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General Covariance in Algebraic Quantum Field Theory

In this review we report on how the problem of general covariance is treated within the algebraic approach to quantum field theory by use of concepts from category theory. Some new results on net cohomology and superselection structure attained in this framework are included.


The Problem of General Covariance
The question of general covariance of physical theories in space and time can be traced back to the famous debate between Gottfried Wilhelm Leibniz and Samuel Clarke (the latter assisted by Sir Isaac Newton) on the ontological status of space in the years 1715-1716 [1], the central question being if space exists as a substance or as an absolute being and absolute motion is present (Clarke) or if it is constituted only in relation to co-existent things allowing for relativism in motions only (Leibniz). This kind of problems also played an important role when the general theory of relativity was being developed in the years around 1910. While Albert Einstein first characterized generally covariant field equations as inadmissible since they did not determine the metric field uniquely as shown in the hole argument (Lochbetrachtung) in the appendix of [56], he later accepted [55] that all physical laws had to be expressed by equations that are valid in all coordinate systems, i. e., which are covariant (generally covariant) under arbitrary substitutions. 1 The hole argument was dismissed by the reasoning that it is not the spacetime metric that has to be fixed uniquely by the field equations, but only the physical phenomena that occur in spacetime need to be given a unique expression with reference to any description of spacetime. All physical statements are given in terms of spacetime coincidences; measurements result in statements on meetings of material points of the measuring rods with other material points or in coincidences between watch hands and points on the clockface. The introduction of a reference system merely serves the easy description of the totality of all these coincidences (point-coincidence argument) [55, p. 776f]. 2 In this paper we give an account of new results on how this concept of general covariance is approached within the framework of algebraic quantum field 1 "Die allgemeinen Naturgesetze sind durch Gleichungen auszudrücken, die für alle Koordinatensysteme

Algebraic Quantum Field Theory
theory, a task already put forward in the Gibbs lecture on Missed Opportunities held by Freeman Dyson in 1972 [54] (including a warning on the necessity of a "delicate weaving together of concepts from differential geometry, functional analysis, and abstract algebra"). A couple of articles on this problem have been published since then [60,8,9]. But the introduction of notions from category theory into these attempts sheds new light on the problem at hand, opening up a new and promising perspective [23]. We will turn to a presentation of these ideas after a short account of the ideas of algebraic quantum field theory in this Introduction.

Algebraic Quantum Field Theory
The use of algebraic concepts has been one of the great achievements in the development of a theory for the description of microscopical phenomena in the middle of the 1920's, when a consistent description as quantum mechanics was presented by Max Born, Pascual Jordan, Werner Heisenberg and Paul Adrien Maurice Dirac [73,18,42,17]. For quantum mechanics an algebraic generalization of the formalism has been presented in [78]. With respect to physical systems of infinitely many degrees of freedom (quantum field theory), a mathematically sound foundation for an algebraic approach has been given by Irving Segal in the 1940's [107], which later, following the introduction of the concept of weak equivalence of representations of the algebras by James Michael Gardner Fell [57], was supplemented by the principle of locality for physical reasons in the work of Rudolf Haag [66] and Rudolf Haag, Hans-Jürgen Borchers and Bert Schroer [16] and axiomatized in the innovative publication An Algebraic Approach to Quantum Field Theory by Rudolf Haag and Daniel Kastler [70] in 1964.
In the following years the mathematical framework thus acquired proved to be a suitable basis for the formulation and investigation of structural questions of quantum field theory. Among these are counted the understanding of the multiparticle structure of field theory in the Haag-Ruelle scattering theory [67,100], the structure of inner symmetries in the Doplicher-Haag-Roberts (DHR) theory [45,46,47,48] and in the Doplicher-Roberts (DR) duality theory [50,51]. Since the general assumptions of the algebraic approach were not restrictive enough to exclude quantum field theoretic models exhibiting unphysical behaviour, attempts were made to supplement the structure by specific information about the phase-space structure of quantum field theory. These fall into two different groups: First, there are qualitative requirements in terms of compactness conditions [71,61]; second, quantitative requirements using the mathematical notion of nuclearity were introduced [36,34]. The former ones proved to be successful in eliminating quantum field theoretic models without a particle interpretation, while the second group was devised with thermodynamical applications in mind (cf. also [31,30,32]). They are an essential ingredient also in the analysis of scattering states in theories with long-range interactions via the concept of particle weights [35] in the attempt to generalize an ansatz by Araki and Haag [7] for the description of scattering states in terms of local observables (cf. [93,94] for a thorough exposition).
The investigation of thermodynamic equilibrium states within the framework of algebraic quantum field theory is based on the algebraic reformulation of the Gibbs ansatz for statistical mechanics [69] based on the so-called Kubo-Martin-Schwinger (KMS) condition [87,90]. Whereas the Gibbs ansatz requires the spectrum of the Hamiltonian to be discrete (system in a box), the KMS condition is mathematically well-defined without this restriction, giving direct access to the thermodynamic limit. Considerable efforts have been undertaken since then to further generalize this condition to relativistic systems [19] and also to develop a mathematically sound description of states not in equilibrium [33]. Closely connected with these questions is the Tomita-Takesaki modular theory, a cornerstone in the development of the theory of operator algebras and important also in applications to physical problems (cf. [14,15] for an overview).
A comprehensive exposition of the principles and achievements of the algebraic approach can be found in the monograph by Rudolf Haag [68]. To the open problems of algebraic quantum field theory belongs the task to exhibit physically relevant interacting models that meet the axioms (this is taken up in the approach of constructive quantum field theory). Furthermore, the incorporation of gravity into this framework is a major desideratum with all the pertaining questions: description of equilibrium and nonequilibrium states in curved spacetimes, their implications on cosmological questions, and a host of other exciting problems.

Observables, States and Experiments
Information on physical systems is gathered by performing experiments, i. e., one prepares the physical system in a clear-cut way to have it in a certain state α (or, more generally, in a mixture of states as a result of an incomplete preparation) which is the embodiment of all information that can possibly be gained, and then one has it interact with a measuring apparatus, the observable Q, which upon interaction reveals a certain change of its (macroscopic) properties. The measurement results are conveniently represented by real numbers so that formally we have to deal with a mapping (α, Q) → r ∈ Ê (1.1) as the basis of any theoretical attempts to the description of nature. A thorough analysis of this scheme has been given by Huzihiro Araki in his lectures at the ETH Zurich [5,6] and constitutes the basis for the corresponding discussion in [4]. Due to the fact that different procedures might result in the preparation of the same state of the physical system and that different measuring devices might return the same information about this state at hand, these notions refer to equivalence classes of preparation and measurement procedures rather than to individual ones.
Moreover, the science of physics is concerned with phenomena that can be reproduced (the theory of a single event being a futile venture) so that measurements are always concerned with ensembles representing a physical system in the same state. Apart from the limited accuracy of any actual measurement, this fact forces one to look for a probabilistic description of experiments; not the individual measurement result is decisive, but rather the probability distribution as the outcome of many (in principle, indefinitely many) experiments is the fact to be explained by a consistent theory. While in classical physics this indeterminacy of the results is ascribed solely to the limited accuracy in the preparation and measurement procedures, the developed theory giving definite predictions on the outcome of future experiments, this situation is changed drastically in quantum physics, where in principle only statistical assertions are possible.
States of a physical system either being the result of a complete preparation as pure ones or else the upshot of an incomplete preparation as a mixture, i. e., a convex combination of the former pure states, the mathematical structure of the state space Σ as a convex set is evident. The general formulation (1.1) takes the shape of a dual pairing, (1.2) where the structure of the set A of (equivalence classes of) observables is not fixed like that of Σ. One proposal for its structure is that of a Jordan algebra [78] with the product defined by Note that the first expression of A • B only involves scalar multiples, sums and powers of elements of A , not the product of A and B like the second one. A comprehensive presentation of Jordan algebras and their state spaces can be found in [3, Part I] (see also [88]). A theory for propositions about quantum systems has been developed under the term quantum logic. Garrett Birkhoff and John von Neumann revealed their structure of an orthocomplemented lattice [12]. If this is supplemented by the assumptions of semi-or orthomodularity, one can establish that the propositions are isomorphic to the set of orthogonal projection operators on a Hilbert space. A comprehensive presentation of orthocomplemented modular lattices is to be found in [109]. In view of these investigations it is not a far-fetched assumption to consider the structure of (1.2) as that of a dual pair of a * -algebra A with a state space Σ as the normalized part of its (algebraic or topological) dual. Normalization is required here in order for a probabilistic interpretation to be possible.
Introduction system under consideration while the self-adjoint operators stand for the measurements. In this way the set of measurements acquires the structure of the self-adjoint part of a concrete * -algebra of operators on a Hilbert space. Segal [107] had pointed to the fact that questions of physical interest could be answered without having to select a certain Hilbert space, if the measurements were part of a C * -algebra A, i. e., a Banach * -algebra with the additional property This idea was absorbed by Haag and Kastler in 1964 [70] who proposed not to consider just one single C * -algebra but rather a net of C * -algebras associated with bounded regions O of spacetime, claiming that the physical content of a theory is encoded in this mapping. Thereby they stressed the local nature of measurements in much the same spirit as Einstein did in 1916 [55] by stating [70, p. 851] that "ultimately all physical processes are analyzed in terms of geometric relations of (unresolved) phenomena." The net (1.5) of local C * -algebras is supposed to be subject to the following conditions (the existence of the quasi-local algebra is in fact not an assumption, but a consequence of the assumed isotony): • Isotony: For any two bounded regions O 1 and O 2 (1.6a) • Quasi-Local Algebra: The set-theoretic union of all local algebras A(O) generates a normed * -algebra which upon completion is again a C * -algebra A called the quasi-local algebra. 3 • Special Relativistic Covariance: If there is a representation of the proper orthochronous Poincaré group P ↑ + on the quasi-local algebra A by automorphisms, P ↑ + ∋ (Λ, x) → α (Λ,x) ∈ Aut(A), then these automorphisms should act covariantly, i. e.
The actual measurements are represented by the self-adjoint elements of the corresponding C * -algebras, while the states are the positive, normalized, linear functionals ω on the quasi-local algebra A.
• Positivity: For any element • Normalization: The norm of ω as an element of the topological dual In view of the probabilistic interpretation, the real number ω(A) that the state ω returns when applied to a self-adjoint element A of A is considered as the expectation value (mean value) resulting from measuring the corresponding observable in the ensemble represented by ω.
The algebra A is not considered as a concrete algebra, i. e., an algebra of bounded linear operators on a given Hilbert space. Instead it can be represented as such in various ways on different Hilbert spaces, where the term representation refers to a * -homomorphism π : A → B(H π ) (denoted (H π , π)) that associates bounded operators π(A) on a certain representation Hilbert space H π with elements A ∈ A, respecting the algebraic structure of A as well as its * -operation. In general there will exist an abundance of different representations of A that are not unitarily equivalent, i. e. connected by a unitary operator U : H 1 → H 2 , while some of them might be physically equivalent in the weak sense of Fell [57] (cf. [70, p. 851]). In fact, via the GNS (Gel'fand, Naimark, Segal) construction, every state ω on A is associated with its own representation (H ω , π ω ). In the years to come this general framework, supplemented by physically motivated further requirements depending on the investigations to be performed, has proven to constitute a flexible foundation to tackle structural questions of quantum field theory. Among the most important contributions rank investigations on nuclearity and the split property [28,49], equilibrium and nonequilibrium physics [19,33], modular theory [15], conformal field theory models [83], energy inequalities [58] and the approach to noncommutative spacetime [44]. Apart from that, a sound formulation of perturbation theory has been given within the algebraic approach to quantum field theory [21,22]. This review will concentrate on the investigation of general covariance as one of the many examples referred to above for research in this domain.

