal natural transformations.
In the case of K = C, we can define further, for each U ∈ Man, an involution endofunctor (−) : which, as a sheaf, coincides with V , but the C ∞ C,U -module structure is defined via complex conjugation of C-valued functions as v·a := v · a * , for all v ∈ V and a ∈ C ∞ C,U . Clearly, the endofunctor (−) squares to the identity and hence defines an involutive structure on the category Sh C ∞ C (U ), see [Jac12,BSW19a]. Observe that (−) is canonically a strong symmetric monoidal functor with respect to the symmetric monoidal structure on Sh C ∞ C (U ) introduced above. Hence, we obtain that Sh C ∞ C (U ) is an involutive symmetric monoidal category, for every U ∈ Man. Furthermore, for each morphism h : U → U ′ in Man, the symmetric monoidal functor (3.5) is involutive via the coherence isomorphisms Corollary 3.11. For K = C, the stack Sh C ∞ C in Proposition 3.8 is canonically a stack Sh C ∞ C : Man op → ISMCat with values in the 2-category ISMCat of involutive symmetric monoidal categories, involutive strong symmetric monoidal functors and involutive monoidal natural transformations.
With these preparations, it is now straightforward to introduce a (pre)stack * Alg ∞ C that provides a smooth refinement of the ordinary category * Alg C = * Mon rev (Vec C ) of associative and unital * -algebras over C. Using that forming order-reversing * -monoids is a 2-functor * Mon rev : ISMCat → Cat, see [Jac12,BSW19a], we define a prestack (in the sense of Definition 2.1) by the composition * Alg ∞ More explicitly, this prestack assigns, to each manifold U ∈ Man, the category * Alg ∞ C (U ) = * Mon rev (Sh C ∞ C (U )) of order-reversing * -monoids in the involutive symmetric monoidal category are morphisms in Sh C ∞ C (U ) that satisfy the axioms of an associative and unital * -algebra. A morphism κ : U ) that preserves the multiplications, units and involutions. To each morphism h : U → U ′ in Man, the prestack * Alg ∞ C assigns the functor obtained by using the coherence isomorphisms of the involutive symmetric monoidal stack Sh C ∞ C from Corollary 3.11.
Proposition 3.12. The prestack * Alg ∞ C defined in (3.9) is a stack, i.e. it satisfies the descent condition from Definition 2.3.
Proof. Let {U α ⊆ U } be any open cover of any U ∈ Man. The key step is to realize that the descent category , which we endow with the involutive symmetric monoidal structure given by where we have suppressed the coherence isomorphisms (3.7) and (3.8). Fully explicitly, the conjugated cocycle ϕ αβ is given by and similarly for the tensor product cocycle ϕ αβ ⊗ C ∞ C,U αβ ϕ ′ αβ . The functor to the descent category given in (2.5) carries a canonical involutive symmetric monoidal structure and it is an equivalence in ISMCat because Sh C ∞ C is a stack. Applying the 2-functor * Mon rev : ISMCat → Cat that takes order-reversing * -monoids then yields the equivalence of Remark 3.13. Observe that the category * Alg ∞ C ({ * }) of global points { * } → * Alg ∞ C of the stack * Alg ∞ C is the ordinary category * Alg C of associative and unital * -algebras over C. △

The stack AQFT
With these preparations, we are now ready to introduce a natural concept of smooth 1-dimensional AQFTs. Recalling that the category of ordinary 1-dimensional AQFTs is described as the functor category AQFT 1 := Fun(Loc 1 , * Alg C ), we propose the following Definition 3.14. The stack of smooth 1-dimensional AQFTs is defined as the mapping stack (see (2.13)) This very simple definition is incredibly rich and powerful, as we shall explain throughout the rest of this subsection. Before discussing some of its more sophisticated consequences, we believe that it is worth spelling out explicitly what a smooth 1-dimensional AQFT is. By definition, it is a global point { * } → AQFT ∞ 1 of the stack introduced in Definition 3.14 which, by the 2-Yoneda Lemma, is equivalently an object A ∈ AQFT ∞ 1 ({ * }). According to (2.13), which defines mapping stacks, we find that a smooth 1-dimensional AQFT is then simply a stack morphism A : Loc ∞ 1 → * Alg ∞ C . Even more explicitly, this consists of a family of functors for all manifolds U ∈ Man, and natural isomorphisms for all morphisms h : U → U ′ in Man, that satisfy the coherence axioms listed in Definition 2.5. When we interpret, as explained in the previous subsections, Loc ∞ 1 (U ) as the category of smooth U -families of 1-dimensional spacetimes and * Alg ∞ C (U ) as the category of smooth U -families of algebras, the role of the functor A U : Loc ∞ 1 (U ) → * Alg ∞ C (U ) is to capture the response of the observable algebras to "smooth variations of spacetimes". Hence, smooth AQFTs have built in a suitable concept of smooth dependence on smooth variations of spacetimes, which we will illustrate in more detail via simple examples in Section 5. Let us also note that the functor defines an ordinary 1-dimensional AQFT. Hence, every smooth AQFT has an underlying ordinary AQFT and it therefore provides a refinement of the ordinary concept.
