Algebraic quantum field theory on spacetimes with timelike boundary

We analyze quantum field theories on spacetimes $M$ with timelike boundary from a model-independent perspective. We construct an adjunction which describes a universal extension to the whole spacetime $M$ of theories defined only on the interior $\mathrm{int}M$. The unit of this adjunction is a natural isomorphism, which implies that our universal extension satisfies Kay's F-locality property. Our main result is the following characterization theorem: Every quantum field theory on $M$ that is additive from the interior (i.e.\ generated by observables localized in the interior) admits a presentation by a quantum field theory on the interior $\mathrm{int}M$ and an ideal of its universal extension that is trivial on the interior. We shall illustrate our constructions by applying them to the free Klein-Gordon field.


Introduction and summary
Algebraic quantum field theory is a powerful and far developed framework to address modelindependent aspects of quantum field theories on Minkowski spacetime [HK64] and more generally on globally hyperbolic spacetimes [BFV03]. In addition to establishing the axiomatic foundations for quantum field theory, the algebraic approach has provided a variety of mathematically rigorous constructions of non-interacting models, see e.g. the reviews [BD15,BDH13,BGP07], and more interestingly also perturbatively interacting quantum field theories, see e.g. the recent monograph [Rej16]. It is worth emphasizing that many of the techniques involved in such constructions, e.g. existence and uniqueness of Green's operators and the singular structure of propagators, crucially rely on the hypothesis that the spacetime is globally hyperbolic and has empty boundary.
Even though globally hyperbolic spacetimes have plenty of applications to physics, there exist also important and interesting situations which require non-globally hyperbolic spacetimes, possibly with a non-trivial boundary. On the one hand, recent developments in high energy physics and string theory are strongly focused on anti-de Sitter spacetime, which is not globally hyperbolic and has a (conformal) timelike boundary. On the other hand, experimental setups for studying the Casimir effect confine quantum field theories between several metal plates (or other shapes), which may be modeled theoretically by introducing timelike boundaries to the system. This immediately prompts the question whether the rigorous framework of algebraic quantum field theory admits a generalization to cover such scenarios.
Most existing works on algebraic quantum field theory on spacetimes with a timelike boundary focus on the construction of concrete examples, such as the free Klein-Gordon field on simple classes of spacetimes. The basic strategy employed in such constructions is to analyze the initial value problem on a given spacetime with timelike boundary, which has to be supplemented by suitable boundary conditions. Different choices of boundary conditions lead to different Green's operators for the equation of motion, which is in sharp contrast to the well-known existence and uniqueness results on globally hyperbolic spacetimes with empty boundary. Recent works addressing this problem are [Zah15] and [IW03,IW04], the latter extending the analysis of [Wal80]. For specific choices of boundary conditions, there exist successful constructions of algebraic quantum field theories on spacetimes with timelike boundary, see e.g. [DNP16,DF16,DF18,BDFK17]. The main message of these works is that the algebraic approach is versatile enough to account also for these models, although some key structures, such as for example the notion of Hadamard states [DF18,Wro17], should be modified accordingly.
Unfortunately, model-independent results on algebraic quantum field theory on spacetimes with timelike boundary are more scarce. There are, however, some notable and very interesting works in this direction: On the one hand, Rehren's proposal for algebraic holography [Reh00] initiated the rigorous study of quantum field theories on the anti-de Sitter spacetime. This has been further elaborated in [DR03] and extended to asymptotically AdS spacetimes in [Rib07]. On the other hand, inspired by Fredenhagen's universal algebra [Fre90,Fre93,FRS92], a very interesting construction and analysis of global algebras of observables on spacetimes with timelike boundaries has been performed in [Som06]. The most notable outcome is the existence of a relationship between maximal ideals of this algebra and boundary conditions, a result which has been of inspiration for this work.