Preliminaries
The following exposition will be concerned with four-dimensional, globally hyperbolic spacetimes. Therefore, also to fix our notation, we will summarize some of their basic properties; for a thorough discussion the reader is referred to [72,111]. The condition of global hyperbolicity does not seem to be very restrictive on physical grounds, its main purpose being to rule out certain causal pathologies.
A spacetime is denoted by the pair (M , g), where M is a smooth (i. e. C ∞ , Hausdorff, paracompact and connected) four-dimensional manifold and g is a Lorentzian metric on it with signature (+1, −1, −1, −1). Not to overburden the formalism, we will usually only write M for the manifold, implicitly including its metric g. This also applies to other manifolds of the said type, where the metric shares the diacritical signs (primes, subscripts, . . . ) differentiating the manifolds. The spacetimes are moreover assumed to be oriented and timeoriented. Time orientability requires the existence of a C ∞ vector field u on M which is timelike everywhere, i. e., g(u, u) > 0 at all points of M . Nonspacelike tangent vectors t are called future directed or past directed if, in relation to the vector field u, one has g(u, t) > 0 or g(u, t) < 0, respectively. This defines a time direction on M which is globally consistent. A smooth curve γ : I → M , I a connected subset of Ê, with tangent vectorγ is called future directed timelike if g(γ,γ) > 0 and g(u,γ) > 0 for all points of γ; it is called future directed causal if g(γ,γ) 0 and g(u,γ) > 0 on all of γ. γ is said to be past directed timelike or past directed causal if in the above definitions the Lorentz product of u andγ is negative. If the nonspacelike curve γ is future directed and the point lim t→sup I γ(t) exists in M it is called the future endpoint. The definition of a past endpoint requires γ to be past directed and sup replaced by inf. A curve γ is called future inextendible (past inextendible) if the corresponding endpoints do not exist; it is called inextendible if it is future and past inextendible [91]. Given a point x ∈ M , the set I + (x) consists of all points in M that can be connected to x by a future directed timelike curve, while J + (x) consists of all points in M that are connectable to x by some future directed causal curve γ : I → M with x = γ(inf I). Replacing the requirement of future-directedness by pastdirectedness, one gets the sets I − (x) and J − (x), respectively.  An oriented and time-oriented spacetime M is called globally hyperbolic if for each pair of points x, y ∈ M the intersection of J − (y) and J + (x) is compact. This property has been shown to be equivalent to the existence of a smooth foliation of M in terms of Cauchy surfaces by Bernal and Sánchez [11]. A Cauchy surface is a smooth hypersurface of M that any inextendible causal curve intersects exactly once. In globally hyperbolic spacetimes the global Cauchy problem (inital value problem) for linear hyperbolic wave equations is well-posed, and these wave equations possess unique retarded and advanced fundamental solutions. While the property of global hyperbolicity does not refer to any spacetime isometries, the concept of isometric embeddings will be important later. Let (M 1 , g 1 ) and (M 2 , g 2 ) be two globally hyperbolic spacetimes, then we call a mapping ψ : M 1 → M 2 an isometric embedding of M 1 into M 2 if ψ is a diffeomorphism onto its range ψ(M 1 ), i. e., the mapψ : M 1 → ψ(M 1 ) ⊆ M 2 is a diffeomorphism, and if ψ is an isometry, i. e., ψ * g 1 = g 2 ↾ ψ(M 1 ).
Given a globally hyperbolic spacetime M , we want to introduce families of open subsets which will be used as an index set for nets of local algebras according to (1.5) and prove their stability under isometric embeddings.

Globally hyperbolic regions Let
This lemma can be used to show that the only obstruction to the directedness of the family K h (M ), viz. the fact that for any pair Then the causal complement of ψ −1 (O) in M 1 is nonempty, and one arrives at We stress that the collection K h (M ) possesses elements which are not simply connected subsets of M and elements whose causal complement is not arcwise-connected. Problems associated with this topological feature can be avoided by passing to certain subcollections like diamond regions.

Diamond regions
O is said to be based on C , i. e., The set of diamonds K d (M ) is a basis for the topology of M .
Proof. We prove the inclusion ⊇. Let O be a diamond of M 1 , i. e., there exist a spacelike Cauchy surface is a compact, spacelike acausal open set of M with boundaries and with a nonempty complement. By [10], there exists in M a spacelike Cauchy surface C such that ψφ −1 . The proof of the reverse inclusion is very similar.
There are other interesting collections of subsets of M , e. g. the family of regular diamonds used in [65] that contains the family of diamonds introduced above as a subset and has the same stability property.

Quantum Field Theories as Covariant Functors
The investigations to be presented draw upon the theory of categories and functors. The reader not familiar with these concepts can find an introduction to the field in [89]. The two categories to be used in this chapter are (ii) the isometric embedding preserves orientation and time-orientation of the embedded spacetime.
The composition for any morphisms ψ and ψ ′ in hom Loc (M 1 , M 2 ) and hom Loc (M 2 , M 3 ), respectively, is the set-theoretic composition of maps ψ ′ • ψ. ψ ′ • ψ : M 1 → M 3 is thus a well-defined map which obviously is a diffeomorphism onto its range ψ ′ ψ(M 1 ) and is evidently isometric. Likewise, the properties (i) and (ii) are obviously fulfilled so that indeed ψ ′ • ψ ∈ hom Loc (M 1 , M 3 ). The associativity of the composition rule is an immediate consequence of that of the set-theoretic composition of maps.
The requirement for categories that each hom Loc (M , M ) has to possess a unit element is fulfilled by the identity map id M : Obs This category has as objects the class Obj(Obs) formed by all unital C *algebras, while the collection of morphisms between the objects (algebras) A 1 and A 2 are the faithful (injective) unit-preserving * -homomorphisms. Given such morphisms α and α ′ belonging to hom Obs (A 1 , A 2 ) and hom Obs (A 2 , A 3 ), respectively, their composition α ′ • α is again the composition of maps and easily seen to be an element of hom Obs (A 1 , A 3 ).
The unit element of hom Obs (A, A) for any algebra A ∈ Obj(Obs) is the identity id A : A → A, A ∈ A.
Remark 2.5. (A) The first requirement (i) that the morphisms of Loc have to satisfy enforces the induced and intrinsic causal structures to coincide for the embedded spacetime ψ(M 1 ) ⊆ M 2 (cf. [84]). The second condition (ii) could be relaxed so that the possible reversal of orientation in space and time allows for a a discussion of PCT theorems.
(B) The framework presented above is open to variations depending on the problems to be investigated. The algebras that constitute the objects of Obs could be replaced by general * -algebras, Borchers algebras or von Neumann algebras. With respect to Loc, the spacetimes could have less specific properties or even be endowed with additional features like spin structures as in [41,110]. Now we introduce the concept of a locally covariant quantum field theory which encodes the generally covariant principle of locality.
Definition 2.6. (a) A locally covariant quantum field theory is a covariant functor A between the two categories Loc and Obs, i. e., writing α ψ for the image A (ψ) of the morphism ψ under the functor A , the following mappings have to satisfy the covariance properties and all spacetimes M ∈ Obj(Loc).
(b) A locally covariant quantum field theory given by the covariant functor A is called causal if for morphisms ψ j ∈ hom Loc (M j , M ), j = 1, 2, embedding M 1 and M 2 in the common spacetime M such that the sets ψ 1 (M 1 ) and ψ 2 (M 2 ) are causally separated in M , the corresponding algebras in A (M ) commute, i. e., A locally covariant quantum field theory given by the functor A assigns to any globally hyperbolic spacetime a corresponding C * -algebra in such a way that the algebras can be identified when the spacetimes are isometric. This is the mathematically precise way to express the requirement for a unique expression for physical phenomena in spacetime regardless of the selected coordinate system as explained on p. 4. The term local is used here in the sense of geometrically local which is not to be confused with locality in the sense of Einstein causality. Causal properties are specified only in (b) and (c) of Definition 2.6. Here, causality means that elements of the algebras α ψ 1 A (M 1 ) and α ψ 2 A (M 2 ) , respectively, commute in the larger algebra A (M ) when the subregions ψ 1 (M 1 ) and ψ 2 (M 2 ) of M are causally separated with respect to the metric g on M . This property is expected to hold generally for observable quantities which can be localized in certain subregions of spacetimes. The time slice axiom (c) (also called strong Einstein causality, or existence of a causal dynamical law, cf. [110]) states that the algebra of observables on a globally hyperbolic spacetime is already determined by the algebra of observables localized in a neighbourhood of any Cauchy surface.

