ESI The Erwin Schr�odinger International Boltzmanngasse 9Institute for Mathematical Physics A-1090 Wien, AustriaCharged Sectors, Spin and Statisticsin Quantum Field Theoryon Curved SpacetimesD. GuidoR. LongoJ.E. RobertsR. Verch

Vienna, Preprint ESI 723 (1999) June 25, 1999Supported by Federal Ministry of Science and Transport, AustriaAvailable via http://www.esi.ac.at

[ math-ph/9906019 / 22 June 1999 ]Charged Sectors, Spin and Statisticsin Quantum Field Theory on Curved SpacetimesD. Guido�1, R. Longo��1, J.E. Roberts��1, R. Verch���� Dipartimento di Matematica,Universit�a della Basilicata,I-85100 Potenza, Italye-mail: guido@unibas.it�� Dipartimento di Matematica,Universit�a di Roma \Tor Vergata",I-00133 Roma, Italye-mail: longo@mat.uniroma2.it,roberts@mat.uniroma2.it��� Institut f�ur Theoretische Physik,Universit�at G�ottingen,D-37073 G�ottingen, Germanye-mail: verch@theorie.physik.uni-goettingen.deAbstract: The �rst part of this paper extends the Doplicher-Haag-Roberts theory ofsuperselection sectors to quantum �eld theory on arbitrary globally hyperbolic spacetimes.The statistics of a superselection sector may be de�ned as in at spacetime and each chargehas a conjugate charge when the spacetime possesses non-compact Cauchy surfaces. In thiscase, the �eld net and the gauge group can be constructed as in Minkowski spacetime.The second part of this paper derives spin-statistics theorems on spacetimes with ap-propriate symmetries. Two situations are considered: First, if the spacetime has a bifurcateKilling horizon, as is the case in the presence of black holes, then restricting the observ-ables to the Killing horizon together with \modular covariance" for the Killing ow yieldsa conformally covariant quantum �eld theory on the circle and a conformal spin-statisticstheorem for charged sectors localizable on the Killing horizon. Secondly, if the spacetimehas a rotation and PT symmetry like the Schwarzschild-Kruskal black holes, \geometricmodular action" of the rotational symmetry leads to a spin-statistics theorem for chargedcovariant sectors where the spin is de�ned via the SU(2)-covering of the spatial rotationgroup SO(3).1Supported by GNAFA and MURST. 1

Table of Contents1. Introduction 21.1 General Setting 31.2 Superselection Sectors 51.3 Covariant Sectors and Univalence (Spin) 71.4 Tomita-Takesaki Theory and Symmetry 71.5 Modular Inclusion and Conformal Theories on the Circle 81.6 Description of Contents 92. Some Spacetime Geometry 102.1 Generalities 102.2 Appendix to Chapter 2 143. Superselection Structure in Curved Spacetimes 153.1 Introduction 153.2 The Selection Criterion 173.3 Localized Endomorphisms 183.4 The Left Inverse and Charge Transfer 233.5 Sectors of a Fixed-Point Net 253.6 Appendix to Chapter 3 274. The Conformal Spin and Statistics Relationfor Spacetimes with Bifucate Killing Horizon 374.1 Spacetimes with bKh 384.2 Conformal Spin-Statistics Relation 424.3 Appendix to Chapter 4 495. The Spin and Statistics Relation for Spacetimes with Rotation Symmetry 515.1 Geometric Assumptions 515.2 Quantum Field Theories on Spacetimes with Rotation Symmetry 565.3 Appendix. Equivalence between local and global intertwiners in Minkowski spacetime 65Acknowledgements 68References 691 Introduction.General Relativity is a theory of gravitation with a geometric interpretation. Asolution to the Einstein{Hilbert equations describes a curved spacetime manifold,whose curvature is related to the distribution of matter.Quantum Field Theory on the other hand arose as a theory for describing �nitelymany elementary particles and the underlying mathematical structure is that of anet of noncommutative von Neumann algebras of local observables.There have been many attempts to fuse the two theories to obtain a theory ofQuantum Gravity but, as is well known, the basic problems remain unsolved andtheir solution would seem to be still a long way o�.There is however one theory describing the e�ects of gravitation on quantum2

