Algebraic Quantum Field Theory on Spacetimes with Timelike Boundary

Algebraic Quantum Field Theory on Spacetimes with Timelike Boundary
Benini, Marco; Dappiaggi, Claudio; Schenkel, Alexander
2018-05-30 00:00:00
Ann. Henri Poincar´ e Online First c 2018 The Author(s) Annales Henri Poincar´ e https://doi.org/10.1007/s00023-018-0687-1 Algebraic Quantum Field Theory on Spacetimes with Timelike Boundary Marco Benini , Claudio Dappiaggi and Alexander Schenkel Dedicated to Klaus Fredenhagen on the occasion of his 70th birthday. Abstract. We analyze quantum ﬁeld theories on spacetimes M with time- like 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 intM . The unit of this adjunc- tion is a natural isomorphism, which implies that our universal extension satisﬁes Kay’s F-locality property. Our main result is the following char- acterization 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 intM 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. 1. Introduction and Summary Algebraic quantum ﬁeld theory is a powerful and far developed framework to address model-independent aspects of quantum ﬁeld theories on Minkowski spacetime [18] and more generally on globally hyperbolic spacetimes [7]. In addition to establishing the axiomatic foundations for quantum ﬁeld theory, the algebraic approach has provided a variety of mathematically rigorous con- structions of non-interacting models, see e.g., the reviews [1, 3, 4], and more in- terestingly also perturbatively interacting quantum ﬁeld theories, see e.g., the recent monograph [26]. It is worth emphasizing that many of the techniques involved in such constructions, e.g., existence and uniqueness of Green’s opera- tors and the singular structure of propagators, crucially rely on the hypothesis that the spacetime is globally hyperbolic and has empty boundary. M. Benini et al. Ann. Henri Poincar´ e 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 em- ployed 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 [19, 20, 33], the latter extending the analysis of [31]. For speciﬁc choices of boundary con- ditions, there exist successful constructions of algebraic quantum ﬁeld theories on spacetimes with timelike boundary, see e.g., [8, 10–12]. 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 [11, 32], should be modiﬁed accordingly. Unfortunately, model-independent results on algebraic quantum ﬁeld the- ory on spacetimes with timelike boundary are more scarce. There are, how- ever, some notable and very interesting works in this direction: On the one hand, Rehren’s proposal for algebraic holography [25] initiated the rigorous study of quantum ﬁeld theories on the anti-de Sitter spacetime. This has been further elaborated in [13] and extended to asymptotically AdS spacetimes in [27]. On the other hand, inspired by Fredenhagen’s universal algebra [15–17], a very interesting construction and analysis of global algebras of observables on spacetimes with timelike boundaries has been performed in [30]. 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 space- times 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 the- ories. Such questions can be naturally addressed by using techniques from the recently developed operadic approach to algebraic quantum ﬁeld theory [5]. Let us describe rather informally the basic idea of our construction and its implications: Given a spacetime M with timelike boundary, an algebraic AQFT on Spacetimes with Timelike Boundary quantum ﬁeld theory on M is a functor B : R → 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 sub- category of regions in the interior of M by R ⊆R , we may restrict int M M any theory B ∈ QFT(M)toatheory 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 [22]. 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 [5]. In particular, the ext res adjunction investigated in the present paper is one concrete instance of a whole fam- ily 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)ona 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 :extres B → B of the adjunction. Our result in Theorem 5.6 and Corollary 5.7 is that induces an isomorphism ext res B/ ker B 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)onthe 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 M. Benini et al. Ann. Henri Poincar´ e 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 [30] and the concrete constructions in [8, 10–12]. The remainder of this paper is structured as follows: In Sect. 2, we recall some basic deﬁnitions and results about the causal structure of spacetimes with timelike boundaries, see also [9, 29]. In Sect. 3, we provide a precise deﬁnition of the categories QFT(M)and QFT(int M ) by using the ideas of [5]. Our universal boundary extension is developed in Sect. 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 Sect. 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., [12]. We included “Appendix A” to state some basic deﬁnitions and results of category theory which will be used in our work. 2. Spacetimes with Timelike Boundary We collect some basic facts about spacetimes with timelike boundary, following [29, Section 3.1] and [9, Section 2.2]. For a general introduction to Lorentzian geometry, we refer to [2, 24], see also [1, Sections 1.3 and A.5] for a concise presentation. We use the term manifold with boundary to refer to a Hausdorﬀ, second countable, m-dimensional smooth manifold M with boundary, see e.g., [23]. This deﬁnition subsumes ordinary manifolds as manifolds with empty bound- ary ∂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 bound- ary 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 de- velopment. Proposition 2.3. Let S, S ⊆ M be subsets of a time-oriented Lorentzian man- ifold M with boundary. Then, the following holds true: (a) S ⊆ S implies D(S) ⊆ D(S ); AQFT on Spacetimes with Timelike Boundary (b) S ⊆ D(S)= D(D(S)); (c) D(D(S) ∩ D(S )) = D(S) ∩ D(S ). We denote by J (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 (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. Deﬁnition 2.4. Let M be a time-oriented Lorentzian manifold with boundary. + − We say that a subset S ⊆ M is causally convex in M if J (S) ∩ J (S) ⊆ S. M M We say that two subsets S, S ⊆ M are causally disjoint in M if (J (S) ∪ J (S)) ∩ S = ∅. The following properties are simple consequences of these deﬁnitions. Proposition 2.5. Let S, S ⊆ M be two subsets of a time-oriented Lorentzian manifold M with boundary. Then, the following holds true: (a) D(S) and D(S ) are causally disjoint if and only if S and S are causally disjoint; (b) Suppose S and S are causally disjoint. Then, the disjoint union S S ⊆ M is causally convex if and only if both S and S are causally convex. 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 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 . (ii) C ⊆ Mor R is the subset of Cauchy morphisms in R , i.e., inclusions M M M i : U → U such that D(U)= D(U ). (iii) R ⊆R is the full subcategory whose objects are contained in the int M M interior int M . We denote by C ⊆C the Cauchy morphisms between int M M objects of R . int M Proposition 2.8. Let M be a spacetime with timelike boundary. For each subset S ⊆ M and each object U ∈R , i.e., a causally convex open subset U ⊆ M, the following holds true: ± ± (a) I (S) is the largest open subset of J (S); M M ± ± ± ± ± (b) J (I (S)) = I (S)= I (J (S)); M M M M M (c) S ⊆ int M implies D(S) ⊆ int M; (d) D(U ) ⊆ M is causally convex and open, i.e., D(U ) ∈R . M M. Benini et al. Ann. Henri Poincar´ e Proof. (a) and (b): These are standard results in the case of empty bound- ary, see e.g., [1, 2, 24]. The extension to spacetimes with non-empty timelike boundary can be found in [29, Section 3.1.1]. (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 . (d): D(U ) ⊆ M is causally convex by the deﬁnition of Cauchy develop- ment and by the causal convexity of U ⊆ M . To check that D(U ) ⊆ M is open, we use quasi-limits as in [24, Deﬁnition 14.7 and Proposition 14.8]: First, ob- serve that I (U ) ⊆ M is open by (a). Hence, it is the same to check whether + − a subset of I := I (U ) ∪ I (U)isopenin M or in I (with the induced topol- M M ogy). Indeed, U ⊆ D(U ) ⊆ I because U is open in M . From now on, in place of M , let us therefore consider I, equipped with the induced metric, orientation and time-orientation. By contradiction, assume that there exists p ∈ D(U )\U such that all of its neighborhoods intersect the complement of D(U ). Then, there exists a sequence {α } of inextensible causal curves in I never meeting U such that {α (0)} converges to p. We ﬁx a convex cover of I reﬁning the open + − cover {I (U ),I (U )}. Relative to the ﬁxed convex cover, the construction M M of quasi-limits allows us to obtain from {α } 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 and p , two successive k k+1 limit points for {α } contained in some element of the ﬁxed convex cover. It + − follows that either p or p belongs to J (U ) ∩ J (U ), which is contained k k+1 I I in U by causal convexity. Hence, we found a subsequence {α } of {α } and n n a sequence of parameters {s } such that {α (s )} converges to a point of U j n j (either p or p ). By construction the sequence {α (s )} is contained in k k+1 n j 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 Sect. 6. The following deﬁnition is due to [9, Section 2.2] and [29, Section 3.1.2]. Deﬁnition 2.9. A spacetime M with timelike boundary is called globally hy- perbolic if the following two properties hold true: (i) Strong causality Every open neighborhood of each point p ∈ M contains a causally convex open neighborhood of p. + − (ii) Compact double-cones J (p) ∩ J (q) is compact for all p, q ∈ M . M M AQFT on Spacetimes with Timelike Boundary Remark 2.10. In the case of empty boundary, this deﬁnition agrees with the usual one in [1, 2, 24]. Simple examples of globally hyperbolic spacetimes with m−1 m non-empty timelike boundary are the half space {x ≥ 0}⊆ R , the spatial m−1 m 1 2 m−1 2 m slab {0 ≤ x ≤ 1}⊆ R and the cylinder {(x ) +···+(x ) ≤ 1}⊆ R in Minkowski spacetime R ,for m ≥ 2, as well as all causally convex open subsets thereof. The following results follow immediately from Deﬁnition 2.9 and Propo- sition 2.8. Proposition 2.11. Let M be a globally hyperbolic spacetime with timelike bound- ary. (a) M admits a cover by causally convex open subsets. (b) For each U ∈R , i.e., a causally convex open subset U ⊆ M, both U and D(U ) are globally hyperbolic spacetimes with (possibly empty) boundary when equipped with the metric, orientation and time-orientation induced by M.Ifmoreover U ⊆ int M is contained in the interior, then both U and D(U ) have empty boundary. 3. Categories of Algebraic Quantum Field Theories Let M be a spacetime with timelike boundary. (In this section we do not have to assume that M is globally hyperbolic in the sense of Deﬁnition 2.9). Recall the category R of open and causally convex regions in M and the subset C M M of Cauchy morphisms (cf. Deﬁnition 2.7). Together with our notion of causal disjointness from Deﬁnition 2.4, these data provide the geometrical input for the traditional deﬁnition of algebraic quantum ﬁeld theories on M . Deﬁnition 3.1. An algebraic quantum ﬁeld theory on M is a functor A : R −→ Alg (3.1) 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 : U → U ← U : i , 1 1 2 2 the induced commutator A(i )(−), A(i )(−) : A(U ) ⊗ A(U ) −→ A(U ) (3.2) 1 2 1 2 A(U ) is zero. (ii) Time-slice axiom For all Cauchy morphisms (i : U → U ) ∈C , the map A(i): A(U ) −→ A(U ) (3.3) is an Alg-isomorphism. We denote by qft(M ) ⊆ Alg the full subcategory of the category of func- tors from R 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). M. Benini et al. Ann. Henri Poincar´ e We shall now show that there exists an alternative, but equivalent, de- scription of the category qft(M ) which will be more convenient for the tech- nical constructions in our paper. Following [5, Section 4.1], we observe that the time-slice axiom in Deﬁnition 3.1(ii) is equivalent to considering functors −1 B : R C → Alg that are deﬁned on the localization of the category R at the set of Cauchy morphisms C . See Deﬁnition A.4 for the deﬁni- M M tion of localizations of categories. By abstract arguments as in [5, Section 4.6], one observes that the universal property of localizations implies that the cat- egory qft(M ) is equivalent to the full subcategory of the functor category −1 R C −1 M [ ] Alg whose objects are all functors B : R C → Alg that satisfy −1 the causality axiom for the pushforward orthogonality relation on R C . Loosely speaking, this means that the time-slice axiom in Deﬁnition 3.1(ii) −1 can be hard-coded by working on the localized category R C instead of using the usual category R of regions in M . The aim of the remainder of this section is to provide an explicit model −1 for the localization functor R →R C . With this model, it will be- M M come particularly easy to verify the equivalence between the two alternative descriptions of the category qft(M ). Let us denote by −1 R C ⊆R (3.4a) M M the full subcategory of R whose objects V ⊆ M are stable under Cauchy de- velopment, i.e., D(V )= V where D(V ) ⊆ M denotes the Cauchy development (cf. Deﬁnition 2.2). In the following, we shall always use letters like U ⊆ M for generic regions in R and V ⊆ M for regions that are stable under Cauchy −1 development, i.e., objects in R C . Recall from Deﬁnition 2.7 that each object U ∈R is a causally convex open subset U ⊆ M , hence the Cauchy de- velopment D(U ) ⊆ M is a causally convex open subset by Proposition 2.8(d), which is stable under Cauchy development by Proposition 2.3(b). This shows −1 that D(U ) is an object of R C . Furthermore, a morphism i : U → U in R is an inclusion U ⊆ U , which induces an inclusion D(U ) ⊆ D(U )by −1 Proposition 2.3(a) and hence a morphism D(i): D(U ) → D(U )in R C . We deﬁne a functor −1 D : R −→ R C (3.4b) M M by setting on objects and morphisms U −→ D(U ), (i : U → U ) −→ D(i): D(U ) → D(U ) . (3.4c) Furthermore, let us write −1 I : R C −→ R (3.5) M M for the full subcategory embedding. Lemma 3.2. D and I form an adjunction (cf. Deﬁnition A.1) −1 D : R R C : I (3.6) M M M AQFT on Spacetimes with Timelike Boundary −1 whose counit is a natural isomorphism (in fact, the identity), hence R C is a full reﬂective subcategory of R . Furthermore, the components of the unit are Cauchy morphisms. Proof. For U ∈R ,the U -component of the unit η :id −→ ID (3.7) 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 2.7(ii). For V ∈ −1 R C ,the V -component of the counit : DI −→ id −1 (3.8) R C M [ ] is given by the identity of the object D(V )= V . The triangle identities hold trivially. −1 −1 Proposition 3.3. The category R C and the functor D : R →R C M M M M M deﬁned in (3.4) provide a model for the localization of R at C . M M Proof. We have to check all the requirements listed in Deﬁnition A.4. (a) By Deﬁnition 2.7, for each Cauchy morphism i : U → U one has D(U)= −1 D(U ) and hence D(i)=id is an isomorphism in R C . D(U ) M (b) Let F : R → D be any functor to a category D that sends morphisms in C to D-isomorphisms. Using Lemma 3.2, we deﬁne F := FI : M W −1 R C → D and consider the natural transformation Fη : F → FI D = F D obtained by the unit of the adjunction D I. Because all components of η are Cauchy morphisms (cf. Lemma 3.2), Fη is a natural isomorphism. −1 (c) Let G, H : R C → D be two functors. We have to show that the map Hom G, H −→ Hom R GD, HD (3.9) −1 R C D [ ] is a bijection. Let us ﬁrst prove injectivity: Let ξ, ξ : G → H be two natural transformations such that ξD = ξD. Using Lemma 3.2, we obtain commutative diagrams ξDI ξDI GDI HDI GDI HDI (3.10) G H G H G H G H where the vertical arrows are natural isomorphisms because the counit is an isomorphism. Recalling that by hypothesis ξD = ξD, it follows that ξ = ξ. Hence, the map (3.9) is injective. M. Benini et al. Ann. Henri Poincar´ e 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 GD HD (3.11) GDη HDη GDID HDID χID 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 G H (3.12) −1 GDI HDI χI where we use that 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 −1 orthogonality relation on the localized category R C , both of which are provided by [5, Lemma 4.29]: (1) The pullback orthogonality relation along −1 the full subcategory embedding I : R C →R and (2) the pushforward M M −1 orthogonality relation along the localization functor D : R →R C . M M In our present scenario, both constructions coincide and one concludes that −1 two R C -morphisms are orthogonal precisely when they are orthogonal in R . Summing up, we obtain −1 Lemma 3.4. We say that two morphisms in the full subcategory R C ⊆ R are orthogonal precisely when they are orthogonal in R (i.e., causally M M −1 disjoint, cf. Deﬁnition 2.4). Then, both functors D : R →R C and M M −1 I : R C →R preserve (and also detect) the orthogonality relations. M M 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. −1 R [C ] Deﬁnition 3.5. We denote by QFT(M ) ⊆ Alg the full subcategory −1 whose objects are all functors B : R C → Alg that satisfy the following version of the causality axiom: For all causally disjoint inclusions i : V → 1 1 −1 V ← V : i in R C , i.e., V , V and V are stable under Cauchy develop- 2 2 M 1 2 ment, the induced commutator B(i )(−), B(i )(−) : B(V ) ⊗ B(V ) −→ B(V ) (3.13) 1 2 1 2 B(V ) is zero. AQFT on Spacetimes with Timelike Boundary Theorem 3.6. By pullback, the adjunction D I of Lemma 3.2 induces an adjoint equivalence (cf. Deﬁnition A.2) ∗ ∗ I : qft(M ) ∼ QFT(M): D . (3.14) In particular, the two categories qft(M ) of Deﬁnition 3.1 and QFT(M ) of Deﬁnition 3.5 are equivalent. −1 Proof. It is trivial to check that the adjunction D : R R C : I M M induces an adjunction −1 ∗ R R [C ] ∗ M M I : Alg Alg : D (3.15) ∗ ∗ between functor categories. Explicitly, the unit η :id R → D I has Alg M components ∗ ∗ η := Aη : A −→ D (I (A)) = AID, (3.16) where A : R → Alg is any functor and η :id → ID denotes the unit of M R ∗ ∗ D I. The counit : I D → id has components −1 R C [ ] Alg ∗ ∗ := B : I (D (B)) = BDI −→ B, (3.17) −1 where B : R C → Alg is any functor and : DI → id −1 denotes M R C M [ ] ∗ ∗ the counit of D I. The triangle identities for I D follow directly from those of D I. Next, we have to prove that this adjunction restricts to the claimed source and target categories in (3.14). Given A ∈ qft(M ) ⊆ Alg , the functor ∗ −1 I (A)= A I : R C → Alg satisﬁes the causality axiom of Deﬁnition 3.5 because of Lemma 3.4. Hence, I (A) ∈ QFT(M ). Vice versa, given B ∈ −1 R C M [ ] ∗ QFT(M ) ⊆ Alg , the functor D (B)= B D : R → Alg satisﬁes the causality axiom of Deﬁnition 3.5 because of Lemma 3.4 and the time-slice axiom of Deﬁnition 3.5 because D sends by construction morphisms in C to isomorphisms. Hence, D (B) ∈ qft(M ). Using Lemma 3.2, we obtain that the counit of the restricted adjunc- tion (3.14) is an isomorphism. Furthermore, all components of η are Cauchy morphisms, hence η = 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. Remark 3.7. Theorem 3.6 provides us with a constructive prescription of how to change between the two equivalent formulations of algebraic quan- tum ﬁeld theories given in Deﬁnitions 3.1 and 3.5. Concretely, given any A ∈ qft(M ), i.e., a functor A : R → Alg satisfying the causality and time-slice axioms as in Deﬁnition 3.1, the corresponding quantum ﬁeld the- ory I (A) ∈ QFT(M ) in the sense of Deﬁnition 3.5 reads as follows: It ∗ −1 is the functor I (A)= A I : R C → Alg on the category of regions V ⊆ M that are stable under Cauchy development, which assigns to V ⊆ M the algebra A(V ) ∈ Alg and to an inclusion i : V → V the algebra map M. Benini et al. Ann. Henri Poincar´ e A(i): A(V ) → A(V ). More interestingly, given B ∈ QFT(M ), i.e., a func- −1 tor B : R C → Alg satisfying the causality axiom as in Deﬁnition 3.5, the corresponding quantum ﬁeld theory D (B) ∈ qft(M ) in the sense of Def- inition 3.1 reads as follows: It is the functor D (B)= B D : R → Alg deﬁned on the category of (not necessarily Cauchy development stable) re- gions U ⊆ M , which assigns to U ⊆ M the algebra B(D(U )) corresponding to the Cauchy development of U and to an inclusion i : U → U the al- gebra map B(D(i)) : B(D(U )) → B(D(U )) associated with the inclusion D(i): D(U ) → D(U ) of Cauchy developments. Remark 3.8. It is straightforward to check that the results of this section still hold true when one replaces R with its full subcategory R of regions M int M contained in the interior of M and C with C (cf. Deﬁnition 2.7). This M int M follows from the observation that the Cauchy development of a subset of the interior of M is also contained in int M , as shown in Proposition 2.8(c). We denote by −1 R C int M [ ] int M QFT(int M ) ⊆ Alg (3.18) the category of algebraic quantum ﬁeld theories in the sense of Deﬁnition 3.5 on the interior regions of M . Concretely, an object A ∈ QFT(int M ) is a functor −1 A : R [C ] → Alg that satisﬁes the causality axiom of Deﬁnition 3.5 int M int M for causally disjoint interior regions. 