Aharonov-Bohm superselection sectors

We show that the Aharonov-Bohm effect finds a natural description in the setting of QFT on curved spacetimes in terms of superselection sectors of local observables. The extension of the analysis of superselection sectors from Minkowski spacetime to an arbitrary globally hyperbolic spacetime unveils the presence of a new quantum number labeling charged superselection sectors. In the present paper we show that this"topological"quantum number amounts to the presence of a background flat potential which rules the behaviour of charges when transported along paths as in the Aharonov-Bohm effect. To confirm these abstract results we quantize the Dirac field in presence of a background flat potential and show that the Aharonov-Bohm phase gives an irreducible representation of the fundamental group of the spacetime labeling the charged sectors of the Dirac field. We also show that non-Abelian generalizations of this effect are possible only on space-times with a non-Abelian fundamental group.


Introduction.
The analysis of superselection sectors is one of the central results of algebraic quantum field theory. It allows to derive from first principles particle statistics, the particle-antiparticle correspondence, quantum charges and, as a consequence, the appearance of global gauge symmetries, independently of the model under consideration [15].
In the original formulation, superselection sectors were defined by Doplicher, Haag and Roberts in terms of localized and transportable endomorphisms ρ(o) : A → A of the global observable C * -algebra A, where o is a double cone in Minkowski spacetime. In this way a new representation (sector) π : A → B(H 0 ) , π := π 0 • ρ(o) , is defined starting from a vacuum representation π 0 . The term localized indicates that ρ(o) ↾ A e is the identity on any local algebra A e ⊂ A generated by observables localized in a double cone e causally disjoint from o, while transportable means that, for any double cone a, ρ(o) is unitary equivalent to an endomorphism localized in a. This yields the DHR selection criterion [22].
It was later recognized by Roberts that an equivalent formulation of superselection sectors could be achieved by considering charge transporters, that is, families z of unitaries z(a, o) ∈ A a , o ⊂ a, fulfilling a cocycle relation. Each z defines a family {ρ z (o)} o of localized endomorphisms, such that z(a, o) ρ z (o)(A) = ρ z (a)(A) z(a, o) , A ∈ A .
In Minkowski spacetime this approach turns out to be equivalent to the original one [31,32]. Yet things change in the context of curved spacetimes M , mainly due to topological obstructions that arise when the family of diamonds, the analogues of double cones, is not upward directed under inclusion. A first point is that the notion of an algebra of global observables is not well-defined and must be replaced by the universal algebra, indicated with A [18], having the property of lifting any family of representations A second point is that localized endomorphisms alone cannot encode the whole physical content of sectors, [35,8]. Therefore the use of charge transporters becomes necessary in the more general scenarioof a curved spacetime M .
A condition that must be imposed on z in order to get a well-defined representation π of A is arises. An answer is given in a series of papers, in which the following facts were established: 1. Statistics, charge content and particle-antiparticle correspondence are well defined on generic charge transporters. Thus they have all the properties that allow to assign a physical interpretation to topologically trivial transporters. Moreover, each z defines a representation which affects the transport of localized endomorphisms along loops. The integer n ∈ N is called the topological dimension [8]. The existence of such sectors for a massive Boson field in a 2-dimensional spacetime has been proved in [5].
2. Generic charge transporters do not define representations of A. Rather, they define representations of the net A KM := {A o } o on flat Hilbert bundles over M , which correspond to covariant representations of a universal C * -dynamical system α : π 1 (M ) → autA * encoding parallel transport along loops [37,38,39]. It turns out that A is a quotient of A * , corresponding to the set of representations of A * with trivial action of π 1 (M ) on the Hilbert space.
The physical interpretation of these results was proposed by Brunetti and the second named author in the scenario of the Aharonov-Bohm effect. There, superposition of wavefunctions of charged particles is affected by phases (parallel transports) of type where A is a potential with vanishing electromagnetic field, and it is natural to note the similarity with (1). The suitable model in which such an interpretation can be tested at the level of quantum field theory is the Dirac field, describing charged quantum particles in a curved spacetime M . A step in this direction has been made in [43], where it is proven that introducing a background potential represented by a closed 1-form yields a representation of the observable net A KM of the free Dirac field over a flat Hilbert bundle over M .
In the present paper we complete the above partial results and perform an analysis of the superselection structure of A KM . Our main results show that: • Superselection sectors z define background flat potentials A z , that are, at the mathematical level, flat connections on suitable flat Hermitean vector bundles L z → M whose rank coincides with the topological dimension of z (Theorem 2.4). We shall be interested, in particular, in those sectors with topological dimension 1, so that L z is a line bundle.
• Superselection sectors with topological dimension 1 and "charge 1", in a sense that shall be clarified later, are in one-to-one correspondence with twisted field nets. These are representations of the field net of the Dirac field on flat Hilbert bundles over M with monodromy given by the "second quantization" of the monodromy of L z (Theorem 4.4); • When no torsion appears in the first homology group of the spacetime, L z is topologically trivial. In this case we are able to construct a twisted field solving the Dirac equation with interaction term A z , and the representation (1), with n = 1, takes the form (2) (Theorem 4.3). This occurs in physically relevant scenarios such as de Sitter and anti-de Sitter spacetimes, as well as the spacetime complementary to the ideally infinite solenoid of the Aharanov-Bohm apparatus.
Both the twisted field net and the twisted Dirac field are indistinguishable from the analogous untwisted objects in simply connected regions of M . This is the reason way A KM is able to detect them. The non-trivial parallel transport carried by the family of localized endomorphisms defined by a sector is interpreted in terms of the background flat potential, shedding some light on the role that the corresponding interaction plays on quantum charges.
The paper is organized as follows.
In §2 we recall results from [8,35] describing the superselection structure, say Z 1 (A KM ), of a generic observable net. Moreover, by applying a result by Barrett [3], for any sector z we exhibit a flat u(n)-connection such that (1) is the associated parallel transport. This yields a Chern character defined on Z 1 (A KM ) with values in the odd cohomology H odd (M, R/Q) of the spacetime (Remark 2.3). The first component of the Chern character will be shown to correspond to Aharonov-Bohm phases in the subsequent sections.
In §3 we focus on the case in which A KM is the observable net of the free Dirac field. By applying results in [6], we show that Haag duality holds in the GNS representation π ω of any pure quasi-free state ω of the Dirac-CAR algebra defined on the spinor space of a globally hyperbolic spacetime (Theorem 3.2). This, and the III-property fulfilled by the local Von Neumann algebras living in π ω , [10], yields two key properties for the analysis of sectors of A KM . Finally, we construct a family of topologically trivial sectors of A KM labeling the electron-positron charge (Remark 3.6).
In §4 we construct Dirac fields interacting with background potentials A, which are closed 1-forms (dA = 0). These fields are "twisted", as in [27], namely they are defined on the space of sections of the Dirac bundle tensor a flat line bundle with connection A. We show that these fields are in one-to-one correspondence with sectors in Z 1 (A KM ) such that L z is topologically trivial (Theorem 4.3). The parallel transport defined by A appears, from one side as a byproduct of the interaction term in the Dirac equation, and on the other side as the parallel transport of DHR-charges (localized endomorphisms) of A KM .
In §5 we draw our conclusions and we illustrate further developments, in particular for π 1 (M ) non-Abelian and generalized Dirac fields with non-Abelian gauge group.
In Appendix §A we prove a technical result, allowing to adopt the usual Haag duality instead of the punctured one in the analysis of sectors.
Some of the results of the present paper have been presented in [44] in a simplified form, in particular by avoiding to discuss Haag duality and without going in details on the structure of twisted Dirac fields.

