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Topological sectors for Weyl-algebra net in the Einstein cylindrical universe

This paper is an extended and more detailed version of arXiv:0812.0533. We tackle the problem of constructing explicit examples of topological cocycles of Roberts' net cohomology, as defined abstractly by Brunetti and Ruzzi. We consider the simple case of massive bosonic quantum field theory on the two dimensional Einstein cylinder. After deriving some crucial results of the algebraic framework of quantization, we address the problem of the construction of the topological cocycles. All constructed cocycles lead to unitarily equivalent representations of the fundamental group of the circle (seen as a diffeomorphic image of all possible Cauchy surfaces).

Introduction A lot of research has be done in physics about the influence of topology on physical theories; to make just a few examples, we can cite the Aharonov-Bohm effect and the Berry phase in quantum mechanics [1], and its classical counterpart called Hannay's angle [38]. This work deals with a kind of topological effect arising in quantum field theory (QFT) on curved spacetime.
The underlying conceptual framework is the algebraic approach to QFT, also know as local quantum theory [36] where, as is well know, the physical content of the theory (observables) is encoded by a net of C * -algebras A [O] indexed by a family (poset) of bounded spacetime regions O ⊂ M; states are implemented, via the GNS theorem, by representations of these local abstract algebras as (concrete) operatorial algebras on a fixed Hilbert space. All the relevant physical information is contained in the inclusion relations of these concrete algebras for different spacetime regions.
So, given the importance of representations of local algebras, it's no surprise that considerable efforts have beeen spent in selecting physical meaningful representations of local observable algebras, that is to say physical interesting states. The way this choice is done goes by the name of selection criterion, and a unitary equivalence class of representations satisfying the criterion is denoted as a superselection sector.
A well known and important example of selection criterion is that introduced by Doplicher, Haag and Roberts [28], and the corresponding equivalence class is known as DHR superselection sector ; DHR superselection sectors describe charge localized in bounded regions. DHR theory has been developed on Minkowski spacetime and extended to simply connected globally hyperbolic spacetimes; however, it turns out that the extension to multiple-connected spacetimes is not straightforward, and it requires substantial changes to be of use. These changes involve modifying the selection criterion and generalizing the very notion of representation of a local algebra net; roughly speaking these generalized representations, named (unitary) net-representations, are local version of standard representations, in a sense that will be made more precise later. Heuristically, net-representations are analogous to local coordinate charts on a manifold, while conventional representations are like a global coordinate chart; this time, although, the coordinate space is infinite-dimensional, being a subset of some B(H).
It can be shown [13] that these new notions of representation and selection criterion genuinely extend the old ones, reducing to them when specialized to simple-connected globally hyperbolic spacetimes; however, when the background spacetime isn't topologically trivial (i.e. its Cauchy surfaces are multiple-connected) they give rise to a truly new kind of superselection sectors of topological nature, leading as an aside to non trivial unitary representations of the first homotopy group of the spacetime. A crucial tool in such a construction is a class of objects named 1-cocycles [59], relating netrepresentations with the fundamental group of the spacetime and encoding both the charge and topological content of representations.
A recent article by Brunetti and Ruzzi [13] discusses the general theory of topological superselection sectors for spacetime dimension ≥ 3; the purpose of this research work, instead, is to examine the situation in a lowerdimension context. Our approach is to consider a simple QFT model in two dimensions in order to explicitly work out the details and to gain some insight about what is going on. Actually, we choose the simpler (non trivial) model available out there, namely the free scalar massive (Klein-Gordon) field on the 2-dimensional Einstein cylinder; given the ultrastatic nature of the background spacetime we are able to pursue the calculations at a good depth. On the other end, since the Cauchy surfaces are diffeomorphic to S 1 we expect the appearance of some interesting phenomenon of topological nature. In fact we have proved the existence of a non countable family of topological net-representations, hence a corresponding class of non-trivial unitary representations of the fundamental group of a Cauchy surface (it being the same as the spacetime fundamental group).
However, we have not delved further about the charge or topological structure of the corresponding topological superselection sectors; in fact what we have done is to exhibit a family of (unitary inequivalent) non-trivial 1-cocycles, giving rise from the one hand to non-trivial net-representations of the Weyl local algebra net of observables, on the other hand to non-trivial unitary representations of the fundamental group. Actually, it turns out that some crucial results valid in dimension ≥ 3 fail to be true in our model, and as a consequence even the proofs of various propositions that carry over to our case need essential modifications.
What can be said about the significance of the existence of topological net-representations ? As we'll see later, a topological net-representation cannot be trivialized, namely it's not unitary equivalent to a trivial netrepresentation; in turn, a trivial net-representation is, in fact, a conventional (global) representation of the local observable net. In other terms a topological net-representation cannot be turned into an ordinary representation via local unitary operators. Pursuing the geometric parallel issued before, the situation here is analogous to the impossibility of converting a set of local charts into a global chart by means of local coordinate trasformations.
We can also adopt another perspective. The GNS construction associates an operatorial representation to every state on a given C * -algebra; so a state defined on the whole quasilocal observable algebra induces a global (ordinary) representation of the algebra itself. We could think a netrepresentation {π, ψ} as originating from a family of local states ω O acting on local algebras A(O); then, if the ω O 's were induced by restriction by a global state ω, our net-representation {π, ψ} would be topologically trivial. Summing up, we conclude that a topological net-representation cannot be generated by a global state.

Overview
In the first chapter we take a quick look to net-cohomology, briefly discussing concepts and facts needed for subsequent constructions. We start by introducing purely set-theoretic notions as those of poset, simplicial sets, (poset) paths and the first homotopy group of a generic poset; then we specialize the general theory to the case of interest to us, namely the topological one, and in particular the poset of spacetime diamonds. We conclude the section speaking about the category of 1-cocycles, cocycle equivalence and localization, net-representations, representations of the fundamental group, and the reciprocal relations between these notions.
In chapter 2 we set the stage for the central part of this work, contained in the last chapter. Here we give a rather detailed exposition of our reference QFT model, i.e. the massive free scalar field on the 2-dimensional Einstein universe. Following the logical path towards quantization, first of all we introduce the classical field equation (Klein-Gordon equation) with a special focus on well posedness of the Cauchy problem on globally hyperbolic spacetimes; we also define ultrastatic spacetimes. Then we talk about the symplectic structure of the solution space of the field equation, leading to the construction of the net of local observable algebras via the Weyl correspondence. After a digression on spacetime symmetries, we define quasifree states on a general ultrastatic spacetime; then we specialize to our case, introducing the reference vacuum state and its associated vacuum representation. The central part of the chapter deals about standard properties of the von Neumann local net induced by the vacuum representation: additivity, local definiteness, factoriality, Haag duality, punctured Haag duality, Reeh-Schlieder property. We also give proofs, some of them are original; in particular our proof of Haag duality carries over to more general cases than our specific model. We conclude with an analysis of DHR sectors on the Einstein cylinder (showing that in fact there is only the vacuum one) and finally extending as far as possible the local net's properties from the spatial poset on the circle to the spacetime poset of diamonds.
Chapter 3 is the heart of this work; after some preparation we state the topological selection criterion [13] and we explicitly exhibit a family of topologically non-trivial 1-cocycles giving rise, as we said, from the one hand to a family of topological net-representations (that is to say topological superselection sectors), from the other hand to a family of non-trivial unitary representations of the fundamental group of the spacetime.
In the appendices we collect some rather technical proofs and statements from previous chapters, in order to avoid distracting the reader from the main line of reasoning. It's worth mentioning that in appendix C we proved the uniqueness of the universal algebra in the general case, and determined its structure in the particular case of the Einstein cylinder.

Chapter 1 Posets and net-cohomology
To formulate the main results of this work we need some background on homotopy of posets and net-cohomology. We will follow closely the exposition given in [59], omitting proofs and details and focusing on the main definitions and results. While summarizing this topic we also fix notations and terminology used in the rest of the work.
The basic idea behind homotopy of posets is to reformulate standard topological concepts lying on the notion of "paths", i.e. continuos curves on a topological space, in terms of open sets belonging to the same topological space. Roughly speaking, this can be accomplished by approximating a curve with a chain of open sets lying on it (see fig. 1.1).
The motivation for this kind of construction comes from the fact that we would like to study the relation between our net of local algebras encoding the quantum field and the topology of the background spacetime, in particular with respect to homotopy theory; unfortunately there is an obstruction here, due to the fact that local algebras are indexed by open sets while homotopy theory relies on the notion of paths, i.e. continuos curves.
The goal of poset homotopy theory is indeed to smoothly join these different mathematical models, in order to overcome the aforementioned obstruction. To start with we recall that a poset (P, ≤) is a partially ordered set, namely a set P endowed with a reflexive, antisimmetric, transitive relation ≤; a standard example of poset is the family 2 X of all the subsets of Then we introduce the notion of (singular) n-simplex. The point here is to express in terms of open sets notions like that of a line segment, a triangle, and so on. We recall that the standard n-simplex ∆ n is defined as: ∆ n := (λ 0 , . . . , λ n ) ∈ R n+1 |λ 0 + . . . λ n = 1, λ i ∈ [0, 1] (1.1) Then we see that ∆ 0 is a point, ∆ 1 is a closed interval and so on. We also note that we can embed a n-simplex into a n+1-simplex in a obvious manner via inclusion maps d n i : ∆ n−1 → ∆ n : d n i (λ 0 , . . . , λ n−1 ) = (λ 0 , λ 1 , . . . , λ i−1 , 0, λ i , λ n−1 ) (1.2) for n ≥ 1 and 0 ≤ i ≤ n. Observing that a standard n-simplex can be viewed as a partially ordered set with respect to the inclusion of its subsimplices, we could define a n-simplex built upon open sets as an order-preserving map from a standard n-simplex to the poset made of open sets with respect to set inclusion. Having in mind this interpretation, we initially adopt a more general approach, constructing our homotopy theory for a generic poset and forgetting for a while every other topology-related details.
So, fixing a poset (P, ≤) we define a singular n-simplex on P as an order-preserving map f : ∆ n → P. We denote by Σ n (P) the collection of singular n-simplices on P and by Σ * (P) the collection of all singular simplices on P, named the simplicial set of P.
The inclusion maps d n i between standard simplices induce by duality the maps ∂ n i : Σ n → Σ n−1 , called boundaries from their geometric meaning, by setting ∂ n i f := f • d n i . For the sake of notational simplicity from now on we will omit the superscript from the symbol ∂ n i , and denote 0-simplices by the letter a, 1-simplices by b, 2-simplices by c and so on. Note that a 0-simplex a is just an element of P, i.e. a point in our simplicial set; a 1-simplex b is formed by an element |b| of P, the support of b, and two 0-simplices ∂ 0 b, ∂ 1 b such that ∂ 0 , ∂ 1 ≤ |b|; so we can view |b| as a segment and ∂ 0 b, ∂ 1 b as its endpoints. Similarly, a 2-simplex c is made out of its support |c|, namely a "triangle", and three 1-simplices ∂ 0 c, ∂ 1 c, ∂ 2 c, the sides of the triangle.
The symbol δ stands for ∂.
Chaining 1-simplices we obtain a path; formally, given a 0 , a 1 ∈ Σ 0 (P), a path from a 0 to a 1 is a finite ordered sequence p = {b n , . . . , b 1 } of 1-simplices satisfying the relations These conditions express the fact that the 1-simplices making the path must be chained with each other starting at a 0 , the starting point ∂ 1 p of p, and ending at a 1 , the ending point ∂ 0 p of p. We will denote by P(a 0 , a 1 ) the set of paths from a 0 to a 1 , and by P(a 0 ) the set of loops (i.e. closed paths) based on a 0 . The poset P is said to be pathwise connected if for every pair a 0 , a 1 of 0-simplices the set P(a 0 , a 1 ) is nonempty. The support of the path p is the collection |p| ≡ {|b i |, i = 1, . . . , n}, and given a subset P of P we will write |p| ⊆ P if |b i | ∈ P for all i.

Causal disjointness
Until now we developed our poset machinery having in mind only the topological structure of the spacetime, but we know that every spacetime is endowed also with a causal structure induced by its Lorentzian metric, and this is an essential ingredient in quantum field theory. So, we need to implement the causal structure in the poset framework; this can be done for a general poset P, not only for the collection of open sets in a spacetime, introducing a causal disjointness relation on the poset P, i.e. a symmetric binary relation ⊥ on P satisfying the following properties: These two properties encode the causal structure on the poset P, and let us define such thing as the causal complement of an element of the poset. Actually, it's better to take a slightly more general approach, defining the causal complement of a whole family of elements in the poset, so that to be able to give a sense to the causal complement of sets not contained in the poset.
Given a subset P ⊆ P, the causal complement of P is the subset P ⊥ of P defined as From the definition it follows immediately that P 1 ⊆ P implies P ⊥ ⊆ P ⊥ 1 .
P(a 0 , a 1 ), thenp = b 1 , . . . ,b n ∈ P(a 1 , a 0 ). Then we come to homotopy of paths. We recall that from the topological point of view two (topological) paths are homotopic if and only if one of them can be continuosly deformed into the other; also the poset notion of homotopy is based on deformation of paths, but it's a deformation of a discrete kind, made up of a finite sequence of elementary deformations. This situation is ultimately due to the fact that (poset) paths are discrete in nature, being made up of a finite number of 1-simplices. We don't insist here on the notion of elementary deformation, for which we refer to [59]; figure 1.3 should give an idea of the situation. Now, given a 0 , a 1 ∈ Σ 0 (P), a homotopy of paths in P(a 0 , a 1 ) is a map h : {1, . . . , n} → P(a 0 , a 1 ) such that h(i) is an elementary deformation of h(i − 1) for 1 < i ≤ n. Two paths p, q ∈ P(a 0 , a 1 ) are said to be homotopic, p ∼ q, if there exists a homotopy of paths h in P(a 0 , a 1 ) such that h(1) = p and h(n) = q.
We will denote by b(a 0 ) the 1-simplex degenerate to a 0 . Below we summarize some basic properties of the notions just discussed. Lemma 1. If p, q, p i , q i are paths, then the following relations hold, whenever they make sense: • Path composition is associative, i.e. p 1 * (p 2 * p 3 ) = (p 1 * p 2 ) * p 3 , • Homotopy of paths is an equivalence relation on paths with the same endpoints, • If p 1 ∼ q 1 and p 2 ∼ q 2 , then p 2 * p 1 ∼ q 2 * q 1 , Finally we are ready to define the first homotopy group of a poset. Fix a base 0-simplex a 0 and consider the set P(a 0 ) of loops, i.e. closed paths, based at a 0 ; then the composition and the reverse are internal operations on P(a 0 ) and b(a 0 ) ∈ P(a 0 ). The first homotopy group π 1 (P, a 0 ) of P is P(a 0 ) quotiented with respect to the homotopy equivalence relation ∼: Let [p] denote the homotopy class of an element p of P(a 0 ); the product is associative and makes π 1 (P, a 0 ) into a group, the group identity being [b(a 0 )] and [p] −1 ≡ [p]. In general the first homotopy group depends on the base point where it is calculated; however, at least in the "topological" case, if the space is arcwise connected all homotopy groups are isomorphic. This situation carries over to the poset case, but arcwise connectedness must be replaced by pathwise connectedness. In fact, given another base point a 1 , let q be a path from a 0 to a 1 ; then the map is a group isomorphism. We can summarize the above discussion giving the following Definition 1. With the above notations, we call π 1 (P, a 0 ) the first homotopy group of P based on 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 the fundamental group is trivial, i.e. π 1 (P) = 1, we'll say that P is simply connected.
We conclude this section citing an useful result. Proposition 1. If P is directed 1 , then it is pathwise and simply connected.