In the present paper we shall analyze quantum field theories on spacetimes with timelike boundary from a model-independent perspective. We are mainly interested in understanding and proving structural results for whole categories of quantum field theories, in contrast to focusing on particular theories. Such questions can be naturally addressed by using techniques from the recently developed operadic approach to algebraic quantum field theory [BSW17]. Let us describe rather informally the basic idea of our construction and its implications: Given a spacetime M with timelike boundary, an algebraic quantum field theory on M is a functor B : R M → Alg assigning algebras of observables to suitable regions U ⊆ M (possibly intersecting the boundary), which satisfies the causality and time-slice axioms. We denote by QFT(M ) the category of algebraic quantum field theories on M . Denoting the full subcategory of regions in the interior of M by R int M ⊆ R M , we may restrict any theory B ∈ QFT(M ) to a theory res B ∈ QFT(int M ) defined only on the interior regions. Notice that it is in practice much easier to analyze and construct theories on int M as opposed to theories on the whole spacetime M . This is because the former are postulated to be insensitive to the boundary by Kay's F-locality principle [Kay92]. As a first result we shall construct a left adjoint of the restriction functor res : QFT(M ) → QFT(int M ), which we call the universal extension functor ext : QFT(int M ) → QFT(M ). This means that given any theory A ∈ QFT(int M ) that is defined only on the interior regions in M , we obtain a universal extension ext A ∈ QFT(M ) to all regions in M , including those that intersect the boundary. It is worth to emphasize that the adjective universal above refers to the categorical concept of universal properties. Below we explain in which sense ext is also "universal" in a more physical meaning of the word.
It is crucial to emphasize that our universal extension ext A ∈ QFT(M ) is always a bona fide algebraic quantum field theory in the sense that it satisfies the causality and time-slice axioms. This is granted by the operadic approach to algebraic quantum field theory of [BSW17]. In particular, the ext ⊣ res adjunction investigated in the present paper is one concrete instance of a whole family of adjunctions between categories of algebraic quantum field theories that naturally arise within the theory of colored operads and algebras over them.
A far reaching implication of the above mentioned ext ⊣ res adjunction is a characterization theorem that we shall establish for quantum field theories on spacetimes with timelike boundary. Given any theory B ∈ QFT(M ) on a spacetime M with timelike boundary, we can restrict and universally extend to obtain another such theory ext res B ∈ QFT(M ). The adjunction also provides us with a natural comparison map between these theories, namely the counit ǫ B : ext res B → B of the adjunction. Our result in Theorem 5.6 and Corollary 5.7 is that ǫ B induces an isomorphism ext res B/ ker ǫ B ∼ = B of quantum field theories if and only if B is additive from the interior as formalized in Definition 5.5. The latter condition axiomatises the heuristic idea that the theory B has no degrees of freedom that are localized on the boundary of M , i.e. all its observables may be generated by observables supported in the interior of M . Notice that the results in Theorem 5.6 and Corollary 5.7 give the adjective universal also a physical meaning in the sense that the extensions are sufficiently large such that any additive theory can be recovered by a quotient. We strengthen this result in Theorem 5.10 by constructing an equivalence between the category of additive quantum field theories on M and a category of pairs (A, I) consisting of a theory A ∈ QFT(int M ) on the interior and an ideal I ⊆ ext A of the universal extension that is trivial on the interior. More concretely, this means that every additive theory B ∈ QFT(M ) may be naturally decomposed into two distinct pieces of data: (1) A theory A ∈ QFT(int M ) on the interior, which is insensitive to the boundary as postulated by F-locality, and (2) an ideal I ⊆ ext A of its universal extension that is trivial on the interior, i.e. that is only sensitive to the boundary. Specific examples of such ideals arise from imposing boundary conditions. We shall illustrate this fact by using the free Klein-Gordon theory as an example. Thus, our results also provide a bridge between the ideas of [Som06] and the concrete constructions in [DNP16,DF16,DF18,BDFK17].
The remainder of this paper is structured as follows: In Section 2 we recall some basic definitions and results about the causal structure of spacetimes with timelike boundaries, see also [CGS09,Sol06]. In Section 3 we provide a precise definition of the categories QFT(M ) and QFT(int M ) by using the ideas of [BSW17]. Our universal boundary extension is developed in Section 4, where we also provide an explicit model in terms of left Kan extension. Our main results on the characterization of additive quantum field theories on M are proven in Section 5. Section 6 illustrates our construction by focusing on the simple example of the free Klein-Gordon theory, where more explicit formulas can be developed. It is in this context that we provide examples of ideals implementing boundary conditions and relate to analytic results, e.g. [DNP16]. We included Appendix A to state some basic definitions and results of category theory which will be used in our work.