Recovering Algebraic Quantum Field Theory
In this section we will demonstrate that and how the setting of algebraic quantum field theory on a fixed globally hyperbolic spacetime as described in Section 1.4 can be regained from a locally covariant quantum field theory given by a covariant functor A with the properties listed in the preceding Section 2.2. The general assumptions on the net of local algebras (1.5), i. e. isotony (1.6a) and locality (1.6b), are supplemented here by the requirement that all the local algebras A(O) as well as the quasi-local algebra A contain a common unit ½.
where α M ,O is an abbreviation for α ι M,O . Then the following statements hold: (ii) Assume that there exists a group G of isometric diffeomorphisms κ : M → M (i. e. κ * g = g) preserving orientation and time orientation. Then there also exists a representation G ∋ κ →α κ of the group G by C * -algebra automorphismsα κ : are causally separated from each other the corresponding algebras commute, Proof. (i) The proof of the first statement uses the covariance properties of the functor A . To demonstrate isotony, let O 1 and O 2 be elements of by the very properties of the functor A .
(ii) To prove the second statement, let κ : M → M be a diffeomorphism preserving the metric as well as time orientation and orientation. The functor A assigns an automorphism α κ : Since A ⊆ A (M ), the definition ofα κ as the restriction of α κ to A yields an automorphism with the required properties. The feature of a group representation is an immediate consequence of the covariance properties of the functor which imply α κ 1 • α κ 2 = α κ 1 •κ 2 for any pair of elements κ 1 , κ 2 ∈ G together with (2.5). Therefore, one concludes that indeedα Proposition 2.7 shows that the Haag-Kastler framework of quantum field theory can be recovered within the functorial approach presented in Section 2.2.

Quantum Fields as Natural Transformations
In the previous section a quantum field theory has been defined in terms of a covariant functor: an algebra is mapped into another algebra via the endomorphism α ψ = A (ψ), but a priori nothing is said about how specific elements of the algebras are mapped onto each other by that transformation. In this section the possibility to define locally covariant fields is considered, their importance lying in the potential construction of fields that only locally depend on the geometry. The physical meaning underlying this construction is the idea that such fields might serve as carriers of information from one point of a spacetime to another in absence of global isometries like translations, or even from one spacetime to another one. The definition of locally covariant quantum fields to be given below generalizes the approach of Gårding and Wightman, who characterized quantum fields as operator-valued distributions. Here the distributions are not specified explicitly in this way, but can take values in a topological * -algebra instead.
We consider a family A(M ) of topological * -algebras that are indexed by all spacetimes M in Obj(Loc). For each spacetime a quantum field is defined as a generalized algebra-valued distribution, i. e. a map Φ M : which is supposed to be continuous, but not necessarily linear. The collection of all these mappings constitutes the family Φ . = Φ M . In addition, we demand that for any ψ ∈ hom Loc (M 1 , M 2 ) there exists a continuous endomorphism The family Φ M with these covariance conditions is called a locally covariant quantum field. This simple description has a functorial translation to be outlined next.
Again we consider the category Loc and furthermore introduce the category TAlg consisting of topological * -algebras with unit elements as objects and of continuous * -endomorphisms as morphisms. This means that α : if it is a continuous, unit-preserving, injective * -morphism. Let Test denote the category containing all possible test function spaces over Loc as objects, i. e., the objects are all spaces C ∞ 0 (M ) of smooth, compactly supported test functions on M for all M ∈ Obj(Loc), and the morphisms are all possible push-forwards ψ * of isometric embeddings ψ : M 1 → M 2 . The action of any push-forward ψ * on an element of a test function space has been defined in the preceding paragraph, and it clearly satisfies the requirements for morphisms between test function spaces. Now, let the locally covariant quantum field theory A be defined as a functor according to Definition 2.6 with the modification that the category TAlg replaces the category Obs; again α ψ stands for A (ψ) whenever ψ is any morphism in Loc. Moreover, let D be the covariant functor between Loc and Test assigning to each M ∈ Obj(Loc) the test function space D (M ) = C ∞ 0 (M ) and to each morphism ψ of Loc its push-forward: D (ψ) = ψ * . Test as well as TAlg are regarded as subcategories of the category of all topological spaces, Top, leading to the following definition. Explicitly, which is the requirement of covariance for fields.
This definition is open to extensions. The test function spaces C ∞ 0 (M ) could be replaced by smooth, compactly supported sections of vector bundles, and as endomorphisms of these test section spaces one takes suitable pull-backs of vector-bundle endomorphisms. One can also include conditions on the wave front set of the field operators (cf. Definition 3.2 below).
(B) The notion of causality is introduced in an obvious way: A locally covari- (C) Admitting nonlinear fields in Definition 2.8 opens up the possibility to apply it to more general objects, e. g. the definition of a locally covariant S-matrix patterned according to the definition of a local S-matrix of Epstein and Glaser [22] (see also [20]).
(D) We give some examples in Chapter 3.

On the Notion of State Space
We have seen previously that part of the physical description of a theory proceeds through the selection (preparation) of a suitable class of states. Here we attempt at a definition of a state space suitable for the description of a generally covariant quantum field theory.
Indeed, suppose that our theory is given in terms of a covariant functor A . The question arises what the concept of a state might be in this case. The first, quite natural idea is to say that a state is a family ω M M ∈ Obj(Loc) indexed by the members in the object class Loc, where each ω M is a state on the C * -algebra A (M ). One might wonder if there are families of states ω M M ∈ Obj(Loc) that are distinguished by a property which in our framework would correspond to local diffeomorphism invariance, viz., However, there is a simple argument that shows that the above property will, in general, not be physically realistic. We refer to [23] for a thorough explanation.
A crucial question is whether there exists a more general concept of invariance that can be attributed to families of states ω M M ∈ Obj(Loc) for a locally covariant quantum field theory given by a functor A . We will argue that there is a positive answer to this question. To arrive at an explanation, let us fix some concepts that will turn out to be useful also in the subsequent chapters.

Folium of a representation Let
A be a C * -algebra and π : A → B(H ) be a * -representation of A by bounded linear operators on a Hilbert space H . The folium of π, denoted F(π), is the set of all states ω ′ on A which can be written as where Tr denotes the trace in H . In other words, the folium of a representation consists of all density matrix states in that representation.
Local quasi-equivalence and local normality Let A be a locally covariant quantum field theory and let, for fixed M , ω andω be two states on A (M ). We call these states (or their GNS representations, denoted π andπ, respectively) Moreover, we say that ω is locally normal toω (or to the corresponding GNS It is known that quasifree states of the free scalar field on globally hyperbolic spacetimes which fulfill the microlocal spectrum condition (cf. Section 3.1) are locally quasi-equivalent. We also note that the property of a state to fulfill the microlocal spectrum condition is a locally covariant property (owing to the covariant behaviour of wavefront sets of distributions under diffeomorphisms [77]). Thus, for a locally covariant quantum field theory it is natural to assume that, if ω ′ M fulfills (any suitable variant of) the microlocal spectrum condition, then so does ω ′ M • α ψ for any ψ ∈ hom Loc (M , M ′ ). In the case where also the folia of states (i. e. the folia of their GNS representations) satisfying the microlocal spectrum condition coincide locally, one thus obtains the invariance of local folia under local diffeomorphisms, more precisely, at the level of the GNS representations of ω M and ω ′ M , one has All these properties are known to hold for quasifree states of the free scalar field fulfilling the microlocal spectrum condition on globally hyperbolic spacetimes.
Thus one sees that local diffeomorphism invariance really occurs at the level of local folia of states for A . In this light it appears natural to give a functorial description of the space of states taking this form of local diffeomorphism invariance into account. To this end, it is convenient to first introduce a new category, the category of the set of states.
Sts An object S ∈ Obj(Sts) is a set of states on a C * -algebra A. Morphisms between members S ′ and S of Obj(Sts) are positive maps γ * : S ′ → S.
In the present work, γ * always arises as the dual map of a faithful C *algebra endomorphism γ : The category Sts is therefore dual to the category Obs. The composition rules for morphisms are thus obvious.
Now we can define a state space for a locally covariant quantum field theory in a functorial manner.
Definition 2.10. Let A be a locally covariant quantum field theory.
(a) A state space for A is a contravariant functor S between Loc and Sts: together with the requirement that unit morphisms are mapped to unit morphisms.
(b) We say that a state space S is locally quasi-equivalent if (2.6) holds for any pair of states ω,ω ∈ S (M ) (with GNS representations π,π, respectively) whenever M ∈ Loc and O ∈ K h (M ).
(c) A state space S is called locally normal if there exists a locally quasiequivalent state spaceS so that for each ω ∈ S (M ) there is someω ∈S (M ) (with GNS representationπ) so that (2.7) holds for all M ∈ Loc and O ∈ K h (M ).
Some straightforward consequences of these definitions can be found in [23]. Further properties are outlined in the last chapter.