systems and this is Quantum Field Theory on a Curved Spacetime, where the gravi-tational �eld is treated as a background �eld so that the backreaction of the quantumsystem is ignored. Of course, this approximation cannot be expected to remain validdown to distances comparable to the Planck length.Progress in the �eld was initially hampered not only by the di�culties of handlinginteractions, well known from Minkowski space, but also through using a mathemati-cal formalism which was not general enough. Nor did it help that no really interestingphysical e�ects were found. This last point changed dramatically with the adventof Black Hole Thermodynamics and more particularly with the well known Hawkinge�ect whereby a quantum e�ect causes a black hole to radiate thermally [35, 62].More recently, the �eld has evolved rapidly on the mathematical side, too, primar-ily thanks to adopting methods and concepts from algebraic quantum �eld theory ase.g. in the work of [27, 40, 34, 46, 62]. But there have been other important develop-ments, too. In particular, the discovery by Radzikowski that the Hadamard conditionis equivalent to a wavefront set condition [50, 13] is worth mentioning. This has ledto ambitious rigorous work on perturbative quantum �eld theory in curved space-time by Brunetti and Fredenhagen [12]. Very recent work in algebraic quantum �eldtheory [19, 51] contributes to clarifying the structure of quantum �eld theories onanti-de Sitter spacetime and its conformal boundary, an issue which has nowadaysattracted great attention.The DHR analysis of superselection sectors in Minkowski spacetime is a good il-lustration of the e�ectiveness of algebraic quantum �eld theory in treating structuraland conceptual problems. The aim of this paper is to lay the foundations of superse-lection theory in quantum �eld theory on curved spacetimes and to derive some �rstresults.We �nd it advantageous to proceed by recalling, for the bene�t of the non-expertreader, the basic ideas and features of algebraic quantum �eld theory relevant to thetwo main themes of this paper: the general theory of superselection selection sec-tors and the connection between Tomita-Takesaki modular theory of von Neumannalgebras and spacetime symmetries, particularly in the context of covariant supers-election sectors. Our presentation will be simpli�ed, with full details appearing inthe main body of the paper. Readers familiar with superselection theory and therelations between modular theory and symmetry in algebraic quantum �eld theorymay wish to turn directly to the outline of the contents in Sec. 1.6 where relations toother papers are indicated.1.1 Algebraic Quantum Field Theory on Curved Space-times: General SettingIn formulating algebraic quantum �eld theory on a curved spacetime one assumes theunderlying spacetime to be described by a smooth manifoldM (of any dimension� 2)together with a Lorentzian metric g. The quantum system in question is supposedto be described by an inclusion preserving map K 3 O 7! A(O) assigning to eachmember O in a collection K of subregions of M a C�-algebra A(O). Usually, K ischosen to be a base for the topology of M (we will specify K later on).3

The motivating idea is that A(O) contains the observables which can be measuredat times and locations within the spacetime region O and that the way these algebrasrelate to each other for di�erent regions O essentially �xes the physical content of thetheory [34]The collection K of subregions of M need not be directed under inclusion, but weshall nevertheless refer to K 3 O 7! A(O) as a net of local algebras. If K is directed,then one can form the \quasilocal algebra", i.e. the smallest C�-algebra containingall the local algebras A(O). It is the norm closure of the union of the local algebras,SOA(O). In the generic case where K is not directed, this possibility is denied to us.But one can still expect Hilbert space representations of the inclusion-preserving mapK 3 O 7! A(O). More precisely, we say that a representation of K 3 O 7! A(O) is aconsistent family f�OgO2K of representations of the local algebra A(O) by boundedoperators on a common Hilbert space H�, i.e. �O1 � A(O) = �O whenever O1 � O.For the known examples of quantum �eld theories on globally hyperbolic space-times and (conformal) quantum �eld theories on S1, such representations exist inabundance. (There are indications to the contrary for non-globally-hyperbolic space-times [39, 38]. The present paper is restricted to quantum �eld theory on globallyhyperbolic spacetimes and the above notion of representation su�ces.) Every repre-sentation f�OgO2K yields states on the local algebras A(O) since each normal state! on B(H�) restricts to a state!O(A) := !(�O(A)) ; A 2 A(O) :of the local algebra. Not every consistent family of local states corresponds to a physi-cal state of the system; nor can all representations of the observable net be consideredas physical so one needs criteria to select physical representations. In practice, onebegins with some collection of physical representations and uses them to constructothers. In what follows, we compile a brief list of criteria to be ful�lled by such aninitial collection P of physical representations of the net K 3 O 7! A(O) of localobservables on a curved spacetime (M;g).1) �O, O 2 K is faithful for each f�OgO2K 2 P. Otherwise the description of thesystem by the net of local algebras K 2 O 7! A(O) would contain redundancies.2) Locality: The algebras �O(A(O)) and �O0(A(O0)) commute elementwise if theregions O and O0 cannot be connected by a causal curve.3) Irreducibility and Duality: P consists of irreducible representations, i.e. represen-tations f�OgO2K ful�lling 1 f [O2K�O(A(O)) g0 = C 1 :These representations are required to ful�ll essential duality, i.e. the net,K 3 O 7! Ad�(O) :=\O1 �O1(A(O1))0 ;1A0 = fB 2 B(H) : BA = AB 8A 2 Ag denotes the commutant of A � B(H).4