Spacetimes with timelike boundary
We collect some basic facts about spacetimes with timelike boundary, following [Sol06, Section 3.1] and [CGS09, Section 2.2]. For a general introduction to Lorentzian geometry we refer to [BEE96,ONe83], see also [BGP07, Sections 1.3 and A.5] for a concise presentation.
We use the term manifold with boundary to refer to a Hausdorff, second countable, mdimensional smooth manifold M with boundary, see e.g. [Lee13]. This definition subsumes ordinary manifolds as manifolds with empty boundary ∂M = ∅. We denote by int M ⊆ M the submanifold without the boundary. Every open subset U ⊆ M carries the structure of a manifold with (possibly empty) boundary and one has int U = U ∩ int M .
Definition 2.1. A Lorentzian manifold with boundary is a manifold with boundary that is equipped with a Lorentzian metric.
Definition 2.2. Let M be a time-oriented Lorentzian manifold with boundary. The Cauchy development D(S) ⊆ M of a subset S ⊆ M is the set of points p ∈ M such that every inextensible (piecewise smooth) future directed causal curve stemming from p meets S.
The following properties follow easily from the definition of Cauchy development.
Proposition 2.3. Let S, S ′ ⊆ M be subsets of a time-oriented Lorentzian manifold M with boundary. Then the following holds true: We denote by J ± M (S) ⊆ M the causal future/past of a subset S ⊆ M , i.e. the set of points that can be reached by a future/past directed causal curve stemming from S. Furthermore, we denote by I ± M (S) ⊆ M the chronological future/past of a subset S ⊆ M , i.e. the set of points that can be reached by a future/past directed timelike curve stemming from S. The following two definitions play an essential role in our work.
Definition 2.6. A spacetime with timelike boundary is an oriented and time-oriented Lorentzian manifold M with boundary, such that the pullback of the Lorentzian metric along the boundary inclusion ∂M ֒→ M defines a Lorentzian metric on the boundary ∂M .
Definition 2.7. Let M be a spacetime with timelike boundary.
(i) R M denotes the category whose objects are causally convex open subsets U ⊆ M and whose morphisms i : U → U ′ are inclusions U ⊆ U ′ ⊆ M . We call it the category of regions in M . ( (iii) R int M ⊆ R M is the full subcategory whose objects are contained in the interior int M . We denote by C int M ⊆ C M the Cauchy morphisms between objects of R int M . (c): We show that if D(S) contains a boundary point, so does S: Suppose p ∈ D(S) belongs to the boundary of M . By Definition 2.6, the boundary ∂M of M can be regarded as a time-oriented Lorentzian manifold with empty boundary, hence we can consider a future directed inextensible causal curve γ in ∂M stemming from p. Since ∂M is a closed subset of M , γ must be inextensible also as a causal curve in M , hence γ meets S because it stems from p ∈ D(S). Since γ lies in ∂M by construction, we conclude that S contains a boundary point of M . Relative to the fixed convex cover, the construction of quasi-limits allows us to obtain from {α n } an inextensible causal curve λ through p ∈ D(U ). Hence, λ meets U , say in q. By the construction of a quasi-limit, q lies on a causal geodesic segment between p k and p k+1 , two successive limit points for {α n } contained in some element of the fixed convex cover. It follows that either p k or p k+1 belongs to J + I (U ) ∩ J − I (U ), which is contained in U by causal convexity. Hence, we found a subsequence {α n j } of {α n } and a sequence of parameters {s j } such that {α n j (s j )} converges to a point of U (either p k or p k+1 ). By construction the sequence {α n j (s j )} is contained in I \ U , however its limit lies in U . This contradicts the hypothesis that U is open in I.
The causal structure of a spacetime M with timelike boundary can be affected by several pathologies, such as the presence of closed future directed causal curves. It is crucial to avoid these issues in order to obtain concrete examples of our constructions in Section 6. The following definition is due to [  Definition 3.1. An algebraic quantum field theory on M is a functor with values in the category Alg of associative and unital * -algebras over C, which satisfies the following properties: (i) Causality axiom: For all causally disjoint inclusions i 1 : U 1 → U ← U 2 : i 2 , the induced commutator is zero.