Topological posets
So far we considered a generic poset P; now we restrict our attention to the case of interest to us, that is when P is a basis for a topological space X. This way we can establish a relation between the poset notions discussed so far and the topological ones, our goal being to understand how topology affects net-cohomology. So let us consider a Hausdorff topological space X, and take as the poset P a topological basis of X ordered under set inclusion ⊆. As we said, our goal is to link classical topological concepts, particularly those of continuos curves and the (topological) first group of homotopy built on them, to poset ones, like paths and the (poset) first group of homotopy; the technical tool needed here is the notion of approximation of a curve by a poset path. By a curve in X we mean as usual a continuos map γ : [0, 1] → X.
Definition 2. Given a (continuos) curve γ in X, 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: We denote the set of poset approximations of γ by App(γ).
Since P is a topological basis for X, we have that App(γ) = ∅ for any curve γ; it can also be shown [59] that approximations of curves behave well with respect to operations defined on curves (composition, reversal).
The next logical step is to introduce an order relation in the set of approximations of a given curve, so that we can talk about "better" or "worse" approximations.
Definition 3. Given p, q ∈ App(γ), we say that q is finer than p whenever p = {b n , . . . , b 1 } and q = q n * · · · * q 1 , where q i are paths satisfying We will write p ≺ q to denote that q is a finer approximation than p.
It's easy to see from the definition that (App(γ), ≺) is a directed poset (i.e. for any p, q ∈ App(γ) there exists p 1 ∈ App(γ) such that p, q ≺ p 1 ; this follows from the fact that P is a basis for X. We already saw that we can find an approximation for any curve γ; the converse, namely that for a given path p there is a curve γ such that p ∈ App(γ) is also true, provided that elements of P are arcwise connected (with respect to the topological space X). Furthermore there is a relation between connectedness for posets and connectedness for topological spaces: if the elements of P are arcwise connected, then an open set Y ⊆ X is arcwise connected in X iff the poset P Y defined as The following lemma relates poset homotopy of paths and topological homotopy of curves.
Lemma 2. Assume that elements of P are arcwise and simply connected subsets of X. Take two curves β and γ with the same endpoints, and let p, q ∈ P(a 0 , a 1 ) be, respectively, approximations of β anf γ; then p and q are homotopic if and only if β and γ are homotopic.
Since first homotopy groups (both the poset and topological ones) are built from curves and paths using only the notion of homotopy, it's clear that they coincide, under suitable assumptions, as stated in the main theorem of this section: Theorem 1. Let X be an Hausdorff, arcwise connected topological space, and let P be a topological basis for X whose elements are arcwise and simply connected subsets of X; then π 1 (X) ≃ π 1 (P).
A consequence of this theorem will be useful later. Corollary 1. Let X and P be as in the previous theorem; if X is nonsimply connected, then P is not directed under inclusion.

Index sets
Let X be a topological space. As we'll see later, if we want to develop netcohomology on it we need objects like nets of local algebras, cocycles and so on, whose in turn rely on the choice of a poset endowed with a causal disjointness relation, namely a triple (P, ≤, ⊥). Of course we want our netcohomology to be related to the topological structure of X, so we assume that P is a subset of the family O(X) of open sets in X, and that the order relation ≤ coicides with set inclusion ⊆; for the moment we make no hypotheses on the causal disjointness relation ⊥.
When we come to net-cohomology, we know that the poset P is used to index local algebras, so we may ask what requirements P must fulfill in order to be considered a "good" index set. Of course it would be desiderable to avoid the introduction of "artificial" topological obstructions, namely we would like to have π 1 (X) ≃ π 1 (P), so in view of theorem 1 we require X to be Hausdorff and arcwise connected and the following definition is motivated: Definition 4. Let X be a Hausdorff, arcwise connected topological space and ⊥ a causal disjointness relation defined on O(X). We say that P ⊆ O(X) is a good index set associated with (O(X), ⊆, ⊥) if P is a topological basis for X whose elements are arcwise and simply connected subsets of X with non-empty causal complements. We denote by I (X, ⊥) the collection of good index sets associated with (O(X), ⊆, ⊥).
Note that in general I (X, ⊥) may be empty, although this doesn't happen for the application we have in mind.

Spacetime posets
Now we come to quantum field theory. For reasons that will become clearer in chapter 2, we restrict ourselves to consider only globally hyperbolic background spacetimes. So let M be a globally hyperbolic spacetime of dimension d = n + 1 ≥ 2; as we said, in order to develop net-cohomology we need to choose a poset as an index set for nets of local algebras. Following the considerations exposed above, we will take a subposet of O(M); the causal disjointness relation is the ordinary spacetime causal disjointness 2 , namely It's easy to see that K is indeed a good choice for an index set. We also note that for a globally hyperbolic spacetime the (topological) first group of homotopy π 1 (M) coincide with π 1 (C ), where C is an arbitrary Cauchy surface of M; this follows from the fact that π 1 (X × Y ) ∼ π 1 (X) × π 1 (Y ) for each pair of topological spaces X, Y , and M ∼ C × R.

Net-cohomology
Algebraic quantum field theory is based on nets of local algebras, and local algebras are indexed by open sets in the spacetime. Actually, this is the very motivation for developing the poset machinery above: to relate the physical content of the theory, encoded by local algebras, with spacetime topology.
As we said, this goal requires to reformulate topological concepts, like curves and the first homotopy group, in terms of paths made up of open sets. We saw that this can be accomplished in a very general framework, where the only structure required is an order relation on a set. Interestingly enough, this poset approach is sufficient to develop the local algebras part of the machinery, at least for the first stages.
As it's well known, in algebraic quantum field theory local algebras come in two flavours: abstract and concrete. Nets of local abstract algebras define a correspondence between spacetime sets and abstract C * -algebras (hence the name); in contrast, nets of local concrete algebras associate to each spacetime set a von Neumann algebra, namely a closed 4 subalgebra of the C * -algebra B(H) of bounded operators acting on some (fixed) Hilbert space H. To be more precise, let (P, ≤, ⊥) be a poset endowed with a causal disjointness relation ⊥; we also assume P to be pathwise connected. A net of local algebras indexed by P is a correspondence associating to any element O in the poset an algebra A(O), and satisfying where with the symbol A(O) ′ we denote the algebraic commutant of the algebra A(O). Depending on the nature of the algebras A(O), we'll talk about abstract/concrete nets of local algebras. It's worth noting that properties like isotony and causality, for abstract algebras, only make sense if we can embed "smaller" algebras into "larger" ones; in other terms for ev- must be defined; then, as in differential geometry, we can identify A with We previously defined the causal complement of an element O ∈ P as a family of elements in the poset, so what do we mean by A(O ⊥ ) ? The answer is: the algebra generated by all members of the family 6 . In symbols:

The category of 1-cocycles
At this point we introduce a technical device which turn out to be useful in constructing unitary representations of the first homotopy group: the category of 1-cocycles. First of all, though, we need to talk about net representations.

Net representations
So let P be a poset with a causal disjointness relation ⊥, and let A P : P ∋ O → A(O) be an irreducible net of local (abstract) algebras; from now on we refer to it as the reference net of observables.
If O ⊆ O, the natural isometric * -homomorphisms given by inclusion maps of A( O) into A(O) will be denoted by j O e O and named the inclusion morphisms as in [13]. The coherence requirement Now we come to the central concept, namely net representations. The idea here is to set up a local representation structure for the chosen reference net of observables; the word "local" here means that the representation map changes as we consider different local (abstract) C * -algebras; as a result we have a family of representations labeled by elements of the poset P. As 6 As before, for abstract algebras we need to assume the existence of an "environment" algebra containing (via embeddings) all relevant local algebras in order to make sense of this definition. 7 See previous footnote.
a matter of principle, Hilbert spaces carrying the representations can vary too, but for our purposes we fix one common Hilbert space and stick to it (we'll see in section 2.3 that this will be the one carrying the vacuum representation).
Of course, to be of any use in modeling quantum field theory, the various local representations need to be related to each other; a minimal request is the validity of an obvious compatibility condition with respect to the ordering of poset elements. Fix an infinite-dimensional, separable Hilbert space H: a net representation on H (for the observable net A P ) is a pair {π, ψ}, where π denotes a function that associates to any O ∈ P a representation We denote the set of intertwiners from {π, ψ} to {ρ, φ} by the symbol ({π, ψ}, {ρ, φ}), and say that two net representations are unitarily equivalent if they admit a unitary intertwiner T , that is T O is a unitary operator for any O ∈ P. {π, ψ} is irreducible when unitary elements of ({π, ψ}, {π, ψ}) are of the form c·1 with c ∈ C and |c| = 1.
Remark. The construction outlined above, concerning local representations for a net of local abstract algebras, follows closely a recurrent pattern in differential geometry, where a global object is often constructed up by pasting together locally defined objects satisfying suitable compatibility conditions. From this perspective it can be said that we are dealing with a kind of non-commutative geometry, where objects are modeled after those living in a (commutative) manifold. In this scheme we can regard the local C * -algebras as open sets on the manifold, net representations as coordinate charts on B(H), the linear operators ψ O e O as "transition functions" between local coordinate charts, and equivalent net representations as compatible atlas belonging to a given differentiable structure. Obviously these considerations are valid only at an heuristic level, but neverthless can be useful to understand what's going on.

Introducing cocycles
As we said, 1-cocycles are a technical tool useful for constructing representations. The basic idea here is to associate a non-commutative object (an operator on some Hilbert space) to each poset path in a homotopy-invariant manner, so that the operator depends only on the equivalence class the path belongs to; this behaviour can be achieved by requesting the map to be invariant with respect to elementary deformations of paths (see section 1.2), and is espressed by a condition called 1-cocycle identity.
The formal definition is as follows: given a complex, infinite dimensional, separable Hilbert space H, a (unlocalized) 1-cocycle z on P valued in B(H) is a field z : Σ 1 (P) ∋ b → z(b) ∈ B(H) of unitary operators satisfying the 1-cocycle identity: (1.6) Relations between 1-cocycles are described by intertwiners. An intertwiner t between a pair of 1-cocycles z, z 1 is a field of operators t : Σ 0 (P) ∋ a → t a ∈ B(H) satisfying the relation

CHAPTER 1. POSETS AND NET-COHOMOLOGY
We denote by (z, z 1 ) the set of intertwiners between z and z 1 . En passant we note the analogy in the definitions of cocycle intertwiners and net representations intertwiners. It turns out that the set of 1-cocycles has a quite rich structure. The category of (unlocalized) 1-cocycles is the category Z 1 (P, B(H)) whose objects are 1-cocycles and whose arrows are the corresponding set of 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 · s) a := t a · s a , ∀ a ∈ Σ 0 (P).
With these operations and the composition "·", the set (z, z) is an algebra with identity 1 z . The category Z 1 (P, B(H)) has an adjoint * , defined as the identity, z * = z, on the objects, while the adjoint t * ∈ (z 1 , z) of an arrow t ∈ (z, z 1 ) is defined as where (t a ) * stands for the operator adjoint of t a in B(H). Now let · be the operator norm in B(H). Given t ∈ (z, z 1 ), it's easy to see [13] that t a = t a 1 for any pair a, a 1 of 0-simplices, since P is pathwise connected. Therefore by putting t := t a , a ∈ Σ 0 (P), we see that (z, z 1 ) is a complex Banach space for any z, z 1 ∈ Z 1 (P, B(H)), and (z, z) is a C * -algebra for any z ∈ Z 1 (P, B(H)). This entails that Z 1 (P, B(H)) is a C * -category.
Needless to say, the expression "unlocalized 1-cocycle" suggests the existence of localized 1-cocycles, as it's the case. The only additional property of this new cocycle's flavour is localization: the operator obtained by evaluating a localized cocycle on a 1-simplex belongs to the local (concrete algebra) living on the simplex' support. In other terms, let R P be a net of local concrete algebras valued in B(H) (for example that arising from a net representation of the reference net A P ); localization amounts to add the following locality condition to the definition given above for unlocalized 1-cocycles: A parallel notion of localized cocycle intertwiners can be made: the intertwiner t is said to be localized if t a ∈ R(a), ∀ a ∈ Σ 0 (P).
The category of localized 1-cocycles on the local algebras net R P will be denoted by the symbol Z 1 (R P ). Finally, we say that a 1-cocycle z is a coboundary if it can be written as , for some field of unitaries Σ 0 (P) ∋ a → W (a) ∈ B(H).