Another interesting consequence of Definition 3.14 is that it introduces a natural concept of "smooth curves of smooth AQFTs", or more generally of smoothŨ -families of smooth AQFTs, for every manifoldŨ ∈ Man. By definition, a smoothŨ -family of smooth AQFTs is aŨ -point U → AQFT ∞ 1 of the stack from Definition 3.14 which, by the 2-Yoneda Lemma, is equivalently an object B ∈ AQFT ∞ 1 (Ũ ). From the definition of mapping stacks (2.13), we obtain that this is simply a stack morphism Loc ∞ 1 ×Ũ → * Alg ∞ C , or equivalently a stack morphism to the mapping stack fromŨ to * Alg ∞ C . Even more explicitly, using again the 2-Yoneda Lemma, this is a family of functors for all manifolds U ∈ Man, and natural isomorphisms for all morphisms h : U → U ′ in Man, that satisfy the coherence axioms listed in Definition 2.5. The role of the functor B U : Loc ∞ 1 (U ) → * Alg ∞ C (U ×Ũ ) is now twofold: Firstly, it captures the response of the observable algebras to "smooth U -variations of spacetimes". Secondly, it captures the response of the observable algebras to "smoothŨ -variations of the smooth AQFT itself". Again, this concept is best illustrated via simple examples, see Section 5.
As another interesting consequence of Definition 3.14, let us note that every smooth AQFT A : { * } → AQFT ∞ 1 has a smooth automorphism group. (We refer to [Few13] for automorphism groups in ordinary AQFT, which in general are not smooth groups.) This can be defined in terms of the loop stack which is a bicategorical pullback in the 2-category St of stacks of categories. 1 By a direct computation of this bicategorical pullback, one finds that Aut(A) : Man op → Set ⊂ Cat is equivalent to a sheaf of sets (i.e. discrete categories), which due to the universal property of bicategorical pullbacks comes endowed with a group structure. This implies that Aut(A) : Man op → Grp is a sheaf of groups on Man, i.e. a smooth group from the functor of points perspective.
Let us briefly explain how this concept of smooth automorphism groups is related to the more practical concept of smooth AQFTs with a smooth action of a Lie group. Given any Lie group G, we use the 2-Yoneda embedding to define a group object G ∈ St in the 2-category of stacks and construct the quotient stack Because G(U ) is a discrete category, i.e. it only has identity morphisms, A2 U is simply a family of AQFT ∞ 1 (U )-isomorphisms A2 U g : AU ( * ) → AU ( * ) labeled by elements g ∈ G(U ) = C ∞ (U, G). The compatibility conditions then state that this labeling is compatible with the point-wise group structure on G(U ) = C ∞ (U, G), i.e. A2 U g·g ′ = A2 U g • A2 U g ′ , for all g, g ′ ∈ G(U ), and A2 U e = id, for the identity element e ∈ G(U ).
with a single object * and morphisms the smooth functions to the Lie group. (Composition of morphisms is given by the point-wise group structure of C ∞ (U, G).) On Man-morphisms h : U → U ′ this prestack acts via pullback of functions [{ * }/G] pre (h) := h * . Because * Alg ∞ C is a stack by Proposition 3.12, the universal property of stackyfication implies that the datum of a stack morphism A eq : [{ * }/G] → AQFT ∞ 1 is equivalent to a pseudo-natural transformation [{ * }/G] pre → AQFT ∞ 1 between prestacks, or equivalently a pseudo-natural transformatioñ (3.24) We will show in Subsection 5.3 that the latter perspective on G-equivariant smooth AQFTs is not very complicated to describe in concrete examples.

Smooth canonical quantization
The construction of free field theories in ordinary AQFT crucially relies on the existence of canonical (anti-)commutation relation quantization functors, see e.g. [BGP07,BG12]. The goal of this section is to show that these quantization functors admit a smooth refinement, which will allow us to construct both Bosonic and Fermionic examples of smooth 1-dimensional AQFTs in Section 5.

Canonical commutation relations
Ordinary canonical commutation relation (CCR) quantization is described by a functor CCR : PoVec R → * Alg C from the category of Poisson vector spaces to the category of associative and unital * -algebras. Recall that an object in the category PoVec R is a tuple (W, τ ), where W ∈ Vec R is a real vector space and τ : W ⊗ R W → R is an antisymmetric morphism in Vec R , and that a morphism ψ : where I CCR (W,τ ) is the 2-sided * -ideal generated by the canonical commutation relations w ⊗ w ′ − w ′ ⊗ w = i τ (w, w ′ ), for all w, w ′ ∈ W , where i ∈ C denotes the imaginary unit. The * -involution on CCR(W, τ ) is specified by w * = w, for all w ∈ W . Let us reformulate (4.1) in a slightly more abstract language. For this it is useful to observe that the construction of CCR algebras (4.1) consists of three steps: 1. Complexify the real vector space W ∈ Vec R to the complex vector space W ⊗ R C ∈ Vec C , which may be endowed with a * -involution id ⊗ R * : W ⊗ R C → W ⊗ R C = W ⊗ R C determined by complex conjugation on C. Hence, (W ⊗ R C, id ⊗ R * ) ∈ * Obj(Vec C ) defines a * -object in the involutive symmetric monoidal category of complex vector spaces.
3. Implement the canonical commutation relations associated with the Poisson structure τ by a coequalizer in the category * Alg C .