(ii) Time-slice axiom: For all Cauchy morphisms (i : is an Alg-isomorphism. We denote by qft(M ) ⊆ Alg R M the full subcategory of the category of functors from R M to Alg whose objects are all algebraic quantum field theories on M , i.e. functors fulfilling the causality and time-slice axioms. (Morphisms in this category are all natural transformations.) We shall now show that there exists an alternative, but equivalent, description of the category qft(M ) which will be more convenient for the technical constructions in our paper. Following by setting on objects and morphisms Furthermore, let us write for the full subcategory embedding.

Lemma 3.2. D and I form an adjunction (cf. Definition A.1)
whose counit is a natural isomorphism (in fact, the identity), hence R M [C −1 M ] is a full reflective subcategory of R M . Furthermore, the components of the unit are Cauchy morphisms.
is given by the inclusion U ⊆ D(U ) of U into its Cauchy development, which is a Cauchy morphism, see Proposition 2.3 (b) and Definition 2.
is given by the identity of the object D(V ) = V . The triangle identities hold trivially.  where the vertical arrows are natural isomorphisms because the counit ǫ is an isomorphism.
It remains to prove that (3.9) is also surjective. Let χ : GD → HD be any natural transformation. Using Lemma 3.2, we obtain a commutative diagram where the vertical arrows are natural isomorphisms because the components of the unit η are Cauchy morphisms and D assigns isomorphisms to them. Let us define a natural transformation ξ : G → H by the commutative diagram where we use that ǫ is a natural isomorphism (cf. Lemma 3.2). Combining the last two diagrams, one easily computes that ξD = χ by using also the triangle identities of the adjunction D ⊣ I. Hence, the map (3.9) is surjective.
We note that there exist two (a priori different) options to define an orthogonality relation on the localized category Proof. For I this holds trivially, while for D see Proposition 2.5 (a).
With these preparations we may now define our alternative description of the category of algebraic quantum field theories.
is zero.
Theorem 3.6. By pullback, the adjunction D ⊣ I of Lemma 3.2 induces an adjoint equivalence (cf. Definition A.2) (3.14) In Next, we have to prove that this adjunction restricts to the claimed source and target categories in (3.14). Using Lemma 3.2, we obtain that the counit ǫ of the restricted adjunction (3.14) is an isomorphism. Furthermore, all components of η are Cauchy morphisms, hence η A = Aη is an isomorphism for all A ∈ qft(M ), i.e. the unit η is an isomorphism. This completes the proof that (3.14) is an adjoint equivalence.

Universal boundary extension
The goal of this section is to develop a universal construction to extend quantum field theories from the interior of a spacetime M with timelike boundary to the whole spacetime. (Again, we do not have to assume that M is globally hyperbolic in the sense of Definition 2.9.) Loosely speaking, our extended quantum field theory will have the following pleasant properties: (1) It describes precisely those observables that are generated from the original theory on the interior, (2) it does not require a choice of boundary conditions, (3) specific choices of boundary conditions correspond to ideals of our extended quantum field theory. We also refer to Section 5 for more details on the properties (1) and (3).