Cocycle equivalence
Now we introduce an equivalence relation between 1-cocycles. Two (localized or not) 1-cocycles z, z 1 are said to be equivalent (or cohomologous) if there exists a unitary arrow t ∈ (z, z 1 ) between them; a 1-cocycle is trivial if it's equivalent to the identity cocycle i, defined as i(b) = 1 for any b ∈ Σ 1 (P); this is equivalent to say that z is a coboundary. Note that if the cocycles z, z 1 are localized, the above definition requires the intertwiner t ∈ (z, z 1 ) to be localized, too. This imply that, for localized cocycles, two different equivalence relations are available, depending on the fact we require or not the operator-valued field t to be localized. In the former case we simply speak about equivalent cocycles; in the latter we say that z, z 1 are equivalent in B(H). It's worth observing that for localized cocycles unitary equivalence is stronger than equivalence in B(H), since localization is required (for unlocalized cocycles only the latter one makes sense).
We denote by Z 1 t (R P ) the set of (localized) 1-cocycles trivial in B(H). From a topological point of view, triviality in B(H) means path indepen-dence. In details, if the evaluation of a 1-cocycle z on the path p = {b n , . . . , b 1 } is defined as z is said to be path-independent on a subset P ⊆ P whenever z(p) = z(q) ∀ p, q ∈ P(a 0 , a 1 ) such that |p|, |q| ⊆ P, for any a 0 , a 1 ∈ Σ 0 (P). Then from pathwise connectedness of P it descends that a 1-cocycle is trivial in B(H) if and only if it is path-independent on all P.
We close this section with a result relating unitary net representations and 1-cocycles. We start by putting out a definition mimicking that given above for net representations: a 1-cocycle z ∈ Z 1 (P, B(H)) is said to be irreducible if there are no unitary intertwiners in (z, z) barring those of the form c·1 with c ∈ C and |c| = 1. Given a unitary net representation {π, ψ} of A P over H define As usual |b| ∈ Σ 0 (P) denotes the support of the symplex b. One can check that ζ π is a 1-cocycle in Z 1 (P, B(H)). {π, ψ} is said to be topologically trivial if ζ π is trivial. It can be proven [13]

Representations of the first homotopy group
At this point we are in the right position to understand the usefulness of the cocycle machinery: it can be used to construct unitary representation of the first homotopy group. In fact, the main result is that there is a one-to-one correspondence between 1-cocycles and unitary representations, modulo suitable equivalence relations. Let us consider a pathwise connected poset P equipped with a causal disjointness relation ⊥, and let R P be an irreducible net of local (concrete) algebras. To start with, we list some preliminary results about cocycles and paths, in particular invariance of 1-cocycles for homotopic paths.
Now we come to the main theorem. Fix a base 0-simplex a 0 ; given z ∈ Z 1 (R P ) define the following map sending an equivalence class of (poset) paths into an operator in B(H): The definition is well posed due to cocycle invariance with respect to path homotopy.
Theorem 2. The correspondence Z 1 (R P ) ∋ z → π z maps 1-cocycles z, equivalent in B(H), into equivalent unitary representations π z of π 1 (P) in H; up to equivalence this map is injective. If π 1 (P) = 1, then Z 1 (R P ) = Z 1 t (R P ). The previous theorem is important because it sheds some light on this question: can we find path-dependent 1-cocycles ? Suppose we are in the topological case, namely P ∈ I (X, ⊥) where X is a topological (Hausdorff, arcwise connected) space; then if X is simply connected it follows from theorem 1 that π 1 (P) = 1, so as a trivial consequence of theorem 2 we have Corollary 2. If X is simply connected, any 1-cocycle is trivial in B(H), namely path-independent. In other words Z 1 (R P ) = Z 1 t (R P ). So we conclude that the only topological obstruction to path-independence of 1-cocycles can be non-simple connectedness of the topological space X; actually this is a real obstruction since in chapter 3 we will exhibit a topological (i.e. path-dependent) 1-cocycle living in the cylindric flat universe.

QFT on the cylindric Einstein universe
In this section we discuss the QFT model on the top of which we'll construct, in the last chapter, our topological cocycles. After developing the model we introduce the vacuum state and prove various properties of the reference vacuum representation, including DHR sectors anf Haag duality.

The Cauchy problem for the field equation
On the road to quantization the first step is to choose a classical wave equation on the background spacetime, namely that satisfied by the classical field; applying to it a suitable quantization scheme gives rise, at least in the algebraic approach [36], to the net of local algebras encoding the fundamental observables of the theory. For our purposes we choose to start from the scalar massive Klein-Gordon equation, whose intrinsic form is: Here the symbol denotes the d'Alembertian operator which is given in local coordinates by the expression: where |g| := | det {g µν } | is the metric determinant and m 2 > 0 is the mass parameter.
An essential condition for later developments is the well posedness of the Cauchy problem for the equation (2.1); but what does it mean well posedness for a wave equation in a general geometric context ? Loosely speaking, we require that, given suitable initial conditions on a suitable class of spacetime hypersurfaces, there exists only one solution of the equation satisfying these initial conditions, for every hypersurface in that class.
This well-posedness requirement forces us to restrict our scope to a particular family of background spacetime manifolds, namely the globally hyperbolic ones. We recall that a connected time-oriented Lorentzian manifold M is said to be globally hyperbolic if M admits a Cauchy hypersurface C , i.e. a subset such that every inextendible timelike curve in M meets it at exactly one point. It turns out [52] that a globally hyperbolic spacetime M admits a (not unique) smooth foliation by Cauchy hypersurfaces; that is to say M is isometric to R × C with metric −βdt 2 ⊕ g t , where β is a smooth positive function, g t is a Riemannian metric on C depending smoothly on t ∈ R and each set {t} × C is a smooth spacelike Cauchy hypersurface in M.
As we said, we restrict ourselves to consider only globally hyperbolic spacetimes so the Cauchy problem for the Klein-Gordon equation is well posed. Actually, we focus on a very special class of globally hyperbolic spacetimes, called ultrastastic spacetimes (further details can be found e.g in [64]). Roughly speaking, ultrastastic spacetimes are a product space × time, so are relatively easy to deal with.

as a Riemannian manifold) M is globally hyperbolic and its natural foliation is made up of spacelike Cauchy surfaces.
The well-posedness of the Cauchy problem for the Klein-Gordon equation implies [4] the existence of two continuos 1 linear maps E ± : C ∞ 0 (M) → C ∞ (M) uniquely determined by the following properties: They are called the advanced (+) and retarded ( So we see that E maps C ∞ 0 (M) to the set S of smooth solutions of the Klein-Gordon equation compactly supported on each Cauchy surface (and one can show [23] that this map is surjective).

Well posedness
As we said, a crucial fact for the theory we are going to develop is wellposedness 2 of the Klein-Gordon equation in a globally hyperbolic spacetime; that is to say that Cauchy data on a Cauchy surface uniquely determine a solution of (2.1). In details, let M be a globally hyperbolic spacetime, and select a Cauchy surface C belonging to a foliation: if M = Σ × R, then C = Σ t for some t ∈ R. Then the vector field ∂ t is a timelike, future-pointing vector field normal to every Σ t , so applying it to a spacetime function gives its normal derivative with respect to C . Now, if ϕ is smooth solution of the Klein-Gordon equation, its Cauchy data on C are defined as the initial "position" ϕ |C and "velocity" ∂ t ϕ |C of the solution itself. So we can define a "projection" operator P C : mapping a solution of equation (2.1) to its Cauchy data: 1 With respect to the standard locally convex topologies on C ∞ 0 (M, R) and C ∞ (M, R). 2 Well-posedness is a consequence of tha classical energy-estimate for solutions of second order hyperbolic partial differential equations; see e.g. [4].
Well posedness of equation (2.1) means exactly that this map is bijective: there exists exactly one smooth solution ϕ of the Klein-Gordon equation such that Furthermore, solutions of equation (2.1) propagate with finite speed, i.e. if Cauchy data are supported on a set G ⊂ C , then the corresponding solution is supported on the causal development of this set, J(G). From this fact it follows that if a solution is compactly supported on a Cauchy surface C then it is compactly supported on any other Cauchy surface C ′ .

Classical field quantization 2.2.1 Symplectic spaces
Now we finally come to quantization. The key fact here is that the space of smooth solutions of the wave equation has a natural symplectic structure, which fact makes it possible to define local algebras, as we'll see shortly.
Actually, we can introduce several equivalent versions of this space. To start with, we consider the space S of all real-valued smooth solutions of the Klein-Gordon equation such that their Cauchy data have compact support on one (hence every) Cauchy surface, endowed with the symplectic form: where dη C denotes the volume element induced by the (Riemannian) metric on C . It's easy to see that the right hand side is independent on the choice of C , and that σ is nondegenerate; so the pair (S, σ) is a real symplectic space indeed.
Another option is to consider the space of Cauchy data living on an arbitrary but fixed Cauchy surface C and to introduce essentially the same symplectic form as above: We obtain another symplectic space (D C , δ C ), clearly isomorphic to the former; they are related by the symplectic map: P C : S → D C . It can also be noted that the Cauchy data spaces associated to different Cauchy surfaces, say C and C ′ , are isomorphic. In fact the map: C is a symplectomorphism, being the composition of two symplectomorphism. What we do here is: "take a pair of Cauchy data on C , make them evolve to the solution ϕ and finally determine its corresponding Cauchy data on C ′ ".
The definition of the third space requires a little more work. Start by taking the space of test functions on the entire spacetime and view two of them as equal if the causal propagator assumes the same value on them; in other words take the quotient K := C ∞ 0 (M, R) \ ker(E). Then consider the following bilinear form: is the quotient map and dη is the metric-induced volume measure on M. It's easy to see that κ is well defined and a nondegenerate symplectic form on K. We already said that the map f → Ef is surjective, so the map [f ] → Ef , mapping (K, κ) to (S, σ), is a bijection; moreover it's a symplectomorphism, so we conclude that (K, κ) and (S, σ) are isomorphic.
So far we have seen that (S, σ), (D C , δ C ) and (K, κ) are just different implementations of the same algebraic object, the solution space of the wave equation; it's a global object, living on the entire spacetime. For the sake of algebraic quantization, though, we need local objects, in primis local observable algebras, so we would like to introduce local versions of those symplectic spaces. It's better to start with Cauchy data, because in this context it's clear what localization means: given an open set G ⊆ C with compact closure, a pair of Cauchy data So we can consider the symplectic subspace of (D C , δ C ) made up of Cauchy data supported in G, (D G , δ G ). This family of symplectic spaces indexed by (precompact) open subsets of a fixed Cauchy surface induces, via the symplectic map P C , a corresponding family of symplectic subspaces of the solution space S, say S G . What about (K, κ) ? We could repeat the previous reasoning, this time using the map f → Ef , thus obtaining a family of symplectic subspaces of the global space (K, κ). How we can characterize these subspaces ? We could be tempted to identify them with the sets Actually this identification fails to be true, and we need to restrict ourselves to a suitable subfamily of open sets, the collection of diamonds based on Cauchy surfaces (see definition 5). To be more specific, fix a Cauchy surface C , and let G be a (relatively compact) open subset of C ; then the symplectic spaces (D G , δ G ) and (K(3(G)), κ |K(3(G)) ) are isomorphic. This descends from the fact, proved by Dimock in [23], that if N is an open neighborhood (in M) of G it holds: and from the remark that 3(G) is a globally hyperbolic spacetime on its own right, equipped with the restriction of the spacetime metric g.

The cylindric flat universe
After these general preliminaries, now we discuss free quantum field theory for our wave equation (2.1) on the cylindric Einstein universe, namely the ultrastatic spacetime M foliated by the 1-dimensional torus S 1 equipped with the standard euclidean metric; in other words M = S 1 × R, and denoting by θ ∈ [−π, π] (with identified endpoints) the standard coordinate over S 1 and t ∈ R, the metric reads: In explicit form the equation of motion is thus: Let C ∼ = S 1 be a Cauchy surface in the chosen foliation; then ∂ t is a timelike smooth field, normal to C . For future convenience we also fix a positive rotation orientation for S 1 .
Our choice of the background spacetime is dictated by a straightforward principle: we need a manifold simple enough to explicitly working out the details. In this respect, the cylindric flat universe is the simplest choice after Minkowski spacetime: in fact it's just a spatial-compactified Minkowski spacetime. As we anticipated above, we can easily quantize the classical Klein-Gordon field by considering the symplectic space S of real smooth solutions of the equation of motion (2.2).
We have at our disposal three different flavours of this space: we choose to work with the Cauchy data version (D C , δ C ). If ϕ ∈ S, then we denote by (Φ, Π) its Cauchy data on C , i.e. Φ = ϕ ↾C , Π = ∂ t ϕ ↾C ; then the symplectic form reads: Then we have to fix a poset indexing local observable algebras. For the sake of simplicity we take a two-steps approach: first we develop the theory with respect to a "spatial" poset, i.e. a family of index sets lying on the chosen Cauchy surface C ; subsequently we construct a "spacetime" poset from the spatial one and extend the results obtained in the spatial case, at least as far as possible.
What about the spatial poset R ? We choose the simplest one, namely the class of open proper intervals of S 1 . A proper interval of S 1 is a connected subset I ⊂ S 1 such that both Int(I) and Int(S 1 \ I) are nonempty. If I ∈ R, D I will denote the symplectic subspace of D C of Cauchy data supported in I. It's worth noting here that R is endowed with a causal disjointness relation: two intervals I, I 1 are causally disjoint, I ⊥ I 1 , if their intersection is empty. Specializing definitions given in section 1.1 to the present case we obtain: Definition 7. Let I ∈ R; we define the causal complement of I as the family of sets Note that the open set C I ⊥ := ∪ I 1 ∈I ⊥ I 1 coincides with the interior of the ordinary set complement of I, namely C I ⊥ = Int(S 1 \ I). For the sake of simplicity, we'll adopt henceforth the shortened notation I ′ ≡ C I ⊥ . By construction I ′′ = I and I ′ ∈ R when I ∈ R.
Remark. The support properties of Cauchy data of solutions of the Klein-Gordon equation depend on the chosen Cauchy surface: if, for example, the Cauchy data (Φ, Π) of ϕ ∈ S belongs to D I for some I ∈ R with respect to a Cauchy surface C , Cauchy data of the same solution ϕ generally fail to fulfill this condition when referring to another Cauchy surface C ′ sufficiently far in time from the former.

Local algebras
In the previous section we have constructed a poset-indexed family D I of symplectic spaces; now we use these spaces to construct a poset-indexed family of C*-algebras, namely the Weyl algebras built on them. We recall that, given a symplectic space (V, ω), there exists a C * -algebra A and a map W : V → A such that, for all ϕ, ψ ∈ V : (iv) A is generated, as a C * -algebra, by the elements W (ϕ).
It turns out [8] that the pair (A, W ) is unique up to isomorphisms and is called the CCR-representation of (V, ω), denoted CCR(V, ω). A is said to be the Weyl algebra built from (V, ω). Now, coming back to our case, we see that each symplectic space D I gives rise to a C * -algebra W(I) localized in I, namely its associated Weyl algebra; in the quantization scheme discussed e.g in [23] these W(I) play the role of local (abstract) observable algebras. If we denote by W KG the global algebra associated to D C , we see that each Notice that all the subalgebras share the same unit element and the following two properties are valid: Strictly speaking the family {W(I)} I∈R isn't a net of C * -algebras 3 because the poset R is not directed with respect to the partial order relation given by set inclusion (there are pairs I, J ∈ R with K ⊃ I, J for every K ∈ R), and thus it is not possible to take the inductive limit defining the overall quasilocal (C * -) algebra containing every W(I). Anyway an universal algebra A generated by {W(I)} I∈R can be defined and A ⊃ W KG (see appendix C).