Before we can generalize this construction to the context of stacks, we have to find a smooth refinement of the category PoVec R . As explained in Subsection 3.2, we consider the stack Sh C ∞ R of sheaves of C ∞ R -modules as a smooth refinement of the category Vec R , hence a smooth refinement of the category PoVec R should be built from this stack. Concretely, we define the (pre)stack PoVec ∞ R : Man op → Cat by the following data. To each manifold U ∈ Man, it assigns the category PoVec ∞ R (U ) whose objects are tuples (W, τ ) with W ∈ Sh C ∞ R (U ) and τ : W ⊗ C ∞ R,U W → C ∞ R,U an antisymmetric morphism in Sh C ∞ R (U ), called Poisson structure. The morphisms ψ : (W, τ ) → (W ′ , τ ′ ) in this category are Sh C ∞ R (U )-morphisms ψ : that assigns to (W, τ ) ∈ PoVec ∞ R (U ′ ) the object in PoVec ∞ R (U ) determined by the object h * W ∈ Sh C ∞ R (U ) (see (3.5b)) and the Poisson structure where ∼ = are the coherence isomorphisms in (3.7).
Proposition 4.1. The prestack PoVec ∞ R : Man op → Cat defined above is a stack, i.e. it satisfies the descent condition from Definition 2.3.
Proof. This follows from the fact that Sh C ∞ R is a stack, see Proposition 3.8. Indeed, spelling out descent for objects (W, τ ) ∈ PoVec ∞ R (U ), one observes that it involves descent for the underlying objects W ∈ Sh C ∞ R (U ) and also for the underlying morphisms, which are again both consequences of descent for the stack Sh C ∞ R and of the fact that the descent data have this property.
Adopting an analogous three step construction as in the case of the ordinary CCR functor, we shall now define a stack morphism that provides a smooth refinement of CCR quantization. By Definition 2.5, this consists of functors for each manifold U ∈ Man, together with coherence isomorphisms. Regarding the first step, we observe that, for each U ∈ Man, there exists an adjunction (4.6) The left adjoint functor L U assigns to we denote the restriction of V ∈ Sh C ∞ C (U ) to a sheaf of C ∞ R,U -modules via the morphism C ∞ R,U → C ∞ C,U from real to complex-valued functions.
Regarding the second step, we observe that, for each U ∈ Man, there exists an adjunction (4.7) The right adjoint functor G U assigns to an associative and unital * -algebra (A, µ, η, * ) in Sh C ∞ C (U ) its underlying * -object (A, * ), i.e. it forgets the multiplication µ and unit η. The left adjoint functor F U is the free order-reversing * -monoid functor. Explicitly, it assigns to a * -object (V, * ) the free order-reversing * -monoid F U (V, * ) := n≥0 V ⊗n , where tensor products and coproducts are formed in the symmetric monoidal category Sh C ∞ C (U ). The order-reversing * -structure of F U (V, * ) is defined by the canonical extension of the * -structure on the generators (V, * ).
With these preparations, we can now define the values of (4.5) on objects by carrying out the third step. Explicitly, given any object (W, τ ) ∈ PoVec ∞ R (U ), i.e. W ∈ Sh C ∞ R (U ) and by a coequalizer in * Alg ∞ C (U ). The relations r 1 , r 2 are defined in terms of their adjuncts under the adjunctions in (4.6) and (4.7) by and where we suppressed for notational convenience the subscripts U on the functors. Here µ (op) denotes the (opposite) multiplication and η the unit element in F L(W ). Because the coequalizer in (4.8) is clearly functorial with respect to morphisms ψ : (W, τ ) → (W ′ , τ ′ ) in PoVec ∞ R (U ), we have successfully defined the desired functor in (4.5).
Remark 4.2. For U = { * } a point, (4.8) gives precisely the usual CCR algebra (4.1). △ To complete our construction of the desired stack morphism CCR : PoVec ∞ R → * Alg ∞ C , it remains to define coherence isomorphisms (see Definition 2.5) for all morphisms h : U → U ′ in Man. These can be built from the analogous coherence isomorphisms for the left adjoint functors in (4.6) and (4.7), i.e.
Explicitly, for W ∈ Sh C ∞ R (U ′ ), the isomorphism L h is given by where in the second step we have used that h * preserves coproducts because it is a left adjoint functor and in the third step we have used the coherence isomorphisms of the involutive symmetric monoidal stack Sh C ∞ C from Corollary 3.11. Pasting the natural isomorphisms in (4.11) defines a natural isomorphism ( descends to the CCR algebras (4.8) and thereby defines the natural isomorphism CCR h in (4.10).
Proposition 4.3. The construction above defines a stack morphism CCR :
2. Implement the canonical anti-commutation relations associated with ·, · by a coequalizer in the category * Alg C .
To obtain a smooth refinement of the category IPVec C , we follow the same strategy as in Subsection 4.1. We define a (pre)stack IPVec ∞ C : Man op → Cat by the following data. To each manifold U ∈ Man, it assigns the category IPVec ∞ C (U ) whose objects (V, * , ·, · ) consist of a * -object (V, * ) ∈ * Obj(Sh C ∞ C (U )) and a symmetric * -morphism ·, · : (V, that assigns to (V, * , ·, · ) ∈ IPVec ∞ C (U ′ ) the object in IPVec ∞ C (U ) determined by the object h * (V, * ) ∈ * Obj(Sh C ∞ C (U )) and the morphism where we have used the coherence isomorphisms of the involutive symmetric monoidal stack Sh C ∞ C from Corollary 3.11. The proof of the following statement is completely analogous to the one of Proposition 4.1.