is local, where the intersection is taken over all O1 2 K causally disjoint from O.This property is stronger than locality but not as strong as Haag duality whichdemands that �O(A(O))00 = Ad�(O) for all O 2 K. This latter property means that thevon Neumann algebras �O(A(O))00 cannot be enlarged by adding elements of B(H�)without violating the locality condition.4) Local Equivalence: Whenever f�OgO2K and f�0OgO2K are two members of P, thereis for each O 2 K a unitary UO :H� ! H�0 such thatUO�O(A) = �0O(A)UO ; A 2 A(O) :5) Covariance: For each f�OgO2K 2 P there is an (anti-)unitary 2 G 3 7! U�( )of a (subgroup of) the spacetime isometry group G on H� so thatU�( )�O(A(O))U�( )� = � O(A( O)) ; 2 G; O 2 K :Obviously, if the underlying spacetime (M;g) has a trivial isometry group, this con-dition is void.If (M;g) is Minkowski spacetime, there is typically a distinguished vacuum repre-sentation �vac in P which is irreducible and covariant and possesses a cyclic vacuumvector vac 2 H�vac invariant under the action of U�vac . Moreover, a vacuum rep-resentation ful�lls the spectrum condition, i.e. the time-translations in any Lorentzframe have positive generator. In more general spacetimes, one can usually not selecta distinguished vacuum representation by similar requirements since, in the absenceof a su�ciently large isometry group, there is no analogue of the vacuum vector norof the spectrum condition. However, one expects that a collection of physical rep-resentations P can still be selected in quantum �eld theory on curved spacetimes,even if there is no single preferred representation. For a Klein-Gordon �eld on anyfour-dimensional globally hyperbolic spacetime the representations induced by purequasifree Hadamard states have been shown to form a collection P satisfying theconditions listed above [59].1.2 Superselection SectorsWe assume now that a curved spacetime (M;g), a net K 3 O 7! A(O) of localalgebras on this spacetime background and a collection P of physical representationsful�lling the conditions stated above has been given. To simplify notation, we denotea representation f�OgO2K of the net of local algebras simply by �.Picking an irreducible physical representation �0 2 P as reference, another irre-ducible representation � (not necessarily belonging toP) is said to satisfy the selectioncriterion for localizable charges if, given O 2 K, there is a unitary VO between therepresentation Hilbert spaces H� and H�0 such thatVO�O1(A) = �0O1(A)VO ; A 2 A(O1) ;2That is, U�( ) is anti-unitary if reverses the time-orientation, otherwise it is unitary5

for all regions O1 2 K causally disjoint from O. Irreducible representations whichful�ll this selection criterion and are globally unitarily equivalent are said to carrythe same charge, or to de�ne the same superselection sector.The selection criterion thus selects representations � di�ering from the referencerepresentation by some \charges" which can be localized in any spacetime region O(and are then not detectable in spacetime regions situated acausally to O). Thisform of localizability does not apply to all kinds of charges, e.g. electric charge is notlocalizable in this way (cf. [34] and references therein for further discussion). Yet forcertain general types of charges, like avours in strong interactions, this descriptionis appropriate and hence a useful starting point.The notion of localized charge and superselection sector now apparently dependson the chosen reference representation �0 (typically the vacuum representation inthe case of at spacetime), but as physical representations are required to be locallyequivalent, the charge structure, being given by the structure of the space of inter-twining operators of representations ful�lling the selection criterion, is expected tobe independent of that choice. Here, a bounded operator T : H� ! H�0 is called anintertwiner for the representations � and �0 of K 3 O 7! A(O) ifT�O(A) = �0O(A)T ; A 2 A(O) ; O 2 K :A crucial point is that the space of intertwiners admits a product having the for-mal properties of a tensor product. The statistics of the charges in the theory re ectsthe behaviour of this product under interchange of factors. Under certain generalconditions, e.g. if the Cauchy surfaces of the spacetime are not compact, each chargehas a conjugate charge and then the statistics of each charge can be characterizedby a number, its statistics parameter. This number can be split into its phase andmodulus being, respectively, the statistics phase and the inverse of the statistical di-mension. (The latter is de�ned to be 1 if the statistics parameter equals 0 andone says that the superselection sector has in�nite statistics. We shall only considersuperselection sectors having �nite statistics.) If the statistics phase takes the values�1, then the (para- )Bose/Fermi alternative holds in that there is a conventionaldescription in terms of Bose and Fermi �elds commuting or anticommuting when lo-calized in causally disjoint regions. This is the generic situation in physical spacetimedimension. In lower spacetime dimension, braid group statistics may occur and thestatistics phase may take values di�erent from �1.In previous papers [25, 24] it was shown that, in Minkowski spacetime, one canconstruct a �eld net together with a unitary action of a compact (global) gauge groupcontaining the observable net A as �xed points so that the superselection sectorscorrespond naturally to the equivalence classes of irreducible representations of thegauge group. A similar result will turn out to hold in curved spacetime as well. As solittle input is used (essentially only the physically motivated selection criterion andlocal commutativity of the observables) this result clearly demonstrates how e�ectivethe operator algebraic approach to quantum �eld theory can be.6