Spacetime symmetries
A word is in order here about spacetime symmetries. Consider a Cauchy surface C belonging to the natural foliation of M; then C is metrically invariant under the action of R viewed as a C -isometry group: r ∈ R induces the isometry β r : The existence of such {α r } r∈R follows immediately from the fact that σ is invariant under every β * r (see e.g. [8, Proposition (5.2.8)]) . Now let ϕ be a real smooth solution of the wave equation and take s ∈ R; ϕ s is the future-translation of ϕ by a time interval s, in the sense that ϕ s (t, θ) := ϕ(t − s, θ) for all t ∈ R and θ ∈ S 1 . Notice that ϕ s is again solution of the Klein-Gordon equation because the spacetime is static. Passing to the Cauchy data (on the same Cauchy surface at t = 0), this procedure induces a one-parameter group of transformations µ s : D C → D C such that µ s (Φ, Π) are the Cauchy data of ϕ s when (Φ, Π) are those of ϕ. The maps {µ s } s∈R preserve the symplectic form due to the invariance of the metric under time displacements. As a consequence we have a oneparameter group of * -isomorphisms {τ s } s∈R acting on W KG and uniquely defined by the requirement (2.5) The groups {α r } r∈R and {τ s } s∈R can be combined into an Abelian group of Therefore one concludes that if (Φ, Π) is supported in I ∈ R, µ s (Φ, Π) is supported in the interval I s ⊂ S 1 constructed as follows. Passing to the new variable θ ′ := θ+c for some suitable constant c ∈ R, one can always represent I as (−a, a) with 0 < a < π. In this representation I s := (−a − |s|, a + |s|) taking the identification −π ≡ π into account. Notice in particular that, for I ∈ R, one has I s ∈ R if and only if |s| < π − ℓ(I)/2 (where ℓ(I) is the length of I ∈ R when ℓ(S 1 ) = 2π), whereas it turns out that I s = S 1 whenever |s| > π − ℓ(I)/2.

Vacuum representations.
Now we want to introduce an important class of states on Weyl algebras, namely quasifree states. They turn out to enjoy remarkable properties and, above all, the privileged vacuum state we are going to discuss later is a quasifree state. En passant we also introduce the notion of a two-point function, although we won't exploit it in what follows.

Quasifree states
Our subsequent constructions rely heavily on the concept of GNS representation; although it's a well established notion, for the sake of completeness we recall here the statement of the main theorem; further details can be found in [7].

Theorem 3 (GNS representation).
Let ω be a state over the C * -algebra A; then there exists a cyclic representation (H ω , π ω , Ω ω ) (called the GNS representation of A induced by the state ω) such that: for all A ∈ A and, consequently, Ω ω 2 = ω = 1. Moreover this representation is unique up to unitary equivalence.
So, let ω be a state on the Weyl algebra A[Z, ξ] associated to some symplectic space (Z, ξ) via the map W : Z → A[Z, ξ] and let (H ω , π ω , Ω ω ) be the GNS representation of ω. Assume that for every z ∈ Z the unitary one-parameter group t → π ω (W (tz)), t ∈ R, is strongly continuos and that Ω ω belongs to the domain of definition of its generator Φ ω (z). Then we call the C-bilinear form λ ω on Z defined as: the spacetime two-point function of ω. It can be shown by direct inspection that Λ ω is a bi-solution of the Klein-Gordon equation, i.e.
Now assume that the two-point function λ ω of ω exists; then the following definition holds.
Quasifree states admit a complete characterization [46] in terms of the notion of one-particle structure. First of all, we need some notation: given a complex Hilbert space H, consider its (symmetric) Fock space F + (H) and for all χ ∈ H construct the operator where a * , a are the creation/annihilation operators acting on F + (H). Ω F ≡ 1 ⊕ 0 ⊕ 0 . . . will denote the Fock vacuum.
where H is a complex Hilbert space and k : Z → H a real-linear injective map, with the properties (for all z,z ∈ Z): We are now in the position to give the quasifree states characterization we have spoken about: Then there exists, uniquely up to unitary equivalence, a one-particle Hilbert space structure (k, H) for ω, such that the GNS representation (H ω , π ω , Ω ω ) is given by Moreover, ω is pure if and only if the range of k is dense in H.
If L is a real-linear subspace of H we write where the double prime accent denotes the weak operatorial closure in . Now let us specialize this construction to QFT. Consider a Cauchy surface C in the globally hyperbolic spacetime (M, g), and suppose that the global algebra W KG is represented as the Cauchy-data space Weyl algebra A[D C , δ C ]. Then from theorem 4 we know that a quasifree state ω on A[D C , δ C ] is characterized by a one-particle Hilbert space structure (k, H), and it gives rise to the correspondence: where I denotes an element of the spatial poset on C . In terms of local von Neumann algebras it turns out that: where we set R(I) := π F (A[D I , δ I ]) ′′ . Finally we restrict ourselves to ultrastastic spacetimes. In this context we can explicitly construct a quasifree, time-invariant state, that we identify with the (unique) vacuum in the theory. So consider a d-dimensional complete Riemannian manifold (Σ, γ) with Laplace-Beltrami operator ∆ γ and metric-induced measure µ γ ; then for fixed m > 0 the Klein-Gordon differential operator is essentially selfadjoint [17]; we denote its closure by A. Let (M, g) be the ultrastastic spacetime foliated by (Σ, γ), and Σ(t) ≡ C t , t ∈ R, the canonical foliation. For each t ∈ R define a quasifree state ω t on A[D Ct , δ Ct ] by setting its one-particle Hilbert space structure to (k t , H t ), where It turns out that ω t is pure and invariant under time translations τ s on W KG . So we may drop the superscript t denoting ω t by ω, and calling it the canonical vacuum state on the Weyl algebra W KG of the (mass m) Klein-Gordon field on the ultrastastic spacetime (M, g).
It's time to introduce the vacuum state for the cylindric flat universe; since it's an ultrastastic spacetime, we have just seen that there exists a unique time-invariant quasifree state, and we also know how to construct it: we have to exploit the generator of the equation of motion, namely eq. (2.2). It's the positive symmetric operator acting on the complex Hilbert space L 2 (S 1 , dθ) C . It is essentially self-adjoint since C ∞ (S 1 , C) contains a dense set of analytic vectors made of exponentials θ → e inθ , n ∈ Z, which are the eigenvectors of the operator. The unique self-adjoint extension of this operator, i.e. its closure, will be denoted by Notice that A is strictly positive (being m > 0) and thus its real powers A α , α ∈ R, are well-defined.
The next proposition concerns some basic properties of A and its powers.
enjoy the following properties: Proof. Consider the family of operators B α : theory. Using the unitary operator U : L 2 (S 1 , dθ) C → ℓ 2 (Z) which associates a function with its Fourier series, one realizes that U −1 B 1 U extends − d 2 dθ 2 + m 2 I and thus its unique self-adjoint extension A must coincide with The operators B α satisfy the corresponding properties in ℓ 2 (Z) of (a), (b), (d) by construction ((b) follows from the fact that B α = (B α ) * , Ker(B α ) is trivial and Ran(S) = Ker(S * ) ⊥ for every densely-defined operator S acting on a Hilbert space), so that A α satisfy (a), (b), (d).
The proof of (c) is the following. The first statement is an obvious consequence of the Fourier representations of C and that of A α , B α . Let us come to the second statement. If f ∈ C ∞ (S 1 , C) its Fourier coefficients c n vanish as n → ±∞ faster than any power of n and thus they define an element of each Dom(B α ). Moreover, if ψ ∈ C ∞ (S 1 , C), the Fourier coefficients of A α ψ vanish faster than every powers of n and thus the associated Fourier series converges uniformly with all of θ-derivatives of any order and therefore A α ψ individuates a function in The other inclusion follows trivially applying A −α to both sides and using arbitrariness of α. To extend this result to C ∞ (S 1 , C) it suffices exploiting the fact that A α commute with C. The final statement is now immediate: Following the general theory discussed in section 2.3.1, we exploit the operator A to define a one-particle Hilbert space structure for the vacuum. Fixing a reference Cauchy surface C , for every (Φ, Π) ∈ D C the quantization map K : D C → L 2 (S 1 , dθ) C is defined as, : and, together with the Hilbert space L 2 (S 1 , dθ) C , it's the one-particle structure we looked for. This map will be useful shortly to determine a preferred unitary irreducible (Fock) representation of the Weyl algebra called the vacuum representation.
A natural physical way to introduce K is noticing that the solution ϕ of equation (2.1) with Cauchy data (Φ, Π) ∈ D C , interpreting the t derivative in the sense of L 2 (S 1 , dθ) C topology, can be written as The right-hand side of (2.7) turns out to be (t, p)-jointly smooth and the t derivative coincides with that in the L 2 sense [67]. Thus, by the uniqueness theorem for solutions of the Klein-Gordon equation with compactly supported data in globally hyperbolic spacetimes, varying t ∈ R the right-hand side of (2.7) defines the proper solution of the wave equation individuated by the Cauchy data (Φ, Π). From (2.7), interchanging A ±1/4 with e −ıtA 1/2 it arises that A 1/2 can be seen as the Hamiltonian generator of Killing time displacements, acting on the Hilbert space of the quantum wavefunctions K(Φ, Π) associated with the classical solutions with Cauchy data (Φ, Π); that Hilbert space is the so called one-particle space. This is the perspective necessary to understand the construction presented in theorem 5 from a physical point of view.
The following fundamental statement about the range of K holds true.
Proposition 4. With the given definitions for D C and K the following facts are valid.
as a consequence K is injective.
Proof. (a) is a straightforward consequence of (c) in proposition 3. The proof of (b) is obtained by direct inspection. Injectivity of K descends immediately from (2.8) and non-degenerateness of σ.
Let us construct the vacuum GNS representation using proposition 4. Let us remind some terminology. In the following, if {α g } g∈G is a representation of a group G in terms of * -automorphisms of an unital * -algebra A, a state λ : A → C will be said to be invariant under {α g } g∈G if one has λ (α g (a)) = λ(a) for all g ∈ G and a ∈ A. Moreover a representation {U g } g∈G , where every U g is a unitary operator defined over the GNS Hilbert space H λ of λ, is said to implement {α g } g∈G if π λ (α g (a)) = U g π λ (a)U * g for all g ∈ G and a ∈ A, where π λ is the GNS representation of A induced by λ.
Theorem 5. With the given definitions for D C and K, the following facts are valid.
(a) There is a pure quasifree state ω 0 : W KG → C uniquely induced by linearity and continuity by , called the vacuum representation of W KG , is constructed as follows (up to unitarities): (ii) the representation π ω 0 is isometric and is induced by linearity and continuity by: where the generators P ⊗ , H ⊗ are respectively given by the tensorialization of the operators P , Proof. We'll give here only a sketch of the proof. Points (a) and (b) are immediate consequences of proposition 3.1 and lemma A.2 in [46]. The fact that ω 0 is pure follows from the cited propositions because K(D C ) = L 2 (S 1 , dθ) C = H ω 0 as a consequence of Proposition 4. Invariance of ω 0 under γ arises by direct inspection. A known theorem (see [2]) establishes, as a consequence, the existence of a unique unitary representation of γ which implements γ leaving Ψ ω 0 invariant. The remaining statements are based on standard arguments valid for QFT in static spacetimes and trivial generalizations of the analogs in Minkowski spacetime.

From abstract algebras to concrete algebras
Having selected a reference state, namely the vacuum state ω 0 , we are now in the position to realize our abstract (Weyl) local algebras as operator algebras on a fixed Hilbert space. All we need to do is to use the GNS representation (H 0 , π 0 , Ψ 0 ) of the vacuum state and to take, as the concrete algebras, the images π 0 (W(I)) of the local algebras W(I) under the representation π 0 itself. Actually, we'd like to work with von Neumann algebras, so we need to take the (weak operatorial) closure of these algebras; so we end up with a poset-indexed family of von Neumann algebras R(I) := π 0 (W(I)) ′′ , I ∈ R.

Properties of local nets 2.4.1 Introduction
In this section we discuss some remarkable properties of the vacuum GNS representation (H 0 , π 0 , Ψ 0 ) and the associated class of von Neumann algebras, {R(I)} I∈R where R(I) := π 0 (W(I)) ′′ . In the following R KG := R(S 1 ) = π 0 (W KG ) ′′ and B(K) will denote the algebra of all bounded op- The proof of irreducibility is quite trivial: first of all notice that R ′ . Notice that it also holds R ′ KG ∩ R KG = C · 1 and hence R KG is a factor (the simplest factor of type I ∞ ).
Our goal is to prove some relevant features of this class of von Neumann algebras. From now on, if J ∈ R and t = (r, s) ∈ R 2 , the set J + t is the subset of S 1 obtained as follows: (1) rotate J by the angle r positively obtaining the set I; (2) pass from the set I to the set I s defined in section 2.2.4. The obtained set is J + t by definition.

Additivity properties
Now we come to the central point of this section. First of all we notice that the properties we call spatial additivity and spatial weak additivity hold true. In the following, if {A j } j∈J is a class of sub * -algebras of a given * -algebra A, ∨ j∈J A j denotes the * -algebra finitely generated by all the algebras A j . (a) Spatial additivity: if {I i } i∈L ⊂ R satisfies ∪ i∈L I i = I or respectively ∪ i∈L I i = S 1 , then i∈L π 0 (W(I i )) ′′ = R(I) and respectively i∈L π 0 (W(I i )) ′′ = R KG .
Proof. By construction we have the von Neumann algebras identity i∈L π 0 (W(I i )) ′′ = i∈L π 0 (W(I i )) ′′ , and the latter equals π 0 i∈L W(I i ) ′′ and thus π 0 i∈L W(I i ) ′′ by conti- Since supp(f ) is compact, for some finite number of intervals I i 1 , . . . , I in , supp(f ) ⊂ ∪ n j=1 I i j . Then, employing a suitable smooth partition of the unit on the union of the intervals I i 1 , 1 = n j=1 g j with supp(g j ) ⊂ I i j , one has f = n j=1 g j ·f where g j ·f ∈ C ∞ 0 (S 1 , R) and supp(g j ·f ) ⊂ I i j . In this way, using Weyl relations, one finds that every generator of W(I) can be written, up to a factor in C, as a product of generators of each W(I i j ), therefore i∈L W(I i ) ⊃ W(I). The case ∪ i∈L I i = S 1 has the same proof. (b) The proof of (a) is valid also if I = S 1 no matter if S 1 ∈ R. The class {I r } r∈R with I r := I + (r, 0) satisfies ∪ r∈R I r = S 1 and then the proof of (a) encompasses the proof of the first statement in (b). The second statement can be proved as follows. R KG Ψ 0 is dense in H 0 because R KG ⊃ π 0 (W KG ) and π 0 (W KG )Ψ 0 is dense in H 0 by GNS theorem. Hence, in view of the first statement, r∈R π 0 (W(I + (r, 0))) ′′ Ψ 0 is dense in H 0 .
The last important property we mention is the Reeh-Schlieder property for {R(I)} I∈R .
Proof. See appendix A.