Proposition 4.4. The prestack IPVec ∞ C : Man op → Cat defined above is a stack, i.e. it satisfies the descent condition from Definition 2.3.
We shall now define a stack morphism that provides a smooth refinement of CAR quantization. In analogy to (4.8), we define the component functors CAR U : IPVec ∞ C (U ) → * Alg ∞ C (U ), for all U ∈ Man, by the coequalizer where the relations s 1 , s 2 are defined in terms of their adjuncts under (4.7) by and where we suppressed for notational convenience the subscripts U on the functors. The coherence isomorphisms

Illustration through free theories
We shall illustrate our formalism by constructing concrete examples of smooth 1-dimensional AQFTs. The models we study are smooth refinements of the Bosonic and Fermionic free field theories discussed in e.g. [BG12,BGP07]. Similarly to the ordinary case, our Bosonic models will be described by stack morphisms obtained as the composition of a stack morphism L b assigning the linear observables with their Poisson structure and the CCR-quantization stack morphism developed in Subsection 4.1. The Fermionic models will be described similarly by stack morphisms factorizing through the CAR-quantization stack morphism developed in Subsection 4.2.
Inspired by the standard constructions in ordinary AQFT [BG12,BGP07], we shall obtain examples of the stack morphisms L b/f assigning linear observables by using a suitable smooth refinement of the concept of retarded/advanced Green operators G ± to be developed in Subsection 5.1 below. Recall that the role of such Green operators is to determine the Poisson structure τ of a Bosonic theory and the bilinear map ·, · of a Fermionic theory. In Subsection 5.2 we will spell out this construction for the simplest case of a 1-dimensional massive scalar field, which is equivalent to the harmonic oscillator 3 . We shall even construct a smoothŨ -family of smooth AQFTs (in the sense of (3.17)) that describes a family of 1-dimensional massive scalar fields with a smoothly varying mass parameter m ∈ C ∞ (Ũ , R >0 ). In Subsection 5.3 we construct the 1-dimensional massless Dirac field as a smooth AQFT and show that its global U (1)-symmetry is realized in terms of smooth automorphisms in the sense of (3.19).

Green operators, solutions and initial data
Let us consider a manifold U ∈ Man and a smooth U -family of 1-dimensional spacetimes (π : M → U, E) ∈ Loc ∞ 1 (U ). We introduce the functor Together with the C ∞ R,U -module structure induced by pullback of functions along the projection map π : M → U , this defines an object C ∞ π ∈ Sh C ∞ R (U ) that we shall interpret as the field configuration space of a real scalar field on the smooth family of spacetimes (π : M → U, E) ∈ Loc ∞ 1 (U ). More generally, the configuration space of vector-valued fields is given by where n ∈ Z ≥1 is the number of field components, and we take K = R for real fields and K = C for complex fields. As equation of motion we will consider a Sh C ∞ K (U )-morphism P : C ∞ π ⊗ K n → C ∞ π ⊗ K n given by a vertical differential operator on π : M → U , i.e. a differential operator on M that differentiates only along the fibers of π : M → U . See our Examples 5.9 and 5.11 below.
In order to define a concept of Green operators for such P , we introduce certain subsheaves of the sheaf of functions C ∞ π on π : M → U that describe functions with restrictions on their vertical support. In the following definition, we shall use that the fiber bundle π : M → U underlying any object (π : M → U, E) ∈ Loc ∞ 1 (U ) admits sections because the fibers are open intervals, see e.g. [Ste51, Sections 12.2 and 6.7]. Furthermore, given any subset S ⊆ M of the total space, we denote by J ± v (S) ⊆ M the vertical future/past of S, i.e. the subset of all points that can be reached from S by future/past directed vertical curves with respect to the orientation induced by E ∈ Ω 1 v (M ).
). We say that ϕ ∈ C ∞ π (U ′ ) is vertically compactly supported if it is both vertically past and future compactly supported, i.e. there exist two sections σ 1 , σ 2 : Remark 5.2. Note that our definition of vertically compactly supported functions uses manifestly the fact that we consider smooth families of 1-dimensional spacetimes. In this 1-dimensional case, we have bundles π : M → U whose fibers are intervals, hence it makes sense to define vertical compactness through vertical boundedness from above and below. A dimension-independent definition for ϕ ∈ C ∞ π (U ′ ) to be vertically compactly supported is given by the condition that supp(ϕ) ∩ π −1 (K) is compact, for all K ⊆ U ′ compact. Upon sheafification (see Definition 5.3 below), this coincides in the 1-dimensional case with our more practical Definition 5.1. △ From this definition we obtain sub-presheaves C ∞ π vc , C ∞ π vpc and C ∞ π vfc of C ∞ π that assign vertically compactly supported, vertically past compactly supported and vertically future compactly supported functions. Note that these presheaves are separated, but they do not satisfy the descent condition for sheaves and hence have to be sheafified.
Definition 5.3. We denote by C ∞ π vc , C ∞ π vpc , C ∞ π vfc ∈ Sh C ∞ R (U ) the sheafifications of the presheaves C ∞ π vc of vertically compactly supported functions, C ∞ π vpc of vertically past compactly supported functions and C ∞ π vfc of vertically future compactly supported functions.