1.3 Covariant Sectors and Univalence (Spin)Our notion of spin on curved spacetime involves a group G of isometries althoughthere ought to be a more general notion not involving symmetries. For this reason,we assume covariance of our reference representation �0.A superselection sector described by a representation � is covariant if there existsan (anti-)unitary representation eG 3 � 7! eU�(�) of the universal covering group of Gon H� with � O�� = Ad eU�(�)��O ; � 2 eG ; O 2 K ;where � 7! denotes the covering projection.We may now consider continuous curves [0; 2�] 3 t 7! �(t) whose projection[0; 2�] 3 t 7! (t) is a cycle, i.e. a closed curve possessing no closed sub-curves.eU�(�(2�)) may be di�erent from 1, but as � is irreducible, eU�(�(2�)) = s� � 1 wheres� is a complex number of modulus 1. When the cycle ([0; 2�]) has the geometricinterpretation of a \spatial rotation by 2�", then it is appropriate to refer to the phasefactor s� as the \spin", or more precisely, the univalence of the charge representedby �. 3 Then, the spin-statistics connection is said to hold if, for all covariantsuperselection sectors of the theory, the univalence equals the statistics phase.1.4 Tomita-Takesaki Theory and SymmetryLet us next summarize some basic points of the modular theory for von Neumannalgebras by Tomita and Takesaki [58]. Given a von Neumann algebra N on a Hilbertspace H together with a cyclic and separating unit vector 2 H, the antilinearoperator S : N! N de�ned by S(A) := A� admits a minimal closed extensionwith polar decomposition S = J�1=2 where J is anti-unitary. J is referred to asmodular conjugation and f�itgt2Ras modular unitary group associated with the pairN;; one refers to AdJ as the antilinear modular morphism associated with N; andusually denotes it by j.These modular objects satisfy JNJ = N0 and �itN��it = N,t 2 R. Moreover, a state ! on a C�-algebra A is a KMS-state (thermal equilibriumstate) at inverse temperature � with respect to a one-parametric group f�tgt2R ofmodular automorphisms of A if and only if�!��t = Ad��i�t=2� � �!where �! is the GNS-representation of ! and f�itgt2R is the modular group of�!(A)00;!, ! being the GNS-vector. Thus the modular group may, in certainsituations, have a physical (dynamical) signi�cance.Furthermore, Bisognano and Wichmann showed [4] that, in Wightman's settingof quantum �elds in Minkowski spacetime, the modular objects associated with pairsA(W );, whereA(W ) is the von Neumann algebra of observables in a certain \wedge-3We do not wish to discuss how s� depends on the di�erent possible \rotations". It su�ce to saythat in the relevant cases the above procedure assigns an invariant s� to any covariant superselectionsector. 7

region" 4 W and the vacuum vector, induce spacetime transformations. That is,if JW ; f�itWgt2R denote the corresponding modular objects, then there are elementsjW , �W;t in the Poincar�e group so thatAdJW A(O) = �jW (A(O)) = A(jW (O)) ; (1.1)Ad�itW A(O) = ��W;t(A(O)) = A(�W;t(O)); (1.2)for all open subregions O of Minkowski spacetime, all t 2 R and all wedge-regions W .Further investigations (e.g. [5, 30, 17, 15, 8]) relate spacetime symmetries andmodular objects and indicate that vacuum states in Minkowski spacetime can possi-bly be characterized through the geometricmeaning of the modular objects associatedwith A(W ); for a certain class of wedge-regions W . This idea has been pursued innon- at spacetimes with a su�ciently rich group of isometries and a suitable class ofwedge-regions, such as de Sitter spacetime and, to some extent, Schwarzschild-Kruskalspacetime, too [56, 10, 9]. There are indications that physical states of quantum �eldtheory on arbitrary spacetime manifolds can be distinguished by the \geometricalaction" of the corresponding modular objects for a certain class of regions, under-stood in su�cient generality. The reader is referred to [18] and references therein forconsiderable further discussion.In Minkowski spacetime, the geometric action of the modular objects associatedwith wedge-algebras A(W ) and the vacuum vector has important consequencesfor the relation between spin and statistics. It can be derived either from geometricmodular action [42], i.e. the geometric action of the modular conjugations as in (1.1),or from modular covariance [31], meaning the geometric action of the modular groupas in (1.2). Similarly, for conformal quantum �eld theories on the circle S1 wheremodular objects and conformal symmetry are intimately related, there is a spin-statistic relation, as will be brie y summarized in the next section.1.5 Modular Inclusion and Conformal Theories on the CircleIn this section we summarize the connection between conformally covariant theorieson the circle S1 and halfsided modular inclusions established by Wiesbrock [64, 65,66, 67].We brie y recall what is meant by a conformally covariant theory on the circleS1 (see e.g. [30, 32] for further details). This is a net (or precosheaf) I 7! M(I)taking proper open subintervals I of S1 to von Neumann algebras M(I) on a Hilbertspace HM so that locality holds, i.e. M(S1nI) � M(I)0. Moreover, there exists aunitary strongly continuous positive energy representation U of PSL(2;R) actingcovariantly, U(g)M(I)U(g)� = M(gI), and preserving a unit vector M, cyclic forthe von Neumann algebra generated by the M(I)'s. (In other words, the theory isgiven in a reference \vacuum representation".)The theory may be equivalently described as a net of von Neumann algebras in-dexed by intervals on the real line, identi�ed as the circle with one point removed.4A wedge region is any Poincar�e transform of the set f(x0; : : : ; xn) : 0 < x1; 0 � jx0j < x1g inMinkowski spacetime. 8