Haag duality.
In this section we establish the validity of a key property, namely (spatial) Haag duality; the proof uses an extension (not very straightforward due to technical difficulties) of the approach by Leyland-Roberts-Testard [48]. Afterwards we'll consider (spatial) local definiteness and (spatial) punctured Haag duality. Haag duality in our theory states that: In more general contexts I ′ does not belong to the class R labeling the local algebras and thus the right-hand side of equation (2.11) has to be defined, imposing spatial additivity, as the von Neumann algebra generated by all local algebras R(J) with J ∈ R and J ⊆ I. However in our case there are no problems since I ′ ∈ R.
To prove Haag duality we need some preliminary definitions and results. Consider a generic complex Hilbert space H; as we saw earlier, for all vectors ψ ∈ H the unitary operator for every real closed subspace M of H. Another important result is that [48]: for any pair of closed real subspaces M, N ⊂ H.
We now specialize to the case where H is the one-particle space L 2 (S 1 , dθ) C . If On the other hand, since the R-linear map K : D C → H is injective, by construction it turns out that π 0 (W (Φ, is strongly continuous (see for instance [8]), we finally obtain We are now ready to state the main theorem.
Theorem 6 (Spatial Haag duality). For every I ∈ R it holds: Proof. By spatial locality one obtains immediately that R(I ′ ) ⊂ R(I) ′ . It remains to show that R(I) ′ ⊂ R(I ′ ). To this end we want to use eq. (2.14) when H 0 = L 2 (S 1 , dθ) C is the one-particle space for the vacuum representation π 0 . In particular F + (H 0 ) = H 0 . Therefore the inclusion R(I) ′ ⊂ R(I ′ ) can be re-written as In view of eq. (2.14), to prove Haag duality is enough to establish that (2. 16) This is true in view of the subsequent lemma.
(Applying definition 2.6 one proves straightforwardly that the linear functionals defined above are continuous in the sense of distributions. In the case of Π ψ one has that the functional individuated -varying f -by ψ, A −1/4 f = A −1/4 ψ, f is trivially continuous. In the case of Φ ψ notice that ψ, A 1/4 f = lim n→+∞ A 1/4 ψ n , f for some sequence Dom(A 1/4 ) ∋ ψ n → ψ independent from f . As each linear functional A 1/4 ψ n , · is a distribution, Φ ψ is a distribution as well. By construction the distributions Φ ψ and Π ψ have supports contained in I ′ because from the definitions descends that Φ ψ (θ)f (θ) dθ = 0 and Π ψ (θ)f (θ) dθ = 0 for ψ ∈ (M I ) ′ , whenever f is supported in I. Now consider ρ ∈ C ∞ 0 (R, R) supported in (−ǫ/2, ǫ/2) and define (using weak operatorial topology) ψ * ρ := R ρ(r)e −irP ⊗ ψ dr . Fubini-Tonelli theorem and the fact that e −irP ⊗ commutes with A α (it can be proved immediately passing to Fourier-series representation) entails that Φ ψ * ρ = Φ ψ * ρ and Π ψ * ρ = Π ψ * ρ, where * in the right-hand side denotes the standard convolution so that Φ ψ * ρ and Π ψ * ρ are smooth functions supported in I ′ + (−ǫ, ǫ) and thus ψ * ρ ∈ M I ′ +(−ǫ,ǫ) . Therefore, assuming the existence of a suitable sequence {ρ k } of real smooth functions supported in (−ǫ/2, ǫ/2), with ψ * ρ k → ψ, the sequence of pairs (Φ k , Π k ) := (Φ ψ * ρ k , Φ ψ * ρ k ) turns out to be made of real smooth functions supported in I ′ + (−ǫ, ǫ), and K(Φ k , Π k ) → ψ as requested, proving that ψ ∈ M I ′ +(−ǫ,ǫ) . It's only left to prove that such a sequence {ρ k } does exist. Consider a family of smooth functions ρ k ≥ 0 with supp(ρ k ) ⊂ [−1/k, 1/k] and with R ρ k (r)dr = 1. In our hypotheses ψ * ρ can be re-written as and the last term vanishes as k → +∞ because r → e −irP ⊗ is strongly continuous. Q.E.D. Proof of II. We have to establish the validity of (2.18). The proof is based on the following technical proposition. (This is a quite difficult point whose proof technically differentiates QFT on S 1 from that in Minkowski space as done in [48].) Proposition 7. Take J ∈ R and assume J ≡ (−a, a) ⊂ (−π, π] ≡ S 1 with a suitable choice of the origin of S 1 . There is a class of operators D λ : Proof. See appendix A. Notice that the requirement J ≡ (−a, a) does not imply any true restriction since all the theory is invariant under rotations of the circle. To go on with the main proof, one sees by direct inspection that, for λ ∈ (0, 1), there is ǫ λ > 0 with λ (J + (−ǫ λ , ǫ λ )) ⊂ J (c).
First we shall be concerned with spatial local definiteness: it simply means that the algebra of observables associated with a single point p ∈ S 1 is the trivial one {c·1} c∈C , 1 being the unit element of R KG . Since {p} ∈ R, the algebra associated with {p} is obtained by taking the intersection of algebras R λ (I) for all I ∈ R with I ∋ p.
Secondly we shall examine the validity of spatial punctured Haag duality, i.e. spatial Haag duality valid on the restricted space S 1 \ {p} for every fixed p ∈ S 1 . In details, fix p ∈ S 1 and, for every I ∈ R with p ∈ cl(I), define I ′ 1 , I ′ 2 ∈ R as the disjoint, not containing p, sets such that With this definition validity of spatial punctured Haag duality means that for every I ∈ R it holds: Finally we shall focus on factoriality. Validity of factoriality for {R λ (I)} I∈R means that each R λ (I) is a factor, for every I ∈ R. The proofs are based on the following important fact.
Proof. First of all we consider the case of λ = ω 0 . We have to demonstrate the validity of (i), (iii) and (iv).
(i) We have to show that J∈R,J∋p R(J) = C · 1. It is clear that C · 1 ⊂ J∈R,J∋p R(J), so we have to prove the converse. Fix S 1 ∋ q = p and consider the disjoint sets I 1 , I 2 ∈ R uniquely determined by assuming that ∂I 1 = ∂I 2 = {p, q}. If A ∈ J∈R,J∋p R(J) it must in particular commute with π 0 (W (Φ, Π)) for every choice of (Φ, Π) whose supports are both contained either in I 1 or I 2 . Therefore A ∈ π 0 (W(I 1 )) ′ = π 0 (W(I 1 )) ′′′ = R(I 1 ) ′ and A ∈ π 0 (W(I 2 )) ′ = π 0 (W(I 2 )) ′′′ = R(I 2 ) ′ . We conclude that , where we employed (2.14) and (2.15). Now, using lemma 4, we may assert that A ∈ R[M I 1 ∩M I 2 ] using the fact that M ′ (iv) We have to establish that every R(I) is a factor, i.e. R(I) ′ ∩R(I) = C·1. The proof of (i) shows, in particular, that each A ∈ R(I 1 ) ′ ∩ R(I 2 ) ′ must be of the form c·1 if I 1 and I 2 are disjoint and fulfill ∂I 1 = ∂I 2 = {p, q} for arbitrarily fixed p, q ∈ S 1 . This applies to the case I 1 := I ∈ R, I 2 = I ′ . Therefore we know that R(I) ′ ∩ R(I ′ ) ′ = C · 1. However, by Haag duality it can be re-written R(I) ′ ∩ R(I) = C · 1, that is R(I) is a factor.
(i) Spatial local definiteness is an immediate consequence of the definition of local unitary equivalence and closedness of von Neumann algebras with respect to strong operatorial topology.
(ii) Let us prove Haag duality for the state λ. Fix I ∈ R and consider J 1 , J 2 ∈ R such that J 1 ∪ J 2 = S 1 and I ⊂ J 1 ∩ J 2 . As I ⊂ J 1 , we have (see the remark after (2.21)) U As a consequence: Taking the double commutant of both members we achieve: Since J 1 ∪ J 2 = S 1 and (I ′ ∩ J 1 ) ∪ (I ′ ∩ J 2 ) = I ′ , additivity implies that the identity written above can be re-written as: Finally R λ (S 1 ) = B(H λ ) because λ is pure and thus its GNS representation is irreducible. We have obtained that R λ (I) ′ = R λ (I ′ ) as wanted.
(iii) The proof of punctured Haag duality is now exactly the same as given for π 0 using spatial additivity, local definiteness, Haag duality and locality. (v) To conclude let us prove the Borchers property. We start from the vacuum case; referring to appendix D, we need to show that, given I ∈ R, there exists R ∋ I 1 I such that every non-zero projection E ∈ R(I 1 ) is equivalent to 1 in R(I). So, choose an interval I ∈ R, take I 1 ∈ R to be an arbitrary subinterval of I, and fix a projection E = 0 in R(I 1 ). If we can prove that the von Neumann algebras inclusion R(I 1 ) ⊂ R(I) satisfies standard split inclusion 15, proposition 15 implies property B so E ∼ 1 in R(I) and we have done. In turn, standard split inclusion holds true if the following two conditions are satisfied: • the inclusion R(I 1 ) ⊂ R(I) is split; • there exists a vector Ω ∈ H 0 cyclic and separating for R(I 1 ), R(I) and R(I 1 ) ′ ∩ R(I).
The former condition holds since the generator of rotations on S 1 satisfies the trace class condition that implies the split property for the observable net (see [20,Th. 3.2]. The latter follows at once from Reeh-Schlieder property for the vacuum (theorem 6); in fact the vacuum vector Ψ 0 is obviously cyclic and separating for R(I 1 ) and R(I); concerning R(I 1 ) ′ ∩ R(I 1 ), we have the following chain of inclusions: for some J ∈ R, taking into account Haag duality. This ends the proof for the vacuum case. The general case λ = ω 0 follows straightforwardly, from local unitary equivalence of local algebras.

DHR sectors
To conclude we prove that, referring to the vacuum representation π 0 , there are no DHR sectors for the massive Klein-Gordon theory on S 1 we are discussing. DHR sectors were introduced [26,27,28,29] to describe localized charges; a representation belongs to a DHR sector if it coincides with the reference (vacuum) representation everywhere but on bounded regions of the spacetime. To be more precise, we shall say that an irreducible representation π of W KG valued on B(H 0 ) and satisfying (spatial) Haag duality fulfills the DHR selection criterion if, for every I ∈ R, there is a unitary operator U I ′ : H 0 → H 0 with U I ′ π(W(I ′ ))U * I ′ = π 0 (W(I ′ )). Notice that in our case, as I ′ ∈ R and the map R ∋ I → I ′ ∈ R is bijective, the mentioned requirement is equivalent to say that π and π 0 are locally unitarily equivalent. A DHR sector is a global unitary equivalence class of irreducible representations π verifying the DHR selection criterion.
Theorem 8. Every representation π of W KG valued on B(H 0 ), locally unitarily equivalent to the vacuum representation π 0 and satisfying spatial Haag duality, is (globally) unitarily equivalent to π 0 itself. As a consequence there is only one DHR sector, i.e. that containing the vacuum representation.
Proof. Define R(I) := π(W(I)) ′′ . We first prove that the unitaries U I which verify U I π(W(I))U * I = π 0 (W(I)) in view of local unitary equivalence, also satisfy U I ′ = e ıγ I U I , for every I ∈ R and some γ I ∈ R. Interchanging the role of I and I ′ one finds in the same way that U I U * I ′ ∈ R(I ′ ) = R(I) ′ , which produces U I ′ U * I ∈ R(I) ′ by hermitean conjugation. Since U I ′ U * I ∈ R(I) ∩ R(I) ′ and factoriality holds, it has to be U I ′ U * I = c· for some c ∈ C. As U I ′ U * I is unitary, it arises |c| = 1 and (2.26) follows immediately. To go on fix I ∈ R: we shall prove that there is a character χ : D C → U (1) with U I π(W (Φ, Π))U * I = χ(Φ, Π)π 0 (W (Φ, Π)) =: π χ (W (Φ, Π)), (2.27) for every (Φ, Π) ∈ D C . Thus π is unitarily equivalent to a character representation π χ . As π is locally unitarily equivalent to π 0 , π χ is such. Since π χ is irreducible it is the GNS representation of a pure state ω χ which can be fixed as any unitary vector of H 0 . Local unitary equivalence, by (d) in theorem 12, entails that π χ is globally unitarily equivalent to π 0 and thus π is such proving the thesis of the theorem. Let us demonstrate (2.27) to conclude. Consider a partition of the unity f I , f I ′ , f F subordinated to the covering I, I ′ , F of S 1 , where F is the disconnected union of two neighborhoods J 1 , J 2 ∈ R of p 1 , p 2 ∈ ∂I respectively. Decomposing (Φ, Π) making use of this partition of the unity, taking advantage of (2.26) and employing Weyl relations one realizes that there is β ∈ C (depending on (Φ, Π) and the chosen partition of the unity) with: Notice that the left-hand side does not depend on F . If (Φ 1 , Π 1 ) has compact support contained either in I or I ′ , taking F sufficiently shrinked about p 1 and p 2 one achieves by direct inspection that the right hand side must commute with π 0 (W (Φ 1 , Π 1 )) (which coincides with U I π (W (Φ 1 , Π 1 )) U * I up to a phase). Therefore U I π (W (−Φ, −Π)) U * I π 0 (W (Φ, Π)) ∈ R(I) ′ ∩R(I ′ ) ′ = C· 1, by Haag duality and factoriality. Since U I π (W (−Φ, −Π)) U * I π 0 (W (Φ, Π)) is unitary we have found that there is a map χ : D C → U (1) with π 0 (W (Φ, Π)) = χ(Φ, Π) U I π (W (Φ, Π)) U * I .
The fact that χ is additive, i.e. a character, follows immediately from Weyl relations.