4. Universal Boundary Extension The goal of this section is to develop a universal construction to extend quan- tum 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 quan- tum 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 Sect. 5 for more details on the properties (1) and (3). The starting point for this construction is the full subcategory inclusion R ⊆R deﬁned by selecting only the regions of R that lie in the int M M M interior of M (cf. Deﬁnition 2.7). We denote the corresponding embedding functor by j : R −→ R (4.1) int M M and notice that j preserves (and also detects) causally disjoint inclusions, i.e., j is a full orthogonal subcategory embedding in the terminology of [5]. Making use of Proposition 3.3, Lemma 3.2 and Remark 3.8, we deﬁne a functor J : −1 −1 R [C ] →R C on the localized categories via the commutative int M M int M M AQFT on Spacetimes with Timelike Boundary diagram −1 −1 R [C ] R [C ] (4.2a) int M M int M M I D R R int M M Notice that J is simply an embedding functor, which acts on objects and morphisms as V ⊆ int M −→ V ⊆ M, (i : V → V ) −→ (i : V → V ). (4.2b) 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 [5, Section 5.3] (see also [6] for details how to treat ∗- algebras) then imply that J induces an adjunction ext : QFT(int M ) QFT(M):res (4.3) between the category of quantum ﬁeld theories on the interior int M (cf. Re- mark 3.8) and the category of quantum ﬁeld theories on the whole spacetime M . The right adjoint res := J : QFT(M ) → QFT(int M ) is the pullback −1 −1 along J : R [C ] →R C , i.e., it restricts quantum ﬁeld theories int M M int M M deﬁned on M to the interior int M . The left adjoint ext : QFT(int M ) → QFT(M ) should be regarded as a universal extension functor which extends quantum ﬁeld theories on the interior int M to the whole spacetime M.The goal of this section is to analyze the properties of this extension functor and to develop an explicit model that allows us to do computations in the sections below. An important structural result, whose physical relevance is explained in Remark 4.2 below, is the following proposition. Proposition 4.1. The unit η :id −→ res ext (4.4) QFT(int M ) of the adjunction (4.3) is a natural isomorphism. Proof. This is a direct consequence of the fact that the functor J given in (4.2) is a full orthogonal subcategory embedding and the general result in [5, Proposition 5.6]. Remark 4.2. The physical interpretation of this result is as follows: Let A ∈ QFT(int M ) be a quantum ﬁeld theory deﬁned only on the interior int M of M and let B := ext A ∈ QFT(M ) denote its universal extension to the whole spacetime M.The A-component η : A −→ res ext A (4.5) of the unit of the adjunction (4.3) allows us to compare A with the restriction res B of its extension B =ext A. Since η is an isomorphism by Propo- sition 4.1, restricting the extension B recovers our original theory A up to M. Benini et al. Ann. Henri Poincar´ e isomorphism. This allows us to interpret the left adjoint ext : QFT(int M ) → QFT(M ) as a genuine extension prescription. Notice that this also proves that the universal extension ext A ∈ QFT(M ) of any theory A ∈ QFT(int M)on the interior satisﬁes F-locality [22]. We next address the question how to compute the extension functor ext : QFT(int M ) → QFT(M ) explicitly. A crucial step toward reaching this goal is to notice that ext may be computed by a left Kan extension. Proposition 4.3. Consider the adjunction −1 −1 R [C ] R [C ] int M M ∗ int M M Lan : Alg Alg : J , (4.6) −1 corresponding to left Kan extension along the functor J : R [C ] → int M int M −1 R C . Then, the restriction of Lan to QFT(int M ) induces a functor M J Lan : QFT(int M ) −→ QFT(M ) (4.7) that is left adjoint to the restriction functor res : QFT(M ) → QFT(int M ) in (4.3). Due to uniqueness (up to unique natural isomorphism) of adjoint functors (cf. Proposition A.3), it follows that ext Lan , i.e., (4.7) is a model for the extension functor ext in (4.3). Proof. A general version of this problem has been addressed in [5, Section 6]. Using in particular [5, Corollary 6.5], we observe that we can prove this propo- −1 sition by showing that every object V ∈R C is J -closed in the sense of [5, Deﬁnition 6.3]. In our present scenario, this amounts to proving that for all −1 causally disjoint inclusions i : V → V ← V : i with V ,V ∈R [C ] 1 1 2 2 1 2 int M int M −1 in the interior and V ∈R C not necessarily in the interior, there ex- ists a factorization of both i and i through a common interior region. Let 1 2 us consider the Cauchy development D(V V ) of the disjoint union and 1 2 the canonical inclusions j : V → D(V V ) ← V : j . As we explain be- 1 1 1 2 2 2 −1 low, D(V V ) ∈R [C ] is an interior region and j ,j provide the 1 2 int M 1 2 int M desired factorization: Since the open set V V ⊆ int M is causally con- 1 2 vex by Proposition 2.5(b), D(V V ) is causally convex, open and contained 1 2 in the interior int M by Proposition 2.8(c–d). It is, moreover, stable under Cauchy development by Proposition 2.3(b), which also provides the inclusion V ⊆ V V ⊆ D(V V ) inducing j ,for k =1, 2. Consider now the chain k 1 2 1 2 k of inclusions V ⊆ V V ⊆ V corresponding to i ,for k =1, 2. From the k 1 2 k stability under Cauchy development of V , V and V , we obtain also the chain 1 2 of inclusions V ⊆ D(V V ) ⊆ V ,for k =1, 2, that exhibits the desired k 1 2 factorization V (4.8) i i 1 2 D(i i ) 1 2 V D(V V ) V 1 1 2 2 j j 1 2 which completes the proof. AQFT on Spacetimes with Timelike Boundary We shall now brieﬂy review a concrete model for left Kan extension along full subcategory embeddings that was developed in [5, Section 6]. This model is obtained by means of abstract operadic techniques, but it admits an intuitive graphical interpretation that we explain in Remark 4.4 below. It allows us to compute quite explicitly the extension ext A =Lan A ∈ QFT(M)ofa quantum ﬁeld theory A ∈ QFT(int M ) deﬁned on the interior int M to the −1 whole spacetime M . The functor ext A : R C → Alg describing the −1 extended quantum ﬁeld theory reads as follows: To V ∈R C it assigns a quotient algebra ext A(V )= A(V ) ∼ (4.9) i:V →V that we will describe now in detail. The direct sum (of vector spaces) in (4.9) −1 runs over all tuples i : V → V of morphisms in R C , i.e., i =(i : V → M 1 1 V,...,i : V → V ) for some n ∈ Z , with the requirement that all sources n n ≥0 −1 V ∈R [C ] are interior regions. (Notice that the regions V are not k int M k int M 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 |V | A(V ):= A(V ), (4.10) k=1 where |V | is the length of the tuple. (For the empty tuple, we set A(∅)= C). This means that the (homogeneous) elements (i,a) ∈ A(V ) (4.11) i:V →V are given by pairs consisting of a tuple of morphisms i : V → V (with all V 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 (i,a)(i ,a ):= (i,i ),a ⊗ a , (4.12) where (i,i )=(i ,...,i ,i ,...,i ) is the concatenation of tuples. The unit 1 n 1 m element in (4.11)is 1 := (∅, 1), where ∅ is the empty tuple and 1 ∈ C, and the ∗-involution is deﬁned by ∗ ∗ (i ,...,i ),a ⊗ ··· ⊗ a := (i ,...,i ),a ⊗ ··· ⊗ a (4.13) 1 n 1 n n 1 n 1 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 ,...,i ,a ⊗ ··· ⊗ a − i, A(i ) a ⊗ ··· ⊗ A(i ) a ∈ A(V ) , 1 n 1 n 1 1 n n i:V →V (4.14) for all tuples i : V → V of length |V | = n ≥ 1 (with all V in the interior), all −1 tuples i : V → V of R [C ]-morphisms (possibly of length zero), for k int M k k int M M. Benini et al. Ann. Henri Poincar´ e k =1,...,n, and all a ∈ A(V ), for k =1,...,n. The tuple in the ﬁrst term k k of (4.14) is deﬁned by composition i i ,...,i := i i ,...,i i ,...,i i ,...,i i (4.15a) 1 11 1 1|V | n n1 n n|V | 1 n 1 n and the expressions A(i) a in the second term are determined by A(i): A(V ) −→ A(V ),a ⊗ ··· ⊗ a −→ A(i ) a ··· A(i ) a , (4.15b) 1 n 1 1 n n where multiplication in A(V ) is denoted by juxtaposition. To a morphism −1 −1 i : V → V in R C , the functor ext A : R C → Alg assigns the M M M M algebra map ext A(i ) : ext A(V ) −→ ext A(V ), [i,a] −→ i (i),a , (4.16) 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 intu- itive graphical interpretation: We shall visualize the (homogeneous) elements (i,a)in(4.11) by decorated trees ··· a a 1 n (4.17) where a ∈ A(V ) is an element of the algebra A(V ) associated to the interior k k k region V ⊆ 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 ∈ A(V ). The product (4.12) is given by concatenation of the k k inputs of the individual decorated trees, i.e., V V V ··· ··· ··· a a a 1 n 1 a a a 1 m m (4.18) 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 ∈ A(V ). Finally, the ∗-ideal in (4.14) implements the fol- k k lowing relations: Assume that (i,a) is such that the sub-family of embeddings (i ,i ,...,i ):(V ,V ,...,V ) → V factorizes through some common in- k k+1 l k k+1 l terior region, say V ⊆ V . Using the original functor A ∈ QFT(int M ), we may form the product A(i )(a ) ··· A(i )(a ) in the algebra A(V ), which we k k l l AQFT on Spacetimes with Timelike Boundary denote for simplicity by a ··· a ∈ A(V ). We then have the relation k l V V ··· · ·· ··· ··· · ·· a a a a a a ··· a a 1 k l n 1 k l n (4.19) which we interpret as follows: Whenever (i ,i ,...,i ):(V ,V ,...,V ) → k k+1 l k k+1 l V is a sub-family of embeddings that factorizes through a common interior re- gion 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 . 5. 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 (4.3) in order to restrict it to a theory res B ∈ QFT(int M ) on the interior of M . Applying now the extension func- tor, we obtain another quantum ﬁeld theory ext res B ∈ QFT(M)onthe whole spacetime M , which we would like to compare to our original theory B ∈ QFT(M ). A natural comparison map is given by the B-component of the counit :extres → id of the adjunction (4.3), i.e., the canonical QFT(M ) QFT(M )-morphism :extres B −→ B. (5.1a) Using our model for the extension functor given in (4.9) (and the formulas −1 following this equation), the R C V -component of explicitly reads M B as ( ) : B(V ) ∼−→ B(V ), [i,b] −→ B(i) b . (5.1b) B V i:V →V In order to establish positive comparison results, we have to introduce the concept of ideals of quantum ﬁeld theories. Deﬁnition 5.1. An ideal I ⊆ B of a quantum ﬁeld theory B ∈ QFT(M)isa −1 functor I : R C → Vec to the category of complex vector spaces, which satisﬁes the following properties: M. Benini et al. Ann. Henri Poincar´ e −1 (i) For all V ∈R C , I(V ) ⊆ B(V ) is a two-sided ∗-ideal of the unital ∗-algebra B(V ). −1 (ii) For all R C -morphisms i : V → V , the linear map I(i): I(V ) → I(V ) is the restriction of B(i): B(V ) → B(V ) to the two-sided ∗-ideals () () I(V ) ⊆ B(V ). Lemma 5.2. Let B ∈ QFT(M ) and I ⊆ B any ideal. Let us deﬁne B/I(V ):= −1 B(V )/I(V ) to be the quotient algebra, for all V ∈R C ,and B/I(i): B/I(V ) → B/I(V ) to be the Alg-morphism induced by B(i): B(V ) → −1 B(V ), for all R C -morphisms i : V → V .Then B/I ∈ QFT(M ) is a quantum ﬁeld theory on M which we call the quotient of B by I. −1 Proof. The requirements listed in Deﬁnition 5.1 ensure that B/I : R C → Alg is an Alg-valued functor. It satisﬁes the causality axiom of Deﬁnition 3.5 because this property is inherited from B ∈ QFT(M ) by taking quotients. Lemma 5.3. Let κ : B → B be any QFT(M )-morphism. Deﬁne the vec- tor space ker κ(V):=ker κ : B(V ) → B (V ) ⊆ B(V ), for all V ∈ −1 R C ,and ker κ(i):ker κ(V ) → ker κ(V ) to be the linear map induced −1 by B(i): B(V ) → B(V ), for all R C -morphisms i : V → V .Then −1 ker κ : R C → Vec is an ideal of B ∈ QFT(M ), which we call the kernel of κ. Proof. The fact that ker κ deﬁnes a functor follows from naturality of κ.Prop- erty (ii) in Deﬁnition 5.1 holds true by construction. Property (i) is a conse- quence of the fact that kernels of unital ∗-algebra morphisms κ : 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 ext res B B (5.2) ext res B ker where both the projection π and the inclusion λ are QFT(M )-morphi- B B sms. 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 [14, Deﬁnition 2.3] for a notion of additivity on globally hyperbolic spacetimes. Deﬁnition 5.5. A quantum ﬁeld theory B ∈ QFT(M ) on a spacetime M with timelike boundary is called additive (from the interior) at the object −1 V ∈R C if the algebra B(V ) is generated by the images of the Alg- −1 morphisms B(i ): B(V ) → B(V ), for all R C -morphism i : V → int int M int int −1 V whose source V ∈R [C ] is in the interior int M of M . We call int int M int M B ∈ QFT(M ) additive (from the interior) if it is additive at every object AQFT on Spacetimes with Timelike Boundary −1 V ∈R C . The full subcategory of additive quantum ﬁeld theories on M add is denoted by QFT (M ) ⊆ QFT(M ). We can now prove our ﬁrst characterization theorem for boundary quan- tum ﬁeld theories. Theorem 5.6. Let B ∈ QFT(M ) be any quantum ﬁeld theory on a (not necessarily globally hyperbolic) spacetime M with timelike boundary and let −1 V ∈R C . Then the following are equivalent: (1) The V -component (λ ) :extres B(V ) ker (V ) −→ B(V ) (5.3) B V B of the canonical inclusion in (5.2) is an Alg-isomorphism. −1 (2) B is additive at the object V ∈R C . −1 Proof. Let i : V → V be any R C -morphism whose source V ∈ int int M int −1 R [C ] is in the interior int M of M . Using our model for the extension int M int M functor given in (4.9) (and the formulas following this equation), we obtain an Alg-morphism [i , −]: B(V ) −→ ext res B(V ),b −→ [i ,b]. (5.4) int int int Composing this morphism with the V -component of given in (5.1), we obtain a commutative diagram ( ) B V ext res B(V ) B(V ) (5.5) [i ,−] B(i ) int int B(V ) int for all i : V → V with V in the interior. int int int Next we observe that the images of the Alg-morphisms (5.4), for all i : V → V with V in the interior, generate ext res B(V ). Combining int int int this property with (5.5), we conclude that ( ) is a surjective map if and B V only if B is additive at V . Hence, (λ ) given by (5.2), which is injective by B V construction, is an Alg-isomorphism if and only if B is additive at V . Corollary 5.7. λ :extres B ker → B given by (5.2) is a QFT(M )- B B add isomorphism if and only if B ∈ QFT (M ) ⊆ QFT(M ) is additive in the sense of Deﬁnition 5.5. We shall now reﬁne this characterization theorem by showing that add QFT (M ) is equivalent, as a category, to a category describing quantum ﬁeld theories on the interior of M together with suitable ideals of their uni- versal extensions. The precise deﬁnitions are as follows. Deﬁnition 5.8. Let B ∈ QFT(M ). An ideal I ⊆ B is called trivial on the −1 interior if its restriction to R [C ] is the functor assigning zero vector int M int M −1 spaces, i.e., I(V ) = 0 for all V ∈R [C ] in the interior int M of M . int int int M int M M. Benini et al. Ann. Henri Poincar´ e Deﬁnition 5.9. Let M be a spacetime with timelike boundary. We deﬁne the category IQFT(M ) as follows: • Objects are pairs (A, I) consisting of a quantum ﬁeld theory A ∈ QFT(int M ) on the interior int M of M and an ideal I ⊆ ext A of its universal extension ext A ∈ QFT(M ) that is trivial on the interior. • Morphisms κ :(A, I) → (A , I )are QFT(M )-morphisms κ :ext A → ext A between the universal extensions that preserve the ideals, i.e., κ restricts to a natural transformation from I ⊆ ext A to I ⊆ ext A . There exists an obvious functor add Q : IQFT(M ) −→ QFT (M ), (5.6a) which assigns to (A, I) ∈ IQFT(M ) the quotient add Q(A, I):=ext A I ∈ QFT (M ). (5.6b) Notice that additivity of ext A I follows from that of the universal exten- sion ext A (cf. the arguments in the proof of Theorem 5.6) and the fact that quotients preserve the additivity property. There exists also a functor add S : QFT (M ) −→ IQFT(M ), (5.7a) which ‘extracts’ from a quantum ﬁeld theory on M the relevant ideal. Explic- add itly, it assigns to B ∈ QFT (M ) the pair SB := res B, ker . (5.7b) Notice that the ideal ker ⊆ ext res B is trivial on the interior: Applying the restriction functor (4.3)to , we obtain a QFT(int M )-morphism res : res ext res B −→ res B. (5.8) Proposition 4.1 together with the triangle identities for the adjunction ext res in (4.3) then imply that (5.8) is an isomorphism with inverse given by η :res B → res ext res B. In particular, res has a trivial kernel and res B B hence ker ⊆ ext res B is trivial on the interior (cf. Deﬁnition 5.8). Our reﬁned characterization theorem for boundary quantum ﬁeld theories is as follows. Theorem 5.10. The functors Q and S deﬁned in (5.6) and (5.7) exhibit an equivalence of categories add QFT (M ) IQFT(M ). (5.9) add Proof. We ﬁrst consider the composition of functors QS : QFT (M ) → add add QFT (M ). To B ∈ QFT (M ), it assigns QSB =ext res B ker . (5.10) The QFT(M )-morphisms λ :ext res B/ ker → B given by (5.2) deﬁne a B B natural transformation λ : QS → id add , which is a natural isomorphism QFT (M ) due to Corollary 5.7. AQFT on Spacetimes with Timelike Boundary Let us now consider the composition of functors SQ : IQFT(M ) → IQFT(M ). To (A, I) ∈ IQFT(M ), it assigns SQ(A, I)= res ext A I , ker = res ext A, ker , (5.11) ext A/I ext A/I where we also used that res ext A I = res ext A because I is by hypothesis trivial on the interior. Using further the QFT(int M )-isomorphism η : A → res ext A from Proposition 4.1, we deﬁne a QFT(M )-morphism q via the (A,I) diagram (A,I) ext A ext A I (5.12) ext η A = ext A/I ext res ext A ext res ext A I Using the explicit expression for given in (5.1) and the explicit expres- ext A/I sion for η given by (η ) : A(V ) −→ res ext A(V )= A(V ) ∼, int int A V int i:V →V int a −→ [id ,a], (5.13) int −1 for all V ∈R [C ], one computes from the diagram (5.12) that q is int int M (A,I) int M the canonical projection π :ext A → ext A/I. Hence, the QFT(M )- isomorphisms ext η :ext A → ext res ext A induce IQFT(M )-isomorphisms ext η :(A, I) −→ SQ(A, I), (5.14) which are natural in (A, I), i.e., they deﬁne a natural isomorphism ext η : id → SQ. IQFT(M ) Remark 5.11. The physical interpretation of this result is as follows: Every add additive quantum ﬁeld theory B ∈ QFT (M ) on a (not necessarily glob- ally hyperbolic) spacetime M with timelike boundary admits an equivalent description in terms of a pair (A, I) ∈ IQFT(M ). Notice that the roles of A and I are completely diﬀerent: On the one hand, A ∈ QFT(int M)isa quantum ﬁeld theory on the interior int M of M and as such it is independent of the detailed aspects of the boundary. On the other hand, I ⊆ ext A is an ideal of the universal extension of A that is trivial on the interior, i.e., it only captures the physics that happens directly at the boundary. Examples of such ideals I arise by imposing speciﬁc boundary conditions on the universal ex- tension ext A ∈ QFT(M ), i.e., the quotient ext A/I describes a quantum ﬁeld theory on M that satisﬁes speciﬁc boundary conditions encoded in I. We shall illustrate this assertion in Sect. 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 cap- tures 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 M. Benini et al. Ann. Henri Poincar´ e 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. 