Aharonov-Bohm effects in terms of sectors
Aim of the present section is to describe the Aharanov-Bohm effect in terms of superselection sectors of the net of local observables on a 4-d globally hyperbolic spacetime M . From the intrinsic viewpoint of the electromagnetic field and of the vector potential, this effect has been studied instead in [4,41].
We start by recalling some basic facts on globally hyperbolic spacetimes and we introduce the net of local observables defined in a reference representation. We then discuss DHR-charges of the observable net in terms of charge transporters. These are cocycles depending on a suitable family of regions of the spacetime, diamonds, taking values in the unitary group of the observable net. The key observation is that diamonds encode the fundamental group π 1 (M ) of the spacetime M and cocycles define representations of π 1 (M ). The Aharonov-Bohm effect is manifested in cocycles which define nontrivial representations of the fundamental group of the spacetime. We shall prove that these cocycles are nothing but the holonomy of smooth flat connections acting on a DHR charge. The form of the action is ruled by the charge quantum numbers.
The material appearing in the present section is essentially a convenient summary of [8,35]. Exceptions are Lemma 2.2 and Theorem 2.4, where the interpretation in terms of flat connections is given.

Local observables in a globally hyperbolic spacetime
Let M denote a 4-dimensional, connected, globally hyperbolic spacetime. Causal disjointedness relation is a symmetric binary relation ⊥ defined on subsets of M as follows: In the present paper we are interested in describing the observable net in a reference representation playing the same rôle as the vacuum representation in Minkowski spacetime. This is defined in terms of a net A KM given by the assignment of a von Neumann algebra A o ⊂ B(H 0 ) for any a ∈ KM . We assume the following standard properties: where A ′ a stands for the commutant of A a , • (The Borchers property) for any inclusion cl(o) ⊂õ, if E is a non vanishing orthogonal projection of A o then there is an isometry V ∈ Aõ such that V V * = E; Meaningful examples satisfying the above properties are the the observable net of the free scalar field [45] and, as we shall see, and that of the free Dirac field, in representations induced by pure quasi-free Hadamard states. Anyway the above properties are expected to hold for any Wightman field over M .

1-cocycles of the observable net and topology
We introduce the mathematical structure underlying the charges studied in [8]. The main fact is that the theory is encoded by charge transporters which define representations of the observable net by means of Haag duality. A crucial mathematical property is that charge transporters satisfy a cocycle equation with respect to the partially ordered set (poset) KM and, as a consequence, give a representation of the fundamental group of the spacetime. In the present paper we give an equivalent but simplified exposition of cocycles which do not relies on simplicial sets.
We start by recalling some elements of the "geometry" of KM , the poset of diamonds ordered under inclusion ⊆. The composition and the operation of taking the opposite extend to arbitrary paths in an obvious way: given p : a → o and q : o →õ then q * p : a →õ , q * p = p * q :õ → a .
A path of the type p : o → o is called a loop over o. It is verified that M is connected if, and only if, KM is pathwise connected meaning that any pair of diamonds a, o can be joined by a path p : a → o [35]. There is a homotopy equivalence relation ∼ on the set of paths: the quotient by ∼ of the set of loops over a fixed a ∈ KM gives the homotopy group π 1 (KM, a) which does not depend, up to isomorphism, on a, because of pathwise connectedness. The isomorphism class is called the fundamental group of KM , written π 1 (KM ). A key result is that, given o ∈ KM and x o ∈ o, there is an isomorphism where π 1 (M, x o ) is the homotopy group of M based at x o ∈ o [35]. In the previous expression p γ is a path-approximation of the closed curve γ : [0, 1] → M : that is, a path such that there is a partition t 0 = 0 < t 1 · · · < t n−1 < t n = 1 : Remark 2.1. We highlight a few consequences of the previous result which are of relevance to this paper. If KM is directed under inclusion, then π 1 (KM ) is trivial [35]. So, if π 1 (M ) is not trivial then KM is not upward directed. This is actually the physical situation we are interested in, since Aharonov-Bohm type effects appear when the spacetime has a non-trivial fundamental group.
We now introduce the set Z 1 (A KM ) of 1-cocycles of the observable net, i.e. the charge transporters. A 1-cocycle of the net A KM is a map assigning to any comparable pair o ≷ a ∈ KM a unitary operator z(a, o) ∈ A |a,o| , fulfilling the property z(a, o) = z(o, a) * and the cocycle equation The Note that z(p) = z(p) * . Any z ∈ Z 1 (A KM ) preserves the homotopy equivalence relation, so that the mapping [p] → z(p), [p] ∈ π 1 (KM, o), is well-defined. By (4), we get the representation where p γ : o → o is a path-approximation of γ. A 1-cocycle z is said to be topologically trivial whenever σ z is the trivial representation.
An intertwiner between two 1-cocycles z,z is a map t a ∈ A a , a ∈ KM , fulfilling the relations t a z(p) =z(p) t o for all p : o → a. The set of intertwiners between z,z is denoted by (z,z). The 1-cocycles z,z are said to be unitarily equivalent if there exists t ∈ (z,z) such that t a is unitary for any a ∈ KM . The 1-cocycle z is said to be irreducible whenever (z, z) = CI, where I ∈ B(H) is the identity, and trivial if it is unitarily equivalent to the trivial 1-cocycle ι(a, o) ≡ I. It is easily seen that cocycles form the set of objects for a category Z 1 (A KM ), with the corresponding sets of intertwiners as arrows. We denote the full subcategory of Z 1 (A KM ) whose objects are topologically trivial 1-cocycles by Z 1 t (A KM )