Spacetime formulation
Until now, we have worked with the "spatial" poset R made of open intervals of S 1 whose set-complement has non-empty interior, partially ordered under set inclusion; we have also endowed this poset with a causal disjointness relation ⊥.
What we would like to do at this point is to reformulate the previous results, as far as possible, in terms of a "spacetime" poset K , i.e. a poset being a topological basis for the entire spacetime; obviously we require K to be compatible with R, in a certain sense to be specified.
First of all we recall a notion belonging to Lorentzian geometry: given a subset S of a time-oriented Lorentzian manifold M, its Cauchy development D(S) is the set of points of M through which every inextendible causal curve in M meets S. For further details, see e.g [4] or [52].
Our choice of the spacetime poset is quite natural: we take the set of diamonds based on elements of the spatial poset. Specializing definition 5, given an element I ∈ R the diamond 3(I) based on I is the set Int(D(I).
Then we need to "translate" these diamonds, based on a specific Cauchy surface, to cover the entire spacetime M. In our case, the cylindric flat universe, the spacetime admits a 1-parameter group of isometries acting as time traslations, sending the point x ≡ (θ, s) to the point x+(0, t) ≡ (θ, t+s); so we can translate the poset R lying on the chosen Cauchy surface C on another Cauchy surface belonging to the same foliation. If we denote with C t the Cauchy surface corresponding to the Cauchy time t (so that by definition C ≡ C t=0 ) we define the translated poset R t on C t as the family of elements of the form I +(0, t), I ∈ R. So we can construct the diamonds based on R t , and it's easy to see that the set K = {3(I) | I ∈ R t , t ∈ R} of the diamonds based on a generic Cauchy surface in the foliation is really a poset, endowed with an order relation (set inclusion) and a causal disjointness relation (that induced by the causal structure of M).
Before we proceed some remarks are in order about the explicit form of geometrical objects (Cauchy surfaces, diamonds and Cauchy developments) on the cylindric flat universe; this kind of information will be of use in what follows. Obviously, what we are going to say is a trivial consequence of Minkowskian geometry, taking into account that in our universe "spatial endpoints" are identified. First of all, the spacetime admits a "natural" smooth foliation in Cauchy surfaces: with respect to canonical coordinates (θ, t) these are the hypersurfaces of equation t = cost, and are of course isometric to S 1 . Diamonds, i.e. members of the spacetime poset K , are very simple: they are squares, with a diagonal coinciding with an element of a spatial poset R t . We note that in this case D(I) is open, so 3(I) ≡ D(I).
Also causal complements of diamonds are simple: they are diamonds, too, based on the (spatial) causal complement of their basis. In symbols: Finally, regarding Cauchy developments it's enough to observe that, by definition: 3(I ′ ) = M \ J(I).
At this point we need to relate the net of local algebras defined on the Cauchy surface(s) with that defined on the entire spacetime. In our case the symplectic space associated with the diamond O ≡ 3(I), I ∈ R t , is given by the (equivalence classes of) test functions supported in O: We previously saw that the casual propagator establishes an isomorphism between the global symplectic space of Cauchy data (on a given Cauchy surface C t ) and (equivalence classes of) test functions on the entire spacetime; consequently, if we restrict ourserlves to the symplectic subspace of Cauchy data supported in a open interval I ∈ R t we obtain a symplectic subspace of test functions on the entire spacetime. Then the question arises of how we can describe this subspace; the following proposition gives an answer, at least in the present case. We start with a technical lemma [67].
Lemma 6. Let M be a globally hyperbolic spacetime, ψ ∈ S a smooth solution of the Klein-Gordon equation with compactly supported Cauchy data on every Cauchy surface, and C 1 , C 2 two Cauchy surfaces belonging to the same foliation; suppose that t is a Cauchy time for the foliation M = C × R and that and take f := −K(χψ), where K is the Klein-Gordon operator; then it's easy to see that supp(f ) ⊆ I + (C 1 ) ∩ I − (C 2 ) ∩ supp(ψ) and is compact. It remains to show that ψ = Ef . We recall that Ef is the difference between the advanced and retarded fundamental solutions with source term f . But is straigthforward to verify that: On the other hand take (Φ, Π) ∈ D I and let ψ be the induced solution of the KG equation ; we want to show that exists [f ] ∈ K(3(I)) such that Ef = ψ; we already know that supp(ψ) ⊆ J(supp(Φ) ∪ supp(Ψ)).
Given two Cauchy surfaces in the standard foliation for M, say C t 1 and C t 2 with t 1 < t 2 , we denote by N the open set delimited by them, i.e. N := J + (C t 1 ) ∩ J − (C t 2 ); then, by lemma 6 it follows that we can find O ⊂ N ∩ supp(ψ) and f ∈ K(O) such that Ef = ψ so the result is proven if we can chose t 1 < t 2 small enough so N ∩J(supp(Φ) ∪ supp(Ψ)) ⊂ 3(I). But in the present case we can see (taking into account the actual geometry of Cauchy surfaces, diamonds and Cauchy developments), that it's true indeed. In conclusion we have shown that K(O) ⊇ D I , and the thesis follows. Proposition 10 (Haag Duality). In the spacetime M, endowed with the net of local (concrete) algebras R K , Haag duality holds, i.e. given O ∈ K the following identity is satisfied: Proof. We already know that spatial Haag duality holds in the cylindric flat spacetime; we'll show that spacetime Haag duality follows from this. Let C be the Cauchy surface where O is based; then In fact we know that ∀ I 1 ∈ I ⊥ , O 1 := 3(I 1 ) ∈ O ⊥ and R[3(I 1 )] = R(I 1 ), so and by isotony we have Applying the algebra commutant to both sides of the inclusion we have Doing some algebra we obtain: given that I 1 ∈ I ⊥ . Repeating this argument for all O 1 ∈ O ⊥ we have, as claimed: Finally we conclude observing that Now we examine punctured Haag duality. We recall [59] that given a point x in M, the causal puncture of the spacetime poset K is the poset K x defined as the collection We also recall that its "topological realization" where C is a (spacelike) Cauchy surface containing x. Considered as a spacetime on its own, M x is globally hyperbolic and K x is a topological basis of M x . Now, for any O ∈ K x we define the causal complement of O in K x as the family of sets: We remark that in general an element of the poset K x it's not a diamond of the spacetime M x , although it's a diamond of the spacetime M by definition.
We are now ready to formulate punctured Haag duality in a spacetime context.
Proposition 11 (Punctured Haag duality). Let x ∈ M, K x the causal puncture of K induced by x, O ∈ K x ; then punctured Haag duality holds. In other words the following identity is true: Then we see it is enough to prove that ]. It's easy to see that each of the subsets V i , i = 1, . . . , 4 making up O ⊥ is a globally hyperbolic spacetime (see lemma 7 below). Take a piece of a "straight" Cauchy surface (namely belonging to the natural foliation of M) around x, say ∆ n , and extend it 4 to a Cauchy . Now transport the local algebra R[O ⊥ ] on C (passing from test functions to Cauchy data, as we have seen in 2.2.1); using a partition of unity, split Cauchy data in three parts, supported respectively in C 1 , C 2 and ∆ n . Passing to local algebras we obtain If we choose ∆ n+1 such that ∆ n+1 ⊂ ∆ n we get R(∆ n+1 ) ⊆ R(∆ n ); because (2.32) is valid for all n ∈ N we finally obtain We conclude this section with some material about Lorentzian geometry used in the proof above; to start with we recall the causality condition for spacetimes.

Topological cocyles
We finally come to the core of this work; in this chapter we exhibit a class of topologically non-trivial cocycles living in the cylindric universe. What about the significance of such a result ? As we'll see presently, the cocycle machinery has a twofold purpose: from the one hand, it allows us to build unitary representations of the first homotopy group of the spacetime (via theorem 2), from the other hand it can be used to construct netrepresentations. So existence of non-trivial 1-cocycles implies both the existence of non-trivial representations of the fundamental group and non-trivial net-representations of the observables net. In this respect, we can say that spacetime topology interacts with algebraic QFT.

Introduction
In this section we show how the non-trivial topology of S 1 allows the existence non-trivial unitary representations of the first homotopy group of the spacetime acting in the Hilbert space of the vacuum. As before, we refer to the (spatial) poset R and the vacuum representation induced by the vacuum state ω 0 , with GNS Hilbert space H 0 := F + (H 0 ) and one-particle space H 0 := L 2 (S 1 , dθ). Hence the reference net of local algebras indexed by R Moreover we shall establish that non-trivial cocycles individuate generalized net representations of the algebra of observables satisfying the (topological) selection criterion defined in [13]. The found situation is however puzzling here, because some of the most important results presented in the mentioned paper (which are valid for dimension of the spacetime ≥ 3) cannot be used here and, as matter of fact, they do not hold true. In particular the theorem of localization of fundamental group, theorem 4.1 in [13], fails to be valid specializing to our definition of cocycles as it can be verified by direct inspection. Therefore, some relevant consequences of that theorem may be false in our low-dimension context, and the actual situation has to be analyzed carefully. In particular, the absence of irreducible cocycles different from characters when the fundamental group of the manifold is Abelian, as established in Corollary 6.8 in [13], no longer holds. Indeed, the irreducible cocycles we are going to present below are a clear counterexample. Furthermore the proof of Theorem 4.3 in [13], which demonstrates equivalence of the category of 1-cocycles and the category of representations of the algebra of observables satisfying the (topological) selection criterion, does not hold since it relies upon theorem 4.1 in [13]. However, it turns out that various results carry over to our model, although their proofs often require substantial modifications.

General theory
First of all we need to set the stage for subsequent constructions. As we said, we choose the poset R of proper, open intervals on S 1 , equipped with the partial order relation ⊆ and the causal disjointness relation ⊥ (see section 2.2.2). Our reference net of observables will be the net of unital C * -algebras (Weyl algebras) The vacuum representation 1 ι on the vacuum GNS Hilbert-Fock space H 0 discussed in section 2.3, induces a unitary net representation {ι, I} of the net of observables W R , namely a family of representations ι I : W(I) → ι I (W(I)) ⊆ B(H 0 ); in this case the "transition functions" I e II are the identity operators for every I, I ∈ R. The unitary net representation {ι, I} gives also rise to a von Neumann algebras net R ∋ I → ι(W(I)) ′′ =: R(I) satisfying irreducibility, Haag duality, punctured Haag duality and Borchers property. Finally (ι, I) is topologically trivial since the cocycle ζ ι associated with it is the simplest co-boundary (hence the associated unitary representation of π 1 S 1 is trivial, too). We refer to notation, definitions and results discussed in section 2.3, noting en passant that corollary 1 entails that (R, ⊆) cannot be directed (as we know independently).
Finally it is worth remarking that every irreducible unitary representation of Z, {λ x (n)} n∈Z , is one-dimensional, λ x (n) : C → C, as the group is Abelian. All those representations are labeled by x ∈ R and have the form: Some remarks about the poset R are in order too. In view of section 1.2 and since S 1 is Hausdorff, arcwise connected and R is a topological basis of S 1 , it turns out that π 1 (R) doesn't depend on the basepoint I 0 and coincides with the fundamental group of S 1 , i.e. π 1 (R) = Z in our case. For the sake of clarity, it's worth restating theorem 2 specialized to our case: Theorem 9. Fix a base 0-simplex I 0 ; given z ∈ Z 1 (R R ), define the following map sending an equivalence class of (poset) paths to an operator in B(H 0 ): The correspondence Z 1 (R R ) ∋ z → π z maps 1-cocycles z, equivalent in B(H 0 ), into equivalent unitary representations π z of π 1 (R) in H 0 ; up to equivalence this map is injective.
Notice that, as a consequence, z ∈ Z 1 (R R ) is trivial if and only if the associated representation of π z is the trivial one made of the unit operator only.

Topological superselection sectors
Now we introduce a selection criterion which generalizes DHR selection criterion; we refer to [13] for motivations and further details. Consider as before the unitary net representation (ι, I) of W R over H 0 ; following [13] we These three requirements represent the selection criterion. It turns out [13, p. 14] that W N J is independent from the region N . The unitary equivalence classes of irreducible unitary net representations satisfying the selection criterion are the superselection sectors and the analysis of their charge structure and topological content, in the case of a generic globally hyperbolic spacetime with dimension ≥ 3 was the scope of the work [13].
We are dealing with a (particular) 2-dimensional spacetime, so we expect that some of the results found there cannot apply.

Localized cocycles
One of the most important results, established in [13,Th. 4.3], is that, for globally hyperbolic spacetimes with dimension ≥ 3, the C * -category whose objects are sharp excitations of {ι, I} with arrows given by intertwiners, is equivalent to the subcategory of localized 1-cocycles Z 1 (R R ) (with respect to ι and the associated net of von Neumann algebras R R : R ∋ I → ι(I) ′′ =: R(I)).
Although this C * -category can be defined without any modifications in our 2-dimensional model, there is no guarantee for the validity of the equivalence theorem. In any cases, the proof fails to go through in the same form as for larger dimension, since an important lemma exploited in the proof of Theorem 4.3, Theorem 4.1 of [13] is not valid in our spacetime.
Remark. The absence of irreducible cocycles different from characters when the fundamental group of the manifold is Abelian, as established in Corollary 6.8 in [13], no longer holds in low dimension. Indeed, the irreducible cocycles we are going to present are a counterexample.