Microlocal Analysis
This section serves the purpose of giving a self-contained introduction to definitions and results of Hörmander's microlocal analysis that will be needed in the sequel. After having defined the wave front set of a distribution, we recall Hörmander's results on the multiplication of distributions which extends to the composition of distribution-valued operators. Further details of this mathematical theory can be found in Hörmander's monograph [77] and in the original sources.
The theory of wave front sets was developed in the 1970's by Hörmander and Duistermaat [76,52], following the work of Sato [105,106]. Mathematicians use wave front sets (WF) mainly as a tool in partial differential equations. These sets are refinements of the notion of the singular support of a distribution. One advantage of the use of wave front sets over singular supports is their providing a simple characterization for the existence of products of distributions, eliminating the difference between local and global results. Duistermaat and Hörmander [52] recognized a link between microlocal analysis and quantum field theory, but these results were rarely used in the physics literature.
In microlocal analysis the study of singularities is shifted from the base space to the cotangent bundle by localizing the distribution around the singularity followed by an analysis of the result in Fourier space. Let u ∈ D ′ (Ê n ) be a distribution and let φ ∈ C ∞ 0 (V) be a smooth function with support in V ⊆ Ê n .
By a well-known argument from the theory of distributions, the Fourier transform of φu yields a smooth function in frequency space with the following relation, φu(ξ) = u , e −i . , ξ φ , where . , . denotes dual pairing. This result implies The singular support, sing supp u, of u ∈ D ′ (V) is the complement of the largest open subset of V where u is smooth. Motivated by the previous lemma, the notion of wave front sets is a refinement of that of the singular support, taking into account the direction in which the Fourier transform does not strongly decay.
Note that a conic set Σ has the property that with (x, ξ) also the pair (x, tξ) belongs to Σ for all t > 0. The following remarks can easily be proved using Lemma 3.1. (D) For any distribution v with wave front set WF(v) the wave front sets of its partial derivatives are contained in WF(v).
be a smooth function, then its wave front set is empty: WF( f ) = ∅.
(2) Consider the Dirac δ-distribution on Ê 2 . Its wave front set is WF δ( It is worth recalling that the set of normal coordinates (x 1 , . . . , x n , ξ 1 , . . . , ξ n ) of the cotangent bundle T * V over the base coordinates (x 1 , . . . , x n ) in V allows for the identification of V × Ê n with T * V and to consider WF(u) as a subset of the cotangent bundle. Since the definition of wave front sets is local, it can be lifted to manifolds. An intrinsic definition of wave front sets will be given below.
In this chapter the manifolds considered are assumed to be n-dimensional carrying a smooth Riemannian or semi-Riemannian metric.

Examples
Now, the wave front set, of the set of all regular directed points of u.
A useful application of wave front sets is the definition of products of distributions. Wave front sets provide a simple characterization for the existence of such products, which turns out to be sequentially continuous provided the wave front sets of the corresponding distributions are contained in a suitable cone in T * M \ {0}.
if and only if the following conditions hold: extends to a unique sequentially continuous operator from the set D ′ In particular, the wave front sets are related by Remark 3.8. Note that for the product of two distributions u and v to exist it is sufficient that WF(u) ⊕ WF(v) does not contain terms of the form (x, 0).

The Scalar Klein-Gordon Field on a Fixed Spacetime Background
We describe quantum fields propagating on a four-dimensional Lorentzian manifold (M , g) in terms of the general theory of quantized fields (cf. [59,68,112]). The manifold is assumed to be globally hyperbolic, i. e., it admits spacelike Cauchy hypersurfaces. We deal with the Gårding-Wightman approach to quantum fields [113] and its algebraic formulation by Borchers and Uhlmann [13,108]. To each differentiable manifold M is assigned a topological * -algebra .
where the elements of the sequence on the right-hand side aref n (x 1 , . . . , x n ) . = f n (x n , . . . , x 1 ) with the bar denoting complex conjugation. The unit element is 1 .
= (1, 0, 0, . . . ). This algebra can be equipped with a fairly natural locally convex topology with respect to which it is complete. See [13,108] for a further discussion of the Borchers-Uhlmann algebra (and also [60,104] Recall that a GNS quadruple (H ω , D ω , φ ω , Ω ω ) satisfies the following properties, usually referred to as the (curved spacetime) Gårding-Wightman axioms: (A) H ω is a separable Hilbert space, D ω is a dense subspace of H ω , and the GNS vacuum Ω ω is a distinguished vector in H ω .
(B) The fields φ ω are operator-valued distributions, i. e., for all Φ, Ψ ∈ D ω the linear mapping The subspace D ω is generated by applying finitely many smeared field operators to Ω ω .
In flat Minkowski spacetime these axioms are supplemented by requiring covariance with respect to the Poincaré (inhomogeneous Lorentz) group and a condition on the spectrum stating that the Fourier transformω(p 1 , . . . , p m ) of the m-point function is concentrated at p k + · · · + p m ∈ −V + for k = 2, 3, . . . , m.
The simplest and best studied example of a quantum field theory in curved spacetime is the scalar Klein-Gordon field. As shown by Dimock [40], its local C * -algebras can easily be constructed on each globally hyperbolic spacetime. As mentioned before, global hyperbolicity of the spacetime (M , g) entails wellposedness of the global Cauchy problem for the scalar Klein-Gordon equation, (for smooth, real-valued ϕ), where ∇ is the covariant derivative of g, m ≥ 0 and ξ ≥ 0 are constants, and R is the scalar curvature of g. Moreover, there exist uniquely determined advanced and retarded fundamental solutions of the Klein-Gordon equation, E adv / ret : It can be shown [40] that the definition with dµ g the metric-induced volume form on M , endowes R with a symplectic form, so that (R, σ) is a symplectic space.
If the field φ ω arising from Theorem 3.9 satisfies the Klein-Gordon equation and its commutator is given by we call ω a state of the Klein-Gordon field over M . Not all states ω are believed to be physically meaningful. A condition expected to be satisfied by physically reasonable states is the Hadamard condition [39] which has been studied by various authors (cf. the references in Fulling's book [63]). A mathematically precise definition of the Hadamard condition in terms of boundary values of certain complex-valued functions has been given by Kay and Wald in [85].
Radzikowski discovered [95] that an equivalent characterization of Hadamard states is possible in terms of their wave front sets. Using his results, Junker [79,80] (note also the erratum to [80] in [81]) constructed Hadamard states for free scalar fields on arbitrary globally hyperbolic spacetimes. Radzikowski's result can be recast in the following definition.
where P denotes a partition of the set {1, . . . , m} into subsets which are pairings of points, labelled by r. The ordering of points in ω 2 is preserved, e. g., (r, 1) < (r, 2), and two arguments never coincide so that the product r exists whenever ω 2 (x i , x j ) are distributions.
Using Theorem 3.7 in connection with (3.4), the wave front set of ω m is seen to satisfy where Q denotes a nonempty set of disjoint pairs and ω r 2 is the 2-point distribution in the variables x (r,1) , x (r,2) that, considered as a distribution on M n , has the wave front set WF(ω r 2 ) = (x 1 , 0; . . . ; x (r,1) , k (r,1) ; . . . ; x (r,2) , k (r,2) ; . . . ; x n , 0) (x (r,1) , k (r,1) ; x (r,2) , k (r,2) ) ∈ WF(ω 2 ) . (3.6) A generalization of the usual spectrum condition in Minkowski space to curved spacetimes can be given in terms of the wave front sets of a quantum field on the globally hyperbolic spacetime M . Its formulation requires some definitions from graph theory: Let G n denote the set of all finite graphs with vertices {1, . . . , n}, such that for every element G ∈ G n all edges occur in both admissible directions. An immersion of a graph G ∈ G n into M is an assignment of the vertices of G to points in M , ν → x(ν), and of the edges of G to piecewise smooth curves in M , e → γ(e) with source s(γ(e)) = x(s(e)) and range r(γ(e)) = x(r(e)), respectively, together with a covariantly constant causal covector field k e on γ, i. e. ∇k e = 0, such that: (A) if e is an edge from ν to ν ′ , then γ(e) connects x(ν) and x(ν ′ ); (B) if e −1 denotes the edge having the opposite direction compared to e, then the corresponding curve γ(e −1 ) is the inverse of γ(e); (C) for every edge e from ν to ν ′ , k e is directed towards the future whenever ν < ν ′ ; (D) k e −1 = −k e .
With this notation the microlocal spectrum condition for field theories over a globally hyperbolic manifold (substituting the usual Minkowski space spectrum condition) reads: The existence of nontrivial states which satisfy the Microlocal Spectrum Condition is the statement of the following proposition.