6. Example: Free Klein–Gordon Theory In order to illustrate and make more explicit our abstract constructions de- veloped 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 are globally hyperbolic spacetimes with empty int M boundary, see Proposition 2.11. This allows us to use the standard techniques of [1, Section 3] on such regions. −1 6.1. Deﬁnition on R [C ]: int M int M Let M be a globally hyperbolic spacetime with timelike boundary, see Def- −1 inition 2.9. The free Klein–Gordon theory on R [C ] is given by the int M int M following standard construction, see e.g., [3, 4] for expository reviews. On the interior int M , we consider the Klein–Gordon operator 2 ∞ ∞ P := + m : C (int M ) −→ C (int M ) , (6.1) where is the d’Alembert operator and m ≥ 0 is a mass parameter. When −1 restricting P to regions V ∈R [C ], we shall write int M int M ∞ ∞ P : C (V ) −→ C (V ). (6.2) It follows from [1] that there exists a unique retarded/advanced Green’s oper- ator ± ∞ ∞ G : C (V ) −→ C (V ) (6.3) V c −1 for P because every V ∈R [C ] is a globally hyperbolic spacetime V int M int M with empty boundary, cf. Proposition 2.11. The Klein–Gordon theory K ∈ QFT(int M ) is the functor −1 K : R [C ] → Alg given by the following assignment: To any V ∈ int M int M −1 R [C ] it assigns the associative and unital ∗-algebra K(V ) that is freely int M int M generated by Φ (f ), for all f ∈ C (V ), modulo the two-sided ∗-ideal gener- ated by the following relations: • Linearity Φ (αf + βg)= α Φ (f)+ β Φ (g), for all α, β ∈ R and V V V f, g ∈ C (V ); ∗ ∞ • Hermiticity Φ (f ) =Φ (f ), for all f ∈ C (V ); V V • Equation of motion Φ (P f ) = 0, for all f ∈ C (V ); V V • Canonical commutation relations (CCR) Φ (f)Φ (g) − Φ (g)Φ (f ) V V V V =i τ (f, g) 1, for all f, g ∈ C (V ), where τ (f, g):= fG (g)vol (6.4) V V V V AQFT on Spacetimes with Timelike Boundary + − with G := G − G the causal propagator and vol the canonical vol- V V V V ume form on V . (Note that τ is antisymmetric, see e.g., [1, Lemma 4.3.5]). −1 −1 To a morphism i : V → V in R [C ], the functor K : R [C ] → int M int M int M int M Alg assigns the algebra map that is speciﬁed on the generators by pushforward along i (which we shall suppress) K(i): K(V ) −→ K(V ), Φ (f ) −→ Φ (f ). (6.5) V V The naturality of τ (i.e., naturality of the causal propagator, cf. e.g., [1, Sec- tion 4.3]) entails that the assignment K deﬁnes a quantum ﬁeld theory in the sense of Deﬁnition 3.5. 6.2. Universal Extension Using the techniques developed in Sect. 4, we may now extend the Klein– Gordon theory K ∈ QFT(int M ) from the interior int M to the whole space- time M . In particular, using (4.9) (and the formulas following this equa- tion), one could directly compute the universal extension ext K ∈ QFT(M ). The resulting expressions, however, can be considerably simpliﬁed. We there- fore prefer to provide a more convenient model for the universal extension ext K ∈ QFT(M ) by adopting the following strategy: We ﬁrst make an ‘ed- ext ucated guess’ for a theory K ∈ QFT(M ) which we expect to be the uni- versal extension of K ∈ QFT(int M ). (This was inspired by partially simpli- fying the direct computation of the universal extension). After this, we shall ext prove that K ∈ QFT(M ) satisﬁes the universal property that characterizes ext ext K ∈ QFT(M ). Hence, there exists a (unique) isomorphism ext K = K in ext QFT(M ), which means that our K ∈ QFT(M ) is a model for the universal extension ext K. −1 ext Let us deﬁne the functor K : R C → Alg by the following assign- −1 ment: To any region V ∈R C , which may intersect the boundary, we ext assign the associative and unital ∗-algebra K (V ) that is freely generated by Φ (f ), for all f ∈ C (int V ) in the interior int V of V , modulo the two-sided ideal generated by the following relations: • Linearity Φ (αf + βg)= α Φ (f)+ β Φ (g), for all α, β ∈ R and V V V f, g ∈ C (int V ); ∗ ∞ • Hermiticity Φ (f ) =Φ (f ), for all f ∈ C (int V ); V V • Equation of motion Φ (P f ) = 0, for all f ∈ C (int V ); V int V • Partially-deﬁned CCR Φ (f)Φ (g) − Φ (g)Φ (f)=i τ (f, g) 1,for V V V V V int −1 all interior regions V ∈R [C ] with V ⊆ int V and f, g ∈ int int M int int M C (int V ) with supp(f ) ∪ supp(g) ⊆ V . int Remark 6.1. We note that our partially-deﬁned CCR are consistent in the () −1 following sense: Consider V ,V ∈R [C ] with V ⊆ int V and int int M int int M int () f, g ∈ C (int V ) with the property that supp(f ) ∪ supp(g) ⊆ V .Us- c int ing the partially-deﬁned CCR for both V and V , we obtain the equality int int ext ext i τ (f, g) 1 =i τ (f, g) 1 in K (V ). To ensure that K (V ) is not the V V int int zero-algebra, we have to show that τ (f, g)= τ (f, g). This holds true V V int int due to the following argument: Consider the subset V ∩ V ⊆ int M . This int int M. Benini et al. Ann. Henri Poincar´ e is open, causally convex and by Proposition 2.3(c) also stable under Cauchy −1 development, hence V ∩ V ∈R [C ]. Furthermore, the inclusions int int M int int M () −1 V ∩ V → V are morphisms in R [C ]. It follows by construction int int M int int int M that supp(f ) ∪ supp(g) ⊆ V ∩ V and hence due to naturality of the τ ’s we int int obtain τ (f, g)= τ (f, g)= τ (f, g). (6.6) V V ∩V V int int int int Hence, for any ﬁxed pair f, g ∈ C (int V ), the partially-deﬁned CCR are independent of the choice of V (if one exists). int −1 −1 ext To a morphism i : V → V in R [C ], the functor K : R C → M M M M Alg assigns the algebra map that is speciﬁed on the generators by the push- forward along i (which we shall suppress) ext ext ext K (i): K (V ) −→ K (V ), Φ (f ) −→ Φ (f ). (6.7) V V ext Compatibility of the map (6.7) with the relations in K is a straightforward check. −1 −1 Recalling the embedding functor J : R [C ] →R C given in int M M int M M (4.2), we observe that the diagram of functors −1 R [C ] Alg (6.8) int M int M ext J K −1 R [C ] ext commutes via the natural transformation γ : K → K J with components speciﬁed on the generators by the identity maps ext γ : K(V ) −→ K (V ), Φ (f ) −→ Φ (f ), (6.9) int int V V V int int int −1 for all V ∈R [C ]. Notice that γ is a natural isomorphism because int int M int M int V = V and the partially-deﬁned CCR on any interior region V coin- int int int cides with the CCR. −1 Theorem 6.2. (6.8) is a left Kan extension of K : R [C ] → Alg along int M int M −1 −1 J : R [C ] →R C . As a consequence of uniqueness (up to unique int M M int M M natural isomorphism) of left Kan extensions and Proposition 4.3,itfollows ext ext that K ext K, i.e., K ∈ QFT(M ) is a model for our universal extension ext K ∈ QFT(M ) of the Klein–Gordon theory K ∈ QFT(int M ). Proof. We have to prove that (6.8) satisﬁes the universal property of left Kan −1 extensions: Given any functor B : R C → Alg and natural transfor- mation ρ : K → B J , we have to show that there exists a unique natural ext transformation ζ : K → B such that the diagram K B J (6.10) ζJ ext K J AQFT on Spacetimes with Timelike Boundary commutes. Because γ is a natural isomorphism, it immediately follows that ζJ is uniquely ﬁxed by this diagram. Concretely, this means that the components −1 ζ corresponding to interior regions V ∈R [C ] are uniquely ﬁxed V int int M int int M by −1 ext ζ := ρ γ : K (V ) −→ B(V ). (6.11) int int V V V int int int It remains to determine the components ext ζ : K (V ) −→ B(V ) (6.12) −1 ext for generic regions V ∈R C . Consider any generator Φ (f)of K (V ), M V where f ∈ C (int V ), and choose a ﬁnite cover {V ⊆ int V } of supp(f)by −1 interior regions V ∈R [C ], together with a partition of unity {χ } α int M α int M subordinate to this cover. (The existence of such a cover is guaranteed by the assumption that M is a globally hyperbolic spacetime with timelike boundary, see Proposition 2.11). We deﬁne ζ Φ (f ) := B(i ) ζ Φ (χ f ) , (6.13) V V α V V α α α where i : V → V is the inclusion. Our deﬁnition (6.13) is independent of the α α choice of cover and partition of unity: For any other {V ⊆ int V } and {χ }, β β we obtain B(i ) ζ Φ (χ f ) = B(i ) ζ Φ (χ χ f ) β V V αβ V ∩V V ∩V α β α α β β β β β β α,β = B(i ) ζ Φ (χ f ) , (6.14) α V V α α α where i : V → V and i : V ∩ V → V are the inclusions. In particular, this β αβ α β β implies that (6.13) coincides with (6.11) on interior regions V = V .(Hint: int Choose the cover given by the single region V together with its partition of int unity). ext We have to check that (6.13) preserves the relations in K (V ). Preserva- tion of linearity and Hermiticity is obvious. The equation of motion relations are preserved because ζ Φ (P f ) = B(i ) ζ Φ (χ P f ) V V α V V α int V α α V = B(i ) ζ Φ (χ P χ f ) αβ V ∩V V ∩V α β α β α β V ∩V α β α,β = B(i ) ζ Φ (P χ f ) =0. (6.15) β V V β β β V −1 Regarding the partially-deﬁned CCR, let V ∈R [C ] with V ⊆ int int M int int M int V and f, g ∈ C (int V ) with supp(f ) ∪ supp(g) ⊆ V .Wemay choose int the cover given by the single region i : V → V together with its partition of int M. Benini et al. Ann. Henri Poincar´ e unity. We obtain for the commutator ζ Φ (f ) ,ζ Φ (g) = B(i) ζ Φ (f ) , B(i) ζ Φ (g) V V V V V V V V int int int int =i τ (f, g) 1, (6.16) int which implies that the partially-deﬁned CCR are preserved. Naturality of the components (6.13) is easily veriﬁed. Uniqueness of the ext resulting natural transformation ζ : K → B is a consequence of uniqueness of ext ext ζJ and of the fact that the Alg-morphisms K (i ): K (V ) K(V ) → int int int ext ext K (V ), for all interior regions i : V → V , generate K (V ), for all V ∈ int int −1 R C . This completes the proof. 6.3. Ideals from Green’s Operator Extensions The Klein–Gordon theory K ∈ QFT(int M ) on the interior int M of the glob- ally hyperbolic spacetime M with timelike boundary and its universal exten- ext sion K ∈ QFT(M ) depend on the local retarded and advanced Green’s oper- ± −1 ∞ ∞ ators G : C (V ) → C (V ) on all interior regions V ∈R [C ] int int int M c int V int M int in M . For constructing concrete examples of quantum ﬁeld theories on globally hyperbolic spacetimes with timelike boundary as in [12], one typically imposes suitable boundary conditions for the ﬁeld equation in order to obtain also global retarded and advanced Green’s operators on M . Inspired by such examples, we + − shall now show that any choice of an adjoint-related pair (G ,G ) consisting of a retarded and an advanced Green’s operator for P on M (see Deﬁnition 6.3 ext below) deﬁnes an ideal I ± ⊆ K ∈ QFT(M ) that is trivial on the interior ext (cf. Deﬁnition 5.8). The corresponding quotient K I ± ∈ QFT(M ) then may be interpreted as the Klein–Gordon theory on M , subject to a speciﬁc choice of boundary conditions that is encoded in G . Deﬁnition 6.3. A retarded/advanced Green’s operator for the Klein–Gordon ± ∞ ∞ operator P on M is a linear map G : C (int M ) → C (int M ) which satisﬁes the following properties, for all f ∈ C (int M ): (i) PG (f)= f , (ii) G (Pf)= f,and (iii) supp(G (f )) ⊆ J (supp(f )). + − A pair (G ,G ) consisting of a retarded and an advanced Green’s operator + − for P on M is called adjoint-related if G is the formal adjoint of G , i.e., + − G (f ) g vol = fG (g)vol , (6.17) M M M M for all f, g ∈ C (int M ). Remark 6.4. In contrast to the situation where M is a globally hyperbolic spacetime with empty boundary [1], existence, uniqueness and adjoint- relatedness of retarded/advanced Green’s operators for the Klein–Gordon op- erator P is in general not to be expected on spacetimes with timelike boundary. Positive results seem to be more likely on globally hyperbolic spacetimes with AQFT on Spacetimes with Timelike Boundary non-empty timelike boundary, although the general theory has not been de- veloped yet to the best of our knowledge. Simple examples of adjoint-related pairs of Green’s operators were constructed, e.g., in [12]. −1 Given any region V ∈R C in M , which may intersect the boundary, we use the canonical inclusion i : V → M to deﬁne local retarded/advanced Green’s operators ∞ ∞ C (int V ) C (int V ) (6.18) ∗ i ∞ ∞ C (int M ) C (int M ) where i denotes the pushforward of compactly supported functions (i.e., ex- tension by zero) and i the pullback of functions (i.e., restriction). Since V ⊆ M ± ± is causally convex, it follows that J (p) ∩ V = J (p) for all p ∈ V . Therefore M V G satisﬁes the axioms of a retarded/advanced Green’s operator for P on V . V V (Here, we regard V as a globally hyperbolic spacetime with timelike bound- ary, see Proposition 2.11. J (p) denotes the causal future/past of p in the −1 spacetime V ). In particular, for all interior regions V ∈R [C ]in M , int int M int M by combining Proposition 2.11 and [1, Corollary 3.4.3] we obtain that G int as deﬁned in (6.18) is the unique retarded/advanced Green’s operator for the restricted Klein–Gordon operator P on the globally hyperbolic spacetime int V with empty boundary. int + − Consider any adjoint-related pair (G ,G ) of Green’s operator for P on −1 ext M . For all V ∈R C ,weset I ± (V ) ⊆ K (V ) to be the two-sided ∗-ideal generated by the following relations: ± ∞ • G -CCR Φ (f)Φ (g) − Φ (g)Φ (f)=i τ (f, g) 1, for all f, g ∈ C V V V V V (int V ), where τ (f, g):= fG (g)vol (6.19) V V V + − with G := G − G the causal propagator and vol the canonical V V V V volume form on V . + − The fact that the pair (G ,G ) is adjoint-related (cf. Deﬁnition 6.3) implies −1 that for all V ∈R C the causal propagator G is formally skew-adjoint, M V hence τ is antisymmetric. ext Proposition 6.5. I ± ⊆ K is an ideal that is trivial on the interior (cf. Deﬁnition 5.8). −1 Proof. Functoriality of I ± : R C → Vec is a consequence of (6.18), G M ext hence I ± ⊆ K is an ideal in the sense of Deﬁnition 5.1. It is trivial on the −1 interior because for all interior regions V ∈R [C ], the Green’s op- int M int int M erators deﬁned by (6.18) are the unique retarded/advanced Green’s operators for P and hence the relations imposed by I ± (V ) automatically hold true G int int ext in K (V ) on account of the (partially-deﬁned) CCR. int M. Benini et al. Ann. Henri Poincar´ e 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 space- time with empty boundary once equipped with the induced metric, orientation and time-orientation. m−1 m Example 6.7. Consider the sub-spacetime M := R × [0,π] ⊆ R of m- dimensional Minkowski spacetime, which has a timelike boundary ∂M = m−1 + − R ×{0,π}. The constructions in [12] 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 ext quantum ﬁeld theory K I ± ∈ 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 con- structed in [12]. 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 BDS ext DNP A (M):= K I ± (M ) with the global algebra A (M ) constructed in [12]. Both algebras are CCR-algebras, however the underlying symplec- tic vector spaces diﬀer: The symplectic vector space underlying our global BDS ∞ ∞ algebra A (M)is C (int M ) PC (int M ) with the symplectic structure c c (6.19). Using that the spatial slices of M = R × [0,π] are compact, we observe DNP that the symplectic vector space underlying A (M ) is given by the space Sol (M)of all solutions with Dirichlet boundary condition on M (equipped Dir with the usual symplectic structure). The causal propagator deﬁnes a sym- ∞ ∞ plectic map G : C (int M ) PC (int M ) → Sol (M ), which however is not Dir c c surjective for the following reason: Any ϕ ∈ C (int M ) has by deﬁnition com- pact support in the interior of M , hence the support of Gϕ ∈ Sol (M)is Dir schematically as follows supp Gϕ supp ϕ x = π x =0 (6.20) 2 2 The usual mode functions Φ (t, x)=cos( k + m t)sin(kx) ∈ Sol (M ), k Dir ∞ ∞ for k ≥ 1, are clearly not of this form, hence G : C (int M ) PC (int M ) → c c Sol (M ) cannot be surjective. As a consequence, the models constructed in Dir ext [12] are in general not additive from the interior and our construction K I ± should be interpreted as the maximal additive subtheory of these examples. AQFT on Spacetimes with Timelike Boundary It is interesting to note that there exists a case where both constructions m−1 m coincide: Consider the sub-spacetime M := R × [0, ∞) ⊆ R 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 BDS the Green’s operators, one may show that our algebra A (M ) is isomorphic to the construction in [12]. Acknowledgements We would like to thank Jorma Louko for useful discussions about boundaries in quantum ﬁeld theory and also Didier A. Solis for sending us a copy of his PhD thesis [29]. The work of M. B. is supported by a research grant funded by the Deutsche Forschungsgemeinschaft (DFG, Germany). Furthermore, M. B. would like to thank Istituto Nazionale di Alta Matematica (INdAM, Italy) for the ﬁnancial support provided during his visit to the School of Mathematical Sciences of the University of Nottingham, whose kind hospitality is gratefully acknowledged. The work of C. D. is supported by the University of Pavia. C. D. gratefully acknowledges the kind of hospitality of the School of Mathematical Sciences of the University of Nottingham, where part of this work has been done. A. S. gratefully acknowledges the ﬁnancial support of the Royal Society (UK) through a Royal Society University Research Fellowship, a Research Grant and an Enhancement Award. He also would like to thank the Research Training Group 1670 “Mathematics inspired by String Theory and Quantum Field Theory” for the invitation to Hamburg, where part of this work has been done. Open Access. This article is distributed under the terms of the Creative Com- mons Attribution 4.0 International License (http://creativecommons.org/licenses/ by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Appendix A: Some Concepts from Category Theory Adjunctions This is a standard concept, which is treated in any category theory textbook, e.g., [28]. Deﬁnition A.1. An adjunction consists of a pair of functors F : C D : G (A.1) M. Benini et al. Ann. Henri Poincar´ e together with natural transformations η :id → GF (called unit)and : FG → id (called counit ) that satisfy the triangle identities Fη ηG F FGF G GF G (A.2) F G id id F G F G 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 F : C ∼ D : G (A.3) for which both the unit η and the counit are natural isomorphisms. Existence of an adjoint equivalence in particular implies that C = D are equivalent as categories. Proposition A.3. If a functor G : D → C admits a left adjoint F : C → D, then F is unique up to a unique natural isomorphism. Vice versa, if a functor F : C → D admits a right adjoint G : D → C, then G is unique up to a unique natural isomorphism. Localizations Localizations of categories are treated for example in [21, Section 7.1]. In our paper we restrict ourselves to small categories. Deﬁnition A.4. Let C be a small category and W ⊆ Mor C a subset of the set −1 of morphisms. A localization of C at W is a small category C[W ] together −1 with a functor L : C → C[W ] satisfying the following properties: (a) For all (f : c → c ) ∈ W , L(f): L(c) → L(c ) is an isomorphism in −1 C[W ]. 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Quantum Gravity. 20, 3815 (2003). arXiv:gr-qc/0305012 M. Benini et al. Ann. Henri Poincar´ e [20] Ishibashi, A., Wald, R.M.: Dynamics in non-globally-hyperbolic static space- times III: anti-de Sitter space-time. Class. Quantum Gravity 21, 2981 (2004). arXiv:hep-th/0402184 [21] Kashiwara, M., Schapira, P.: Categories and Sheaves. Springer, Berlin (2006) [22] Kay, B.S.: The principle of locality and quantum ﬁeld theory on (non globally hyperbolic) curved spacetimes. Rev. Math. Phys. 4, 167 (1992) [23] Lee, J.M.: Introduction to Smooth Manifolds. Springer, New York (2013) [24] O’Neill, B.: Semi-Riemannian Geometry. Academic Press, New York (1983) [25] Rehren, K.-H.: Algebraic holography. Ann. Henri Poincar´ e 1, 607 (2000). arXiv:hep-th/9905179 [26] Rejzner, K.: Perturbative Algebraic Quantum Field Theory: An Introduction for Mathematicians. Springer, Heidelberg (2016) [27] Ribeiro, P.L.: Structural and Dynamical Aspects of the AdS/CFT Correspon- dence: A Rigorous Approach. Ph.D. Thesis, University of Sao Paulo (2007). arXiv:0712.0401 [math-ph] [28] Riehl, E.: Category Theory in Context. Dover Publications Inc, New York (2016) [29] Solis, D.A.: Global Properties of Asymptotically de Sitter and Anti de Sitter Spacetimes. Ph.D. Thesis, University of Miami (2006). https:// scholarlyrepository.miami.edu/dissertations/2414/ [30] Sommer, C.: Algebraische Charakterisierung von Randbedingungen in der Quan- tenfeldtheorie. Diploma Thesis in German, University of Hamburg (2006). http://www.desy.de/uni-th/lqp/psﬁles/dipl-sommer.ps.gz [31] Wald, R.M.: Dynamics in nonglobally hyperbolic, static space-times. J. Math. Phys. 21, 2802 (1980) [32] Wrochna, M.: The holographic Hadamard condition on asymptotically anti-de Sitter spacetimes. Lett. Math. Phys. 107, 2291 (2017). arXiv:1612.01203 [math- ph] [33] Zahn, J.: Generalized Wentzell boundary conditions and quantum ﬁeld theory. Ann. Henri Poincar´ e 19, 163 (2018). arXiv:1512.05512 [math-ph] Marco Benini Fachbereich Mathematik Universit¨ at Hamburg Bundesstr. 55 20146 Hamburg Germany e-mail: marco.benini@uni-hamburg.de Claudio Dappiaggi INFN, Sezione di Pavia Universit` a di Pavia Via Bassi 6 27100 Pavia Italy e-mail: claudio.dappiaggi@unipv.it and AQFT on Spacetimes with Timelike Boundary Dipartimento di Fisica INFN, Sezione di Pavia Via Bassi 6 27100 Pavia Italy Alexander Schenkel School of Mathematical Sciences University of Nottingham University Park Nottingham NG7 2RD UK e-mail: alexander.schenkel@nottingham.ac.uk Communicated by Karl Henning Rehren. Received: January 3, 2018. Accepted: May 3, 2018.
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Algebraic Quantum Field Theory on Spacetimes with Timelike Boundary