The starting point for this construction is the full subcategory inclusion R int M ⊆ R M defined by selecting only the regions of R M that lie in the interior of M (cf. Definition 2.7). We denote the corresponding embedding functor by and notice that j preserves (and also detects) causally disjoint inclusions, i.e. j is a full orthogonal subcategory embedding in the terminology of [BSW17]. Making use of Proposition 3.3, Lemma 3.2 and Remark 3.8, we define a functor J : Notice that J is simply an embedding functor, which acts on objects and morphisms as From this explicit description it is clear that J preserves (and also detects) causally disjoint inclusions, i.e. it is a full orthogonal subcategory embedding. The constructions in [BSW17, Section 5.3] (see also [BSW18] for details how to treat * -algebras) then imply that J induces an An important structural result, whose physical relevance is explained in Remark 4.2 below, is the following proposition.  , which also provides the inclusion V k ⊆ V 1 ⊔ V 2 ⊆ D(V 1 ⊔ V 2 ) inducing j k , for k = 1, 2. Consider now the chain of inclusions V k ⊆ V 1 ⊔ V 2 ⊆ V corresponding to i k , for k = 1, 2. From the stability under Cauchy development of V 1 , V 2 and V , we obtain also the chain of inclusions V k ⊆ D(V 1 ⊔ V 2 ) ⊆ V , for k = 1, 2, that exhibits the desired factorization that we will describe now in detail. The direct sum (of vector spaces) in (4.9) runs over all tuples int M ] are interior regions. (Notice that the regions V k are not assumed to be causally disjoint and that the empty tuple, i.e. n = 0, is also allowed.) The vector space A(V ) is defined by the tensor product of vector spaces are given by pairs consisting of a tuple of morphisms i : V → V (with all V k in the interior) and an element a ∈ A(V ) of the corresponding tensor product vector space (4.10). The product on (4.11) is given on homogeneous elements by where (i, i ′ ) = (i 1 , . . . , i n , i ′ 1 , . . . , i ′ m ) is the concatenation of tuples. The unit element in (4.11) is ½ := (∅, 1), where ∅ is the empty tuple and 1 ∈ C, and the * -involution is defined by (i 1 , . . . , i n ), a 1 ⊗ · · · ⊗ a n * := (i n , . . . , i 1 ), a * n ⊗ · · · ⊗ a * 1 (4.13) and C-antilinear extension. Finally, the quotient in (4.9) is by the two-sided * -ideal of the algebra (4.11) that is generated by i i 1 , . . . , i n , a 1 ⊗ · · · ⊗ a n − i, A(i 1 ) a 1 ⊗ · · · ⊗ A(i n ) a n ∈ i:V →V A(V ) , (4.14) for all tuples i : -morphisms (possibly of length zero), for k = 1, . . . , n, and all a k ∈ A(V k ), for k = 1, . . . , n. The tuple in the first term of (4.14) is defined by composition i i 1 , . . . , i n := i 1 i 11 , . . . , i 1 i 1|V 1 | , . . . , i n i n1 , . . . , i n i n|V n | (4.15a) and the expressions A(i) a in the second term are determined by , a 1 ⊗ · · · ⊗ a n −→ A(i 1 ) a 1 · · · A(i n ) a n , where we used square brackets to indicate equivalence classes in (4.9).
Remark 4.4. The construction of the algebra ext A(V ) above admits an intuitive graphical interpretation: We shall visualize the (homogeneous) elements (i, a) in (4.11) by decorated trees V a 1 a n · · · (4.17) where a k ∈ A(V k ) is an element of the algebra A(V k ) associated to the interior region V k ⊆ V , for all k = 1, . . . , n. We interpret such a decorated tree as a formal product of the formal pushforward along i : V → V of the family of observables a k ∈ A(V k ). The product (4.12) is given by concatenation of the inputs of the individual decorated trees, i.e.
where the decorated tree on the right-hand side has n + m inputs. The * -involution (4.13) may be visualized by reversing the input profile and applying * to each element a k ∈ A(V k ). Finally, the * -ideal in (4.14) implements the following relations: Assume that (i, a) is such that the sub-family of embeddings (i k , i k+1 , . . . , i l ) : (V k , V k+1 , . . . , V l ) → V factorizes through some common interior region, say V ′ ⊆ V . Using the original functor A ∈ QFT(int M ), we may form the product A(i k )(a k ) · · · A(i l )(a l ) in the algebra A(V ′ ), which we denote for simplicity by a k · · · a l ∈ A(V ′ ). We then have the relation V a 1 a k a l a n · · · · · · · · · V a 1 a k · · · a l a n · · · · · · ∼ (4.19) which we interpret as follows: Whenever (i k , i k+1 , . . . , i l ) : (V k , V k+1 , . . . , V l ) → V is a sub-family of embeddings that factorizes through a common interior region V ′ ⊆ V , then the formal product of the formal pushforward of observables is identified with the formal pushforward of the actual product of observables on V ′ . △

Characterization of boundary quantum field theories
In the previous section we established a universal construction that allows us to extend quantum field theories A ∈ QFT(int M ) that are defined only on the interior int M of a spacetime M with timelike boundary to the whole spacetime. The extension ext A ∈ QFT(M ) is characterized abstractly by the ext ⊣ res adjunction in (4.3). We now shall reverse the question and ask which quantum field theories B ∈ QFT(M ) on M admit a description in terms of (quotients of) our universal extensions.