Proposition 3.14. Let ω denote a quasifree Hadamard state for the Klein-Gordon field on a globally hyperbolic manifold (M , g), then ω satisfies the µSC.
Proof. All odd m-point distributions vanish by assumption, hence ω m satisfies the µSC trivially for odd m. The wave front set of the 2-point distribution ω 2 is explicitly given by (3.3) and obviously satisfies the µSC. A general even mpoint distributions has the representation (3.4), which states that ω m is a sum of tensor products of 2-point distributions. Hence, there exists a disconnected graph G m ∈ G m together with an immersion (x, γ, k) satisfying (a) and (b) of Definition 3.11, i. e., G m consists of subgraphs G 2 ∈ G 2 such that the immersion (x, γ, k) restricted to these subgraphs is compatible with the wave front set of the corresponding 2-point distribution.  [86,37].
(B) Junker and Schrohe [82] have recently constructed states with adiabatic finiteness condition related to Sobolev order. They proved that, besides their existence, the states of finite "Sobolev order" are not Hadamard, while those with infinite order indeed are. They made extensive use of Sobolev wave front sets.
States satisfying the µSC obey the important properties laid down in the following theorems. Proof. By Theorem 3.7, it is sufficient to show that the sums of WF(ω i m ), i = 1, 2, do not intersect the zero section in order to prove that the products of the corresponding n-point distributions exist. Now, by assumption, WF(ω 1 m ) and WF(ω 2 m ) are both contained in the set Γ m which is stable under addition, according to Lemma 3.13. Hence, WF(ω 1 m ) ⊕ WF(ω 2 m ) ⊆ Γ m ⊆ T * M m \ {0}, as required for the products to exist. Moreover, this implies that they satisfy the µSC.
In order to prove that these new m-point distributions yield a state, i. e. satisfy Wightman positivity, we consider the tensor product of ω 1 and ω 2 . This is a state on the Borchers-Uhlmann algebra of two commuting scalar fields. Positivity of this state means that for all test functions Choose a family of real test functions {g ǫ } 0<ǫ 1 ⊆ C ∞ 0 (M 2 , Ω 2 1 ) such that the limit for ǫ → 0 is just the Dirac δ-distribution. Inserting them into (3.7), the limit for ǫ → 0 that exists by the above consideration is m,n (ω 1 ) n+m (ω 2 ) n+m (x n , . . . , From sequential continuity according to Theorem 3.7 we conclude that this expression is nonnegative, which is the desired positivity. Let ω be a state on the Borchers-Uhlmann algebra B(M ) with associated GNS quadruple (H ω , D ω , φ ω , Ω ω ). We recall from the previous chapter that the folium of ω consists of finite convex linear combinationsω of states induced by vectors in D ω :ω where Tr denotes the trace in H ω , π ω is the representation of B(M ) associated with ω and is some density matrix. Contrary to the usual spectrum condition, the µSC does not characterize a distinguished state, but is a property of the full folium instead. We recall that this characterization has been helpful for the functorial description of the state space in the previous chapter. Therefore, it is sufficient to show that for all m ∈ AE Moreover, since by assumption ω l+m satisfies the microlocal spectrum condition, to every element (y 1 , 0; . . . ; x i , k i ; . . . ; y l , 0) ∈ WF(ω l+m ) there correspond a graph G l+m ∈ G l+m and an immersion (x, γ, k) such that the covector fields k e are zero whenever γ(e) does not connect two points in {x 1 , . . . , x m }. For this statement note that the direction associated to y 1 vanishes by (3.10). Moreover, all causal covector fields associated to curves γ starting at y 1 are directed towards the future by the definition of an immersion, hence, using property (b) of Definition 3.11, k e = 0 whenever γ starts or ends at y 1 .
Consider now the point y 2 . By assumption, the direction associated to it is again zero. Using the properties of the immersion and the previous result for covector fields along curves ending at y 1 , one sees that the covector fields k e for all curves starting at y 2 are either future directed or zero. As in the previous case this implies k e = 0 for all curves starting or ending at y 2 . By induction, this result extends to all points up to y k and, analogously, to all points from y l down to y k+1 . Therefore, all points y 1 , . . . , y k , y k+1 , . . . , y l together with all lines starting or ending at them can be removed from the graph G l+m . The result is another graph G m ∈ G m together with an immersion (x, γ, k) such that (a) and (b) of Definition 3.11 are satisfied. This completes the proof.
The following theorem shows that the microlocal spectrum condition is compatible with the usual Minkowski space spectrum condition. It is not yet known whether, vice versa, the microlocal spectrum condition µSC is also sufficient for the usual spectrum condition to hold in Minkowski spacetime.

The Free Scalar Klein-Gordon Field as a Natural Transformation
In this section we give an example for fields as natural transformations by use of the Borchers-Uhlmann algebra B(M ) associated with the spacetime M (cf. Section 3.1).
An endomorphism ψ ∈ hom Loc (M 1 , M 2 ) between two spacetimes can be lifted to an algebraic endomorphism α ψ : B(M 1 ) → B(M 2 ) of the corresponding Borchers-Uhlmann algebras by setting where ψ (n) * denotes the n-fold push-forward defined via ψ

Interacting Fields
One of the most recent crucial results is the possibility to cast the usual formalism of perturbation theory into the setting we are reviewing.
It has always been a source of diffidence among physicists that algebraic quantum field theory was only (apparently, in retrospective) managing structural aspects and never addressed the task of constructing explicitly interacting models that form the core of modern quantum field theory as QED, the standard model, or QCD. One of the authors, in collaboration with Fredenhagen [21,22], launched a program which aimed at showing that algebraic quantum field theory can indeed be used to construct models at the perturbative level. It has been proved that the usual classification of the renormalization program can be obtained within the algebraic setting and that, moreover, one is able to produce results that were sought for a long time, like the ultraviolet classification of perturbative quantum field theory on curved spacetimes. Naively speaking, this last aspect was expected on the ground of the equivalence principle, since at very small scales (ultraviolet) one knows that the singularities arising in the renormalization procedure are essentially the same as in flat Minkowski spacetime. However, the details hide several difficulties, and the strategy for establishing the result required some ingenuity and devices that went far beyond the usual bag of tools that theoretical physicists commonly use. Indeed, one of the main problems is that the usual renormalization procedure makes use of the Feynman propagator (i. e. the Feynman graphic language) which is an ambiguous concept on curved spacetimes (and also in time-dependent external fields in general), not being uniquely defined. One of the commonly envisaged ways out was to use artificial boundary conditions, but in the general case not even this can be used. Notice that the same criticism applies to the path integral approach.
The use of microlocal analysis to overcome these difficulties was crucial. Indeed, wave front set properties of Wightman functions, the use of distinguished parametrices, Fourier integral operators and other tools form the essential language that allowed for the classification. Recently, Hollands and Wald [74,75] have refined to a large extent the structure of the renormalization program. They introduced as a basic input the principle of local covariance according to Definition 2.6 (plus some further technicalities) with the remarkable result that they can prove that the interacting fields are indeed locally covariant, i. e. natural transformation as in the free case considered in the previous section.
Since the bulk of these results can by themselves constitute a (long) review paper, we have to restrict ourselves to informing the interested reader about the relevant points in the literature.

Sharply Localized Charges and Particle Weights
One of the aims of local quantum physics is to understand the nature and the features of charges of elementary particles in terms of superselection sectors of the observable net. The term charges here not only refers to charges of electromagnetic type, but also to the baryonic number, the leptonic number and the isospin. A common feauture of these charges is that they can be added, and that to any charge there corresponds an opposite one according to the particleantiparticle symmetry. Furthermore, every charge has a statistics (Fermi-Bose alternative), that of the particle carrying that charge. Superselection sectors are the unitary equivalence classes of irreducible representations of the observable net, and quantum numbers are the labels distinguishing different sectors. The main task is to single out, from the infinite number of representations of the observable net, those describing the charges of particles.

DHR analysis
The first class of superselection sectors, the DHR sectors, has been established in a series of papers by Doplicher, Haag and Roberts [47,48], a method that came to be known as DHR analysis. Let A K dc (Å 4 ) be the observable net indexed by the set K dc (Å 4 ) of double cones in 4-dimensional Minkowski space representations π of the observable net which can be characterized as "sharp excitations" of the vacuum representation π 0 in the sense that, when restricted to the algebra A(O ′ ) pertaining to the spacelike complement O ′ of the double cone O, π is unitarily equivalent to π 0 , i. e., This family of representations together with the corresponding intertwining operators forms a C * -category that turns out to be equivalent to a C * -category of endomorphisms ρ of the observable net defined in the vacuum representation. This is a crucial fact, because it is in this category that the charge structure of sectors arises. In fact, it is possible to introduce a tensor product ⊗ representing the property of composition of charges. The tensor product has a permutation symmetry which encodes the possible statistics of sectors. The statistics of an irreducible object ρ is classified by means of a number λ, the statistics parameter, which is an invariant of the equivalence class of ρ. λ is the product of two invariants, In the case of finite d < +∞, the possible statistics are classified by the statistical phase χ distinguishing para-Bose (1) and para-Fermi (−1) statistics and by the statistical dimension d giving the order of the parastatistics. Ordinary Bose and Fermi statistics correspond to d = 1. Any object ρ with finite statistics has a conjugate: there exists an object ρ with the same statistics as ρ such that ρ ⊗ ρ contains the vacuum sector. Furthermore, if one consider covariant sectors, it is possible to construct asymptotic multiparticle states associated with charged sectors and to establish a clear-cut connection between the statistics of sectors and the spin of the corresponding particle state. Finally, in the case of infinite statistics, d = +∞, there is no charge interpretation since these sectors do not possess conjugates.
The criterion (4.1) excludes charges of electromagnetic type. Because of Gauß' law, the flux of the electric field of a localized electric charge is nonvanishing in any double cone containing the charge; however, it was believed by Doplicher, Haag and Roberts that (4.1) should hold in theories with a mass gap. Buchholz and Fredenhagen [29] have shown that this is not true. Sectors which describe the charge of particles in purely massive theories correspond to a localization in a cone which extends to spacelike infinity. These sectors are known as BF sectors. Except for the localization region, the charge structure of DHR sectors also applies to these BF sectors.
It is a deep result by Doplicher and Roberts that the superselection structure determines a compact global gauge group G Å 4 acting on a field net F Å 4 [51].
The observables turn out to be the gauge-invariant elements of the field net, while the sectors are in 1-1 correspondence with the irreducible representations of the gauge group, a result that holds true both for DHR and BF sectors.