Using the Cayley transform, conformal transformations on the circle correspond tofractional linear transformations on the line. Modular transformations have a geomet-ric meaning and Haag duality holds for any conformal theory on the circle, namelyM(S1 n I) = M(I)0 [14]. Haag duality on the line, M(R n I) = M(I)0, holds pre-cisely when the net I 7!M(I) is strongly additive[33], i.e. if M(I) = M(I1) _M(I2)whenever the union of I1 and I2 yields I up to at most a single point.We recall that a �hsm inclusion (N � M;) is given by a pair N � M of vonNeumann algebras on some Hilbert space together with a unit vector , cyclic andseparating for both N and M, such that �itN��it � N for all �t � 0, where �it,t 2 R, is the modular group ofM;. A�hsm inclusion (N �M;) is called standardif is cyclic for N0 \M, too (hsm abbreviates \half sided modular").An interesting result of Wiesbrock ([64, 65] see also [33]) asserts that there is aone-to-one correspondence between strongly additive conformally covariant theorieson S1 and standard �hsm inclusions.The rotations of S1 form a subgroup of the covering group of PSL(2;R). Let� be a Hilbert space representation of a covariant superselection sector and eU� theassociated unitary representation of the covering group of PSL(2;R). Assuming thateU� has positive energy, the generator of rotations in the unitary representation eU�has a lowest eigenvalue L�. Then the conformal spin of the superselection sector,or rather, its univalence, is de�ned by s� = e2�iL� . For superselection sectors withpositive energy in a conformally covariant theory on S1, the univalence equals thestatistics phase, which may be any complex number of modulus 1 [32].1.6 Description of ContentsWe now describe the contents of the subsequent chapters.In Chapter 2 we summarize several notions of spacetime geometry needed here.Lemma 2.2, of relevance to superselection theory, asserts that the set of pairs ofcausally separated points in a globally hyperbolic spacetime is connected.Chapter 3 contains the general framework for superselection theory in curvedspacetimes, patterned conceptually on the DHR analysis in Minkowski spacetime([23], cf. also [34, 53] and references given there). It will be formulated for netsK 3 O 7! A(O) of operator algebras in a reference representation with generalindex sets K possessing a partial ordering and a causal disjointness relation. Thusquantum �elds on arbitrary globally hyperbolic spacetimes in any dimensions, withcompact or non-compact Cauchy surfaces, as well as quantum �eld theory on thecircle, can be treated on an equal footing. The existence of statistics is establishedin this generality. If the index set K is directed, all the other basic results known forsuperselection theory on Minkowski spacetime, classi�cation of statistics, existenceof charge conjugation and construction of �eld algebra and gauge group (cf. [25]) canagain be shown to hold.Chapter 4 begins with a summary of the geometry of spacetimes with a bifurcateKilling horizon following Kay and Wald [40]. We introduce a family of wedge-regionsRa, a > 0 which are copies of the canonical right wedge shifted by a in the a�negeodesic parameter on the horizon (a similar construction can be carried out for the9