Given any quantum field theory B ∈ QFT(M ) on the whole spacetime M , we can use the right adjoint in (5.1a) Using our model for the extension functor given in (4.9) (and the formulas following this equation), In order to establish positive comparison results, we have to introduce the concept of ideals of quantum field theories. Proof. The fact that ker κ defines a functor follows from naturality of κ. Property (ii) in Definition 5.1 holds true by construction. Property (i) is a consequence of the fact that kernels of unital * -algebra morphisms κ V : B(V ) → B(V ′ ) are two-sided * -ideals.
Remark 5.4. Using the concept of ideals, we may canonically factorize (5.1) according to the diagram where both the projection π B and the inclusion λ B are QFT(M )-morphisms. △ As a last ingredient for our comparison result, we have to introduce a suitable notion of additivity for quantum field theories on spacetimes with timelike boundary. We refer to [Few13, Definition 2.3] for a notion of additivity on globally hyperbolic spacetimes.  We can now prove our first characterization theorem for boundary quantum field theories. (1) The V -component of the canonical inclusion in (5.2) is an Alg-isomorphism.
(2) B is additive at the object is in the interior int M of M . Using our model for the extension functor given in (4.9) (and the formulas following this equation), we obtain an Alg-morphism (5.4) Composing this morphism with the V -component of ǫ B given in (5.1), we obtain a commutative diagram ext res B(V ) Next we observe that the images of the Alg-morphisms (5.4), for all i int : V int → V with V int in the interior, generate ext res B(V ). Combining this property with (5.5), we conclude that (ǫ B ) V is a surjective map if and only if B is additive at V . Hence, (λ B ) V given by (5.2), which is injective by construction, is an Alg-isomorphism if and only if B is additive at V . We shall now refine this characterization theorem by showing that QFT add (M ) is equivalent, as a category, to a category describing quantum field theories on the interior of M together with suitable ideals of their universal extensions. The precise definitions are as follows. Using the explicit expression for ǫ ext A/I given in (5.1) and the explicit expression for η A given by We shall illustrate this assertion in Section 6 below using the explicit example given by the free Klein-Gordon field.
Let us also note that there is a reason why our universal extension captures only the class of additive quantum field theories on M . Recall that ext A ∈ QFT(M ) takes as an input a quantum field theory A ∈ QFT(int M ) on the interior int M of M . As a consequence, the extension ext A can only have knowledge of the 'degrees of freedom' that are generated in some way out of the interior regions. Additive theories in the sense of Definition 5.5 are precisely the theories whose 'degrees of freedom' are generated out of those localized in the interior regions. △

Example: Free Klein-Gordon theory
In order to illustrate and make more explicit our abstract constructions developed in the previous sections, we shall consider the simple example given by the free Klein-Gordon field. From now on M will be a globally hyperbolic spacetime with timelike boundary, see Definition 2.9. This assumption implies that all interior regions R int M are globally hyperbolic spacetimes with empty boundary, see Proposition 2.11. This allows us to use the standard techniques of [BGP07, Section 3] on such regions.
Let M be a globally hyperbolic spacetime with timelike boundary, see Definition 2.9. The free Klein-Gordon theory on R int M [C −1 int M ] is given by the following standard construction, see e.g. [BD15,BDH13] for expository reviews. On the interior int M , we consider the Klein-Gordon operator where is the d'Alembert operator and m ≥ 0 is a mass parameter. When restricting P to regions V ∈ R int M [C −1 int M ], we shall write It follows from [BGP07] that there exists a unique retarded/advanced Green's operator int M ] it assigns the associative and unital *algebra K(V ) that is freely generated by Φ V (f ), for all f ∈ C ∞ c (V ), modulo the two-sided * -ideal generated by the following relations: To a morphism i : Alg assigns the algebra map that is specified on the generators by pushforward along i (which we shall suppress) (6.5) The naturality of τ (i.e. naturality of the causal propagator, cf. e.g. [BGP07, Section 4.3]) entails that the assignment K defines a quantum field theory in the sense of Definition 3.5.