Charge quantum numbers
The physical content of 1-cocycles is encoded in the charge quantum numbers associated to their equivalence classes. In this section we shall recall how these quantum numbers arise and explain the nature of these charges. As already mentioned, these results have been proved in [35,8]. There is, however, an important difference: in the present paper we assume Haag duality, whilst in the above references a stronger form of duality is used, the punctured Haag duality. In the appendix we show succinctly that Haag duality suffices.
Using the defining properties of the observable net one can introduce a tensor product × (charge composition) and a permutation symmetry ε (statistics), making Z 1 (A) and Z 1 t (A) symmetric, tensor C * -categories. In particular, one can identify a subset of 1-cocycles called objects with finite statistics which turn out to be a finite direct sum of irreducible 1-cocycles. The unitary equivalence class of any such irreducible object z is classified by the following charge quantum numbers: • the statistical phase κ(z) ∈ {−1, 1} distinguishing between Fermi and Bose statistics; • the statistical dimension d(z) ∈ N giving the order of the (para)statistics; • the charge conjugation assigning to z a conjugated irreducible 1-cocyclez which has the same statistical phase and dimension as z.
We stress that objects with statistical dimension 1 obey the Bose/Fermi statistics according to their statistical phase. We denote the subcategory of 1-cocycles with finite statistics by Z 1 AB (A) and its subcategory of topologically trivial 1-cocycles by Z 1 DHR (A). To complete the physical interpretation we need to observe that any 1-cocycle z ∈ Z 1 AB (A) splits in two parts: the charged and the topological component. To see this we need a path frame P e consisting of a collection of arbitrary paths p ae : e → a joining a fixed diamond e, the pole, with any a, and such that p ee is the trivial path (e, e). Once a path frame P e is given, the charged where p eã is the opposite pã e of pã e . This cocycle encodes the charge content of z as the maps preserving the tensor product, the permutation symmetry and, hence, the conjugation 1 . On the other end, setting one gets a 1-cocycle u z , the topological component of z, taking values in the algebra of the chosen pole e. i.e. u z (ã, a) ∈ A e for allã ≷ a. It encodes the topological content of z since it defines the same representation (6). Finally, the elements of Z 1 AB (A KM ) are completely characterized by their topological and the charged component since where αã e (A) := z c (pã e ) A z c (pã e ) * . The composition ✶ is called the join. These constructions do depend neither on the choice of the pole e nor on that of the path frame P e : different choices lead to equivalent, in the corresponding categories, charged and topological components. Now, any 1-cocycle z defines representations of the observable net which are sharp excitations of the reference one. To be precise, given a diamond o define where p : e → o is an arbitrary path such that e ⊂ a ⊥ . By (70), given in appendix, this definition is independent of p, and This amount to saying that the family in the sense of [37]. Furthermore, always using the property (70), we find that and transportable i.e.
where the above equation should be read as Equation (13) says that ρ z (o) describes a charge localized in o, since it equals the reference representation in the causal complement of o. On the contrary equation (14) illustrates the role of z as a charge transporter along paths. The important point is that the charges described by AB and DHR cocycles are the same since ρ z (o) = ρ zc (o) for any o ∈ KM . This is an immediate consequence of the path independence of the definition of ρ z (o) and of the definition of the charged component z c . However an important difference arises when the charge is transported from o to a along to different paths p, q : o → a. If z is of DHR type then the transport is path-independent, Hence z(p * q)ρ z (a) depends on the homotopy equivalence class of the loop p * q. In particular z(p * q) = 1 if p * q ∼ 1. We conclude that the Aharonov-Bohm effect manifests itself in sectors of AB type and, as we shall prove in the next section, it is due to a presence of a background flat potential.