Remark 5.4. These sheaves admit the following explicit description as subsheaves of C ∞ π . To each open subset U ′ ⊆ U , the sheaf C ∞ π v(p/f)c assigns the subset C ∞ π v(p/f)c (U ′ ) ⊆ C ∞ π (U ′ ) consisting of all functions ϕ ∈ C ∞ π (U ′ ) that satisfy the following local support condition: For every point x ∈ U ′ , there exists an open neighborhood U x ⊆ U ′ of x such that the restriction ϕ| Ux ∈ C ∞ π (U x ) is vertically (past/future) compactly supported in the sense of Definition 5.1. △ With these preparations we can now introduce a concept of Green operators.
Definition 5.5. Let P : C ∞ π ⊗ K n → C ∞ π ⊗ K n be a Sh C ∞ K (U )-morphism that is determined from a vertical differential operator on π : M → U , i.e. a differential operator on M that differentiates only along the fibers of π : M → U . A retarded/advanced Green operator for P is a Sh C ∞ K (U )morphism G ± : C ∞ π vpc/vfc ⊗ K n → C ∞ π vpc/vfc ⊗ K n that satisfies the following properties: (i) G ± is the inverse of the restriction P : C ∞ π vpc/vfc ⊗ K n → C ∞ π vpc/vfc ⊗ K n of P to the subsheaves of vertically past/future compactly supported functions.
We refer to the Sh C ∞ K (U )-morphism G := G + − G − : C ∞ π vc ⊗ K n → C ∞ π ⊗ K n as the causal propagator.
Remark 5.6. Observe that, as a consequence of item (i), retarded and advanced Green operators are unique, provided they exist. Their existence is instead a condition on P , namely the restrictions P : C ∞ π vpc/vfc ⊗ K n → C ∞ π vpc/vfc ⊗ K n must be invertible and their inverses must fulfill also item (ii). Examples 5.9, 5.10 and 5.11 present vertical differential operators that fulfill these conditions. △ The usual exact sequence for P and G, see e.g. [BGP07], generalizes to our context.
Proposition 5.7. Let P : C ∞ π ⊗ K n → C ∞ π ⊗ K n be a Sh C ∞ K (U )-morphism that is determined from a vertical differential operator on π : M → U and G ± : C ∞ π vpc/vfc ⊗ K n → C ∞ π vpc/vfc ⊗ K n retarded/advanced Green operators for P . Then the associated sequence is exact. Even stronger, the corresponding sequence of presheaves is exact, i.e. for each open subset U ′ ⊆ U , the sequence Proof. We prove the second (stronger) statement, which implies the first. Let U ′ ⊆ U be any open subset. To prove exactness at the first node, consider any ϕ ∈ C ∞ π vc (U ′ ) ⊗ K n such that P ϕ = 0 and note that 0 = G ± P ϕ = ϕ by Definition 5.5 (i). For the second node, let ϕ ∈ C ∞ π vc (U ′ ) ⊗ K n be such that Gϕ = 0. Then G + ϕ = G − ϕ =: ρ ∈ C ∞ π vc (U ′ ) ⊗ K n because of the support properties of Green operators and the definition of vertically compact support. Hence, P ρ = P G ± ϕ = ϕ by Definition 5.5 (i).
Remark 5.8. As a direct consequence of this proposition, we obtain that the cokernel sheaf C ∞ π vc ⊗K n P (C ∞ π vc ⊗K n ) := coker P : C ∞ π vc ⊗ K n → C ∞ π vc ⊗ K n ∈ Sh C ∞ K (U ) (5.7a) may be computed as a presheaf quotient, i.e.
for every open subset U ′ ⊆ U . Furthermore, this sheaf is isomorphic via the causal propagator to the solution sheaf Sol π := ker(P :

△
Let us now illustrate these concepts by examples.
Example 5.9. Consider any U ∈ Man and any smooth U -family of 1-dimensional spacetimes (π : M → U, E) ∈ Loc ∞ 1 (U ). As equation of motion we take the vertical differential operator where m ∈ (0, ∞) is a fixed parameter and * v d v * v d v is the vertical Laplacian, which is obtained from the vertical de Rham differential d v on π : M → U and the vertical Hodge operator * v induced by E ∈ Ω 1 v (M ). This differential operator describes a smooth U -family of 1-dimensional scalar fields (or equivalently harmonic oscillators) with a fixed mass/frequency parameter m on time intervals whose geometry (i.e. length) depends on the point x ∈ U . To prove that (5.9) admits a retarded and an advanced Green operator, it is sufficient to prove existence of local retarded and advanced Green operators G ± α : C ∞ π vpc/vfc | Uα → C ∞ π vpc/vfc | Uα for an arbitrary choice of open cover {U α ⊆ U }. This is because uniqueness of retarded/advanced Green operators entails that the family {G ± α } satisfies the relevant compatibility conditions on all overlaps U αβ and hence, recalling from Proposition 3.8 that Sh C ∞ R is a stack, it defines a global retarded/advanced Green operator.