Topological cocycles
In this section we show explicitly a topological 1-cocycle; actually, we'll construct an entire family of (localized) 1-cocycles with respect to the reference vacuum representation {ι, I}. To achieve this result we need a preliminary construction. First of all, fix an orientation on S 1 and, in the following, refer to that orientation the former and the latter endpoints of 0-simplices. Afterwards, define an assignment of smooth functions to 0-simplices such that: (ii) χ (a) (θ) = 0 in a neighborhood of the former endpoint of a, (iii) χ (a) (θ) = 1 in a neighborhood of the latter endpoint of a.
Now we pass to 1-simplices. Consider a 1-simplex b; extend χ (∂ 1 b) and χ (∂ 0 b) smoothly and uniquely as constant functions over |b| \ ∂ 1 b and |b| \ ∂ 0 b; the functions so extended over the whole |b| will be denoted by χ (∂ 1 b) and χ (∂ 0 b) again; finally, for every b ∈ Σ 1 (R) we define the function χ (b) ∈ C ∞ (|b|, R) Notice that this function vanishes in a neighborhood of each endpoint of |b|, because it must hold can be extended uniquely to a smooth function defined on the whole circle S 1 and supported in |b|. We shall denote by χ (b) again this unique extension. Let us come to 1-cocycles. Adopting the notation (2.13), we define (where as before ι is the vacuum representation of W R ) for (f, g) ∈ D C . In the following, to define a 1-cocycle localized at b ∈ Σ 1 (R), we shall replace the arguments f and g with χ (b) -smeared restrictions of those functions to 0-simplices |b| for any 1-simplex b. The restriction is necessary in order to fulfill the localization requirement of 1-cocycles. The smearing procedure is necessary too, at least for the entry of A 1/4 , whose domain generally does not include elements χ |b| g, χ |b| being the characteristic function of the set |b|. It, however, includes every smoothed function We are now in place to state our result defining, in fact, a family of 1-cocycles labeled by χ, f , g. The following theorem also establishes independence from χ, up to equivalence of the defined 1-cocycles, irreducibility of cocycles and the fact that they are inequivalent if f = f ′ or g = g ′ .
Theorem 10. Fix an orientation of S 1 , an assignment χ : Σ 0 (R) ∋ a → χ (a) as in (3.4) and define χ (b) as in (3.5). For every choice of (f, g) ∈ D C the map z (χ) is a 1-cocycle of Z 1 (R R ). The following further facts hold.
For fixed (f, g) ∈ D C , but different assignments χ 1 , χ 2 , the cocycles z (d) If the assignment χ : Σ 0 (R) ∋ a → χ (a) is covariant 2 with respect to the isometry group of θ-displacement on S 1 : then, for every (f, g) ∈ D C and for every r ∈ R and b ∈ Σ 1 (R), 0) is the one-parameter unitary group implementing θ-displacements β r and leaving the vacuum invariant, introduced in theorem 3, and β * r is the pull-back action of θ-displacements on functions defined on S 1 .
Proof. Let us prove that (3.7) defines a localized 1-cocycle. First we notice that the unitary operator is an element of R(|b|) since supp(χ (b) ) ⊂ |b| as previously noticed. So, only the identity (1.6) remains to be proved. It can be established as follows. Consider a 2-cocycle c. To simplify the notation we define b k := ∂ k c for k = 0, 1, 2. Since |c| cannot coincide with the whole circle (and this is the crucial point), all functions χ (∂ 1 b j ) can uniquely and smoothly be extended to functions defined on |c| by defining them as constant functions outside their original domain. The extension procedure does not affect the definition of the functions χ (b i ) . We shall exploit this extension from now on. We have to show that z Now notice that, in view of the very definition of a 2-cocycle, so that the left-hand side of (3.9) can be recast as where all functions χ (∂ i b j ) are now defined on the whole |c| and the differences are defined everywhere on S 1 and compactly supported in |c|. Finally, making use of Weyl relations, taking the definition (3.6) of Z(f, g) into account, we find that the terms ±χ (∂ 0 b 2 ) cancel each other in the final exponent, and the left-hand side of (3.9) is: Since the phase vanishes, we have found the very right-hand side of (3.9) as wanted.
(a) Let us pass to the irreducibility property of the defined cocycles. Let V : Σ 0 (R) ∋ I → V I ∈ R(I) be a field of unitary operators such that where r k , s k are smooth real functions supported in ∂ 1 b, c k ∈ C, and the series converges in the strong operatorial topology. Therefore, using Weyl relations, for some reals ϕ k . The final series converges in the strong operatorial topology, too. Since ι(W (r k , s k )) ∈ R (∂ 1 b) for hypotheses, c k ι(W (r k , s k )) exp{ıϕ k } ∈ R(∂ 1 b) for every k, and thus we also have z ) is closed with respect to the strong operatorial topology. As a consequence, it must hold Since ∂ 0 b, ∂ 1 b ∈ Σ 0 (R) and V ∂ 0 b are generic, we have found that V a ∈ I∈R R(I) = C · 1 in view of the irreducibility property of the net. in the open set a, so that it can uniquely and smoothly be extended to a smooth function over S 1 supported in a. As usual, we indicate by ∆χ (a) this unique extension. Now, define the field of unitaries V : Σ 0 (R) ∋ a → V a := Z ∆χ (a) f, ∆χ (a) g ∈ R(a). For every b ∈ Σ 1 (R), we get where, passing from the first to the second line, we have omitted a phase arising from Weyl relations, since it vanishes as before. With an analogous computation we similarly find: following an argument as that in the proof of (a), we achieve V ∂ 1 b ∈ R(∂ 1 b) and so can be computed and, in view of Weyl relations, it finally arises where ϕ ∈ R depends on f, f ′ , g, g ′ and χ. Now consider two real smooth functions r, s supported in I b . The Weyl generator Z(r, s) belongs to R(I b ) and thus it commutes with both V ∂ 1 b * and V ∂ 0 b so that (3.15) produces Comparing with (3.15) we conclude that it must be exp ı Since r, s are arbitrary we have that f ′ − f = 0, g ′ − g = 0 on I b . Since the procedure can be implemented fixing I b as a sufficiently small neighborhood of every point on S 1 , we conclude that f = f ′ and g = g ′ everywhere on S 1 .
(d) Referring to theorem 5 one finds where we used the fact that e −ırP and the spectral measure of A commute as one can easily prove exploiting the definition of A and the fact that P is the unique self-adjoint extension of the essentially self-adjoint operator −ı d dθ : C). On the other hand, by direct inspection and working in Fourier representation, one finds that (e −ırP h)(θ) = h(θ − r) =: (β * r (h))(θ) for every h ∈ L 2 (S 1 , dθ). This, joined to (3.8), implies immediately the validity of the thesis.
3.3.1 Topological representations of π 1 (S 1 ) Int this section we state and prove some properties of the unitary representations of π 1 S 1 associated to the found cocycles.
Theorem 11. Consider the 1-cocycle z (χ) (f,g) defined in theorem 10 and the associated representation (3.2) of π 1 S 1 ≡ Z. In this case the representation reads π (f,g) : Z ∋ n → Z(nf, ng), (3.17) where it is manifest that the representation does not depend on the choice of χ; moreover, it enjoys the following properties: the unitary representations π (f,g) and π (f ′ ,g ′ ) are unitarily equivalent.
(c) For every (f, g) ∈ D C , every r ∈ R and every n ∈ Z, it holds where U (r,0) is the one-parameter unitary group implementing θ-displacements β r and leaving the vacuum invariant, and β * r is the pull-back action of θdisplacements on functions defined on S 1 .
with H x := C and dx being the Lebesgue measure on R.
Proof. Let us first prove (3.17) for n = 1. Since we know that (3.2) gives rise to a group representation of π 1 (S 1 ) when z (χ) (f,g) is a cocycle, to prove (3.17) for n = 1 ∈ Z = π 1 S 1 i.e., it is enough to prove it for a fixed path p ∈ 1, because the result depends only on the equivalence class. To this end, if S 1 = [−π, π] where −π ≡ π, consider the path p ∈ 1 made of the simplices b, with |b| := (− π 2 − ǫ, π 2 + ǫ), We have established (3.19), i.e. (3.2) for n = 1. Let us generalize the result for n ∈ Z. By the definition of Z and making use of Weyl commutation relations one gets Z(nf, ng)Z(mf, mg) = Z((n + m)f, (n + m)g), ∀ n, m ∈ Z. (3.20) Using the fact that π z (χ) as defined in (3.2) is a group representation of π 1 S 1 = Z, which is Abelian and generated by 1, one has that (3.20) and (3.19) together yield eq. (3.2) in the general case. Let us pass to prove (a). As a consequence of (3.17), it is clear that this representation is trivial, that is z (χ) (f,g) is such due to theorem 3.2, if and only if Z(f, g) = 1. It is equivalent to say W 2 −1/2 (A 1/4 f + ıA −1/4 g) = 1. By theorem 3 we know that for all f, g ∈ C ∞ 0 (S 1 , R). Since Ψ 0 = 1 we have finally that Z(f, g) = 1 entails f, A 1/2 f + g, A −1/2 g = 0 and so f, g = 0 because A −1/4 and A 1/4 are strictly positive. We have found that triviality of π (f,g) implies f, g = 0. The converse is obvious and so the proof of (a) is concluded. Let us demonstrate (b). Assume that (the case "= 0" being obvious). Defining ψ 1 := 2 −1/2 (A 1/4 f + ıA −1/4 g) we can complete this vector to a maximal orthogonal system {ψ n } n∈N of L 2 (S 1 , dθ) where ψ n = a for every n ∈ N. Similarly, defining φ 1 := 2 −1/2 (A 1/4 f ′ + ıA −1/4 g ′ ) we can complete this vector to a maximal orthogonal system {φ n } n∈N of L 2 (S 1 , dθ), where ψ n = a for every n ∈ N. There is a unique unitary operator U : L 2 (S 1 , dθ) → L 2 (S 1 , dθ) completely individuated by imposing the conditions U ψ n = φ n for every n ∈ N. It is a known property of Weyl generators W [ψ] = e a(ψ)−a * (ψ) that where the unitary operator V ⊗ in the Fock space is defined by tensorialization of the unitary operator V in the one-particle space, with the requirement that V ⊗ reduces to the identity acting on the vacuum vector. As a conse- ] or, equivalently, U ⊗ Z(f, g)U * ⊗ = Z(f ′ , g ′ ) and thus U ⊗ Z(nf, ng)U * ⊗ = Z(nf ′ , ng ′ ), making use of (3.20); we have found that π (f,g) and π (f ′ ,g ′ ) are unitarily equivalent. Let us pass to the the case 0 = 2 −1/2 (A 1/4 f + ıA −1/4 g) = 2 −1/2 (A 1/4 f ′ + ıA −1/4 g ′ ) = 0 and define the real number With the procedure used in the former case one achieves the existence of a unitary operator V on the Fock space such that To conclude it is sufficient to establish the existence of a second unitary operator E (depending on f , g and r) with This is an immediate consequence of the following result proved in appendix A. (3.21) The proof of (c) follows immediately from (d) in theorem 10 taking the independence from χ into account. Finally we prove (d). Fix (f, g) ∈ D C . In view of Weyl commutation relations, the unitary operators fulfill the one-dimensional Weyl relations From the well-known uniqueness result, due to Stone, von Neumann and Mackey [61], the space H 0 decomposes into a direct sum of pairwise orthogonal closed subspaces H k where each H k is unitarily equivalent to L 2 (R, dx) and the relevant unitary map satisfies X, P being the standard position and impulse operators on the real line (aX + bP is defined on the core given by the Schwartz space). As a consequence with c = 2 −1/2 (A 1/4 f + ıA −1/4 g) > 0. Then the spectral decomposition of cX gives rise to (3.18) immediately. To end the proof the only thing remained to show is that the number of spaces H k is infinite. Since H 0 is separable that infinity must be countable. It is known by the general theory of Weyl algebras on finite-dimensional symplectic spaces that the H k 's can be obtained as follows. Using weak operator topology, define the operator which turns out to be a nonvanishing orthogonal projector. If {φ k } k∈G is a Hilbert basis for the subspace P (H 0 ), for any fixed k ∈ G, H k is the closed space generated by all of U (a, b)φ k as a, b ∈ R. To conclude it is sufficient to prove that G must be infinite. To this end consider a Hilbert basis in H 0 , ψ 1 = (A 1/4 f + ıA −1/4 g)/ A 1/4 f + ıA −1/4 g , ψ 2 , ψ 3 , . . . and an associated orthonormal (not necessarily complete) system in H 0 : Ψ 1 := Ψ 0 (the vacuum), Ψ 2 := a * (ψ 2 )Ψ 0 , Ψ 3 := a * (ψ 3 )Ψ 0 , . . .. By construction, one can verify that Therefore, up to normalization, P Ψ 1 , P Ψ 2 , . . . ∈ P (H 0 ) is an infinite orthonormal system in P (H 0 ). This means that P (H 0 ) admits an infinite Hilbert basis.