Particle Weights
As mentioned before, in contradistinction to sharply localized charges, there exist other variants of charges that elude a treatment within the Doplicher-Haag-Roberts approach. A prominent example are electrically charged particles; the amount of their charge can be determined by measuring the electric flux passing through an arbitrarily large closed surface surrounding them. Therefore, these states can be discerned from the vacuum by measurements at appropriate large distances [29]. This problem, associated with the long range of the electromagnetic interaction, is also connected with the problem to exhibit a clear-cut theoretical scheme for the assignment of a definite mass and spin to these charges that are inevitably acompanied by clouds of soft photons.
Usually, the theoretical basis for assigning mass and spin to a particle is the analysis of all unitary representations of the Poincaré group by Eugene Wigner [114]. These representations are characterized by two parameters, m and s, their real respectively integer and half-integer values being interpreted as mass and spin of a particle whose state is described as a vector in the corresponding representation space. Since Lorentz symmetry is broken in theories with longrange interactions [26], this approach is no longer useful under these circumstances, giving rise to the so-called infraparticle problem (in discrimination to the Wigner particles aforementioned). A unified treatment of both Wigner particles and infraparticles has been proposed by Buchholz [27] and elaborated in [35].
The idea is to single out elements of the quasi-local algebra A that can be interpreted as particle detectors. These elements are thus required to be insensitive to the vacuum with concurrent appropriate localization properties. Due to the Reeh-Schlieder Theorem [96], a strictly local operator, i. e. an element of a local algebra A(O), cannot annihilate the vacuum. Therefore attention is restricted to almost local vacuum annihilation operators, characterized by their annihilating states with energy below a certain threshold together with the property that they can be approximated by local operators in such a way that the norm of their dislocalized part (outside a region of radius r) falls off more rapidly than any power of r. When supplemented with an assumption on their infinite differentiability with respect to Poincaré transformations, these elements constitute a subspace L 0 of A, while multiplication from the left by quasi-local operators yields a left ideal L in A. The algebra C of detectors is then spanned by elements of the form L 1 * L 2 , L 1 , L 2 ∈ L, i. e. by quasilocal operators A ∈ A that are sandwiched with almost local vacuum annihilation operators L 0 , K 0 ∈ L 0 : C . = L 0 * AK 0 . Note that this algebra does not contain a unit! Now, on the algebra C of detectors thus defined one considers physical states of bounded energy and the expectation values that they return when applied to detectors. As in the approach of Araki and Haag [7], which was limited to massive theories, it can be established in the present more general setting that the limits of these expectation values exist at asymptotic times (in the future as well as in the past). In view of the construction of the algebra of detectors, the existence of these limits is interpreted as the signature of an asymptotic stable constellation of particles. Since, due to their construction, the corresponding asymptotic positive functionals would take the value ∞ when evaluated on the unit ½ of A, they have the characteristics of weights (or extended positive functionals) that have been introduced by Jacques Dixmier into the theory of C * -algebras [43,Section I.4.2]. Therefore, the sesquilinear forms on L × L that these asymptotic functionals define are called particle weights.
These sesquilinear forms exhibit properties expected from asymptotic mixtures of stable particles; in fact they are interpreted as mixtures of pure particle weights pertaining to a definite energy-momentum. One should have in mind at this point Dirac's notion of improper energy-momentum eigen-states. These unnormalizable ket vectors |p 0 , p become decent Hilbert space vectors when acted upon (localized) by application of a quasi-local operator A ∈ A: |p 0 , p → A|p 0 , p ∈ H . In a similar way, a general particle weight . | . : L × L → is a singular object, while, upon localization with an operator L ∈ L, the mapping A ∋ A → L|AL ∈ is a bounded linear functional on the C * -algebra A. Generic particle weights allow for a GNS construction yielding a highly reducible representation of the quasi-local algebra. It has been demonstrated that this representation can be decomposed in terms of a direct integral of irreducible representations (in fact a standard result of the theory of C * -algebras), where the sesquilinear forms arising in this way are in fact again particle weights, i. e., they exhibit all their characteristic features. These are the pure particle weights already mentioned at the beginning of this paragraph. They are assigned a definite energy-momentum and permit a clear-cut definition of mass and spin, this time for Wigner particles as well as infraparticles. A thorough presentation of the theory of particle weights can be found in [93,94].
The particle weights considered as sesquilinear forms on L constitute a positive cone in an infinite-dimensional locally convex space. Given a convex base of this cone (defined by evaluation on certain elements of C), it is natural to ask for the decomposition of a generic particle weight in terms of extremal ones. The result of this barycentric decomposition will be an integral with support on the extreme boundary (Choquet disintegration) [2,92]. It is an obvious question if and how the pure particle weights arising from the spatial disintegration of the preceding paragraph correspond to the extremal elements of the cone. In this guise, the particle spectrum of a quantum field theory is expected to be encoded in the geometrical structure of the positive cone of particle weights. How this viewpoint can be related to the analyses expounded in other parts of our review is an open question.

Net Cohomology
Net cohomology arose as an equivalent approach to the theory of superselection sectors in Minkowski space [97]. The basic idea is that the physical content of DHR sectors is completely encoded in the charge transporters. These turn out to be 1-cocycles of the partially ordered set (poset) formed by the set of indices of the observable net when ordered by inclusion. The relevance of this cohomological approach becomes clear in the extension of the theory of superselection sectors to curved spacetimes, where one has to deal with spacetimes having nontrivial topologies. In this context net cohomology appears as a very powerful tool, both for analysing the charge structure of sectors and, in particular, to establish a clear connection between topology and sectors.
By the term net cohomology we mean the cohomology of a poset P with coefficients in a net of local algebras A P (the observable net) indexed by elements of P. So it is a nonabelian cohomology, and the language of n-categories is an essential ingredient if one is interested in degrees of the cohomology higher than 1. It is a matter of fact that, up to now, only 1-cocycles have a direct physi-cal interpretation. Therefore, this section is focused on 1-cocycles, and the term net cohomology refers to 1-cohomology. After some preliminaries the main topics are discussed in full generality in the following two subsections, viz. the first homotopy group of a poset, the connection between homotopy and net cohomology, and the behaviour of net cohomology under a change of the index set. The last subsection is devoted to study the case of a poset being the basis for the topology of a topological space. References for this section are [98,65,99,102].
Note that ∆ 0 is a point, ∆ 1 a closed interval etc.The inclusion maps d n i between standard simplices are maps d n i : ∆ n−1 → ∆ n defined as d n i (λ 0 , . . . , λ n−1 ) for n 1 and 0 i n − 1. Now, regarding a standard n-simplex as a partially ordered set with respect to the inclusion of its subsimplices, a singular n-simplex of a poset P is an order preserving map f : ∆ n → P. We denote by Σ n (P) the collection of singular n-simplices of P and by Σ * (P) the collection of all singular simplices of P. Σ * (P) is the simplicial set of P. The inclusion maps d n i between standard simplices are extended to maps ∂ n i : Σ n (P) → Σ n−1 (P) called boundaries by setting ∂ n i f One can easily check, by definition of d n i , that the following relations hold: From now on we will omit the superscript from the symbol ∂ n i and denote the composition ∂ i • ∂ j by ∂ ij , 0-simplices by the letter a, 1-simplices by b and 2simplices by c. Note that a 0-simplex a is nothing but an element of P. A 1-simplex b is formed from two 0-simplices ∂ 0 b, ∂ 1 b and an element |b| of P, |b|; a 2-simplex c is formed from three 1-simplices ∂ 0 c, ∂ 1 c, ∂ 2 c, whose 0-boundaries are chained according to (4.2), and from a 0-simplex |c|, the support of c, such that ∂ 0 c, ∂ 1 c, ∂ 2 c |c|. The reverse of a 1-simplex b is the 1-simplex b defined by Finally, a 1-simplex b is said to be degenerate to a 0-simplex a 0 whenever We will denote by b(a 0 ) the 1-simplex degenerate to a 0 .
Paths Given a pair a 0 , a 1 of 0-simplices a path from a 0 to a 1 is a finite ordered sequence p = {b n , . . . , b 1 } of 1-simplices with the relations The starting point of p, ∂ 1 p, is the 0-simplex a 0 , while the endpoint of p, ∂ 0 p, is the 0-simplex a 1 . We denote by P(a 0 , a 1 ) the set of paths from a 0 to a 1 . The poset P is said to be pathwise-connected if P(a 0 , a 1 ) = ∅ for any pair a 0 , a 1 of 0simplices. The set of paths is equipped with the following operations: Consider a path p = {b n , . . . , b 1 } ∈ P(a 0 , a 1 ). Its reverse, p, is the path 1 , a 0 ).
Note that forming the reverse is an involutive mapping, while the composition * is associative.