left wedge). We suppose that we are given a net of von Neumann algebras O 7! A(O)in the representation of a state which is, in restriction to the subnet of observableswhich are localized on the horizon, a KMS-state at Hawking temperature for theKilling ow. Thus on the horizon we have modular covariance and are consequentlyin Wiesbrock's situation of half-sided modular inclusion [64]. Using Haag duality andadditivity of the net, it follows that the maximal subnet of observables localized onthe horizon is a conformally covariant family of von Neumann algebras. Restrict-ing the original net of von Neumann algebras to the Killing horizon thus yields aconformal quantum �eld theory on S1. A conformal spin is therefore assigned to asuperselection sector of the original theory, localizable on the horizon, and the confor-mal spin-statistics connection [32] holds. This approach has, however, the drawbackof applying only to horizon-localizable charges, and this may be quite restrictive.In Chapter 5 we introduce a class of spacetimes with a special rotation symmetryand certain adapted wedge-regions. Essentially we assume that there is a group ofsymmetries, to be viewed as rotations, generated by pairs of time-reversing wedge-re ections mapping wedge-regions onto each other. In the Schwarzschild-Kruskalspacetime, for example, these wedge-regions can be envisaged as the causal comple-tions of \halves" of the canonical Cauchy-surfaces chosen so that rotating by � abouta suitable axis maps each such half onto its causal complement. These wedge-regionsdi�er from the usual canonical \right" and \left" wedges (R and L in Chapter 4)and lie in a sense transversal to the latter. Then we consider a net of von Neumannalgebras O 7! A(O) over such a spacetime in a representation where the full isome-try group acts covariantly. Moreover we suppose that there is an isometry-invariantstate and that the modular conjugations associated with the vacuum vector and thevon Neumann algebras A(W ) for the said class of wedges W induce the geometricaction of the wedge-re ections. This form of geometric modular action will allow usto de�ne the rotational spin of a covariant superselection sector and to derive thespin and statistics connection using a variant of arguments presented in [45].2 Some Spacetime Geometry2.1 GeneralitiesIn the present section we summarize some notions about causal structure of Lorent-zian manifolds, thereby establishing our notation. Standard references for this sectioninclude [3, 36, 49, 61].We begin by recalling that a curved spacetime (M;g) is a 1 + s-dimensional(s 2 N), Lorentzian manifold. In other words, it is a 1 + s-dimensional orientable,Hausdor�, second countable C1-manifold equipped with a smooth Lorentzian metricg having signature (+;�; : : : ;�).A continuous, (piecewise) smooth curve : I !M , de�ned on a connected subsetI of R and having tangent _ , is called a timelike curve whenever g( _ ; _ ) > 0, a causalcurve if g( _ ; _ ) � 0, and a lightlike curve if g( _ ; _ ) = 0 while _ 6= 0, for all parametervalues t. 10

A spacetime (M;g) is called time-orientable if there exists a global timelike (non-vanishing) vector�eld � on M . Such a vector �eld induces a time-orientation: acausal curve is called future-directed or past-directed according as g(�; _ ) > 0 org(�; _ ) < 0. We shall henceforth tacitly assume our spacetimes to be time-orientablewith a given time-orientation.A future-directed causal curve : I !M is said to have a future (past)-endpointif (t) converges to some point inM as the parameter t approaches sup I (inf I). Cor-respondingly one de�nes the past (future)-endpoints of past-directed causal curves.A future (past)-directed causal curve is said to start at a point p 2M provided that pis the past (future)-endpoint of . Moreover, one calls a future (past)-directed causalcurve future (past)-inextendible if it possesses no future (past)-endpoint.For any subset O of M one de�nes the sets J�(O) as consisting of all pointsin M lying on future(+)/past({)-directed causal curves that start at some pointin O. Then J�(O) are called the causal future(+)/causal past({) of O. The setJ(O) := J+(O) [ J�(O) is then referred to as the causal set of O. The subsetsD�(O) of M are, for given O � M , de�ned as the collection of all those pointsp 2 M such that every past(+)/future({)-inextendible causal curve starting at pmeets O. One calls D�(O) the future(+)/past({)-domain of dependence of O, andD(O) := D+(O) [D�(O) the domain of dependence of O.One says that two points p and q in M are causally disjoint, in symbols p ? q, ifthere are open neighbourhoods U of p and V of q such that there is no causal curveconnecting U and V (i.e. U \ J(V ) = ; = V \ J(U)). Correspondingly one calls twosubsets P and Q of M causally disjoint if p ? q holds for all pairs p 2 P and q 2 Q;this will be abbreviated as P ? Q.In the present paper we will primarily be interested in globally hyperbolic space-times. A spacetime (M;g) is globally hyperbolic if it can be smoothly foliated inacausal Cauchy surfaces. Here, an acausal Cauchy surface C is a smooth hypersur-face in M such that each causal curve in (M;g) without endpoints meets C exactlyonce. This implies that C is indeed acausal, i.e. p ? q holds for all distinct p; q 2 C.By a (smooth) foliation of (M;g) in acausal Cauchy surfaces we mean a di�eomor-phism F : R � � ! M where � is an s-dimensional smooth manifold such thatF (ftg � �) is, for each t 2 R, an acausal Cauchy surface in (M;g), and the curvesR 3 t 7! F (t; q), q 2 �, are timelike and endpointless. Thus, the foliation-parametert plays the role of a \time-parameter". One may give a broader de�nition of Cauchysurfaces which are not necessarily acausal, by de�ning a Cauchy surface as a C0 hy-persurface C such that C \ intJ�(C) = ; and D(C) = M . With this de�nition, aCauchy surface is allowed to have lightlike parts. Such a broader de�nition of Cauchysurfaces is often useful. However, it is a remarkable fact that the existence of a sin-gle, not necessarily acausal Cauchy surface in (M;g) already implies that (M;g) isglobally hyperbolic in the above sense [29, 21, 61].Whilst the question of whether physical spacetime models are necessarily globallyhyperbolic has been discussed in the literature (see [20, 61, 63] and references giventhere), it is certainly the case that a great number of the prominent spacetime modelsare globally hyperbolic, like Schwarzschild-Kruskal, deSitter, the Robertson-Walkermodels, and many others, including of course Minkowski spacetime. One may there-11