Universal extension:
Using the techniques developed in Section 4, we may now extend the Klein-Gordon theory K ∈ QFT(int M ) from the interior int M to the whole spacetime M . In particular, using (4.9) (and the formulas following this equation), one could directly compute the universal extension ext K ∈ QFT(M ). The resulting expressions, however, can be considerably simplified. We therefore prefer to provide a more convenient model for the universal extension ext K ∈ QFT(M ) by adopting the following strategy: We first make an 'educated guess' for a theory K ext ∈ QFT(M ) which we expect to be the universal extension of K ∈ QFT(int M ). (This was inspired by partially simplifying the direct computation of the universal extension.) After this we shall prove that K ext ∈ QFT(M ) satisfies the universal property that characterizes ext K ∈ QFT(M ). Hence, there exists a (unique) isomorphism ext K ∼ = K ext in QFT(M ), which means that our K ext ∈ QFT(M ) is a model for the universal extension ext K.
Let us define the functor K ext : R M [C −1 M ] → Alg by the following assignment: To any region V ∈ R M [C −1 M ], which may intersect the boundary, we assign the associative and unital * -algebra K ext (V ) that is freely generated by Φ V (f ), for all f ∈ C ∞ c (int V ) in the interior int V of V , modulo the two-sided ideal generated by the following relations: Remark 6.1. We note that our partially-defined CCR are consistent in the following sense: int . Using the partially-defined CCR for both V int and V ′ int , we obtain the . This holds true due to the following argument: It follows by construction that supp(f ) ∪ supp(g) ⊆ V int ∩ V ′ int and hence due to naturality of the τ 's we obtain Hence, for any fixed pair f, g ∈ C ∞ c (int V ), the partially-defined CCR are independent of the choice of V int (if one exists). △ To a morphism i : Alg assigns the algebra map that is specified on the generators by the pushforward along i (which we shall suppress) (6.7) Compatibility of the map (6.7) with the relations in K ext is a straightforward check.
Recalling the embedding functor J : given in (4.2), we observe that the diagram of functors commutes via the natural transformation γ : K → K ext J with components specified on the generators by the identity maps Notice that γ is a natural isomorphism because int V int = V int and the partially-defined CCR on any interior region V int coincides with the CCR.
Regarding the partially-defined CCR, let We may choose the cover given by the single region i : V int → V together with its partition of unity. We obtain for the commutator (6.16) which implies that the partially-defined CCR are preserved.
Naturality of the components (6.13) is easily verified. Uniqueness of the resulting natural transformation ζ : K ext → B is a consequence of uniqueness of ζJ and of the fact that the Alg- This completes the proof.
Given any region V ∈ R M [C −1 M ] in M , which may intersect the boundary, we use the canonical inclusion i : V → M to define local retarded/advanced Green's operators where i * denotes the pushforward of compactly supported functions (i.e. extension by zero) and i * the pullback of functions (i.e. restriction). Since V ⊆ M is causally convex, it follows that J ± M (p) ∩ V = J ± V (p) for all p ∈ V . Therefore G ± V satisfies the axioms of a retarded/advanced Green's operator for P V on V . (Here we regard V as a globally hyperbolic spacetime with timelike boundary, see Proposition 2.11. J ± V (p) denotes the causal future/past of p in the spacetime V .) In particular, for all interior regions V int ∈ R int M [C −1 int M ] in M , by combining Proposition 2.11 and [BGP07, Corollary 3.4.3] we obtain that G ± V int as defined in (6.18) is the unique retarded/advanced Green's operator for the restricted Klein-Gordon operator P V int on the globally hyperbolic spacetime V int with empty boundary.
Consider any adjoint-related pair (G + , G − ) of Green's operator for P on M . For all V ∈ R M [C −1 M ], we set I G ± (V ) ⊆ K ext (V ) to be the two-sided * -ideal generated by the following relations: with G V := G + V − G − V the causal propagator and vol V the canonical volume form on V .