Background flat potentials and the AB effect
In this section we complete the picture drawn in the preceding sections aimed at explaining the Aharonov-Bohm effect in terms of superselection sectors of local observables. In particular we shall prove that the topological component of 1-cocycles are nothing but the holonomy of a flat connection. To this end we shall use a result that has been rediscovered by several authors 2 . We refer in particular to a paper by Barrett [3].
Before proceeding it is convenient to specify the notion of flat connection. Let G be a compact Lie group with Lie algebra g and P → M denote a smooth G-principal bundle. We assume that P is trivialized on any diamond, and that it is locally constant in the sense that it admits a set of locally constant transition maps Thus dg oa ≡ 0, and following [28, §I.1-2] we define a flat connection on P as a collection A of of When G = U(1), each g oa is a phase and A a ↾ o∩a = A o ↾ o∩a ; thus the forms A o can be glued and give a u(1)-valued 1-form that we denote again by A. Because of flatness, and since u(1) ≃ R, we may regard A as a closed real 1-form, in symbols Now, following [8] for some unitary operator U : H 0 → H 0 ⊗ C n . The integer n is an invariant of the equivalence class of z which is called the topological dimension of z and it is denoted by τ (z).
where U is the unitary realizing (19) and hol A z (ℓ (ã,a) ) ∈ U(n) is the holonomy over any closed curve ℓ (ã,a) : Proof. As observed in the previous section z and its topological component define the same representation σ z (6) of the fundamental group of M . As the algebra A(M, o) is a type I n factor, there exists an irreducible n-dimensional representation σ of the fundamental group of M such that where U is the unitary realizing (19). The key observation is that the mapping ℓ → σ([ℓ]) assigning to any closed curve ℓ over x o the unitary σ([ℓ]) ∈ U(n) is a holonomy map in the sense of Barrett [3, cfr. Section 2.4]. By Barrett's reconstruction theorem [3], ℓ → σ([ℓ]) is the parallel transport of a flat connection A z on a smooth U(n)-principal bundle P z , that is, σ([ℓ]) = hol A z (ℓ) for any closed curve ℓ : x e → x e . Explicitly, whereM is the universal covering of M carrying the well-known right π 1 (M )-action, whilst Hence by (6) and (21) we have that for any loop p ℓ and any closed curve ℓ such that p ℓ is a path approximation of ℓ. Given a comparable pairã ≷ a, if we take the loop p oã * (ã, a) * p ao and a closed curve ℓ (ã,a) : , the proof follows from (21), (23) and from the definition of the topological component.
It is easily seen that a change of the path frame P o used in the previous proof yields a representation σ ′ : π 1 (M, x o ′ ) → U(n) unitarily equivalent to σ. This implies that we find a flat connection A ′ z providing, up to conjugation, the same parallel transport as A z . Remark 2.3 (The Chern character on Z 1 AB (A KM )). Let z denote an irreducible cocycle and let σ : π 1 (M ) → U(n), n := τ (z), denote the associated representation of the fundamental group (21). By standard results in differential geometry, σ defines a flat Hermitean vector bundle defined analogously to (22). Using the results in [9], we can define the Chern character The interpretation of ccs(z) is that of a generalized Aharonov-Bohm phase, as we shall see in the next sections.
From now on, to emphasize the geometric aspects, we shall write instead of the expression appearing in (20). We can state the main result of this section.
where A z is the flat connection of the U(n)-principal bundle P z defined by z.
Proof. In general, since where the r.h.s. is a C * -algebra formed by the component e of the intertwiners of (z, z). If m < d(z), then the dimension of (z, z) e would be smaller that d(z) 2 because any ζ k is irreducible. This leads to a contradiction because of the above inclusion and because the dimension of A(M, e) is τ (z) 2 and τ (z) = d(z) by assumption. This implies that all ζ k have statistical dimension 1 and, on account of the same dimensional argument just used, that all ζ k are equivalent. Thus the proof follows by applying the previous theorem to (10).
We shall refer to A z as the background flat potential associated with z.
Hence, an irreducible 1-cocycle z with statistical dimension d(z) = τ (z) is the transporter of charge of Fermionic/Bosonic type in the presence of a background field A z . If, in particular, d(z) > 1 then we see that the charge z c is a multiplet, and the background field acting upon this multiplet, realizing the AB effect, is non-Abelian. The background field quantifies how the charge depends on the homotopy equivalence of paths when transported along paths. Precisely, referring to (16) so the final representation ρ z (o) differs from ρ z (a) by the factor Hol A z (p * q) (see (15) for the notation).
The next result proves that non-Abelian AB effects can appear only in spacetimes having a non-Abelian fundamental group.
Proof. It is enough to observe that if τ (z) = 1 the topological component takes values in U(1) and the joining reduces to the multiplication by the so-defined phase. Moreover, if π 1 (M ) is Abelian, then any irreducible representation of π 1 (M ) is a phase.
It is known that a principal U(1)-bundle P with a flat connection A is trivial if, and only if, That such a P is trivial shall be proved in the following §4.
This always holds true when H 2 (M ) has no torsion classes.
By well-known results, the above condition of H 2 (M ) being torsion free is equivalent to H 1 (M ) being torsion free (see [19,Sec.3

On the observable net of the free Dirac field
In the previous section we have shown that the nontrivial topology of the spacetime induces on the observable net superselection sectors affected by the topology of the spacetime. These sectors factorize in a part which describes a quantum charge (DHR charges) and in a topological part ruling the behaviour of the quantum charge when moved along a path in the same way as it occurs in the Aharonov-Bohm effect.
Our final aim is to show that this is not merely an analogy but the real physical Aharonov-Bohm effect: we want to show that the quantization of the free Dirac field in the presence of a background flat potential gives rise to the same type of superselection sectors as those described in the abstract analysis of the previous section.
As a preliminary step, in this section we analyze the free Dirac field in a 4-d globally hyperbolic spacetime without the presence of a background flat potential. We show that the net of local observables in the representation defined by a pure quasi-free Hadamard state satisfies all the properties assumed in section 2.1, in particular Haag duality and (a strengthening) of the Borchers property. Then we provide an explicit construction of the superselection sectors of the Dirac field and we show, as expected, that these are DHR cocycles obeying the Fermi statistics.

The Dirac operator
It is convenient to recall some notions on spin structures. The existence of any such structure is guaranteed if and only the second Stiefel-Whitney class of the underlying manifold w 2 (M ) ∈ H 2 (M, Z 2 ) vanishes. This is always the case if we consider four dimensional globally hyperbolic spacetimes and, in addition the corresponding Dirac bundle is trivial, namely DM ≃ M × C 4 , cf. The vector spaces whose elements are spinors and co-spinors are isomorphic via the Dirac adjoint defined in terms of the gamma matrices as Here we are taking the gamma matrices in the standard representation: γ * 0 = γ 0 and γ * k = −γ k for k = 1, 2, 3.