fore regard the class of globally hyperbolic spacetimes as being su�ciently generaland comprising many examples of physical interest. Note that global hyperbolicityin no way presupposes the presence of spacetime symmetries.At this point we recall some properties of causal sets; for their proof and furtherdiscussion, we refer to the indicated references. Whenever N � M and (M;g) isglobally hyperbolic, then: N compact implies J�(N) closed, N compact implies thatJ(N)\C is compact for each Cauchy surface C, N compact impliesD(N) compact.Furthermore, J+(N+) \ J�(N�) is empty or compact for all compact N+; N� �M . Moreover, in (time-orientable) spacetimes (M;g), a time-orientation preservingisometry � of (M;g) satis�es � (J�(O)) = J�(� (O)) ; (2.1)for O � M . It is moreover worth mentioning that for any two subsets P and Q of aglobally hyperbolic spacetime (M;g) we have P ? Q if and only if P � Q?, wherethe causal complement Q? of Q � M is de�ned by Q? := MnJ(Q), see e.g. [41,Prop. 8.1].We need to consider special regions of a globally hyperbolic spacetime (M;g)namely those causally closed regions generated by an open subset of a Cauchy surface.More particularly we are interested in regular diamonds de�ned as follows. A set ofthe form O = intD(G) is a regular diamond provided O? is non-void and(i) G is an open subset of an acausal Cauchy-surface C, and G is compact andcontractible to a point in G,(ii) @G, the boundary ofG, is a (possibly multiply connected) locally at embedded,two-sided topological submanifold of C which is an embedded C1-submanifoldnear to points in each of its connected components.We refer to [11, 60] for the precise de�nition of locally at embeddings and two-sidedness. Intuitively, these two conditions are substitutes for the existence of anoriented normal vector �eld over @G. These regularity properties serve to prove thefollowing assertion:Lemma 2.1. Let O be a regular diamond and p 2 O?. Then there exists anotherregular diamond O1 with O [ fpg � O1 :A rough sketch of the proof will be given in Sec. 2.2, the Appendix to this chapter.The reader is referred to [60] for a detailed proof.A double cone in Minkowski space is, of course, a regular diamond. Doublecones may be generalized easily to curved spacetime. They are sets of the formint (J�(fv+g) \ J+(fv�g)) with v+ 2 intJ+(fv�g). However, double cones need nothave the property analogous to Lemma 2.1, think e.g. of a spacelike strip in Minkowskispacetime. Nor is it clear that a double cone is a regular diamond. For this reason,it is not clear, even for simple free �elds, whether duality is satis�ed for such regions.We expect the requirement of essential duality (cf. Sec. 1.1) to be realistic for regular12

diamonds, in particular, as their bases are assumed contractible. Furthermore, dualityfor regular diamonds has already been established for the Klein-Gordon �eld[57] andcan presumably be veri�ed for other free �elds. For these reasons, we have chosento use the collection K of regular diamonds rather than the collection of doublecones whose causal complement has non-empty interior as an index set in a globallyhyperbolic spacetime.Given a spacetime (M;g), we introduce the setXM;g := f(x; y) 2M �M : x ? yg (2.2)of pairs of causally disjoint points in M . According to the de�nition of causal dis-jointness, this set is an open subset of M � M . The subsequent assertion aboutXM;g will prove to be important in discussing the statistics of superselection sectorsin the next chapter. It may be known to experts, but as we have not found it in theliterature, we put it on record here.Lemma 2.2. Let (M;g) be a globally hyperbolic spacetime then XM;g is pathwiseconnected except when its Cauchy surfaces are noncompact and 1{dimensional inwhich case there are precisely two path{components corresponding to x being causallyto the left or to the right of y.Proof. Let F : R � � ! M be a foliation in acausal Cauchy surfaces and writeC := F (f0g � �). We �rst show that it su�ces to restrict one's attention to theCauchy surface C. More precisely, we show thatY := f(x; y) 2 C � C : x ? ygis a strong deformation retract of XM;g. In fact, using F to parametrize M andde�ning h : XM;g � I ! XM;g byh(t; �; t0; �0; s) := ((1� s)(t+ s(t0 � t)); �; (1� s)t0; �0)we have a homotopy of the identity on XM;g onto the projection, (t; �; t0; �0) 7!(0; �; 0; �0), onto C leaving C �xed. The only non{trivial point is to show that theimage of h lies in XM;g and this is where the causal structure enters. However, tworemarks su�ce: �rst, causal disjointness reduces to disjointness on an acausal Cauchysurface and hence is preserved if we pass from one acausal Cauchy surface to anotherby changing the value of t. Secondly, if we take causally disjoint points xi = F (ti; �i),i = 1; 2 with distinct values of t then the curve : [infft1; t2g; supft1; t2g] 3 t 7!F (t; �1) is timelike and connects x1 with that Cauchy surface of the foliation con-taining x2. Its range must lie in fx2g? or there would be a causal curve comingarbitrarily close to connecting x1 and x2, contrary to assumption. We now knowthat the inclusion of Y in XM;g induces an isomorphism in homotopy and, in par-ticular, an isomorphism of path-components. Now unless C is one dimensional andnon{compact, the complement of a point of C is path{connected and Y is then alsopath{connected. If C is one dimensional and non{compact it is isomorphic to R sothat Y has two path{components. 13