The fact that the pair (G + , G − ) is adjoint-related (cf. Definition 6.3) implies that for all V ∈ R M [C −1 M ] the causal propagator G V is formally skew-adjoint, hence τ V is antisymmetric.
Proposition 6.5. I G ± ⊆ K ext is an ideal that is trivial on the interior (cf. Definition 5.8).
Proof. Functoriality of I G ± : R M [C −1 M ] → Vec is a consequence of (6.18), hence I G ± ⊆ K ext is an ideal in the sense of Definition 5.1. It is trivial on the interior because for all interior regions V int ∈ R int M [C −1 int M ], the Green's operators defined by (6.18) are the unique retarded/advanced Green's operators for P V int and hence the relations imposed by I G ± (V int ) automatically hold true in K ext (V int ) on account of the (partially-defined) CCR.
Remark 6.6. We note that the results of this section still hold true if we slightly weaken the hypotheses of Definition 2.9 by assuming the strong causality and the compact double-cones property only for points in the interior int M of M . In fact, int M can still be covered by causally convex open subsets and any causally convex open subset U ⊆ int M becomes a globally hyperbolic spacetime with empty boundary once equipped with the induced metric, orientation and time-orientation. △ Example 6.7. Consider the sub-spacetime M := R m−1 ×[0, π] ⊆ R m of m-dimensional Minkowski spacetime, which has a timelike boundary ∂M = R m−1 × {0, π}. The constructions in [DNP16] define an adjoint-related pair (G + , G − ) of Green's operators for P on M that corresponds to Dirichlet boundary conditions. Using this as an input for our construction above, we obtain a quantum field theory K ext I G ± ∈ QFT(M ) that may be interpreted as the Klein-Gordon theory on M with Dirichlet boundary conditions. It is worth to emphasize that our theory in general does not coincide with the one constructed in [DNP16]. To provide a simple argument, let us focus on the case of m = 2 dimensions, i.e. M = R × [0, π], and compare our global algebra A BDS (M ) := K ext I G ± (M ) with the global algebra A DNP (M ) constructed in [DNP16]. Both algebras are CCR-algebras, however the underlying symplectic vector spaces differ: The symplectic vector space underlying our global algebra A BDS (M ) is C ∞ c (int M ) P C ∞ c (int M ) with the symplectic structure (6.19). Using that the spatial slices of M = R × [0, π] are compact, we observe that the symplectic vector space underlying A DNP (M ) is given by the space Sol Dir (M ) of all solutions with Dirichlet boundary condition on M (equipped with the usual symplectic structure). The causal propagator defines a symplectic map G : C ∞ c (int M ) P C ∞ c (int M ) → Sol Dir (M ), which however is not surjective for the following reason: Any ϕ ∈ C ∞ c (int M ) has by definition compact support in the interior of M , hence the support of Gϕ ∈ Sol Dir (M ) is schematically as follows supp Gϕ (6.20) The usual mode functions Φ k (t, x) = cos( √ k 2 + m 2 t) sin(kx) ∈ Sol Dir (M ), for k ≥ 1, are clearly not of this form, hence G : C ∞ c (int M ) P C ∞ c (int M ) → Sol Dir (M ) cannot be surjective. As a consequence, the models constructed in [DNP16] are in general not additive from the interior and our construction K ext I G ± should be interpreted as the maximal additive subtheory of these examples. It is interesting to note that there exists a case where both constructions coincide: Consider the sub-spacetime M := R m−1 × [0, ∞) ⊆ R m of Minkowski spacetime with m ≥ 4 even and take a massless real scalar field with Dirichlet boundary conditions. Using Huygens' principle and the support properties of the Green's operators one may show that our algebra A BDS (M ) is isomorphic to the construction in [DNP16]. together with natural transformations η : id C → GF (called unit) and ǫ : F G → id D (called counit) that satisfy the triangle identities We call F the left adjoint of G and G the right adjoint of F , and write F ⊣ G.
Definition A.2. An adjoint equivalence is an adjunction for which both the unit η and the counit ǫ are natural isomorphisms. Existence of an adjoint equivalence in particular implies that C ∼ = D are equivalent as categories.

Localizations:
Localizations of categories are treated for example in [KS06, Section 7.1]. In our paper we restrict ourselves to small categories.