The First Homotopy Group of a Poset
The logical steps necessary to define the first homotopy group of posets are the same as in the case of topological spaces. We first recall the definition of a homotopy of paths. Then we prove that the reverse of a path and the composition of paths are well-behaved mappings under the homotopy equivalence relation. Finally, the first homotopy group of a poset is defined.
An elementary deformation of a path p consists in replacing a 1-simplex ∂ 1 c of the path by a pair, ∂ 0 c, ∂ 2 c, where c is a singular 2-simplex in Σ 2 (P), or, conversely, in replacing a consecutive pair ∂ 0 c, ∂ 2 c of 1-simplices of p by a single 1-simplex ∂ 1 c.  Two paths with the same endpoints are homotopic if they can be obtained from one another by a finite number of elementary deformations. Homotopy thus defines an equivalence relation ∼ on the set of paths with the same end points.
The reverse and the composition of paths are compatible with the homotopy equivalence relation. To be precise, for any p, q ∈ P(a 0 , a 1 ) and p 1 , q 1 ∈ P(a 1 , a 2 ), we have Furthermore, for any p ∈ P(a 0 , a 1 ), the following relations hold: where b(a 0 ) is the 1-simplex degenerate to a 0 .
We now are in a position to define the first homotopy group of a poset. Fix a base 0-simplex a 0 and consider the set of closed paths P(a 0 ). Note that the composition and the reverse are internal operations of P(a 0 ) and that b(a 0 ) ∈ P(a 0 ). We define π 1 (P, a 0 ) . = P(a 0 )/ ∼ , (4.5) where ∼ is the homotopy equivalence relation. Let [p] denote the homotopy class of an element p of P(a 0 ). We equip π 1 (P, a 0 ) with the product The operation * is associative, and it easily follows from previous results that π 1 (P, a 0 ) endowed with * is a group. The identity 1 of the group is [b(a 0 )] and the inverse . Now assume that P is pathwise-connected. Given a 0-simplex a 1 , let q be a path from a 0 to a 1 . Then the map is a group isomorphism. On the basis of these facts we give the following Definition 4.1. We call π 1 (P, a 0 ) the first homotopy group of P with base a 0 ∈ Σ 0 (P). If P is pathwise-connected, we denote this group by π 1 (P) and call it the fundamental group of P. If π 1 (P) = 1 we will say that P is simply connected.
We have the following result: If P is directed, then P is pathwise-and simply connected.

Net Cohomology of a Poset
We introduce the net cohomology of a poset equipped with a causal disjointness relation. Throughout it is assumed that the poset is pathwise-connected.
Causal disjointness and net of local algebras Given a poset P, a causal disjointness relation on P is a symmetric binary relation ⊥ on P satisfying the following properties: Given a subset P ⊆ P, the causal complement of P is the subset P ⊥ of P defined as Note that if P 1 ⊆ P, then P ⊥ ⊆ P ⊥ 1 . Now, assume that P is a pathwiseconnected poset equipped with a causal disjointness relation ⊥. A net of local algebras indexed by P is a correspondence associating with any O a von Neumann algebra A(O) defined on a fixed Hilbert space H 0 and satisfying isotony: where the prime denotes the commutant of the algebra. The algebra A(O ⊥ ) associated with the causal complement O ⊥ of O ∈ P, is the C * -algebra generated by all the algebras The category of 1-cocycles We refer the reader to [64] for the definition of a C * -category. Let P be a poset with a causal disjointness relation ⊥, and let A P be an irreducible net of local algebras. A 1-cocycle z of P with values in A P is a field z : and the locality condition, z(b) ∈ A(|b|), for any 1-simplex b. An intertwiner t ∈ (z, z 1 ) between a pair of 1-cocycles z and z 1 is a field t : Σ 0 (P) ∋ a → t a ∈ B(H 0 ) satisfying the relation and the locality condition, t a ∈ A(a), for any 0-simplex a. The category of 1cocycles Z 1 (A P ) is the category whose objects are 1-cocycles and whose arrows are the corresponding intertwiners. The composition between s ∈ (z, z 1 ) and t ∈ (z 1 , z 2 ) is the arrow t · s ∈ (z, z 2 ) defined as = t a · s a , a ∈ Σ 0 (P).
Note that the arrow 1 z of (z, z) defined as (1 z ) a .
= ½ for any a ∈ Σ 0 (P) is the identity of (z, z). It turns out that Z 1 (A P ) is a C * -category (for details cf. [102]). Two 1-cocycles z and z 1 are equivalent (or cohomologous) if there exists a unitary arrow t ∈ (z, z 1 ). A 1-cocycle z is trivial if it is equivalent to the identity cocycle ι defined as ι(b) = ½ for any 1-simplex b. Note that, since A P is irreducible, ι is irreducible: (ι, ι) = ½.
Equivalence in B(H 0 ) and path independence A weaker form of equivalence between 1-cocycles is the following. z and z 1 are said to be equivalent in Note that the field V is not an arrow of (z, z 1 ) because it is not required that V satisfies the locality condition. A 1-cocycle is trivial in B(H 0 ) if it is equivalent in B(H 0 ) to the trivial 1-cocycle ι. We denote by Z 1 t (A P ) the set of 1-cocycles that are trivial in B(H 0 ), and with the same symbol we denote the full C *subcategory of Z 1 (A P ) whose objects are the 1-cocycles trivial in B(H 0 ). Triviality in B(H 0 ) is related to the notion of path independence. The evaluation of a 1-cocycle z on a path p = {b n , . . . , b 1 } is defined as z is said to be path-independent whenever z(p) = z(q) for any p, q ∈ P(a 0 , a 1 ). (4.7) As P is pathwise-connected, a 1-cocycle is trivial in B(H 0 ) if and only if it is path-independent [65].
Connection between homotopy and net cohomology Let us consider a poset P equipped with a causal disjointness relation ⊥, and let A P be an irreducible net of local algebras. In this section we establish the relation between π 1 (P) and the set Z 1 (A P ).
To begin with, note that 1-cocycles are invariant for homotopic paths. This means that, given z ∈ Z 1 (A P ) and two paths p, q with the same endpoints, We are now in the position to show the connection between the fundamental group of P and Z 1 (A P ). Then, π z is a unitary representation of π 1 (P, a 0 ) in H 0 . The correspondence z → π z maps 1-cocycles equivalent in B(H 0 ) into equivalent unitary representations of π 1 (P, a 0 ) in H 0 . Up to equivalence, this map is injective. If π 1 (P) = 1, then Proof. Observe that the definition is well-posed as z is invariant for homotopic paths. By (4.9a) and (4.8)), we have π z (1) Hence π z is a unitary representation of So what remains to be proved is that the correspondonce is injective up to equivalence. To this end consider a unitary representation π of π 1 (P, a 0 ) on H 0 . For any 0-simplex a denote by p a a path with ∂ 1 p a = a and ∂ 0 p a = a 0 . Let Given a 2-simplex c, we have Hence z π satisfies the 1-cocycle identity but in general z π ∈ Z 1 (A K ) because z π (b) does not have to belong to A(|b|). However, note that if we consider π z 1 for some z 1 ∈ Z 1 (A K ), then Therefore, z π z 1 is equivalent in B(H 0 ) to z 1 . This entails that, if π z is equivalent to π z 1 , then z is equivalent in B(H 0 ) to z 1 . Finally, assume that π 1 (P) = 1, then z(p) = ½ for any closed path p. Thus z is path-independent on P, hence z is trivial in B(H 0 ).

Change of Index Set
We now study the behaviour of net cohomology under a change of the index set. By a subposet of a poset P we mean a subset P of P equipped with the same order relation as P.
Definition 4.4. Consider a subposet P of P. We will say that P is a refinement of P, if for any O ∈ P there exists O ∈ P such that O O. A refinement P of P is said to be locally relatively connected if, given O ∈ P, for any pair Let P be a pathwise-connected poset and let ⊥ be a causal disjointness relation for P. Let A P be an irreducible net of local algebras indexed by P and defined on a Hilbert space H 0 . If P is a locally relatively connected refinement of P, then it is easily seen that P is pathwise-connected and that ⊥ is a causal disjointness relation on P. Furthermore, the restriction of A P to P is a net of local algebras A P indexed by P. Let Z 1 t A P be the category of 1-cocycles of P, trivial in B(H 0 ), with values in the net A P . Notice that A P need not be irreducible. However, as P is a refinement of P, the trivial 1-cocycleι of Theorem 4.5. Let P be a locally relatively connected refinement of P. Then the categories Z 1 t (A P ) and Z 1 t (A P ) are equivalent.
Proof. For any z ∈ Z 1 t (A P ) and for any t ∈ (z, z 1 ) define R is a covariant and faithful functor from Z 1 t (A P ) into Z 1 t (A P ). We now define a functor from Z 1 t (A P ) to Z 1 t (A P ). To this end, choose a function f : P → P satisfying the following properties: ) whose support is contained in |b|; this is possible because P is a locally relatively connected refinement of P. Given z, Asẑ is path-independent, given c ∈ Σ 2 (P), we have Hence F(ẑ) satisfies the 1-cocycle identity, and it is trivial in B(H 0 ) because so isẑ. Therefore, F(ẑ) ∈ Z 1 t (A P ). In an analogous fashion one can show that F(t) ∈ F(ẑ), F(ẑ 1 ) . Hence, F is a covariant functor from Z 1 t A P to Z 1 t (A P ). Finally, it can easily be checked that the pair R, F constitutes an equivalence between Z 1 t (A P ) and Z 1 t A P (for details cf. [25]).