Topological superselection sectors
In this section we show how it is possible to associate each localized cocycle z (χ) (f,g) with a sharp excitation of the reference vacuum representation {ι, I} in such a way that unitarily inequivalent pair of cocycles individuates unitarily inequivalent pair of unitary net representations fulfilling the selection criterion, and thus two different superselection sectors. The idea is similar to that exploited to define a relevant functor in the proof of theorem 4.3 in [13], although there are two important differences. First of all, here we are dealing with a proper subset of cocycles and not with the whole category Z 1 (R R ). Secondly, as we shall see in details shortly, the map which associates cocycles to net representations in the proof of Theorem 4.3 in [13] either N ∈ R or N = S 1 \ {p} for some p ∈ S 1 ) with O ⊂ N . Fix I ∈ R We can extend t in the right-hand side of (A.2) to complex values: t → z ∈ C 2 provided (ℑm z|q) ≤ 0 for all q ∈ Γ m . Therefore U t is extensible for complex values of z such that ℑm z belongs to the closure of future light-cone Employing standard tools of spectral theory in Hilbert spaces, one obtains that the map z → U z is continuous for ℑm z ∈ V + in the strong-operatorial topology and is holomorphic over V + when viewed as a function valued in the Banach space of bounded operators on H 0 . Finally U z ≤ 1 if ℑm z ∈ V + . Remembering that analyticity implies weak analyticity, one can prove straightforwardly that, for (z 1 , . . . , z n ) ∈ C n the function (notice that is well-defined if (ℑmz 1 , . . . , ℑmz n ) ∈ V + ×· · ·×V + , it is jointly continuous in this domain, analytic in each variable separately for (ℑmz 1 , . . . , ℑmz n ) ∈ V + × · · · × V + (and thus analytic as a function of several variables) and it reduces to the left-hand side of (A.1) when z 1 = t 1 , z 2 = t 2 − t 1 , ... z n = t n − t n−1 for t k ∈ R 2 , k = 1, . . . , n. In this case it must vanish when t k ∈ O, k = 1, . . . , n. Using the "edge of the wedge theorem" we are committed to conclude that the function in (A.3) vanishes everywhere on However since the domain of analyticity of the considered function is connected, it must vanish everywhere therein. Finally, by continuity it must vanish on the boundary of the domain of analyticity too, in particular: This implies that Υ is orthogonal to r∈R π 0 (W(I 1 + (r, 0))) Ψ 0 . By spatial weak additivity that space is dense in H 0 and thus Υ = 0. We conclude the proof by showing that Ψ 0 is separating for every R(I). We have to prove that, for any fixed I ∈ R, if A, B ∈ R(I) and AΨ 0 = BΨ 0 then A = B. Take I 1 ∈ R with I 1 ∩ I = ∅ and thus R(I 1 ) commute with R(I) by locality. If C ∈ R(I 1 ) we have CAΨ 0 = CBΨ 0 and so ACΨ 0 = BCΨ 0 . R(I 1 )Ψ 0 is dense in H 0 due to (a); since C ∈ R(I 1 ) is arbitrary, ACΨ 0 = BCΨ 0 entails A = B on a dense domain and thus everywhere in H 0 by continuity. 2 Proof of lemma 8. In the following λ ∈ R. We complete the unit-norm vector ψ ∈ H to a Hilbert basis of H, pass to the associated Hilbert basis in F + (H) and denote by F the dense subspace of F + (H) containing all the finite linear combinations of the vectors of that basis. Assuming E (ψ) λ = e ıλA , taking the derivative at λ = 0 of the identity (without paying much attention to domain issues) and, finally, making use of (2.12), one gets that if Φ belongs to some suitable domain we shall determine presently. Taking the commutation relation [a(ψ), a * (ψ)] = 1 into account (recall that ψ = 1), we see that a candidate for A is some self-adjoint extension of A := (ı/2)(a(ψ)a(ψ) − a * (ψ)a * (ψ)). A turns out to be symmetric if defined on F . If Φ ∈ F contains exactly k particles in the state ψ one finds A n Φ ≤ (2n + k)!. From that it arises +∞ n=0 λ n A n Φ /n! < +∞ if |λ| < 1/2. Therefore the vectors in F are analytic for A and thus A is essentially selfadjoint on F , A being its unique self-adjoint extension. In particular the commutation relation (A.5) are, in fact, valid for Φ ∈ F and lead to the further commutation relations for all Φ ∈ F . Using (A.6) one easily proves the validity of this identity for |λ| < 1/4 and Φ ∈ F : The series +∞ n=0 (ıλA+λ1) n n! Φ converges for every Φ ∈ F and |λ| < 1/4 as one can establish making use of the bounds (A + ı1) n Φ ≤ 2 n (2n + k)! when Φ ∈ F contains exactly k particles in the state ψ. Therefore closedness of a(ψ) − a * (ψ) imply, via (A.7), that the following two facts hold: can be re-written as where we have also used the fact that (a(ψ) − a * (ψ))Φ ∈ F when Φ ∈ F and thus the exponential e ıλA (a(ψ) − a * (ψ))Φ can be expanded in series.
(A.9) Since e ıλA is unitary, (A.9) entails that, for Φ ∈ F , |λ| < 1/4 and every u ∈ C: where we have used the fact that every Φ ∈ F is analytic (for every value of the parameter u) for ıa(ψ) − ıa * (ψ) as is well known (see [8]). We have found that e ıλA Φ is analytic for e λ (a(ψ) − a * (ψ)). In this context, the identity arising from (A.9) for Φ ∈ F and |λ| < 1/4, can be re-written as e ıλA e a(ψ)−a * (ψ) Φ = e e λ (a(ψ)−a * (ψ)) e ıλA Φ. That is, taking advantage from the fact that F is dense, = e e λ (a(ψ)−a * (ψ)) , where we have defined E (ψ) λ := e ıλA . Finally, employing R-linearity of ψ → a(ψ), a * (ψ), the achieved formula can be re-stated as The restriction |λ| < 1/4 can be dropped by employing iteratively the identity above and noticing that E (ψ) λ is additive in λ ∈ R. Hence the obtained identity holds true for every λ ∈ R. 2 Proof of proposition 7. Choose two intervals J 1 , J 2 ∈ R with (−π, π) ⊃ J 2 , J 2 ⊃ J 1 and J 1 ⊃ J. As a further ingredient we fix an open neighborhood of 1, O = (e −ω , e ω ) with ω > 0 so small that (1) λJ ⊂ J 1 , (2) λJ 2 ⊂ (−π, π) for all λ ∈ O. Notice that λ ∈ O iff λ −1 ∈ O. With these definitions, let χ ∈ C ∞ (S 1 , R) be such that 0 ≤ χ(θ) ≤ 1 for θ ∈ S 1 and χ(θ) = 1 for θ ∈ J 1 but χ(θ) = 0 in S 1 \ J 2 . Now consider the class of operators U λ : L 2 (S 1 , dθ) → L 2 (S 1 , dθ), λ ∈ O, defined by: Exploiting the presence of the smoothing function χ and making a trivial change of variables where appropriate, one proves the following features of U λ : By direct inspection one also finds that: for all θ ∈ (−π, π] and λ ∈ O. From (A.14) properties analogous to those found for U λ can be straightforwardly established: Remark. In view of the definition of U λ and (A. 18) Taking the definition of the quantization map (2.6) into account, one realizes that a candidate for D λ is the following operator, initially defined on C ∞ (S 1 , C): for all λ ∈ O and ψ ∈ M J 0 . The right hand side is in fact well-defined if ψ ∈ K(D L ) with R ∋ L J; A −1/4 ℜe(ψ) and A 1/4 ℑm(ψ) belong to C ∞ 0 (J 0 , R) so both act on vectors belonging to their domains due to (A.10) and (A.15). Moreover it fulfills (a) in the thesis since D (0) λ ψ ∈ K(D λL ) in view of the previous remark. However both operators A 1/4 U λ A −1/4 and A −1/4 U * 1/λ A 1/4 are well defined on C ∞ (S 1 , C). To extend the validity of (a) to every space M L := K(D L ) with L J as requested in the thesis, it is sufficient to prove that the operators A 1/4 U λ A −1/4 and A −1/4 U * 1/λ A 1/4 are bounded on C ∞ (S 1 , C) and to extend them and D (0) λ by continuity on the whole space L 2 (S 1 , dθ). The restriction D λ to M L of this extension will satisfy (a) by construction.
We'll use an argument based on an interpolation theorem. Consider f ∈ C ∞ (S 1 , C) and define χ λ (θ) := χ(λθ). By direct inspection one finds where A λm is A with mass m replaced by λm. One also finds that for λ < 1 A λm g 2 L 2 is bounded by Ag 2 , otherwise by λ 4 Ag 2 . Summarizing We can improve this upper bound expanding A(χ λ f ) as follows: Now, integrating per parts we get: Now notice that A ≥ λ 0 1 where λ 0 > 0 is the least eigenvalue of A and thus Af L 2 ≥ λ 0 f L 2 . Similarly A ≥ − d 2 dθ 2 and thus Af L 2 ≥ d 2 f /dθ 2 L 2 , therefore: Using these estimates in (A.21) we finally obtains: C is finite: it can be proved by shrinking O and noticing that the function (λ, θ) → χ λ (θ) and its derivatives are bounded in the compact O × S 1 since they are continuous. Since C ∞ 0 (S 1 ) is a core for the self-adjoint (and thus closed) operator A, as a byproduct (A.22) implies: The proof is immediate noticing that if f ∈ Dom(A) there is a sequence C ∞ 0 (S 1 ) ∋ f n → f with Af n → Af and , in view of continuity of U λ , {U λ f n } n∈N is Cauchy and, in view of (A. 22), {AU λ f n } n∈N is Cauchy too.
Closedness of A implies that U λ f n → U λ f ∈ Dom(A) and A(U λ f n ) → A(U λ f ). This also proves that (A.22) is still valid in Dom(A) by continuity.
As A ≥ 0 and (A.11) holds, [56, prop. 9, chap. IX.5] used twice implies that so that, since Ran(A −1/4 ) = Dom(A 1/4 ) and Dom(A −1/4 ) is the whole Hilbert space, This concludes the proof of the continuity of the former operator in the right-hand side of (A.20).
Notice that A −1/4 is continuous and so, when ψ ∈ K(D L ), one has where we have used (A. 19) and (A.17) noticing that A 1/4 (ℑm(ψ)) ∈ C ∞ (L) ⊂ L 2 (J, dθ) when ψ ∈ K(D L ). Due to the uniform bound (A.23) the result can be extended to M L := K(D L ). If ψ ∈ M L , let K(D L ) ∋ ψ n → ψ and denote ℑm(ψ n ) and ℑm(ψ) respectively by f n and f ; obviously f n → f . One has (for λ ∈ O so that (A.23) holds) For any fixed ǫ > 0, taking n = n ǫ such that ( To conclude we prove that lim λ→1 B λ ℜe(ψ) = ℜe(ψ) for ψ ∈ M L and L J. Let us indicate ℜe(ψ) by f . As before, first consider the case ψ ∈ K(D L ). This means in particular that f = A 1/4 h for some h ∈ C ∞ 0 (J, R). Now notice that: In our case, as λ → 1 − , due to (A.12) and (A.13): Similarly B λ f 2 → f, f as λ → 1 − because: and from the definition of U λ and U * λ one finds that U * λ U λ h = h, for each h ∈ C ∞ (J, R). Putting all together one concludes that lim λ→1 B λ ℜe(ψ) = ℜe(ψ) when ψ ∈ K(D L ). Extension to the case ψ ∈ M L := K(D L ) is the same as before. 2 Proof of lemma 5. As a general fact it holds D I ⊂ (D I ′ ) ′ and D J ⊂ (D J ′ ) ′ , thus applying the quantization map, taking closures and intersections, we get In other terms, if ψ ∈ M I ∩M J then ℑm ψ, φ = 0 when either φ ∈ M I ′ or φ ∈ M J ′ . In particular, ℑm ψ, K(Φ, Π) = 0 when both the smooth real functions Φ, Π are supported in I ′ or in J ′ . Therefore Since I and J are disjoint proper open segments one has I ∩ J = ∂I ∩ ∂J. Therefore, if ∂I ∩ ∂J = ∅ both distributions ℜe(ψ), A −1/4 · and ℑm(ψ), A 1/4 · vanish and this implies that ψ = 0 since A ±1/4 (C ∞ (S 1 )) = L 2 (S 1 , dθ) as proved in proposition 3. Otherwise ∂I ∩∂J contains two points at most, say p and q. We can assume, without loss of generality, that θ p = 0 and θ q ∈ (0, 2π) (this situation can always be achieved by redefining the origin of coordinate θ on S 1 ). It is a well-known result of distributions theory that distributions with support given by a single point are polynomials of derivatives of Dirac deltas supported on that point (the case of a finite number of points is a trivial extension). Consider first of all ℑm(ψ), A 1/4 f · . In our case there must be a finite number of coefficients a j , b j ∈ R such that, for every f ∈ C ∞ (S 1 , R) it must hold Passing to Fourier transformation, the identity above can be re-written if ψ k and f k are the Fourier coefficients of ℑm(ψ) and f respectively (notice that f k → 0 faster than every power |k| −M so that the right hand side is well defined). Since the functions f are dense in L 2 (S 1 , dθ) C , this is equivalent to say that: where we have defined N := max(N p , N q ) (assuming a j = 0 and b j = 0 for the added coefficients). Let us prove that the right-hand side defines a ℓ 2 (Z) sequence -as it is required by ψ ∈ L 2 (S 1 , dθ) C -only if a j = 0 and b j = 0 for every j. Assume that {ψ k } k ∈ ℓ 2 (Z) so that the right-hand side of (A.29) defines a ℓ 2 (Z) sequence. If c j,k := ℜe (a j + e ıkθq b j )ı j , then The sequence {kθ q } k∈Z in [0, 2π] may be either periodic -and this happens when θq 2π is rational -or it is dense in [0, 2π] -and this arises for θq 2π irrational. In both cases, fixing k 0 ∈ Z \ {0} and ǫ > 0, there is a sequence of integers {k (ǫ) n } n∈Z such that: Moreover, defining M := max j=0,...,N |a j | + |b j | one has c j,k ≥ −M > −∞, therefore a lower bound for the right-hand side of (A.30) is If c N,k 0 = 0 the leading term in the latter expression is so that the right-hand side of (A) diverges to +∞ -and {ψ k } k / ∈ ℓ 2 (Z)unless c N,k 0 − sign(c N,k 0 )ǫ = 0. Arbitrariness of ǫ implies c N,k 0 = 0 that is ℜe (a N + e ık 0 θq b N )ı j = 0. Analogously one sees that ℑm (a N + e ık 0 θq b N )ı j = 0, and thus a N + e ık 0 θq b N = 0. However, since k 0 was arbitary one also has a N = b N = 0. Iterating the procedure one achieves a j = b j = 0, ∀ j = N, N − 1, . . . , 1 . It remains to consider the case j = 0, that is the case of {ψ k } k∈Z ∈ ℓ 2 (Z) with where a 0 , b 0 ∈ R are constant. Now and thus {ψ} k / ∈ ℓ 2 (Z) unless b 0 = ±a 0 . With that choice we have in turn: |ψ k | 2 = 2a 2 0 (k 2 + m 2 ) − 1 2 (1 ± cos(kθ q )).
As the series ∞ k=0 cos(kθq) k converges (θ q = 0 mod 2π by hypotheses), and (k 2 + m 2 ) − 1 2 ∼ 1 k for k → ∞, it arises that ∞ k=0 |ψ k | 2 diverges barring the case a 0 = b 0 = 0. This concludes the proof of the fact that a j = b j = 0 for all j if ψ ∈ L 2 (S 1 , dθ). We have found that the distribution ℑm(ψ), A 1/4 · must vanish. The proof for ℜe(ψ), A −1/4 · is strictly analogous. Since both distributions vanish and A ±1/4 (C ∞ (S 1 )) = L 2 (S 1 , dθ) C , we are commited to admit that ψ = 0, so that M I ∩ M J = {0}. Proof. (a) The first statement has been proved in the appendix of [21] in the general case where D C is a topological vector space. Concerning the uniqueness of f χ , we notice that if there was another such function f ′ χ , the difference g := f χ − f ′ χ would be a continuous linear function g : D C → R (resp. g : D I → R) whose range is a subset of W := {2πk | k ∈ Z}. Since the range must be connected -because of continuity and connectedness of D C (resp. D I ) -Ran(g) must be one of the components of W . Since g(0) = 0 by linearity, the only possible component is {0}. Hence g ≡ 0 and f χ = f ′ χ . (b) Concerning (B.3) one has, using Dom(A −1/4 ) = L 2 (S 1 , dθ), self-adjointness of A, f =f and the fact that A −1/4 commutes with complex conjugation: As a consequence: and so χ f is continuous with respect to µ. The proof of (B.4) is analogous.
(d) We have only to prove that local unitary equivalence implies unitary equivalence, the other implication being trivial. Suppose that ω χ is locally unitarily equivalent to ω 0 and consider I, J ∈ R with I ∪ J = S 1 . Let f I : D I → R and f J : D J → R be the continuous linear functions associated with χ ↾ D I and χ ↾ D J as established in proposition 12. Finally consider two functions g I , g J ∈ C ∞ (S 1 , R) with g I (θ) + g J (θ) = 1 for all θ ∈ S 1 and supp(g I ) ⊂ I, supp(g J ) ⊂ J. If (Φ, Π) ∈ D C one has: χ((Φ, Π)) = χ((g I Φ, g I Π))χ((g J Φ, g J Π)) = e ı[f I ((g I Φ,g I Π)+f J ((g J Φ,g J Π))] .

Remark
(1) Validity of (d) for free QFT in the cylindric flat spacetime implies that there is no straightforward generalization of DHR superselection sectors in our theory referring to the states induced by characters χ ∈ Ch(D C ). This is because any state in a DHR sector different from the vacuum sector must be however locally unitarily equivalent to the vacuum state. Using characters, sectors different from that containing the vacuum may arise in the occurrence of breakdown of local unitarily equivalence only. In Minkowski spacetime the poset R is replaced by a suitable directed set of relatively compact regions of a fixed Cauchy surface and there is no finite number of such regions whose union covers the Cauchy surface; thus the proof given for (c) does not apply.
(2) The unitary intertwiners W ı 2 −1/2 A −1/4 f and W −2 −1/2 A 1/4 g do not belong to π 0 (W KG ) in general but they belong to π 0 (W KG ) ′′ = B(H 0 ) in any case. The next proposition establishes existence and uniqueness of the universal algebra, and shows that it truly extends the notion of inductive limit of a C * -algebras net.
space admits an unique extension to A, it is anyway possible that B A: it is sufficient that B includes a closed two-sided ideal of A (see [24]). Therefore the requirement that the sub algebras A(I) generate A is essential in proving the uniqueness of the universal algebra A.
We conclude with the definition of a C * -category.
Definition 16 (C * -category). A category C is said to be a C * -category if the following conditions are satisfied: • The set of arrows (z, z 1 ) between two elements z, z 1 ∈ C has a complex vector space structure {(z, z 1 ), +, ·}; composition between arrows is bilinear with respect to it; • The space (z, z 1 ) admits a complete norm · satisfying the C * -property (namely r * r = r 2 for each r ∈ (z, z 1 )), making it into a complex Banach space; • There should be an adjoint * , that is an involutive controvariant functor acting as the identity on the objects.
As a consequence, if C is a C * -category then (z, z) is a C * -algebra for every z ∈ C.