Physics; Theoretical, Mathematical and Computational Physics; Dynamical Systems and Ergodic Theory; Quantum Physics; Mathematical Methods in Physics; Classical and Quantum Gravitation, Relativity Theory; Elementary Particles, Quantum Field Theory

Ann. Henri Poincar´ e Online First c 2018 The Author(s) Annales Henri Poincar´ e https://doi.org/10.1007/s00023-018-0687-1 Algebraic Quantum Field Theory on Spacetimes with Timelike Boundary Marco Benini , Claudio Dappiaggi and Alexander Schenkel Dedicated to Klaus Fredenhagen on the occasion of his 70th birthday. Abstract. We analyze quantum ﬁeld theories on spacetimes M with time- like 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 intM . The unit of this adjunc- tion is a natural isomorphism, which implies that our universal extension satisﬁes Kay’s F-locality property. Our main result is the following char- acterization 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 intM 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. 1. Introduction and Summary Algebraic quantum ﬁeld theory is a powerful and far developed framework to address model-independent aspects of quantum ﬁeld theories on Minkowski spacetime [18] and more generally on globally hyperbolic spacetimes [7]. In addition to establishing the axiomatic foundations for quantum ﬁeld theory, the algebraic approach has provided a variety of mathematically rigorous con- structions of non-interacting models, see e.g., the reviews [1, 3, 4], and more in- terestingly also perturbatively interacting quantum ﬁeld theories, see e.g., the recent monograph [26]. It is worth emphasizing that many of the techniques involved in such constructions, e.g., existence and uniqueness of Green’s opera- tors and the singular structure of propagators, crucially rely on the hypothesis that the spacetime is globally hyperbolic and has empty boundary. M. Benini et al. Ann. Henri Poincar´ e 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 em- ployed 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 [19, 20, 33], the latter extending the analysis of [31]. For speciﬁc choices of boundary con- ditions, there exist successful constructions of algebraic quantum ﬁeld theories on spacetimes with timelike boundary, see e.g., [8, 10–12]. 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 [11, 32], should be modiﬁed accordingly. Unfortunately, model-independent results on algebraic quantum ﬁeld the- ory on spacetimes with timelike boundary are more scarce. There are, how- ever, some notable and very interesting works in this direction: On the one hand, Rehren’s proposal for algebraic holography [25] initiated the rigorous study of quantum ﬁeld theories on the anti-de Sitter spacetime. This has been further elaborated in [13] and extended to asymptotically AdS spacetimes in [27]. On the other hand, inspired by Fredenhagen’s universal algebra [15–17], a very interesting construction and analysis of global algebras of observables on spacetimes with timelike boundaries has been performed in [30]. 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 space- times 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 the- ories. Such questions can be naturally addressed by using techniques from the recently developed operadic approach to algebraic quantum ﬁeld theory [5]. Let us describe rather informally the basic idea of our construction and its implications: Given a spacetime M with timelike boundary, an algebraic AQFT on Spacetimes with Timelike Boundary quantum ﬁeld theory on M is a functor B : R → 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 sub- category of regions in the interior of M by R ⊆R , we may restrict int M M any theory B ∈ QFT(M)toatheory 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 [22]. 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 [5]. In particular, the ext res adjunction investigated in the present paper is one concrete instance of a whole fam- ily 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)ona 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 :extres B → B of the adjunction. Our result in Theorem 5.6 and Corollary 5.7 is that induces an isomorphism ext res B/ ker B 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)onthe 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 M. Benini et al. Ann. Henri Poincar´ e 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 [30] and the concrete constructions in [8, 10–12]. The remainder of this paper is structured as follows: In Sect. 2, we recall some basic deﬁnitions and results about the causal structure of spacetimes with timelike boundaries, see also [9, 29]. In Sect. 3, we provide a precise deﬁnition of the categories QFT(M)and QFT(int M ) by using the ideas of [5]. Our universal boundary extension is developed in Sect. 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 Sect. 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., [12]. We included “Appendix A” to state some basic deﬁnitions and results of category theory which will be used in our work. 2. Spacetimes with Timelike Boundary We collect some basic facts about spacetimes with timelike boundary, following [29, Section 3.1] and [9, Section 2.2]. For a general introduction to Lorentzian geometry, we refer to [2, 24], see also [1, Sections 1.3 and A.5] for a concise presentation. We use the term manifold with boundary to refer to a Hausdorﬀ, second countable, m-dimensional smooth manifold M with boundary, see e.g., [23]. This deﬁnition subsumes ordinary manifolds as manifolds with empty bound- ary ∂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 bound- ary 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 de- velopment. Proposition 2.3. Let S, S ⊆ M be subsets of a time-oriented Lorentzian man- ifold M with boundary. Then, the following holds true: (a) S ⊆ S implies D(S) ⊆ D(S ); AQFT on Spacetimes with Timelike Boundary (b) S ⊆ D(S)= D(D(S)); (c) D(D(S) ∩ D(S )) = D(S) ∩ D(S ). We denote by J (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 (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. Deﬁnition 2.4. Let M be a time-oriented Lorentzian manifold with boundary. + − We say that a subset S ⊆ M is causally convex in M if J (S) ∩ J (S) ⊆ S. M M We say that two subsets S, S ⊆ M are causally disjoint in M if (J (S) ∪ J (S)) ∩ S = ∅. The following properties are simple consequences of these deﬁnitions. Proposition 2.5. Let S, S ⊆ M be two subsets of a time-oriented Lorentzian manifold M with boundary. Then, the following holds true: (a) D(S) and D(S ) are causally disjoint if and only if S and S are causally disjoint; (b) Suppose S and S are causally disjoint. Then, the disjoint union S S ⊆ M is causally convex if and only if both S and S are causally convex. 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 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 . (ii) C ⊆ Mor R is the subset of Cauchy morphisms in R , i.e., inclusions M M M i : U → U such that D(U)= D(U ). (iii) R ⊆R is the full subcategory whose objects are contained in the int M M interior int M . We denote by C ⊆C the Cauchy morphisms between int M M objects of R . int M Proposition 2.8. Let M be a spacetime with timelike boundary. For each subset S ⊆ M and each object U ∈R , i.e., a causally convex open subset U ⊆ M, the following holds true: ± ± (a) I (S) is the largest open subset of J (S); M M ± ± ± ± ± (b) J (I (S)) = I (S)= I (J (S)); M M M M M (c) S ⊆ int M implies D(S) ⊆ int M; (d) D(U ) ⊆ M is causally convex and open, i.e., D(U ) ∈R . M M. Benini et al. Ann. Henri Poincar´ e Proof. (a) and (b): These are standard results in the case of empty bound- ary, see e.g., [1, 2, 24]. The extension to spacetimes with non-empty timelike boundary can be found in [29, Section 3.1.1]. (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 . (d): D(U ) ⊆ M is causally convex by the deﬁnition of Cauchy develop- ment and by the causal convexity of U ⊆ M . To check that D(U ) ⊆ M is open, we use quasi-limits as in [24, Deﬁnition 14.7 and Proposition 14.8]: First, ob- serve that I (U ) ⊆ M is open by (a). Hence, it is the same to check whether + − a subset of I := I (U ) ∪ I (U)isopenin M or in I (with the induced topol- M M ogy). Indeed, U ⊆ D(U ) ⊆ I because U is open in M . From now on, in place of M , let us therefore consider I, equipped with the induced metric, orientation and time-orientation. By contradiction, assume that there exists p ∈ D(U )\U such that all of its neighborhoods intersect the complement of D(U ). Then, there exists a sequence {α } of inextensible causal curves in I never meeting U such that {α (0)} converges to p. We ﬁx a convex cover of I reﬁning the open + − cover {I (U ),I (U )}. Relative to the ﬁxed convex cover, the construction M M of quasi-limits allows us to obtain from {α } 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 and p , two successive k k+1 limit points for {α } contained in some element of the ﬁxed convex cover. It + − follows that either p or p belongs to J (U ) ∩ J (U ), which is contained k k+1 I I in U by causal convexity. Hence, we found a subsequence {α } of {α } and n n a sequence of parameters {s } such that {α (s )} converges to a point of U j n j (either p or p ). By construction the sequence {α (s )} is contained in k k+1 n j 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 Sect. 6. The following deﬁnition is due to [9, Section 2.2] and [29, Section 3.1.2]. Deﬁnition 2.9. A spacetime M with timelike boundary is called globally hy- perbolic if the following two properties hold true: (i) Strong causality Every open neighborhood of each point p ∈ M contains a causally convex open neighborhood of p. + − (ii) Compact double-cones J (p) ∩ J (q) is compact for all p, q ∈ M . M M AQFT on Spacetimes with Timelike Boundary Remark 2.10. In the case of empty boundary, this deﬁnition agrees with the usual one in [1, 2, 24]. Simple examples of globally hyperbolic spacetimes with m−1 m non-empty timelike boundary are the half space {x ≥ 0}⊆ R , the spatial m−1 m 1 2 m−1 2 m slab {0 ≤ x ≤ 1}⊆ R and the cylinder {(x ) +···+(x ) ≤ 1}⊆ R in Minkowski spacetime R ,for m ≥ 2, as well as all causally convex open subsets thereof. The following results follow immediately from Deﬁnition 2.9 and Propo- sition 2.8. Proposition 2.11. Let M be a globally hyperbolic spacetime with timelike bound- ary. (a) M admits a cover by causally convex open subsets. (b) For each U ∈R , i.e., a causally convex open subset U ⊆ M, both U and D(U ) are globally hyperbolic spacetimes with (possibly empty) boundary when equipped with the metric, orientation and time-orientation induced by M.Ifmoreover U ⊆ int M is contained in the interior, then both U and D(U ) have empty boundary. 3. Categories of Algebraic Quantum Field Theories Let M be a spacetime with timelike boundary. (In this section we do not have to assume that M is globally hyperbolic in the sense of Deﬁnition 2.9). Recall the category R of open and causally convex regions in M and the subset C M M of Cauchy morphisms (cf. Deﬁnition 2.7). Together with our notion of causal disjointness from Deﬁnition 2.4, these data provide the geometrical input for the traditional deﬁnition of algebraic quantum ﬁeld theories on M . Deﬁnition 3.1. An algebraic quantum ﬁeld theory on M is a functor A : R −→ Alg (3.1) 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 : U → U ← U : i , 1 1 2 2 the induced commutator A(i )(−), A(i )(−) : A(U ) ⊗ A(U ) −→ A(U ) (3.2) 1 2 1 2 A(U ) is zero. (ii) Time-slice axiom For all Cauchy morphisms (i : U → U ) ∈C , the map A(i): A(U ) −→ A(U ) (3.3) is an Alg-isomorphism. We denote by qft(M ) ⊆ Alg the full subcategory of the category of func- tors from R 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). M. Benini et al. Ann. Henri Poincar´ e We shall now show that there exists an alternative, but equivalent, de- scription of the category qft(M ) which will be more convenient for the tech- nical constructions in our paper. Following [5, Section 4.1], we observe that the time-slice axiom in Deﬁnition 3.1(ii) is equivalent to considering functors −1 B : R C → Alg that are deﬁned on the localization of the category R at the set of Cauchy morphisms C . See Deﬁnition A.4 for the deﬁni- M M tion of localizations of categories. By abstract arguments as in [5, Section 4.6], one observes that the universal property of localizations implies that the cat- egory qft(M ) is equivalent to the full subcategory of the functor category −1 R C −1 M [ ] Alg whose objects are all functors B : R C → Alg that satisfy −1 the causality axiom for the pushforward orthogonality relation on R C . Loosely speaking, this means that the time-slice axiom in Deﬁnition 3.1(ii) −1 can be hard-coded by working on the localized category R C instead of using the usual category R of regions in M . The aim of the remainder of this section is to provide an explicit model −1 for the localization functor R →R C . With this model, it will be- M M come particularly easy to verify the equivalence between the two alternative descriptions of the category qft(M ). Let us denote by −1 R C ⊆R (3.4a) M M the full subcategory of R whose objects V ⊆ M are stable under Cauchy de- velopment, i.e., D(V )= V where D(V ) ⊆ M denotes the Cauchy development (cf. Deﬁnition 2.2). In the following, we shall always use letters like U ⊆ M for generic regions in R and V ⊆ M for regions that are stable under Cauchy −1 development, i.e., objects in R C . Recall from Deﬁnition 2.7 that each object U ∈R is a causally convex open subset U ⊆ M , hence the Cauchy de- velopment D(U ) ⊆ M is a causally convex open subset by Proposition 2.8(d), which is stable under Cauchy development by Proposition 2.3(b). This shows −1 that D(U ) is an object of R C . Furthermore, a morphism i : U → U in R is an inclusion U ⊆ U , which induces an inclusion D(U ) ⊆ D(U )by −1 Proposition 2.3(a) and hence a morphism D(i): D(U ) → D(U )in R C . We deﬁne a functor −1 D : R −→ R C (3.4b) M M by setting on objects and morphisms U −→ D(U ), (i : U → U ) −→ D(i): D(U ) → D(U ) . (3.4c) Furthermore, let us write −1 I : R C −→ R (3.5) M M for the full subcategory embedding. Lemma 3.2. D and I form an adjunction (cf. Deﬁnition A.1) −1 D : R R C : I (3.6) M M M AQFT on Spacetimes with Timelike Boundary −1 whose counit is a natural isomorphism (in fact, the identity), hence R C is a full reﬂective subcategory of R . Furthermore, the components of the unit are Cauchy morphisms. Proof. For U ∈R ,the U -component of the unit η :id −→ ID (3.7) 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 2.7(ii). For V ∈ −1 R C ,the V -component of the counit : DI −→ id −1 (3.8) R C M [ ] is given by the identity of the object D(V )= V . The triangle identities hold trivially. −1 −1 Proposition 3.3. The category R C and the functor D : R →R C M M M M M deﬁned in (3.4) provide a model for the localization of R at C . M M Proof. We have to check all the requirements listed in Deﬁnition A.4. (a) By Deﬁnition 2.7, for each Cauchy morphism i : U → U one has D(U)= −1 D(U ) and hence D(i)=id is an isomorphism in R C . D(U ) M (b) Let F : R → D be any functor to a category D that sends morphisms in C to D-isomorphisms. Using Lemma 3.2, we deﬁne F := FI : M W −1 R C → D and consider the natural transformation Fη : F → FI D = F D obtained by the unit of the adjunction D I. Because all components of η are Cauchy morphisms (cf. Lemma 3.2), Fη is a natural isomorphism. −1 (c) Let G, H : R C → D be two functors. We have to show that the map Hom G, H −→ Hom R GD, HD (3.9) −1 R C D [ ] is a bijection. Let us ﬁrst prove injectivity: Let ξ, ξ : G → H be two natural transformations such that ξD = ξD. Using Lemma 3.2, we obtain commutative diagrams ξDI ξDI GDI HDI GDI HDI (3.10) G H G H G H G H where the vertical arrows are natural isomorphisms because the counit is an isomorphism. Recalling that by hypothesis ξD = ξD, it follows that ξ = ξ. Hence, the map (3.9) is injective. M. Benini et al. Ann. Henri Poincar´ e 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 GD HD (3.11) GDη HDη GDID HDID χID 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 G H (3.12) −1 GDI HDI χI where we use that 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 −1 orthogonality relation on the localized category R C , both of which are provided by [5, Lemma 4.29]: (1) The pullback orthogonality relation along −1 the full subcategory embedding I : R C →R and (2) the pushforward M M −1 orthogonality relation along the localization functor D : R →R C . M M In our present scenario, both constructions coincide and one concludes that −1 two R C -morphisms are orthogonal precisely when they are orthogonal in R . Summing up, we obtain −1 Lemma 3.4. We say that two morphisms in the full subcategory R C ⊆ R are orthogonal precisely when they are orthogonal in R (i.e., causally M M −1 disjoint, cf. Deﬁnition 2.4). Then, both functors D : R →R C and M M −1 I : R C →R preserve (and also detect) the orthogonality relations. M M 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. −1 R [C ] Deﬁnition 3.5. We denote by QFT(M ) ⊆ Alg the full subcategory −1 whose objects are all functors B : R C → Alg that satisfy the following version of the causality axiom: For all causally disjoint inclusions i : V → 1 1 −1 V ← V : i in R C , i.e., V , V and V are stable under Cauchy develop- 2 2 M 1 2 ment, the induced commutator B(i )(−), B(i )(−) : B(V ) ⊗ B(V ) −→ B(V ) (3.13) 1 2 1 2 B(V ) is zero. AQFT on Spacetimes with Timelike Boundary Theorem 3.6. By pullback, the adjunction D I of Lemma 3.2 induces an adjoint equivalence (cf. Deﬁnition A.2) ∗ ∗ I : qft(M ) ∼ QFT(M): D . (3.14) In particular, the two categories qft(M ) of Deﬁnition 3.1 and QFT(M ) of Deﬁnition 3.5 are equivalent. −1 Proof. It is trivial to check that the adjunction D : R R C : I M M induces an adjunction −1 ∗ R R [C ] ∗ M M I : Alg Alg : D (3.15) ∗ ∗ between functor categories. Explicitly, the unit η :id R → D I has Alg M components ∗ ∗ η := Aη : A −→ D (I (A)) = AID, (3.16) where A : R → Alg is any functor and η :id → ID denotes the unit of M R ∗ ∗ D I. The counit : I D → id has components −1 R C [ ] Alg ∗ ∗ := B : I (D (B)) = BDI −→ B, (3.17) −1 where B : R C → Alg is any functor and : DI → id −1 denotes M R C M [ ] ∗ ∗ the counit of D I. The triangle identities for I D follow directly from those of D I. Next, we have to prove that this adjunction restricts to the claimed source and target categories in (3.14). Given A ∈ qft(M ) ⊆ Alg , the functor ∗ −1 I (A)= A I : R C → Alg satisﬁes the causality axiom of Deﬁnition 3.5 because of Lemma 3.4. Hence, I (A) ∈ QFT(M ). Vice versa, given B ∈ −1 R C M [ ] ∗ QFT(M ) ⊆ Alg , the functor D (B)= B D : R → Alg satisﬁes the causality axiom of Deﬁnition 3.5 because of Lemma 3.4 and the time-slice axiom of Deﬁnition 3.5 because D sends by construction morphisms in C to isomorphisms. Hence, D (B) ∈ qft(M ). Using Lemma 3.2, we obtain that the counit of the restricted adjunc- tion (3.14) is an isomorphism. Furthermore, all components of η are Cauchy morphisms, hence η = 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. Remark 3.7. Theorem 3.6 provides us with a constructive prescription of how to change between the two equivalent formulations of algebraic quan- tum ﬁeld theories given in Deﬁnitions 3.1 and 3.5. Concretely, given any A ∈ qft(M ), i.e., a functor A : R → Alg satisfying the causality and time-slice axioms as in Deﬁnition 3.1, the corresponding quantum ﬁeld the- ory I (A) ∈ QFT(M ) in the sense of Deﬁnition 3.5 reads as follows: It ∗ −1 is the functor I (A)= A I : R C → Alg on the category of regions V ⊆ M that are stable under Cauchy development, which assigns to V ⊆ M the algebra A(V ) ∈ Alg and to an inclusion i : V → V the algebra map M. Benini et al. Ann. Henri Poincar´ e A(i): A(V ) → A(V ). More interestingly, given B ∈ QFT(M ), i.e., a func- −1 tor B : R C → Alg satisfying the causality axiom as in Deﬁnition 3.5, the corresponding quantum ﬁeld theory D (B) ∈ qft(M ) in the sense of Def- inition 3.1 reads as follows: It is the functor D (B)= B D : R → Alg deﬁned on the category of (not necessarily Cauchy development stable) re- gions U ⊆ M , which assigns to U ⊆ M the algebra B(D(U )) corresponding to the Cauchy development of U and to an inclusion i : U → U the al- gebra map B(D(i)) : B(D(U )) → B(D(U )) associated with the inclusion D(i): D(U ) → D(U ) of Cauchy developments. Remark 3.8. It is straightforward to check that the results of this section still hold true when one replaces R with its full subcategory R of regions M int M contained in the interior of M and C with C (cf. Deﬁnition 2.7). This M int M follows from the observation that the Cauchy development of a subset of the interior of M is also contained in int M , as shown in Proposition 2.8(c). We denote by −1 R C int M [ ] int M QFT(int M ) ⊆ Alg (3.18) the category of algebraic quantum ﬁeld theories in the sense of Deﬁnition 3.5 on the interior regions of M . Concretely, an object A ∈ QFT(int M ) is a functor −1 A : R [C ] → Alg that satisﬁes the causality axiom of Deﬁnition 3.5 int M int M for causally disjoint interior regions. 4. Universal Boundary Extension The goal of this section is to develop a universal construction to extend quan- tum 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 quan- tum 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 Sect. 5 for more details on the properties (1) and (3). The starting point for this construction is the full subcategory inclusion R ⊆R deﬁned by selecting only the regions of R that lie in the int M M M interior of M (cf. Deﬁnition 2.7). We denote the corresponding embedding functor by j : R −→ R (4.1) int M M and notice that j preserves (and also detects) causally disjoint inclusions, i.e., j is a full orthogonal subcategory embedding in the terminology of [5]. Making use of Proposition 3.3, Lemma 3.2 and Remark 3.8, we deﬁne a functor J : −1 −1 R [C ] →R C on the localized categories via the commutative int M M int M M AQFT on Spacetimes with Timelike Boundary diagram −1 −1 R [C ] R [C ] (4.2a) int M M int M M I D R R int M M Notice that J is simply an embedding functor, which acts on objects and morphisms as V ⊆ int M −→ V ⊆ M, (i : V → V ) −→ (i : V → V ). (4.2b) 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 [5, Section 5.3] (see also [6] for details how to treat ∗- algebras) then imply that J induces an adjunction ext : QFT(int M ) QFT(M):res (4.3) between the category of quantum ﬁeld theories on the interior int M (cf. Re- mark 3.8) and the category of quantum ﬁeld theories on the whole spacetime M . The right adjoint res := J : QFT(M ) → QFT(int M ) is the pullback −1 −1 along J : R [C ] →R C , i.e., it restricts quantum ﬁeld theories int M M int M M deﬁned on M to the interior int M . The left adjoint ext : QFT(int M ) → QFT(M ) should be regarded as a universal extension functor which extends quantum ﬁeld theories on the interior int M to the whole spacetime M.The goal of this section is to analyze the properties of this extension functor and to develop an explicit model that allows us to do computations in the sections below. An important structural result, whose physical relevance is explained in Remark 4.2 below, is the following proposition. Proposition 4.1. The unit η :id −→ res ext (4.4) QFT(int M ) of the adjunction (4.3) is a natural isomorphism. Proof. This is a direct consequence of the fact that the functor J given in (4.2) is a full orthogonal subcategory embedding and the general result in [5, Proposition 5.6]. Remark 4.2. The physical interpretation of this result is as follows: Let A ∈ QFT(int M ) be a quantum ﬁeld theory deﬁned only on the interior int M of M and let B := ext A ∈ QFT(M ) denote its universal extension to the whole spacetime M.The A-component η : A −→ res ext A (4.5) of the unit of the adjunction (4.3) allows us to compare A with the restriction res B of its extension B =ext A. Since η is an isomorphism by Propo- sition 4.1, restricting the extension B recovers our original theory A up to M. Benini et al. Ann. Henri Poincar´ e isomorphism. This allows us to interpret the left adjoint ext : QFT(int M ) → QFT(M ) as a genuine extension prescription. Notice that this also proves that the universal extension ext A ∈ QFT(M ) of any theory A ∈ QFT(int M)on the interior satisﬁes F-locality [22]. We next address the question how to compute the extension functor ext : QFT(int M ) → QFT(M ) explicitly. A crucial step toward reaching this goal is to notice that ext may be computed by a left Kan extension. Proposition 4.3. Consider the adjunction −1 −1 R [C ] R [C ] int M M ∗ int M M Lan : Alg Alg : J , (4.6) −1 corresponding to left Kan extension along the functor J : R [C ] → int M int M −1 R C . Then, the restriction of Lan to QFT(int M ) induces a functor M J Lan : QFT(int M ) −→ QFT(M ) (4.7) that is left adjoint to the restriction functor res : QFT(M ) → QFT(int M ) in (4.3). Due to uniqueness (up to unique natural isomorphism) of adjoint functors (cf. Proposition A.3), it follows that ext Lan , i.e., (4.7) is a model for the extension functor ext in (4.3). Proof. A general version of this problem has been addressed in [5, Section 6]. Using in particular [5, Corollary 6.5], we observe that we can prove this propo- −1 sition by showing that every object V ∈R C is J -closed in the sense of [5, Deﬁnition 6.3]. In our present scenario, this amounts to proving that for all −1 causally disjoint inclusions i : V → V ← V : i with V ,V ∈R [C ] 1 1 2 2 1 2 int M int M −1 in the interior and V ∈R C not necessarily in the interior, there ex- ists a factorization of both i and i through a common interior region. Let 1 2 us consider the Cauchy development D(V V ) of the disjoint union and 1 2 the canonical inclusions j : V → D(V V ) ← V : j . As we explain be- 1 1 1 2 2 2 −1 low, D(V V ) ∈R [C ] is an interior region and j ,j provide the 1 2 int M 1 2 int M desired factorization: Since the open set V V ⊆ int M is causally con- 1 2 vex by Proposition 2.5(b), D(V V ) is causally convex, open and contained 1 2 in the interior int M by Proposition 2.8(c–d). It is, moreover, stable under Cauchy development by Proposition 2.3(b), which also provides the inclusion V ⊆ V V ⊆ D(V V ) inducing j ,for k =1, 2. Consider now the chain k 1 2 1 2 k of inclusions V ⊆ V V ⊆ V corresponding to i ,for k =1, 2. From the k 1 2 k stability under Cauchy development of V , V and V , we obtain also the chain 1 2 of inclusions V ⊆ D(V V ) ⊆ V ,for k =1, 2, that exhibits the desired k 1 2 factorization V (4.8) i i 1 2 D(i i ) 1 2 V D(V V ) V 1 1 2 2 j j 1 2 which completes the proof. AQFT on Spacetimes with Timelike Boundary We shall now brieﬂy review a concrete model for left Kan extension along full subcategory embeddings that was developed in [5, Section 6]. This model is obtained by means of abstract operadic techniques, but it admits an intuitive graphical interpretation that we explain in Remark 4.4 below. It allows us to compute quite explicitly the extension ext A =Lan A ∈ QFT(M)ofa quantum ﬁeld theory A ∈ QFT(int M ) deﬁned on the interior int M to the −1 whole spacetime M . The functor ext A : R C → Alg describing the −1 extended quantum ﬁeld theory reads as follows: To V ∈R C it assigns a quotient algebra ext A(V )= A(V ) ∼ (4.9) i:V →V that we will describe now in detail. The direct sum (of vector spaces) in (4.9) −1 runs over all tuples i : V → V of morphisms in R C , i.e., i =(i : V → M 1 1 V,...,i : V → V ) for some n ∈ Z , with the requirement that all sources n n ≥0 −1 V ∈R [C ] are interior regions. (Notice that the regions V are not k int M k int M 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 |V | A(V ):= A(V ), (4.10) k=1 where |V | is the length of the tuple. (For the empty tuple, we set A(∅)= C). This means that the (homogeneous) elements (i,a) ∈ A(V ) (4.11) i:V →V are given by pairs consisting of a tuple of morphisms i : V → V (with all V 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 (i,a)(i ,a ):= (i,i ),a ⊗ a , (4.12) where (i,i )=(i ,...,i ,i ,...,i ) is the concatenation of tuples. The unit 1 n 1 m element in (4.11)is 1 := (∅, 1), where ∅ is the empty tuple and 1 ∈ C, and the ∗-involution is deﬁned by ∗ ∗ (i ,...,i ),a ⊗ ··· ⊗ a := (i ,...,i ),a ⊗ ··· ⊗ a (4.13) 1 n 1 n n 1 n 1 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 ,...,i ,a ⊗ ··· ⊗ a − i, A(i ) a ⊗ ··· ⊗ A(i ) a ∈ A(V ) , 1 n 1 n 1 1 n n i:V →V (4.14) for all tuples i : V → V of length |V | = n ≥ 1 (with all V in the interior), all −1 tuples i : V → V of R [C ]-morphisms (possibly of length zero), for k int M k k int M M. Benini et al. Ann. Henri Poincar´ e k =1,...,n, and all a ∈ A(V ), for k =1,...,n. The tuple in the ﬁrst term k k of (4.14) is deﬁned by composition i i ,...,i := i i ,...,i i ,...,i i ,...,i i (4.15a) 1 11 1 1|V | n n1 n n|V | 1 n 1 n and the expressions A(i) a in the second term are determined by A(i): A(V ) −→ A(V ),a ⊗ ··· ⊗ a −→ A(i ) a ··· A(i ) a , (4.15b) 1 n 1 1 n n where multiplication in A(V ) is denoted by juxtaposition. To a morphism −1 −1 i : V → V in R C , the functor ext A : R C → Alg assigns the M M M M algebra map ext A(i ) : ext A(V ) −→ ext A(V ), [i,a] −→ i (i),a , (4.16) 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 intu- itive graphical interpretation: We shall visualize the (homogeneous) elements (i,a)in(4.11) by decorated trees ··· a a 1 n (4.17) where a ∈ A(V ) is an element of the algebra A(V ) associated to the interior k k k region V ⊆ 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 ∈ A(V ). The product (4.12) is given by concatenation of the k k inputs of the individual decorated trees, i.e., V V V ··· ··· ··· a a a 1 n 1 a a a 1 m m (4.18) 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 ∈ A(V ). Finally, the ∗-ideal in (4.14) implements the fol- k k lowing relations: Assume that (i,a) is such that the sub-family of embeddings (i ,i ,...,i ):(V ,V ,...,V ) → V factorizes through some common in- k k+1 l k k+1 l terior region, say V ⊆ V . Using the original functor A ∈ QFT(int M ), we may form the product A(i )(a ) ··· A(i )(a ) in the algebra A(V ), which we k k l l AQFT on Spacetimes with Timelike Boundary denote for simplicity by a ··· a ∈ A(V ). We then have the relation k l V V ··· · ·· ··· ··· · ·· a a a a a a ··· a a 1 k l n 1 k l n (4.19) which we interpret as follows: Whenever (i ,i ,...,i ):(V ,V ,...,V ) → k k+1 l k k+1 l V is a sub-family of embeddings that factorizes through a common interior re- gion 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 . 5. 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 (4.3) in order to restrict it to a theory res B ∈ QFT(int M ) on the interior of M . Applying now the extension func- tor, we obtain another quantum ﬁeld theory ext res B ∈ QFT(M)onthe whole spacetime M , which we would like to compare to our original theory B ∈ QFT(M ). A natural comparison map is given by the B-component of the counit :extres → id of the adjunction (4.3), i.e., the canonical QFT(M ) QFT(M )-morphism :extres B −→ B. (5.1a) Using our model for the extension functor given in (4.9) (and the formulas −1 following this equation), the R C V -component of explicitly reads M B as ( ) : B(V ) ∼−→ B(V ), [i,b] −→ B(i) b . (5.1b) B V i:V →V In order to establish positive comparison results, we have to introduce the concept of ideals of quantum ﬁeld theories. Deﬁnition 5.1. An ideal I ⊆ B of a quantum ﬁeld theory B ∈ QFT(M)isa −1 functor I : R C → Vec to the category of complex vector spaces, which satisﬁes the following properties: M. Benini et al. Ann. Henri Poincar´ e −1 (i) For all V ∈R C , I(V ) ⊆ B(V ) is a two-sided ∗-ideal of the unital ∗-algebra B(V ). −1 (ii) For all R C -morphisms i : V → V , the linear map I(i): I(V ) → I(V ) is the restriction of B(i): B(V ) → B(V ) to the two-sided ∗-ideals () () I(V ) ⊆ B(V ). Lemma 5.2. Let B ∈ QFT(M ) and I ⊆ B any ideal. Let us deﬁne B/I(V ):= −1 B(V )/I(V ) to be the quotient algebra, for all V ∈R C ,and B/I(i): B/I(V ) → B/I(V ) to be the Alg-morphism induced by B(i): B(V ) → −1 B(V ), for all R C -morphisms i : V → V .Then B/I ∈ QFT(M ) is a quantum ﬁeld theory on M which we call the quotient of B by I. −1 Proof. The requirements listed in Deﬁnition 5.1 ensure that B/I : R C → Alg is an Alg-valued functor. It satisﬁes the causality axiom of Deﬁnition 3.5 because this property is inherited from B ∈ QFT(M ) by taking quotients. Lemma 5.3. Let κ : B → B be any QFT(M )-morphism. Deﬁne the vec- tor space ker κ(V):=ker κ : B(V ) → B (V ) ⊆ B(V ), for all V ∈ −1 R C ,and ker κ(i):ker κ(V ) → ker κ(V ) to be the linear map induced −1 by B(i): B(V ) → B(V ), for all R C -morphisms i : V → V .Then −1 ker κ : R C → Vec is an ideal of B ∈ QFT(M ), which we call the kernel of κ. Proof. The fact that ker κ deﬁnes a functor follows from naturality of κ.Prop- erty (ii) in Deﬁnition 5.1 holds true by construction. Property (i) is a conse- quence of the fact that kernels of unital ∗-algebra morphisms κ : 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 ext res B B (5.2) ext res B ker where both the projection π and the inclusion λ are QFT(M )-morphi- B B sms. 