The Poset as a Basis for a Topological Space
Consider a topological Hausdorff space X . The topics of the previous sections are now investigated in the case that P is a basis for the topology of X ordered under inclusion ⊆. This allows us both to show the connection between the notions for posets and the corresponding topological ones and to understand how topology affects net cohomology.
Homotopy In what follows, by a curve γ in X we mean a continuous function from the interval [0, 1] into X . We recall that the reverse of a curve γ is the curve γ defined as γ(t) Finally, the constant curve e x is the curve e x (t) = x for any t ∈ [0, 1].
Definition 4.6. Given a curve γ. A path p = {b n , . . . , b 1 } is said to be a poset approximation of γ (or simply an approximation) if there is a partition 0 = s 0 < s 1 < · · · < s n = 1 of the interval [0, 1] such that (Fig. 4.3) By App(γ) we denote the set of approximations of γ. Since P is a basis for the topology of X , App(γ) = ∅ for any curve γ. The converse, i. e., that for a given path p there is a curve γ such that p is an approximation of γ, holds if the elements of P are arcwise-connected sets of the topological space X . The approximations of curves have the following properties: where β(1) = σ(0), ∂ 0 q = ∂ 1 p. As already said, we can find an approximation for any curve γ.
We now compare the connectedness relations of X with those of P. If the elements of P are arcwise-connected sets of X , then an open set X ⊆ X is arcwise-connected in X if and only if the poset P X , defined as is pathwise-connected. Note that the set P X is a sieve of P, i. e. a subfamily S of P such that, if O ∈ S and O 1 ⊆ O, then O 1 ∈ S. Now, assume that P is a sieve of P. Then P is pathwise-connected in P if and only if the open set X P , defined as is arcwise-connected in X .
For simply connectedness, assume that the elements of P are arcwise-and simply connected subsets of X . Given two curves γ and β with the same endpoints, let p ∈ App(γ) and q ∈ App(β) have the same endpoints. Then (4.14) By using this approximation result we get Theorem 4.7. Let X be a Hausdorff, arcwise-connected topological space. Assume that there is a basis P for the topology of X whose elements are arcwise-and simply connected subsets of X . Then π 1 (X ) ≃ π 1 (P).
where p is an approximation of γ. By (4.11a) and (4.14), this map is a group isomorphism.
Corollary 4.8. Let X and P be as in the previous theorem. If X is not simply connected, then P is not directed under inclusion.

Net cohomology
The aim here is to apply the results of the previous section to the case where the topological space is a globally hyperbolic spacetime M .
In particular we will consider the set of globally hyperbolic regions K h (M ) and the set of diamonds K d (M ) introduced in Section 2.1. As seen there, both of them are bases for the topology of M consisting of arcwise-connected elements. However, while diamonds are simply connected, this is not true for globally hyperbolic regions. So we conclude: Proof.  hand, charges of Buchholz-Fredenhagen type associated with a particular class of noncompact regions like spacelike cones. In both cases, sectors define a C *category in which the charge structure manifests itself by the existence of a tensor product, a permutation symmetry and a conjugation(see Subsection 4.1.1).

The Reference State Space
Our aim is to study the superselection sectors of Doplicher-Haag-Roberts type in the framework of a locally covariant quantum field theory defined via the functor A . The first step is to introduce the notion of a reference state space taking on the role of the vacuum representation. To this end, note that the vacuum representation serves as a reference representation that singles out charged sectors, and that it is enough to consider a vacuum representation that has the Borchers property and satisfies Haag duality 1 [51].
Definition 4.11. We call a reference state space for A a locally quasi-equivalent state space S 0 on globally hyperbolic regions such that for any M ∈ Loc there is a state ω ∈ S 0 (M ) such that (a) the net ω * A K h (M ) satisfies the Borchers property; As K h (M ) is a basis for the topology of M , the net ω * A K h (M ) is locally definite as well. An example of a locally covariant quantum field theory with a state space satisfying the properties of Definition 4.11 is provided by the Klein-Gordon scalar field and by the space of quasi-free states satisfying the microlocal spectrum condition [23,103]. We stress that we require the existence for any M ∈ Loc of at least one state ω ∈ S 0 (M ) satisfying punctured Haag duality. This property seems to be the correct generalization of Haag duality to deal with the nontrivial topology of arbitrary globally hyperbolic spacetimes [102]. The reason for punctured Haag duality to be assumed for the net indexed by K d (M ) is that K h (M ) contains elements which are not simply connected as well as elements whose causal complement is not arcwise-connected. Therefore, punctured Haag duality (and also Haag duality) might not hold for a net indexed by K h (M ) (cf. [103]).
Let us consider the main properties of such a state space.
First, as a consequence of local quasi-equivalence, for any pair ω, σ ∈ S 0 (M ) the nets ω * A K h (M ) and σ * A K h (M ) are isomorphic, i. e., there is a collection consisting of * -isomorphisms of von Neumann algebras which are compatible with the net structure, It is clear that, by restricting ρ ω,σ to the set of diamonds K d (M ), one gets a net isomorphism ρ ω,σ : Second, note that in S 0 (M ) there is a state ω satisfying punctured Haag duality, the Borchers property and, as seen above, local definiteness. Since the Borchers property and local definiteness are local properties, the fact that for any σ ∈ S 0 (M ) ρ ω,σ is a net isomorphism implies that also the net σ * A K h (M ) satisfies the Borchers property and local definiteness.
Third, consider the mapping ψ ∈ hom Loc (M 1 , M ) and the associated injective C * -morphism α ψ : A (M 1 ) → A (M ). Any state ω ∈ S (M ) induces two different representations of A (M 1 ). On the one hand, if π ω is the GNS representation associated with ω, then π ω α ψ is a representation of A (M 1 ). On the other hand, by local covariance ωα ψ ∈ S 0 (M 1 ) so that the GNS representation π ωα ψ associated with ωα ψ is again a representation of A (M 1 ). As noticed before, any element of S 0 (M ) for arbitrary M ∈ Loc satisfies the Borchers property and local definiteness. Using the Borchers property, it turns out that the representations π ω α ψ and π ωα ψ are locally quasi-equivalent (for details cf. [24]). In particular, if we define τ ω ψ π ω α ψ (A) . = π ωα ψ (A), A ∈ A (M 1 ), (4. 19) it turns out that is a net isomorphism, where ω * A ψ(K h (M 1 )) is the restriction of ω * A K h (M ) to the poset ψ K h (M 1 ) .
At this point some observations concerning the definition of superselection sectors in a locally covariant quantum field theory are in order.
(A) The above definition of superselection sectors in terms of net cohomology is equivalent to the usual one given in terms of representations of the net of local observables which are "sharp excitations" of a reference representation. In particular, it is shown in [24] that for any spacetime M and any σ ∈ S 0 (M ), to any 1-cocycle z ∈ Z 1 t σ, K d (M ) there corresponds a unique representation π z (up to equivalence) of the net of local observables which is a "sharp excitation" of a representation π ω associated with a state ω ∈ S 0 (M ) satisfying punctured Haag duality. is the same as that of M and any diamond has an arcwise-connected causal complement. These two properties belong to the key ingredients in [102] where, in the Haag-Kastler framework, the charge structure of sharply localized sectors in a fixed background spacetime has been established. On the contrary, K h (M ) is simply connected irrespective of the topology of M (Proposition 4.9), and it has elements with a nonarcwise-connected causal complement. On the other hand, Proposition 4.10 states that there is no loss of generality in studying the net isomorphism (4.17). We stress that, in spite of our considering the categories associated with the set K d (M ), the fact that ρ ω,σ is a net isomorphism of the nets indexed by K h (M ) is of crucial importance to establish the claimed isomorphism. Given any pair z, z 1 ∈ Z 1 t ω, K d (M ) and t ∈ (z, z 1 ), define The purpose is to show that F ω,σ : is a * -isomorphism of C * -categories. Theorem 4.14. For any ω ∈ S 0 (M ) the category Z 1 t ω, K d (M ) is a (symmetric tensor) C * -category with left inverses. Any object with finite statistics has conjugates. For any σ ∈ S 0 (M ) is a covariant (symmetric tensor) * -isomorphism.
We will refer to the functor F ω,σ as the flip functor.
where ψ(b) is the 1-simplex of K d (M ) defined as ψ(b) = ψ(|b|), ∂ 0 ψ(b) = ψ(∂ 0 b), ∂ 1 ψ(b) = ψ (∂ 1 b). Then, by using Theorem 4.15, it turns out that this mapping, E ω ψ , is a covariant symmetric tensor * -functor which is full and faithful. We call E ω ψ the embedding functor. By (4.21), the embedding and the flip functor enjoy the following relation: The superselection functor We can now establish the covariant structure of superselection sectors. Let Sym be the category, whose objects are symmetric tensor C * -categories and whose arrows are the full and faithful, symmetric tensor * -functors. According to the approach to locally covariant quantum field theory expounded in this review, the superselection sectors are expected to exhibit the structure of a functor from the category Loc into the category Sym. We know that the superselection sectors of any spacetime M ∈ Loc identify a family of categories within the same isomorphism class, any such category Z 1 t ω, K d (M ) is labelled by an element of the reference state space, ω ∈ S 0 (M ). Since there is no natural way to associate an element of this isomorphism class to the spacetime M , M varying in Loc, we are forced to make a choice.
Given a locally covariant quantum field theory A and a reference state space S 0 , let a choice of states be given by ω .
= ω M M ∈ Loc, ω M ∈ S 0 (M ) . (4.25) We define a map of categories via and call the mapping S ω : Loc → Sym the superselection functor associated with the choice ω. There holds the following theorem. Proof. Let ψ ∈ hom Loc (M 1 , M ). Since S ω (ψ) is defined as the composition of the flip and of the embedding functor, the above discussion of the embedding functor and Theorem 4.14 imply that S ω (ψ) : S ω (M ) → S ω (M 1 ) is a full and faithful, symmetric tensor * -functor. Given φ ∈ hom Loc (M 2 , M 1 ), by (4.24), we have Finally, the definitions of the flip and of the embedding functors imply , so that S ω turns out to be a contravariant functor. For the rest of the proof see [24] This result establishes the covariance of charged superselection sectors. If ψ ∈ Furthermore, z and S ω (ψ)(z) have the same statistics, i. e.,