The (CAR) algebra of the Dirac field
There is a fairly vast literature on the CAR algebra and the Dirac quantum field in a 4dimensional globally hyperbolic spacetime. In the present section we give a brief description of these topics, and refer the reader to the references [14,40,11] for details.
A convenient approach to quantize the Dirac field and to study the corresponding representations is the self-dual approach [1]. We consider the following Whitney sum of bundles In addition we extend the Dirac operators and the corresponding propagators on this space by setting Observe that S( DM ) is equipped with a positive semi-definite sesquilinear form for any f i ⊕ h i ∈ S( DM ), i = 1, 2 (both (·, ·) s and (·, ·) c are positive semi-definite sesquilinear forms on S(DM ) and S(D * M ) respectively). This form (29) annihilates on the image of the extended Dirac operator D. Taking the quotient we get, up to a suitable closure, a Hilbert space (h, (, )). Finally setting one observes that Γ extends to an anti-unitary operators on h: We now are ready to give the definition of the CAR algebra. (ii) Self-duality: (iii) Canonical Anticommutation Relations (CARs): We draw on the consequences of this definition.
(1) The generators of the CAR algebra encode the Dirac field ψ and the conjugated Dirac fieldψ: Notice in fact that, since Ψ • D = 0, the fields ψ andψ satisfy the Dirac equation and thus Ψ[f ⊕ h] = ψ(f ) + ψ(h † ) * =ψ(f † ) * +ψ(h). This implies in particular that the algebra C(M ) can be generated by taking only either the Dirac field or the conjugated one.
(4) Finally, we note that ψ is a C * -valued distribution when S(DM ) is equipped with the topology of test-functions (the same holds true forψ). In fact by the CARs (32) we have and the proof follows by the continuity of (·, ·) s with respect to the test function topology.

Quasi free Hadamard states
We describe the net of the Dirac field in the representation associated with a pure and gauge invariant quasi-free Hadamard state. We show that the net satisfies twisted Haag duality and the local algebras are type III factors. As a consequence the resulting net of local observables in the vacuum (zero charge) representation verifies all the properties assumed in Section 2.1.
To begin with we consider a pure quasi-free state ω of the algebra C(M ) and the corresponding GNS triple (H, π, Ω) (see [1]). Here H is the antisymmetric Fock space associated with a closed subspace of P h defined by an orthogonal projection P , the base projection enjoying the relation ΓP = (1 − P )Γ; π is an irreducible representation of the CAR algebra on the Hilbert space H and Ω is a unit norm vector H such that ω(·) = (Ω, π(·)Ω). To ease notation we keep symbols Ψ, ψ andψ to denote the generator of the CAR algebra, the Dirac field and its conjugate, respectively, in the Fock representation. Finally, if ω is also gauge invariant ω • η = ω then there exists a unitary representation U : U(1) → U(H) such that for any ζ ∈ U(1), it holds U (ζ)Ω = Ω In addition U (ζ)ψ(f ) U (ζ) * = ζψ(f ) and U (ζ)ψ(h) U (ζ) * =ζψ(h). So given a pure and gauge invariant quasi-free state of the CAR algebra and H, π, ω, U as above, the net of local Dirac fields is the correspondence F : KM ∋ o → F o ⊆ B(H) associating the von Neumann algebra to any diamond o. The net F KM satisfies isotony, F o ⊆ Fõ for any o ⊆õ, but not causality because of the CARs. This is replaced by twisted causality where F t o is the von Neumann algebra defined as F t o :=Ũ − F oŨ * − in terms of the twist operator U − :Ũ Theorem 3.2. The net of the Dirac field F KM in a Fock representation induced by a pure and gauge invariant quasi-free state satisfies twisted Haag duality Proof. The proof relies on abstract twisted duality and on a density result. We recall that for the CAR algebra in a Fock representation, abstract twisted duality holds for any closed subspace which is invariant under the conjugation defining the abstract CAR algebra [6]. Observing that the closure h o of S o (DM ) in h is Γ invariant for any o, abstract twisted duality in our case reads as We now need two observations. First, any diamond o is of the form D(G) with G ⊂ Σ where Σ is a spacelike Cauchy surface and G is an open relatively compact subset of G diffeomorphic to an open 3-ball with cl(G) Σ . So, it is easily seen that o ⊥ = I(Σ \ cl(G)) where I stands for the chronological set. Secondly, the Hilbert space h is isomorphic to the Hilbert space L 2 (Σ, C 4 ) ⊕ L 2 (Σ, (C 4 ) * ) with the scalar product (u 1 ⊕ v 1 , u 2 ⊕ v 2 ) := Σ (u † 1 / nu 2 + v 2 / nv † 1 ) dΣ where n µ is the timelike future pointing normal vector to the Cauchy surface. In particular the isomorphism relies on the following identity On these grounds it is enough to prove that the space C ∞ 0 (G, C 4 ) ⊕ C ∞ 0 (Σ \ cl(G), C 4 ) is dense in L 2 (Σ, C 4 ) and similarly for the dual. This follows from a partition of unity argument, similarly to [23, Prop. III.1] for Majorana fields, because G is a relatively compact open subset of Σ with a smooth boundary.
A class of quasi-free states for quantum fields on curved spacetimes having several physically meaningful applications is that of Hadamard states [29] (see also [20]). The existence of pure quasi-free and gauge invariant Hadamard states has been shown in [17]. The local algebras of the Dirac field in a representation defined by such a state are type III factors [23] 4 . This property is stronger than the Borchers property since for any o ∈ KM any projection E ∈ F o is equivalent in F o to the identity i.e. there exists V ∈ F o s.t. V * V = E and V V * = 1 (V is an isometry). Definition 3.3. We take as a reference representation for the Dirac field, the representation defined by a pure quasi-free and gauge invariant Hadamard state.
As shown above the net F KM is isotonous, satisfies twisted Haag duality and the local algebras F o are type III factors.
We define the observable net. To this end we consider the spectral subspaces For the grade n = 0 any F 0 o is a von Neumann algebra which is gauge invariant and such that [T, S] = 0 for T ∈ F 0 o and S ∈ F 0 a with a ⊥ o. This yields the net of local observables which satisfies isotony and causality. The local observable algebras F 0 o inherit from F o the properties of being type III factors, see for instance [23]. Finally we have to take into the game the zero-charge "vacuum" representation of the net of local observables. The vacuum Hilbert space H 0 is the subspace and the vacuum representation π 0 : F 0 KM → B(H 0 ) is defined as defined in terms of the projection E 0 onto H 0 . Since E 0 is an element of (F 0 o ) ′ for any diamond o and since the local algebras F 0 o are type III factors, π 0 is faithful and the von Neumann algebras