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 [14, Deﬁnition 2.3] for a notion of additivity on globally hyperbolic spacetimes. Deﬁnition 5.5. A quantum ﬁeld theory B ∈ QFT(M ) on a spacetime M with timelike boundary is called additive (from the interior) at the object −1 V ∈R C if the algebra B(V ) is generated by the images of the Alg- −1 morphisms B(i ): B(V ) → B(V ), for all R C -morphism i : V → int int M int int −1 V whose source V ∈R [C ] is in the interior int M of M . We call int int M int M B ∈ QFT(M ) additive (from the interior) if it is additive at every object AQFT on Spacetimes with Timelike Boundary −1 V ∈R C . The full subcategory of additive quantum ﬁeld theories on M add is denoted by QFT (M ) ⊆ QFT(M ). We can now prove our ﬁrst characterization theorem for boundary quan- tum ﬁeld theories. Theorem 5.6. Let B ∈ QFT(M ) be any quantum ﬁeld theory on a (not necessarily globally hyperbolic) spacetime M with timelike boundary and let −1 V ∈R C . Then the following are equivalent: (1) The V -component (λ ) :extres B(V ) ker (V ) −→ B(V ) (5.3) B V B of the canonical inclusion in (5.2) is an Alg-isomorphism. −1 (2) B is additive at the object V ∈R C . −1 Proof. Let i : V → V be any R C -morphism whose source V ∈ int int M int −1 R [C ] is in the interior int M of M . Using our model for the extension int M int M functor given in (4.9) (and the formulas following this equation), we obtain an Alg-morphism [i , −]: B(V ) −→ ext res B(V ),b −→ [i ,b]. (5.4) int int int Composing this morphism with the V -component of given in (5.1), we obtain a commutative diagram ( ) B V ext res B(V ) B(V ) (5.5) [i ,−] B(i ) int int B(V ) int for all i : V → V with V in the interior. int int int Next we observe that the images of the Alg-morphisms (5.4), for all i : V → V with V in the interior, generate ext res B(V ). Combining int int int this property with (5.5), we conclude that ( ) is a surjective map if and B V only if B is additive at V . Hence, (λ ) given by (5.2), which is injective by B V construction, is an Alg-isomorphism if and only if B is additive at V . Corollary 5.7. λ :extres B ker → B given by (5.2) is a QFT(M )- B B add isomorphism if and only if B ∈ QFT (M ) ⊆ QFT(M ) is additive in the sense of Deﬁnition 5.5. We shall now reﬁne this characterization theorem by showing that add QFT (M ) is equivalent, as a category, to a category describing quantum ﬁeld theories on the interior of M together with suitable ideals of their uni- versal extensions. The precise deﬁnitions are as follows. Deﬁnition 5.8. Let B ∈ QFT(M ). An ideal I ⊆ B is called trivial on the −1 interior if its restriction to R [C ] is the functor assigning zero vector int M int M −1 spaces, i.e., I(V ) = 0 for all V ∈R [C ] in the interior int M of M . int int int M int M M. Benini et al. Ann. Henri Poincar´ e Deﬁnition 5.9. Let M be a spacetime with timelike boundary. We deﬁne the category IQFT(M ) as follows: • Objects are pairs (A, I) consisting of a quantum ﬁeld theory A ∈ QFT(int M ) on the interior int M of M and an ideal I ⊆ ext A of its universal extension ext A ∈ QFT(M ) that is trivial on the interior. • Morphisms κ :(A, I) → (A , I )are QFT(M )-morphisms κ :ext A → ext A between the universal extensions that preserve the ideals, i.e., κ restricts to a natural transformation from I ⊆ ext A to I ⊆ ext A . There exists an obvious functor add Q : IQFT(M ) −→ QFT (M ), (5.6a) which assigns to (A, I) ∈ IQFT(M ) the quotient add Q(A, I):=ext A I ∈ QFT (M ). (5.6b) Notice that additivity of ext A I follows from that of the universal exten- sion ext A (cf. the arguments in the proof of Theorem 5.6) and the fact that quotients preserve the additivity property. There exists also a functor add S : QFT (M ) −→ IQFT(M ), (5.7a) which ‘extracts’ from a quantum ﬁeld theory on M the relevant ideal. Explic- add itly, it assigns to B ∈ QFT (M ) the pair SB := res B, ker . (5.7b) Notice that the ideal ker ⊆ ext res B is trivial on the interior: Applying the restriction functor (4.3)to , we obtain a QFT(int M )-morphism res : res ext res B −→ res B. (5.8) Proposition 4.1 together with the triangle identities for the adjunction ext res in (4.3) then imply that (5.8) is an isomorphism with inverse given by η :res B → res ext res B. In particular, res has a trivial kernel and res B B hence ker ⊆ ext res B is trivial on the interior (cf. Deﬁnition 5.8). Our reﬁned characterization theorem for boundary quantum ﬁeld theories is as follows. Theorem 5.10. The functors Q and S deﬁned in (5.6) and (5.7) exhibit an equivalence of categories add QFT (M ) IQFT(M ). (5.9) add Proof. We ﬁrst consider the composition of functors QS : QFT (M ) → add add QFT (M ). To B ∈ QFT (M ), it assigns QSB =ext res B ker . (5.10) The QFT(M )-morphisms λ :ext res B/ ker → B given by (5.2) deﬁne a B B natural transformation λ : QS → id add , which is a natural isomorphism QFT (M ) due to Corollary 5.7. AQFT on Spacetimes with Timelike Boundary Let us now consider the composition of functors SQ : IQFT(M ) → IQFT(M ). To (A, I) ∈ IQFT(M ), it assigns SQ(A, I)= res ext A I , ker = res ext A, ker , (5.11) ext A/I ext A/I where we also used that res ext A I = res ext A because I is by hypothesis trivial on the interior. Using further the QFT(int M )-isomorphism η : A → res ext A from Proposition 4.1, we deﬁne a QFT(M )-morphism q via the (A,I) diagram (A,I) ext A ext A I (5.12) ext η A = ext A/I ext res ext A ext res ext A I Using the explicit expression for given in (5.1) and the explicit expres- ext A/I sion for η given by (η ) : A(V ) −→ res ext A(V )= A(V ) ∼, int int A V int i:V →V int a −→ [id ,a], (5.13) int −1 for all V ∈R [C ], one computes from the diagram (5.12) that q is int int M (A,I) int M the canonical projection π :ext A → ext A/I. Hence, the QFT(M )- isomorphisms ext η :ext A → ext res ext A induce IQFT(M )-isomorphisms ext η :(A, I) −→ SQ(A, I), (5.14) which are natural in (A, I), i.e., they deﬁne a natural isomorphism ext η : id → SQ. IQFT(M ) Remark 5.11. The physical interpretation of this result is as follows: Every add additive quantum ﬁeld theory B ∈ QFT (M ) on a (not necessarily glob- ally hyperbolic) spacetime M with timelike boundary admits an equivalent description in terms of a pair (A, I) ∈ IQFT(M ). Notice that the roles of A and I are completely diﬀerent: On the one hand, A ∈ QFT(int M)isa quantum ﬁeld theory on the interior int M of M and as such it is independent of the detailed aspects of the boundary. On the other hand, I ⊆ ext A is an ideal of the universal extension of A that is trivial on the interior, i.e., it only captures the physics that happens directly at the boundary. Examples of such ideals I arise by imposing speciﬁc boundary conditions on the universal ex- tension ext A ∈ QFT(M ), i.e., the quotient ext A/I describes a quantum ﬁeld theory on M that satisﬁes speciﬁc boundary conditions encoded in I. We shall illustrate this assertion in Sect. 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 cap- tures 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 M. Benini et al. Ann. Henri Poincar´ e 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. 6. Example: Free Klein–Gordon Theory In order to illustrate and make more explicit our abstract constructions de- veloped 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 are globally hyperbolic spacetimes with empty int M boundary, see Proposition 2.11. This allows us to use the standard techniques of [1, Section 3] on such regions. −1 6.1. Deﬁnition on R [C ]: int M int M Let M be a globally hyperbolic spacetime with timelike boundary, see Def- −1 inition 2.9. The free Klein–Gordon theory on R [C ] is given by the int M int M following standard construction, see e.g., [3, 4] for expository reviews. On the interior int M , we consider the Klein–Gordon operator 2 ∞ ∞ P := + m : C (int M ) −→ C (int M ) , (6.1) where is the d’Alembert operator and m ≥ 0 is a mass parameter. When −1 restricting P to regions V ∈R [C ], we shall write int M int M ∞ ∞ P : C (V ) −→ C (V ). (6.2) It follows from [1] that there exists a unique retarded/advanced Green’s oper- ator ± ∞ ∞ G : C (V ) −→ C (V ) (6.3) V c −1 for P because every V ∈R [C ] is a globally hyperbolic spacetime V int M int M with empty boundary, cf. Proposition 2.11. The Klein–Gordon theory K ∈ QFT(int M ) is the functor −1 K : R [C ] → Alg given by the following assignment: To any V ∈ int M int M −1 R [C ] it assigns the associative and unital ∗-algebra K(V ) that is freely int M int M generated by Φ (f ), for all f ∈ C (V ), modulo the two-sided ∗-ideal gener- ated by the following relations: • Linearity Φ (αf + βg)= α Φ (f)+ β Φ (g), for all α, β ∈ R and V V V f, g ∈ C (V ); ∗ ∞ • Hermiticity Φ (f ) =Φ (f ), for all f ∈ C (V ); V V • Equation of motion Φ (P f ) = 0, for all f ∈ C (V ); V V • Canonical commutation relations (CCR) Φ (f)Φ (g) − Φ (g)Φ (f ) V V V V =i τ (f, g) 1, for all f, g ∈ C (V ), where τ (f, g):= fG (g)vol (6.4) V V V V AQFT on Spacetimes with Timelike Boundary + − with G := G − G the causal propagator and vol the canonical vol- V V V V ume form on V . (Note that τ is antisymmetric, see e.g., [1, Lemma 4.3.5]). −1 −1 To a morphism i : V → V in R [C ], the functor K : R [C ] → int M int M int M int M Alg assigns the algebra map that is speciﬁed on the generators by pushforward along i (which we shall suppress) K(i): K(V ) −→ K(V ), Φ (f ) −→ Φ (f ). (6.5) V V The naturality of τ (i.e., naturality of the causal propagator, cf. e.g., [1, Sec- tion 4.3]) entails that the assignment K deﬁnes a quantum ﬁeld theory in the sense of Deﬁnition 3.5. 6.2. Universal Extension Using the techniques developed in Sect. 4, we may now extend the Klein– Gordon theory K ∈ QFT(int M ) from the interior int M to the whole space- time M . In particular, using (4.9) (and the formulas following this equa- tion), one could directly compute the universal extension ext K ∈ QFT(M ). The resulting expressions, however, can be considerably simpliﬁed. We there- fore prefer to provide a more convenient model for the universal extension ext K ∈ QFT(M ) by adopting the following strategy: We ﬁrst make an ‘ed- ext ucated guess’ for a theory K ∈ QFT(M ) which we expect to be the uni- versal extension of K ∈ QFT(int M ). (This was inspired by partially simpli- fying the direct computation of the universal extension). After this, we shall ext prove that K ∈ QFT(M ) satisﬁes the universal property that characterizes ext ext K ∈ QFT(M ). Hence, there exists a (unique) isomorphism ext K = K in ext QFT(M ), which means that our K ∈ QFT(M ) is a model for the universal extension ext K. −1 ext Let us deﬁne the functor K : R C → Alg by the following assign- −1 ment: To any region V ∈R C , which may intersect the boundary, we ext assign the associative and unital ∗-algebra K (V ) that is freely generated by Φ (f ), for all f ∈ C (int V ) in the interior int V of V , modulo the two-sided ideal generated by the following relations: • Linearity Φ (αf + βg)= α Φ (f)+ β Φ (g), for all α, β ∈ R and V V V f, g ∈ C (int V ); ∗ ∞ • Hermiticity Φ (f ) =Φ (f ), for all f ∈ C (int V ); V V • Equation of motion Φ (P f ) = 0, for all f ∈ C (int V ); V int V • Partially-deﬁned CCR Φ (f)Φ (g) − Φ (g)Φ (f)=i τ (f, g) 1,for V V V V V int −1 all interior regions V ∈R [C ] with V ⊆ int V and f, g ∈ int int M int int M C (int V ) with supp(f ) ∪ supp(g) ⊆ V . int Remark 6.1. We note that our partially-deﬁned CCR are consistent in the () −1 following sense: Consider V ,V ∈R [C ] with V ⊆ int V and int int M int int M int () f, g ∈ C (int V ) with the property that supp(f ) ∪ supp(g) ⊆ V .Us- c int ing the partially-deﬁned CCR for both V and V , we obtain the equality int int ext ext i τ (f, g) 1 =i τ (f, g) 1 in K (V ). To ensure that K (V ) is not the V V int int zero-algebra, we have to show that τ (f, g)= τ (f, g). This holds true V V int int due to the following argument: Consider the subset V ∩ V ⊆ int M . This int int M. Benini et al. Ann. Henri Poincar´ e is open, causally convex and by Proposition 2.3(c) also stable under Cauchy −1 development, hence V ∩ V ∈R [C ]. Furthermore, the inclusions int int M int int M () −1 V ∩ V → V are morphisms in R [C ]. It follows by construction int int M int int int M that supp(f ) ∪ supp(g) ⊆ V ∩ V and hence due to naturality of the τ ’s we int int obtain τ (f, g)= τ (f, g)= τ (f, g). (6.6) V V ∩V V int int int int Hence, for any ﬁxed pair f, g ∈ C (int V ), the partially-deﬁned CCR are independent of the choice of V (if one exists). int −1 −1 ext To a morphism i : V → V in R [C ], the functor K : R C → M M M M Alg assigns the algebra map that is speciﬁed on the generators by the push- forward along i (which we shall suppress) ext ext ext K (i): K (V ) −→ K (V ), Φ (f ) −→ Φ (f ). (6.7) V V ext Compatibility of the map (6.7) with the relations in K is a straightforward check. −1 −1 Recalling the embedding functor J : R [C ] →R C given in int M M int M M (4.2), we observe that the diagram of functors −1 R [C ] Alg (6.8) int M int M ext J K −1 R [C ] ext commutes via the natural transformation γ : K → K J with components speciﬁed on the generators by the identity maps ext γ : K(V ) −→ K (V ), Φ (f ) −→ Φ (f ), (6.9) int int V V V int int int −1 for all V ∈R [C ]. Notice that γ is a natural isomorphism because int int M int M int V = V and the partially-deﬁned CCR on any interior region V coin- int int int cides with the CCR. −1 Theorem 6.2. (6.8) is a left Kan extension of K : R [C ] → Alg along int M int M −1 −1 J : R [C ] →R C . As a consequence of uniqueness (up to unique int M M int M M natural isomorphism) of left Kan extensions and Proposition 4.3,itfollows ext ext that K ext K, i.e., K ∈ QFT(M ) is a model for our universal extension ext K ∈ QFT(M ) of the Klein–Gordon theory K ∈ QFT(int M ). Proof. We have to prove that (6.8) satisﬁes the universal property of left Kan −1 extensions: Given any functor B : R C → Alg and natural transfor- mation ρ : K → B J , we have to show that there exists a unique natural ext transformation ζ : K → B such that the diagram K B J (6.10) ζJ ext K J AQFT on Spacetimes with Timelike Boundary commutes. Because γ is a natural isomorphism, it immediately follows that ζJ is uniquely ﬁxed by this diagram. Concretely, this means that the components −1 ζ corresponding to interior regions V ∈R [C ] are uniquely ﬁxed V int int M int int M by −1 ext ζ := ρ γ : K (V ) −→ B(V ). (6.11) int int V V V int int int It remains to determine the components ext ζ : K (V ) −→ B(V ) (6.12) −1 ext for generic regions V ∈R C . Consider any generator Φ (f)of K (V ), M V where f ∈ C (int V ), and choose a ﬁnite cover {V ⊆ int V } of supp(f)by −1 interior regions V ∈R [C ], together with a partition of unity {χ } α int M α int M subordinate to this cover. (The existence of such a cover is guaranteed by the assumption that M is a globally hyperbolic spacetime with timelike boundary, see Proposition 2.11). We deﬁne ζ Φ (f ) := B(i ) ζ Φ (χ f ) , (6.13) V V α V V α α α where i : V → V is the inclusion. Our deﬁnition (6.13) is independent of the α α choice of cover and partition of unity: For any other {V ⊆ int V } and {χ }, β β we obtain B(i ) ζ Φ (χ f ) = B(i ) ζ Φ (χ χ f ) β V V αβ V ∩V V ∩V α β α α β β β β β β α,β = B(i ) ζ Φ (χ f ) , (6.14) α V V α α α where i : V → V and i : V ∩ V → V are the inclusions. In particular, this β αβ α β β implies that (6.13) coincides with (6.11) on interior regions V = V .