To prove local existence, consider any open cover {U α ⊆ U } in which the restricted bundles M | Uα → U α admit a trivialization M | Uα ∼ = R × U α . In this trivialization, we have that E| Uα ∼ = ρ dt for a positive function ρ ∈ C ∞ (R × U α , R >0 ), where t ∈ R is a time coordinate on R, and the equation of motion operator reads as P α = ρ −1 ∂ t ρ −1 ∂ t + m 2 . We can simplify this differential operator even further by introducing a new (x ∈ U α dependent) time coordinate T (t, x) such that d v T = ρ dt. Note that in these coordinates the fiber over x ∈ U α is the interval (T (−∞, x), T (∞, x)), i.e. the geometry/length of the interval may depend on x. The equation of motion operator then reads as P α = ∂ 2 T + m 2 , which admits the retarded/advanced Green operator Because ϕ ∈ C ∞ π vpc/vfc | Uα is vertically past/future compactly supported, this integral exists and it depends smoothly on both T ∈ (T (−∞, x), T (∞, x)) and x ∈ U α . ▽ Example 5.10. In order to construct in Subsection 5.2 an example of a smoothŨ -family of smooth AQFTs, we generalize Example 5.9 to the case where the mass parameter is not a constant but rather a smooth positive function m ∈ C ∞ (Ũ , R >0 ) onŨ ∈ Man. Given any U ∈ Man and any smooth U -family of 1-dimensional spacetimes (π : M → U, E) ∈ Loc ∞ 1 (U ), we consider the object (π × id : M ×Ũ → U ×Ũ , pr * M (E)) ∈ Loc ∞ 1 (U ×Ũ ) and define on it the vertical differential operatorP T + m 2 (x), where we made the dependence onx ∈Ũ explicit. This operator admits a retarded/advanced Green operator given by for all ϕ ∈ C ∞ π×id vpc/vfc | Uα×Ũ . ▽ Example 5.11. Consider any U ∈ Man and any smooth U -family of 1-dimensional spacetimes (π : M → U, E) ∈ Loc ∞ 1 (U ). The 1-dimensional massless Dirac field is described by the vertical differential operator where i ∈ C is the imaginary unit and the elements Ψ Ψ ∈ C ∞ π ⊗ C 2 should be interpreted as the Dirac field Ψ and its Dirac conjugate Ψ. The existence of retarded/advanced Green operators can be proven as in the previous examples by a local argument. Indeed, restricting again to a trivializing cover {U α ⊆ U } and introducing the local time coordinate T , the local Dirac operator reads as and its associated retarded/advanced Green operator is given by fiber integration where T (∓∞, x) was defined in Example 5.9. ▽ We conclude this subsection with a few remarks about smoothly parametrized initial value problems. Let us start with the case where the Sh C ∞ K (U )-morphism P : C ∞ π ⊗ K n → C ∞ π ⊗ K n corresponds to a second order vertical differential operator on π : M → U , as it is the case in our Examples 5.9 and 5.10. Choosing any section σ : U → M , we can define a Sh C ∞ K (U )-morphism data 2nd We say that P has a well-posed initial value problem if (5.16) is an isomorphism in Sh C ∞ K (U ). Note that if P has retarded/advanced Green operators and a well-posed initial value problem, it follows by using also Remark 5.8 that is an isomorphism in Sh C ∞ K (U ) and hence that C ∞ π vc ⊗ K n /P (C ∞ π vc ⊗ K n ) ∈ Sh C ∞ K (U ) is a free C ∞ K,U -module of rank 2n. This observation will be useful in Subsection 5.2.
Example 5.12. The equation of motion operators in Examples 5.9 and 5.10 have a well-posed initial value problem. Let us show this for the more general operator (5.11) in the latter example, which reduces forŨ = { * } a point to the operator (5.9). Using again that Sh C ∞ R is a stack, it is sufficient to prove the isomorphism property of the sheaf morphism (5.16) in each patch U α of an arbitrary open cover {U α ⊆ U }. Using as in Example 5.10 a trivializing cover and suitable time coordinates T , we obtain the local equation of motion operatorP α = ∂ 2 T + m 2 (x). The inverse of the restriction of the initial data map data 2nd σ to U α ⊆ U is then given by Remark 5.13. The case of a first order vertical differential operator P : C ∞ π ⊗ K n → C ∞ π ⊗ K n on π : M → U works similarly. The analog of (5.16) is given by the We again say that P has a well-posed initial value problem if (5.19) is an isomorphism. It is easy to check that the Dirac operator from Example 5.11 has a well-posed initial value problem in this sense. △

1-dimensional scalar field
The aim of this subsection is to construct an explicit smoothŨ -family of smooth AQFTs that can be interpreted as a smooth refinement of the 1-dimensional scalar field with a smoothly varying mass parameter m ∈ C ∞ (Ũ , R >0 ). Our construction will be based on Example 5.10 and we will carry out (smooth generalizations of) the usual steps in the construction of Bosonic free field theories, see e.g. [BG12,BGP07]. Since we are interested in smoothŨ -families (see (3.17)), we have to define instead of (5.1) a stack morphism which for the case ofŨ = { * } a point reduces to (5.1). For our example of interest, the stack morphism W : Loc ∞ 1 → Map(Ũ , PoVec ∞ R ) is given by the following data: For each manifold U ∈ Man, we define the functor that assigns to (π : M → U, E) ∈ Loc ∞ 1 (U ) the object whereP (π,E) is the equation of motion operator in (5.11). The Poisson structure reads as τ (π×id,pr * M (E)) = ·,G (π,E) (·) (π×id,pr * M (E)) , (5.22b) σ (h × id) = (h M × id) σ h , hence the maps in (5.27) preserve vertically compact support. Due to naturality of the vertical differential operatorsP in (5.11), we obtain the commutative diagram , which allows us to induce (5.26) to the quotients (5.29) Here we also used that (h × id) * is a left adjoint functor, hence it commutes with the colimit defining these quotients. From the explicit expression (5.27) for (the adjunct of) this morphism and observing that a diagram similar to (5.28) involving retarded/advanced Green operators commutes due to their uniqueness, one checks that (5.29) preserves the relevant Poisson structures and thereby defines the desired It remains to confirm that (5.29) is an isomorphism in PoVec ∞ R (U ′ ×Ũ ). Using the causal propagators (5.8) and the initial data morphisms (5.16) corresponding to any choice of section σ : U ′ ×Ũ → M ×Ũ and its induced section σ h : U ×Ũ → h * M ×Ũ of the pullback bundle, we obtain the commutative diagram where the bottom horizontal isomorphism uses that (h × id) * preserves coproducts (as it is a left adjoint functor) and the symmetric monoidal coherence isomorphism for the monoidal unit in (3.7). By Remark 5.8 and Examples 5.10 and 5.12, all vertical arrows in this diagram are isomorphisms, hence the top horizontal arrow is an isomorphism too. This implies that W h is an isomorphism in PoVec ∞ R (U ×Ũ ). Summing up, the main result of this section is Proposition 5.14. The construction described above defines a stack morphism W : . As a consequence, we obtain an explicit example of a smoothŨ -family of smooth 1-dimensional AQFTs B := Map(Ũ , CCR) • W : Loc ∞ 1 → Map(Ũ , * Alg ∞ C ) describing a smooth refinement of the 1-dimensional scalar field with a smoothly varying mass parameter m ∈ C ∞ (Ũ , R >0 ). In the special case whereŨ = { * } is a point, our construction describes a smooth refinement of the 1-dimensional scalar field with a fixed mass m > 0.

1-dimensional Dirac field
The construction of Subsection 5.2 can be easily adapted to the case of the 1-dimensional Dirac field introduced in Example 5.11. We will spell out the relevant steps to construct the corresponding stack morphism L f : Loc ∞ 1 → IPVec ∞ C such that A f := CAR • L f in (5.2) describes a smooth refinement of the massless 1-dimensional Dirac field. For this we will carry out (smooth generalizations of) the usual steps in the construction of Fermionic free field theories, see e.g. [DHP09]. After that we will show that the smooth automorphism group (3.19) of this model includes the global U (1)-symmetry of the Dirac field.
The stack morphism L f : Loc ∞ 1 → IPVec ∞ C is given by the following data: For each manifold U ∈ Man, we define the functor that assigns to each (π : M → U, E) ∈ Loc ∞ 1 (U ) the object where D (π,E) is the Dirac operator from Example 5.11. The * -involution * (π,E) ψ ψ is given by swapping the components followed by complex conjugation, which descends to the quotient since * (π,E) • D (π,E) = D (π,E) • * (π,E) . The symmetric pairing is given by fiber integration, the causal propagator S (π,E) for D (π,E) and the displayed matrix multiplications. It is easy to check that ·, · (π,E) descends to the quotient in (5.32a) and that it satisfies the compatibility condition (4.13) for * -involutions. The definition of the functor (5.31) on morphisms f : (π : M → U, E) → (π ′ : M ′ → U, E ′ ) is as in (5.24) via pushforward of vertically compactly supported functions. The coherence isomorphisms for Man-morphisms h : U → U ′ are constructed in complete analogy to (5.25). Summing up, we obtain Proposition 5.15. The construction described above defines a stack morphism L f : Loc ∞ 1 → IPVec ∞ C . As a consequence, we obtain another example of a smooth 1-dimensional AQFT A f := CAR • L f : Loc ∞ 1 → * Alg ∞ C describing a smooth refinement of the 1-dimensional massless Dirac field. To conclude this section, we will show that our construction can be refined to define a U (1)equivariant smooth AQFT, showing that the global U (1)-symmetry of the Dirac field is smooth in our sense. Recalling from (3.24) their explicit description, we will define a U (1)-equivariant smooth AQFT by specifying a pseudo-natural transformationL f . For each manifold U ∈ Man, we define the functorL L f U π : M → U, E := L f U π : M → U, E = C ∞ π vc ⊗C 2 D (π,E) (C ∞ π vc ⊗C 2 ) , * (π,E) , ·, · (π,E) .
(The second equality in (5.36) follows from the fact that f * only acts along the fibers where π * (g) is constant.) These maps clearly preserve the quotient in (5.35), the * -involution (5.32b) and the pairing (5.32c), hence they define IPVec ∞ C (U )-morphisms. The coherence isomorphisms for Man-morphisms h : U → U ′ are constructed in complete analogy to our previous examples.
Summing up, we obtain Proposition 5.16. The construction described above defines a pseudo-natural transformatioñ As a consequence, we obtain an example of a U (1)-equivariant smooth 1-dimensional AQFT A f := CAR •L f : Loc ∞ 1 × [{ * }/U (1)] pre → * Alg ∞ C describing a smooth refinement of the 1dimensional massless Dirac field together with its global U (1)-symmetry.

Outlook: Towards higher dimensions and gauge theories
The aim of this section is to outline the way we believe the results of this paper could be generalized to higher-dimensional AQFTs and also to gauge theories. In particular, we shall explain the additional technical challenges and open questions that arise from such a generalization.