Charge transporters of the free Dirac Field
Having established that the observable net A KM of the Dirac field in the vacuum representation satisfies the same properties as those assumed in Section 2.1, now we construct the charge transporters of the Dirac field, which provide the DHR-sectors for A KM .
We start by taking for any o ∈ KM a normalized spinor f o of S o (DM ), that is (f o , f o ) s = 1. With this choice any ψ(f o ) turns out to be a partial isometry of F o since, as it is easily verified by (32), the operators ψ(f o )ψ(f o ) * and ψ(f o ) * ψ(f o ) are projections. Notice, in particular, that ψ(f o ) is a partial isometry of the spectral subspace F 1 o , see (36). We can now make ψ(f o ) equivalent to a unitary of the same spectral subspace. To this end we note that both the projections The operator is a unitary in F o and an element of and similarly for the other CARs. In conclusion, for all o ⊥ a Now we define the charge transporter z associated with the Dirac field as where π 0 is the vacuum representation. Clearly z is localized and z(a, o) ∈ A a for any o ⊆ a, satisfies the 1-cocycle identity for any path p : a → o, and for o = a we find z(p) = 1. Thus z is topologically trivial, z ∈ Z 1 t (A KM ).
is a unitary intertwiner of (z, z ′ ). (ii) It is enough to prove that the symmetry intertwiner ε ∈ (z × z, z × z) is such that ε a = −1 for all a, where ε a := z(q) * × z(p) * · z(p) × z(q) , p : a →õ and q : a → o are paths with o ⊥õ, and × is the tensor product of Z 1 (A KM ) [35,Theorem 4.9]. The latter is defined by the expressions z(q) * × z(p) * := z(q) * z(p 1 )z(p) * z(p 1 ) * , z(p) × z(q) := z(p) z(p 1 )z(q)z(p 1 ) * , with p 1 : o 1 → a, o 1 ⊥ |q|. According to these relations we may take o ⊥ a,õ = a and p : a → a the trivial path. So, z(p) = 1. Moreover, by (43) we have z(q) = π 0 (ϕ * o ϕ a ). Finally, we may take o 1 causally disjoint from o and a and z(p 1 ) = π 0 (ϕ * a ϕ o1 ). In this way, In complete analogy to what has been shown for z, one can prove thatz is independent from the choice of the normalized co-spinor and it is a topologically trivial 1-cocycle obeying the Fermi statistics: d(z) = 1 and κ(z) = −1.
We now prove thatz is the conjugate of z in the sense of the theory of superselection sectors.
where we have used the fact that o 1 is causally disjoint from a and o and that ϕ o ϕ * a is an observable.
Remark 3.6. Performing tensor powers z ×n andz ×n , n ∈ N, we obtain a structure Z 1 ec (A KM ) of DHR-sectors labeled by Z, where positive integers are associated to z ×n and negative integers toz ×n (we define z 0 as the reference representation). Using the techniques explained in the present section it is easily seen that z ×n may be defined starting from products of field operators ψ(f 1 ) · · · ψ(f n ), and similarlyz ×n with products of conjugate field operators. Thus it is natural to interpret Z 1 ec (A KM ) as the charge superselection rule.

Dirac fields in background flat potentials
The Aharonov-Bohm effect concerns the behaviour of electrically charged quantum particles in a space M where a classical background potential A is defined, with the property of having vanishing electromagnetic tensor F := dA. This amounts to saying that A is a closed de Rham 1-form, dA = 0, and we say that A is a flat background potential. Existence of non-trivial flat background potentials is equivalent to the property of having a non-trivial de Rham cohomology, H 1 dR (M ) = 0. By standard results, this implies that π 1 (M ) = 0. Passing to a relativistic setting, the scenario we adopt for describing quantum interactions with the flat background potential A is given a spinor field on a globally hyperbolic spacetime M , fulfilling the Dirac equation with interaction A. In the present section we construct such interacting Dirac fields, and show that they can be interpreted as superselection sectors of a given reference free Dirac field.

Preliminaries on flat potentials
We start by collecting some properties of flat potentials, closed de Rham forms A ∈ Z 1 dR (M ), in particular their relations with flat line bundles, and we establish a mapping A →Â withÂ a real 1-cocycle in the cohomology of KM . This will be useful in the discussion of AB-sectors of the observable net of the free Dirac field.
As a first step we note that, since diamonds o ∈ KM are simply connected, there are C ∞ local primitives φ : o → R such that On these grounds, defining we get a family of locally constant functions fulfilling the cocycle relations g oa g ac = g oc , as it can be trivially verified on each overlap o ∩ a ∩ c = ∅. The next result is standard in geometry: for a proof we refer the reader to [44]: . Then g := {g oa }, defined as in (45), is a set of transition maps for the flat line bundle L A :=M × σ C defined as in (24), where denote local charts such that g oa π a = π o on o ∩ a. The previous Lemma implies that the relation e −iφo π o = e −iφa π a holds for any a ∩ o = ∅. Thus the maps e −iφo π o : L A ↾ o → M × C, o ∈ KM , glue in the correct way and they induce the bundle isomorphism ϑ : L A → M × C defined as Note that ϑ locally looks like the multiplication by e −iφo . Let now a ⊆ o; then φ o − φ a is a constant real function on a and we definê The relationsÂ co +Â oa =Â ca are clearly fulfilled for all a ⊂ o ⊂ c ∈ KM , that is,Â is a real cocycle in the cohomology of KM . Note that by definition (45), we have The cocycleÂ can be used to express the holonomy of A.