(Hint: int Choose the cover given by the single region V together with its partition of int unity). ext We have to check that (6.13) preserves the relations in K (V ). Preserva- tion of linearity and Hermiticity is obvious. The equation of motion relations are preserved because ζ Φ (P f ) = B(i ) ζ Φ (χ P f ) V V α V V α int V α α V = B(i ) ζ Φ (χ P χ f ) αβ V ∩V V ∩V α β α β α β V ∩V α β α,β = B(i ) ζ Φ (P χ f ) =0. (6.15) β V V β β β V −1 Regarding the partially-deﬁned CCR, let V ∈R [C ] with V ⊆ int int M int int M int V and f, g ∈ C (int V ) with supp(f ) ∪ supp(g) ⊆ V .Wemay choose int the cover given by the single region i : V → V together with its partition of int M. Benini et al. Ann. Henri Poincar´ e unity. We obtain for the commutator ζ Φ (f ) ,ζ Φ (g) = B(i) ζ Φ (f ) , B(i) ζ Φ (g) V V V V V V V V int int int int =i τ (f, g) 1, (6.16) int which implies that the partially-deﬁned CCR are preserved. Naturality of the components (6.13) is easily veriﬁed. Uniqueness of the ext resulting natural transformation ζ : K → B is a consequence of uniqueness of ext ext ζJ and of the fact that the Alg-morphisms K (i ): K (V ) K(V ) → int int int ext ext K (V ), for all interior regions i : V → V , generate K (V ), for all V ∈ int int −1 R C . This completes the proof. 6.3. Ideals from Green’s Operator Extensions The Klein–Gordon theory K ∈ QFT(int M ) on the interior int M of the glob- ally hyperbolic spacetime M with timelike boundary and its universal exten- ext sion K ∈ QFT(M ) depend on the local retarded and advanced Green’s oper- ± −1 ∞ ∞ ators G : C (V ) → C (V ) on all interior regions V ∈R [C ] int int int M c int V int M int in M . For constructing concrete examples of quantum ﬁeld theories on globally hyperbolic spacetimes with timelike boundary as in [12], one typically imposes suitable boundary conditions for the ﬁeld equation in order to obtain also global retarded and advanced Green’s operators on M . Inspired by such examples, we + − shall now show that any choice of an adjoint-related pair (G ,G ) consisting of a retarded and an advanced Green’s operator for P on M (see Deﬁnition 6.3 ext below) deﬁnes an ideal I ± ⊆ K ∈ QFT(M ) that is trivial on the interior ext (cf. Deﬁnition 5.8). The corresponding quotient K I ± ∈ QFT(M ) then may be interpreted as the Klein–Gordon theory on M , subject to a speciﬁc choice of boundary conditions that is encoded in G . Deﬁnition 6.3. A retarded/advanced Green’s operator for the Klein–Gordon ± ∞ ∞ operator P on M is a linear map G : C (int M ) → C (int M ) which satisﬁes the following properties, for all f ∈ C (int M ): (i) PG (f)= f , (ii) G (Pf)= f,and (iii) supp(G (f )) ⊆ J (supp(f )). + − A pair (G ,G ) consisting of a retarded and an advanced Green’s operator + − for P on M is called adjoint-related if G is the formal adjoint of G , i.e., + − G (f ) g vol = fG (g)vol , (6.17) M M M M for all f, g ∈ C (int M ). Remark 6.4. In contrast to the situation where M is a globally hyperbolic spacetime with empty boundary [1], existence, uniqueness and adjoint- relatedness of retarded/advanced Green’s operators for the Klein–Gordon op- erator P is in general not to be expected on spacetimes with timelike boundary. Positive results seem to be more likely on globally hyperbolic spacetimes with AQFT on Spacetimes with Timelike Boundary non-empty timelike boundary, although the general theory has not been de- veloped yet to the best of our knowledge. Simple examples of adjoint-related pairs of Green’s operators were constructed, e.g., in [12]. −1 Given any region V ∈R C in M , which may intersect the boundary, we use the canonical inclusion i : V → M to deﬁne local retarded/advanced Green’s operators ∞ ∞ C (int V ) C (int V ) (6.18) ∗ i ∞ ∞ C (int M ) C (int M ) where i denotes the pushforward of compactly supported functions (i.e., ex- tension by zero) and i the pullback of functions (i.e., restriction). Since V ⊆ M ± ± is causally convex, it follows that J (p) ∩ V = J (p) for all p ∈ V . Therefore M V G satisﬁes the axioms of a retarded/advanced Green’s operator for P on V . V V (Here, we regard V as a globally hyperbolic spacetime with timelike bound- ary, see Proposition 2.11. J (p) denotes the causal future/past of p in the −1 spacetime V ). In particular, for all interior regions V ∈R [C ]in M , int int M int M by combining Proposition 2.11 and [1, Corollary 3.4.3] we obtain that G int as deﬁned in (6.18) is the unique retarded/advanced Green’s operator for the restricted Klein–Gordon operator P on the globally hyperbolic spacetime int V with empty boundary. int + − Consider any adjoint-related pair (G ,G ) of Green’s operator for P on −1 ext M . For all V ∈R C ,weset I ± (V ) ⊆ K (V ) to be the two-sided ∗-ideal generated by the following relations: ± ∞ • G -CCR Φ (f)Φ (g) − Φ (g)Φ (f)=i τ (f, g) 1, for all f, g ∈ C V V V V V (int V ), where τ (f, g):= fG (g)vol (6.19) V V V + − with G := G − G the causal propagator and vol the canonical V V V V volume form on V . + − The fact that the pair (G ,G ) is adjoint-related (cf. Deﬁnition 6.3) implies −1 that for all V ∈R C the causal propagator G is formally skew-adjoint, M V hence τ is antisymmetric. ext Proposition 6.5. I ± ⊆ K is an ideal that is trivial on the interior (cf. Deﬁnition 5.8). −1 Proof. Functoriality of I ± : R C → Vec is a consequence of (6.18), G M ext hence I ± ⊆ K is an ideal in the sense of Deﬁnition 5.1. It is trivial on the −1 interior because for all interior regions V ∈R [C ], the Green’s op- int M int int M erators deﬁned by (6.18) are the unique retarded/advanced Green’s operators for P and hence the relations imposed by I ± (V ) automatically hold true G int int ext in K (V ) on account of the (partially-deﬁned) CCR. int M. Benini et al. Ann. Henri Poincar´ e 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 space- time with empty boundary once equipped with the induced metric, orientation and time-orientation. m−1 m Example 6.7. Consider the sub-spacetime M := R × [0,π] ⊆ R of m- dimensional Minkowski spacetime, which has a timelike boundary ∂M = m−1 + − R ×{0,π}. The constructions in [12] 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 ext quantum ﬁeld theory K I ± ∈ 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 con- structed in [12]. 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 BDS ext DNP A (M):= K I ± (M ) with the global algebra A (M ) constructed in [12]. Both algebras are CCR-algebras, however the underlying symplec- tic vector spaces diﬀer: The symplectic vector space underlying our global BDS ∞ ∞ algebra A (M)is C (int M ) PC (int M ) with the symplectic structure c c (6.19). Using that the spatial slices of M = R × [0,π] are compact, we observe DNP that the symplectic vector space underlying A (M ) is given by the space Sol (M)of all solutions with Dirichlet boundary condition on M (equipped Dir with the usual symplectic structure). The causal propagator deﬁnes a sym- ∞ ∞ plectic map G : C (int M ) PC (int M ) → Sol (M ), which however is not Dir c c surjective for the following reason: Any ϕ ∈ C (int M ) has by deﬁnition com- pact support in the interior of M , hence the support of Gϕ ∈ Sol (M)is Dir schematically as follows supp Gϕ supp ϕ x = π x =0 (6.20) 2 2 The usual mode functions Φ (t, x)=cos( k + m t)sin(kx) ∈ Sol (M ), k Dir ∞ ∞ for k ≥ 1, are clearly not of this form, hence G : C (int M ) PC (int M ) → c c Sol (M ) cannot be surjective. As a consequence, the models constructed in Dir ext [12] are in general not additive from the interior and our construction K I ± should be interpreted as the maximal additive subtheory of these examples. AQFT on Spacetimes with Timelike Boundary It is interesting to note that there exists a case where both constructions m−1 m coincide: Consider the sub-spacetime M := R × [0, ∞) ⊆ R 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 BDS the Green’s operators, one may show that our algebra A (M ) is isomorphic to the construction in [12]. Acknowledgements We would like to thank Jorma Louko for useful discussions about boundaries in quantum ﬁeld theory and also Didier A. Solis for sending us a copy of his PhD thesis [29]. The work of M. B. is supported by a research grant funded by the Deutsche Forschungsgemeinschaft (DFG, Germany). Furthermore, M. B. would like to thank Istituto Nazionale di Alta Matematica (INdAM, Italy) for the ﬁnancial support provided during his visit to the School of Mathematical Sciences of the University of Nottingham, whose kind hospitality is gratefully acknowledged. The work of C. D. is supported by the University of Pavia. C. D. gratefully acknowledges the kind of hospitality of the School of Mathematical Sciences of the University of Nottingham, where part of this work has been done. A. S. gratefully acknowledges the ﬁnancial support of the Royal Society (UK) through a Royal Society University Research Fellowship, a Research Grant and an Enhancement Award. He also would like to thank the Research Training Group 1670 “Mathematics inspired by String Theory and Quantum Field Theory” for the invitation to Hamburg, where part of this work has been done. Open Access. This article is distributed under the terms of the Creative Com- mons Attribution 4.0 International License (http://creativecommons.org/licenses/ by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Appendix A: Some Concepts from Category Theory Adjunctions This is a standard concept, which is treated in any category theory textbook, e.g., [28]. Deﬁnition A.1. An adjunction consists of a pair of functors F : C D : G (A.1) M. Benini et al. Ann. Henri Poincar´ e together with natural transformations η :id → GF (called unit)and : FG → id (called counit ) that satisfy the triangle identities Fη ηG F FGF G GF G (A.2) F G id id F G F G 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 F : C ∼ D : G (A.3) for which both the unit η and the counit are natural isomorphisms. Existence of an adjoint equivalence in particular implies that C = D are equivalent as categories. Proposition A.3. If a functor G : D → C admits a left adjoint F : C → D, then F is unique up to a unique natural isomorphism. Vice versa, if a functor F : C → D admits a right adjoint G : D → C, then G is unique up to a unique natural isomorphism. Localizations Localizations of categories are treated for example in [21, Section 7.1]. In our paper we restrict ourselves to small categories. Deﬁnition A.4. Let C be a small category and W ⊆ Mor C a subset of the set −1 of morphisms. A localization of C at W is a small category C[W ] together −1 with a functor L : C → C[W ] satisfying the following properties: (a) For all (f : c → c ) ∈ W , L(f): L(c) → L(c ) is an isomorphism in −1 C[W ]. (b) For any category D and any functor F : C → D that sends morphisms −1 in W to isomorphisms in D, there exists a functor F : C[W ] → D and a natural isomorphism F F L. −1 C[W ] (c) For all objects G, H ∈ D in the functor category, the map Hom −1 G, H −→ Hom C GL,H L (A.4) C[W ] is a bijection of Hom-sets. −1 Proposition A.5. If C[W ] exists, it is unique up to equivalence of categories. References [1] B¨ ar, C., Ginoux, N., Pf¨ aﬄe, F.: Wave Equations on Lorentzian Manifolds and Quantization. European Mathematical Society, Zuric ¨ h (2007) [2] Beem, J.K., Ehrlich, P.E., Easley, K.L.: Global Lorentzian geometry. Marcel Dekker, New York (1996) AQFT on Spacetimes with Timelike Boundary [3] Benini, M., Dappiaggi, C.: Models of free quantum ﬁeld theories on curved back- grounds. In: Brunetti, R., Dappiaggi, C., Fredenhagen, K., Yngvason, J. (eds.) Advances in Algebraic Quantum Field Theory, pp. 75–124. Springer, Heidelberg (2015). arXiv:1505.04298 [math-ph] [4] Benini, M., Dappiaggi, C., Hack, T.-P.: Quantum ﬁeld theory on curved backgrounds—a primer. Int. J. Mod. Phys. A 28, 1330023 (2013). arXiv:1306.0527 [gr-qc] [5] Benini, M., Schenkel, A., Woike, L.: Operads for algebraic quantum ﬁeld theory. arXiv:1709.08657 [math-ph] [6] Benini, M., Schenkel, A., Woike, L.: Involutive categories, colored ∗-operads and quantum ﬁeld theory. arXiv:1802.09555 [math.CT] [7] Brunetti, R., Fredenhagen, K., Verch, R.: The generally covariant locality prin- ciple: A new paradigm for local quantum ﬁeld theory. Commun. Math. Phys. 237, 31 (2003). arxiv:math-ph/0112041 [8] Bussola, F., Dappiaggi, C., Ferreira, H.R.C., Khavkine, I.: Ground state for a massive scalar ﬁeld in the BTZ spacetime with Robin boundary conditions. Phys. Rev. D 96(10), 105016 (2017). arXiv:1708.00271 [gr-qc] [9] Chru´ sciel, P.T., Galloway, G.J., Solis, D.: Topological censorship for Kaluza– Klein space-times. Ann. Henri Poincar´ e 10, 893 (2009). arXiv:0808.3233 [gr-qc] [10] Dappiaggi, C., Ferreira, H.R.C.: Hadamard states for a scalar ﬁeld in anti-de Sit- ter spacetime with arbitrary boundary conditions. Phys. Rev. D 94(12), 125016 (2016). arXiv:1610.01049 [gr-qc] [11] Dappiaggi, C., Ferreira, H.R.C.: On the algebraic quantization of a massive scalar ﬁeld in anti-de-Sitter spacetime. Rev. Math. Phys. 30, 1850004 (2018). arXiv:1701.07215 [math-ph] [12] Dappiaggi, C., Nosari, G., Pinamonti, N.: The Casimir eﬀect from the point of view of algebraic quantum ﬁeld theory. Math. Phys. Anal. Geom. 19, 12 (2016). arXiv:1412.1409 [math-ph] [13] Dutsc ¨ h, M., Rehren, K.-H.: Generalized free ﬁelds and the AdS-CFT correspon- dence. Ann. Henri Poincar´ e 4, 613 (2003). arXiv:math-ph/0209035 [14] Fewster, C.J.: Endomorphisms and automorphisms of locally covariant quantum ﬁeld theories. Rev. Math. Phys. 25, 1350008 (2013). arXiv:1201.3295 [math-ph] [15] Fredenhagen, K.: Generalizations of the theory of superselection sectors. In: Kastler, D. (ed.) The Algebraic theory of Superselection Sectors: Introduction and Recent Results, p. 379. World Scientiﬁc Publishing, Singapore (1990) [16] Fredenhagen, K.: Global observables in local quantum physics. In: Araki, H., Ito, K.R., Kishimoto, A., Ojima, I. (eds.) Quantum and Non-commutative Analysis: Past, Present and Future Perspectives, pp. 41–51. Kluwer Academic Publishers, Dordrecht (1993) [17] Fredenhagen, K., Rehren, K.-H., Schroer, B.: Superselection sectors with braid group statistics and exchange algebras II: geometric aspects and conformal co- variance. Rev. Math. Phys. 4, 113 (1992) [18] Haag, R., Kastler, D.: An algebraic approach to quantum ﬁeld theory. J. Math. Phys. 5, 848 (1964) [19] Ishibashi, A., Wald, R.M.: Dynamics in non-globally-hyperbolic static space- times II: general analysis of prescriptions for dynamics. Class. Quantum Gravity. 20, 3815 (2003). arXiv:gr-qc/0305012 M. Benini et al. Ann. Henri Poincar´ e [20] Ishibashi, A., Wald, R.M.: Dynamics in non-globally-hyperbolic static space- times III: anti-de Sitter space-time. Class. Quantum Gravity 21, 2981 (2004). arXiv:hep-th/0402184 [21] Kashiwara, M., Schapira, P.: Categories and Sheaves. Springer, Berlin (2006) [22] Kay, B.S.: The principle of locality and quantum ﬁeld theory on (non globally hyperbolic) curved spacetimes. Rev. Math. Phys. 4, 167 (1992) [23] Lee, J.M.: Introduction to Smooth Manifolds. Springer, New York (2013) [24] O’Neill, B.: Semi-Riemannian Geometry. Academic Press, New York (1983) [25] Rehren, K.-H.: Algebraic holography. Ann. Henri Poincar´ e 1, 607 (2000). arXiv:hep-th/9905179 [26] Rejzner, K.: Perturbative Algebraic Quantum Field Theory: An Introduction for Mathematicians. Springer, Heidelberg (2016) [27] Ribeiro, P.L.: Structural and Dynamical Aspects of the AdS/CFT Correspon- dence: A Rigorous Approach. Ph.D. Thesis, University of Sao Paulo (2007). arXiv:0712.0401 [math-ph] [28] Riehl, E.: Category Theory in Context. Dover Publications Inc, New York (2016) [29] Solis, D.A.: Global Properties of Asymptotically de Sitter and Anti de Sitter Spacetimes. Ph.D. Thesis, University of Miami (2006). https:// scholarlyrepository.miami.edu/dissertations/2414/ [30] Sommer, C.: Algebraische Charakterisierung von Randbedingungen in der Quan- tenfeldtheorie. Diploma Thesis in German, University of Hamburg (2006). http://www.desy.de/uni-th/lqp/psﬁles/dipl-sommer.ps.gz [31] Wald, R.M.: Dynamics in nonglobally hyperbolic, static space-times. J. Math. Phys. 21, 2802 (1980) [32] Wrochna, M.: The holographic Hadamard condition on asymptotically anti-de Sitter spacetimes. Lett. Math. Phys. 107, 2291 (2017). arXiv:1612.01203 [math- ph] [33] Zahn, J.: Generalized Wentzell boundary conditions and quantum ﬁeld theory. Ann. Henri Poincar´ e 19, 163 (2018). arXiv:1512.05512 [math-ph] Marco Benini Fachbereich Mathematik Universit¨ at Hamburg Bundesstr. 55 20146 Hamburg Germany e-mail: marco.benini@uni-hamburg.de Claudio Dappiaggi INFN, Sezione di Pavia Universit` a di Pavia Via Bassi 6 27100 Pavia Italy e-mail: claudio.dappiaggi@unipv.it and AQFT on Spacetimes with Timelike Boundary Dipartimento di Fisica INFN, Sezione di Pavia Via Bassi 6 27100 Pavia Italy Alexander Schenkel School of Mathematical Sciences University of Nottingham University Park Nottingham NG7 2RD UK e-mail: alexander.schenkel@nottingham.ac.uk Communicated by Karl Henning Rehren. Received: January 3, 2018. Accepted: May 3, 2018.