Let us first discuss possible generalizations of the stack Loc ∞ 1 of 1-dimensional spacetimes from Subsection 3.1 to the case of higher dimensions m ≥ 2. For U ∈ Man a manifold, we can define a smooth U -family of m-dimensional Lorentzian manifolds to be a tuple (π : M → U, g) consisting of a (locally trivializable) fiber bundle π : M → U with typical fiber an m-manifold N and a metric g of signature (+ − · · · −) on the vertical tangent bundle of π : M → U . For illustrative purposes, we note that in a local trivialization M | U ′ ∼ = N ×U ′ and in local coordinates y µ on the fiber N , the vertical metric takes the form g| U ′ ∼ = g µν (y, x) dy µ ⊗ dy ν , i.e. it has only vertical components along N that however are allowed to depend smoothly on x ∈ U ′ ⊆ U . There are obvious notions of vertical orientation o and vertical time-orientation t, hence we can introduce a concept of smooth U -families of m-dimensional oriented and time-oriented Lorentzian manifolds (π : M → U, g, o, t). What is less obvious is the correct generalization of the important concept of global hyperbolicity to this smoothly parametrized context. One could either impose the point-wise condition that each fiber (M | x , g| x ) is globally hyperbolic in the usual sense or seek for a condition that is more uniform on U . The role of this condition should be to ensure that vertical normally hyperbolic operators, such as the vertical Klein-Gordon operator (6.1) admit retarded and advanced Green operators and a well-posed initial value problem, both described in terms of morphisms of sheaves of C ∞ R,U -modules. This can be interpreted saying that both the retarded/advanced Green operators and the initial value problem are smoothly parametrized. Again for illustrative purposes, we note that in a local trivialization M | U ′ ∼ = N ×U ′ and in local coordinates y µ on the fiber N , the vertical differential operator (6.1) takes the form P | U ′ ∼ = g µν (y, x) ∂ 2 ∂y µ ∂y ν + B µ (y, x) ∂ ∂y µ + A(y, x) , (6.2) i.e. there are no derivatives along x ∈ U ′ but the coefficients may be x-dependent. Summing up, we record Open Problem 6.1. Find a suitable generalization of global hyperbolicity to smooth U -families of m-dimensional oriented and time-oriented Lorentzian manifolds (π : M → U, g, o, t) such that vertical normally hyperbolic operators admit smoothly parametrized retarded and advanced Green operators and a well-posed smoothly parametrized initial value problem.
Successfully solving this problem will lead to a sensible definition of a stack Loc ∞ m of mdimensional globally hyperbolic spacetimes. One can then attempt to construct examples of smooth m-dimensional AQFTs in terms of stack morphisms A : Loc ∞ m → * Alg ∞ C by using the same strategy as in (5.1). We note that most of our constructions in Section 5 only rely on the existence (and uniqueness) of Green operators, hence they would generalize directly to the higherdimensional case, provided that Open Problem 6.1 is solved. There is however one exception: In the higher-dimensional case, the space of initial data is infinite-dimensional, hence we can not argue as in (5.30) to conclude that the assignment of linear observables L : Loc ∞ m → PoVec ∞ R is a stack morphism. More specifically, this could lead to the problem that the coherence maps L h are only natural transformations, but not natural isomorphisms, which means that L is only a lax stack morphism. At the moment we do not know whether it will be more convenient to enlarge the 2-category St of stacks to include also lax morphisms or to replace the stack Sh C ∞ R of sheaves of C ∞ R -modules by a stack describing sheaves of topological (or bornological) modules in order to obtain a better control on these infinite-dimensional aspects. Summing up, we record Open Problem 6.2. Find a suitable framework such that the assignment L : Loc ∞ m → PoVec ∞ R of linear observables for a smooth m-dimensional free AQFT is a morphism between stacks. Possible options could be enlarging the 2-category St of stacks to allow for lax morphisms or replacing the stack Sh C ∞ R of sheaves of C ∞ R -modules by a stack describing sheaves of topological (or bornological) modules.
As already emphasized at the beginning of Section 3, higher-dimensional AQFTs are sensitive to the phenomenon of Einstein causality, which means that they are not simply functors but rather algebras over a suitable colored operad [BSW21,BSW19a]. Encoding this aspect in our smooth setting leads to the following Open Problem 6.3. Develop a theory of stacks of colored operads in order to define the stack AQFT ∞ m of smooth m-dimensional AQFTs in terms of a suitable mapping stack between stacks of colored operads.
To conclude, we would like to comment briefly on a potential generalization of our framework to gauge theories. The latter are most appropriately described by the BV-formalism, which is captured by a concept of AQFTs taking values in cochain complexes, see e.g. [FR12,FR13,BSW19b]. This necessarily introduces to an ∞-categorical context because the natural notion of equivalence between cochain complexes is given by quasi-isomorphisms, as opposed to isomorphisms. This in particular means that, instead of the smooth refinement Sh C ∞ K of the ordinary category Vec K from Subsection 3.2, one has to consider a smooth refinement of the ∞-category Ch K of cochain complexes. A natural candidate for this purpose is the ∞-stack Ch(Sh C ∞ K ) of cochain complexes of sheaves of modules.
Open Problem 6.4. Replacing the stack Sh C ∞ K of sheaves of modules by the ∞-stack Ch(Sh C ∞ K ) of cochain complexes of sheaves of modules, show that the relevant definitions and constructions from Section 3 generalize to the context of ∞-stacks.