The Dirac field interacting with a background flat potential
We construct a quantum Dirac field interacting with a background flat electromagnetic potential, and we describe it in terms of a family of Dirac fields locally defined on diamonds. These fields yield a "local coordinate" description of the interacting field, and will play an important role at the level of the associated nets of von Neumann algebras. The interacting field shares with twisted fields in the sense of Isham [26,2] the property of being defined on the space of sections of a twisted bundle. Yet it is not exactly a field of this type as we shall see in the following lines.
Our task is the construction of a Dirac field ψ A such that where / A is defined in local frames as γ µ o A µ (note that here the components of A have lower indices), and since the local Dirac matrices γ µ o can be arranged to form a section of T * M ⊗ endDM , we have / A ∈ S(endDM ).
We note, that if ψ A is given, then for any o ∈ KM and f ∈ S o (DM ) it turns out Thus applying the local gauge transformation ψ A → ψ A,o := ψ A • e iφo , we find that ψ A,o , as a field evaluated on the test space S o (DM ), solves the free Dirac equation. The idea of the previous computation is that A appears as an exact 1-form on each (simply connected) diamond o ∈ KM , thus it can be represented locally as a local gauge transformation making ψ A a free Dirac field. Thus, to construct our field ψ A , it comes natural to reverse the above argument and to start with a free Dirac field ψ : S(DM ) → B(H) and then to define This implies Defining we have that if a ⊆ o then, combining (51) and (54), The previous computation shows that the operator ψ A,a (ς) is independent of the choice of a ∈ KM such that supp(ς) ⊆ a. By (52) we find e −iφa ς a = ϑ(ς), implying that each ψ A,a extends to the well-defined field Remark 4.2. In the previous expression ψ A appears as a field defined on D A M rather than DM . This kind of field was introduced and studied by Isham [26,2]. Note that ψ A fulfills the CARs To evaluate the Dirac equation on ψ A it is convenient to give a description of / ∇ as an operator on sections of S(D A M ) explicitly presented as tensors ς = f ⊗ s, f ∈ S(DM ), s ∈ S(L A ). To this end we define d := ϑ −1 dϑ, where ϑ is defined in (46) and d : C ∞ (M, C) → S(T * M ⊗ C) is the complex exterior derivative. In this way we obtain a connection d on L A compatible with the inner product of S(L A ) in the sense that d( s, s ′ ) = ds, s ′ + s, ds ′ , s, s ′ ∈ S(L A ). The Leibniz rule for the Dirac connection leads to the following extension: Before constructing the associated Dirac operator we introduce some useful objects. We start with the normalized section e iφ ∈ S(L A ) obtained by applying the inverse of (46) to the constant section 1 of M × C. Note that by definition e iφo = π o • e iφ | o for all o ∈ KM (and this justifies our notation). Hence we can write having regarded the complex function e iφ , s as a section of M × C. We can write explicitly the complex 1-form X s := d( e iφ , s ) ∈ S(T * M ⊗ C), having used i∂ µ φ o = iA µ and the fact that · , · is conjugate linear in the first variable. Also note that ds ∈ S(L A ⊗ T * M ), so the pairing e iφ , ds yields a complex 1-form. Applying a contraction with local Dirac matrices γ µ o , we get an operator / X s ∈ S(endDM ), where, in local frames over o ∈ KM , Here ∂ µ has lower indices, thus ∂ µ s lives in L A ⊗ T M and contraction by γ µ o drops components in T * M and T M . Moreover, pairing by e iφ yields e iφ , f ⊗ / ds ∈ S(DM ). With the notation (59), pairing (56) with the Dirac matrices yields In conclusion, having written / A(f ⊗ s) := ( / Af ) ⊗ s and used (57),(58), (60). This implies as expected. Note that if one starts with the field ψ A , then defining for all o ∈ KM we get a family of fields fulfilling (51) (see (53)). Moreover, the chain of identities (50) shows that each field ψ o • e iφo fulfills the free Dirac equation and, again by (53), the family {ψ o • e iφo } glue in the correct way to define a free Dirac field ψ := ψ A • ϑ −1 . In conclusion: • We accomplished our task of constructing a Dirac field ψ A interacting with A, by paying the price of switching from DM to the twisted Dirac bundle D A M ; • ψ A is equivalently described by the family {ψ o } fulfilling (51) and such that each ψ o is gauge-equivalent to a free field ψ.

Charge transporters and Aharonov-Bohm effect
In the present section we compare the net defined by ψ and the ones defined by the interacting fields ψ A , verifying that they all define the same observable net A KM . We show that any interacting field defines an irreducible 1-cocycle in Z 1 AB (A KM ) obeying the Fermi statistics and having topological dimension one.
As a first step, recalling (55), we define Clearly F A,o = F o , where F o is the von Neumann algebra generated by the free field ψ. The additional information carried by A is, instead, encoded by (51), which imposes to define the *-monomorphisms  oa : F A,a → F A,o , where η is the gauge action (33) and U is the corresponding unitary action on the Fock space (34). This definition allows to recover the relation between the locally defined fields ψ o , and by applying (47) and subsequent remarks, we find The pair (F A , ) is called the field net twisted by A and, at the mathematical level, it defines a precosheaf [43]. These objects are more general than nets since, instead of the inclusion maps, we have the non-trivial *-monomorphisms  oa . As a consequence, if T o ∈ F A,o and T a ∈ F A,a , then the correct way to perform their product is We define the gauge-invariant von Neumann algebras This implies  oa (S) = S for all S ∈ F 0 A,a , thus F 0 A is a net, coinciding with the gauge-invariant subnet F 0 of the free Dirac field since F 0 A,o = F 0 o for all o ∈ KM . We now pass to the construction of charge transporters starting from the twisted field net (F A , ). To this end, we note that the operators ϕ o , o ∈ KM , defined in (40), can be regarded as unitaries ϕ o of F A,o carrying charge 1. To construct the associated charge transporters, we invoke (64) and define Recalling the definitions of charged and topological component of a 1-cocycle given in Section 2.2, we can now prove the following theorem which yields a converse for Corollary 2.6.
The charged component z c obeys the Fermi statistics, and the holonomy of z is given by Proof. The holonomy of z is readily computed by using (49), (43) and (65): where p ℓ = (a, o n−1 ) * · · · * (o 1 , a) is a path-approximation of the loop ℓ : x → x, x ∈ a ∈ KM . Concerning the first part of the theorem, given a path frame P e and recalling the definitions of topological and charged component given in Section 2.3 we have Defining the unitary t a := e −iÂp ae I for any a ∈ KM , we find t o z c (o, a) = π 0 (ϕ * o ϕ a )t a for any inclusion a ⊆ o. So z c is unitary equivalent to the DHR cocycle (42) obeying the Fermi statistics. Finally, from the splitting formula (10) and from the above relation we get z(ã, a) = αã e u z (ã, a) z c (ãa) = αã e z(p eã * (ã, a) * p ae ) z c (ãa) We stress that the connection 1-form A z defined by z in the sense of Corollary 2.5 is gauge equivalent to A up to a singular cocycle, that is, A z stands in the same de Rham class of A modulo a cohomology class ξ z ∈ H 1 (M ). In fact, by iterating (26) over the path approximation p ℓ , we obtain having used topological triviality of z c , that is, z c (p ℓ ) = I. Thus by applying (67), we arrive to for all loops ℓ. Since the exponential map has kernel Z, and since any singular 1-cycle can be interpreted as a loop, we find (2πi) −1 c A modZ = (2πi) −1 c A z modZ, for all 1-cycles c. The previous equality says that A and A z define the same differential character in the sense of Cheeger and Simons, thus by the third exact sequence of [9, Theorem 1.1] we conclude that the de Rham cohomology class of A − A z defines by integration the desired class ξ z ∈ H 1 (M ).