Journal

Annales Henri Poincaré
– Springer Journals

Published: May 30, 2018

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References

Wave Equations on Lorentzian Manifolds and Quantization

Bär, C; Ginoux, N; Pfäffle, F

Global Lorentzian geometry

Beem, JK; Ehrlich, PE; Easley, KL

Quantum field theory on curved backgrounds—a primer

Benini, M; Dappiaggi, C; Hack, T-P

Ground state for a massive scalar field in the BTZ spacetime with Robin boundary conditions

Bussola, F; Dappiaggi, C; Ferreira, HRC; Khavkine, I

Topological censorship for Kaluza–Klein space-times

Chruściel, PT; Galloway, GJ; Solis, D

Hadamard states for a scalar field in anti-de Sitter spacetime with arbitrary boundary conditions

Dappiaggi, C; Ferreira, HRC

On the algebraic quantization of a massive scalar field in anti-de-Sitter spacetime

Dappiaggi, C; Ferreira, HRC

The Casimir effect from the point of view of algebraic quantum field theory

Dappiaggi, C; Nosari, G; Pinamonti, N

Endomorphisms and automorphisms of locally covariant quantum field theories

Fewster, CJ

The Algebraic theory of Superselection Sectors: Introduction and Recent Results

Fredenhagen, K

Quantum and Non-commutative Analysis: Past, Present and Future Perspectives

Fredenhagen, K

Superselection sectors with braid group statistics and exchange algebras II: geometric aspects and conformal covariance

Fredenhagen, K; Rehren, K-H; Schroer, B

An algebraic approach to quantum field theory

Haag, R; Kastler, D

Categories and Sheaves

Kashiwara, M; Schapira, P

The principle of locality and quantum field theory on (non globally hyperbolic) curved spacetimes

Kay, BS

Introduction to Smooth Manifolds

Lee, JM

Semi-Riemannian Geometry

O’Neill, B

Perturbative Algebraic Quantum Field Theory: An Introduction for Mathematicians

Rejzner, K

Category Theory in Context

Riehl, E

Dynamics in nonglobally hyperbolic, static space-times

Wald, RM

The holographic Hadamard condition on asymptotically anti-de Sitter spacetimes

Wrochna, M

Generalized Wentzell boundary conditions and quantum field theory

Zahn, J

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