Twisted nets and cocycles
Indeed, the notion of twisted net can be given for arbitrary families σ oa ∈ U(1), a ⊆ o ∈ KM , fulfilling the cocycle relations σ oa σ ae = σ oe . By the results in [34], this is equivalent to giving the morphism σ : Let us now return on the observable net A KM and consider the charge 1 DHR-sector (42), that here we denote by z 1 (o, a) := π 0 (ϕ * o ϕ a ), a ⊆ o.
potentials interacting with Dirac fields. The correspondence is essentially one-to-one, because potentials having the same Aharonov-Bohm phase define the same sector, and the phase is the actual observable quantity. The picture is then completed in terms of twisted field nets, whose inclusion morphisms are designed to reproduce the relative phases that appear when one tries to describe the interacting Dirac field in terms of "local charts" (51). Twisted field nets are, on turns, in one-to-one correspondence with superselection sectors.
For the physical interpretation of our results a crucial point is that the background flat potentials can be reconstructed having merely as input the localized loop observables (6) defined by the sectors. Thus the potential is a byproduct of the sector, which, as well-known, corresponds to a state of the observable net, in general affected by the spacetime topology. We then conclude that the potential is codified in the preparation of the state.
In particular, in the case of the classical Aharonov-Bohm effect, we argue that: • When the experimenter shields the solenoid, the space where the charged particles are confined acquires a non-trivial topology, with fundamental group Z.
• Switching on the magnetic field B inside the solenoid makes the system fall into a superselection sector of AB type, labeled by the Aharonov-Bohm phase γ → exp −i γ A. This fits the abstract discussion by Morchio and Strocchi [30], as well as old results in path-integral quantization (see [25] and references cited therein). Thus the presence of the non-trivial potential A, d A = B, is regarded as part of the preparation of the state: if the experimenter switches off the magnetic field, B = 0, then we have a topologically trivial sector.
The lesson that we learn is that A is interpreted as a generalized DHR-charge of Aharonov-Bohm type, and can be reconstructed using exclusively localized observables.
Finally we remark that for the construction of our interacting field ψ A it is essential that the twisting bundle L A is topologically trivial. Yet in general the twisting bundle L z :=M × σ C n , where σ : π 1 (M ) → U(n) is as usual the holonomy representation defined by z ∈ Z 1 AB (A KM ), is non trivial. This may occur when: 1. n = 1 and the homology H 1 (M ) has torsion 5 ; 2. n > 1, a case that appears for π 1 (M ) non-Abelian or for Dirac fields with non-Abelian gauge group. Under this hypothesis the argument for proving triviality of L A fails. We note that this is the case of explicit physical interest: for example, two parallel solenoids in the Aharanov-Bohm apparatus yield π 1 (M ) isomorphic to the free group with two generators.
Even if there exists no problem in defining the twisted field net (F σ , ) in these cases, we believe that it is desirable from the point of view of physical interpretation to construct the interacting field in correspondence of the sector, thus this point is object of a work in progress. One possible approach may be to embed L z into some trivial bundle M × C m , which always exists for m great enough. As an alternative we may directly take twisted Dirac bundles of the type D z M := DM ⊗ L z , and construct a twisted Dirac field ψ z : S(D z M ) → B(H) in sense of Isham.

A Results on topological sectors
This appendix is devoted to proving that one can avoid using punctured Haag duality in the analysis of superselection sectors; only Haag duality is enough.
We consider the observable net A KM defined in a representation satisfying Haag duality. The key property of 1-cocycles that we need in this appendix is the following: for any pair a, a 1 ∈ KM in the causal complement of o ∈ KM one has that [8, Corollary 1] which is a consequence of homotopy invariance of 1-cocycles and of pathwise connectedness of the causal complement of a diamond. Proof. Take B ∈ A a , A ∈ A ′ a and p as in the statement. By assumption we have thatp : o →ã and that o ⊥ã. By the previous Lemma we have that z(p)Bz(p) ∈ A a ; hence z(p)Az(p)B = z(p)Az(p)Bz(p)z(p) = z(p)z(p)Bz(p)Az(p) = Bz(p)Az(p) , completing the proof.
This result implies that (11) gives an endomorphism ρ z (o)ã : A ′ a → A ′ a for anyã ⊆ a and o ⊥ a and that we can reply the analysis of superselection sectors without making use of punctured Haag duality.