When XM;g has two components, we use the foliation F : R�R!M into acausalCauchy surfaces to distinguish the \right" component from the \left" component asthat containing pairs (x; y), where the spatial component of y is greater than that of x.In fact, this distinction depends only on the nowhere vanishing spacelike vector �eld �induced by the foliation. Given such a �eld �, a spacelike curve I 3 t 7! (t) is calledright-directed if g(�; _ ) > 0 and left-directed if g(�; _ ) < 0 for one and hence all valuesof t. (A di�erent choice of � would at most lead to interchanging \right-directed" and\left-directed" since in two spacetime dimensions the set of spacelike vectors at eachpoint has two components.) The orientation of spacelike curves de�ned in this waycan now be used to specify the two connected components of XM;g in the case of anon-compact Cauchy surface. The right component is that containing ( (0); (1)) forthe endpoints (0) and (1) of some and hence any right-directed spacelike curve .This follows from the previous description in terms of the foliation since the spatialcomponent is strictly increasing along such a curve.2.2 Appendix to Chapter 2Proof of Lemma 2.1 (Sketch)Let O = intD(G) be a regular diamond,G � C where C is an acausal Cauchy-surface,and p 2 O?.Choose a C1-foliation F : R � � ! M of M into smooth, acausal Cauchysurfaces. Then for each y 2 �, the curves t 7! F (t; y) are inextendible, future-directed timelike curves. Therefore, given any acausal Cauchy surface C0, each ofthese curves intersects C0 exactly once, at the parameter value t = �C0(y). Thefunction �C0 : �! R is a smooth function and one has C0 = fF (�C0(y); y) : y 2 �g.Furthermore, the map �C;C0 : C ! C0 induced by F (�C(y); y) 7! F (�C0(y); y) is adi�eomorphism.Using the results of [11], one can show that there is an open neighbourhood U ofG in C possessing the same properties (i) and (ii) as G, i.e. U is the base of a regulardiamond. It is also not di�cult to show (cf. [60]) that there exists an acausal Cauchysurface C0 containing p and with the additional property thatJ(G) \ C0 � �C;C0(U) =: U0 :The latter property means there are acausal Cauchy surfaces C0 passing through pand coming arbitrarily close to G. This entails that O0 := intD(U0) contains O.Since �C;C0 is a di�eomorphism, U0 satis�es (i) and (ii) with respect to the Cauchysurface C0.It remains to show that U0 [ fpg is contained in a subset U1 of C0 satisfying (i)and (ii) with respect to the Cauchy surface C0. This is done by connecting a pointin a smooth part of @U0 by a smooth curve � to p and by attaching to U0 a suitablesmooth deformation of a tubular normal neighbourhood of �. This yields the requiredset U1; properties (i) and (ii) follow by construction as doesO [ fpg � intD(U1) =: O1 :14

3 Superselection Structure in Curved Spacetimes3.1 IntroductionIn this section, we adapt the basic notions and results of the theory of superselectionsectors to curved spacetime, limiting ourselves to globally hyperbolic spacetimes. Aswe shall see, the basic theory goes through smoothly in the case of globally hyperbolicspacetimes with a noncompact Cauchy surface and much of it in the case of a compactCauchy surface. The geometry of spacetime fortunately enters the long analysis onlyin establishing a few speci�c points. We can therefore limit ourselves to clarifyingthese points and otherwise just quoting the consequences.We let K denote the set of regular diamonds in M , ordered under inclusion. IfM is globally hyperbolic with a non-compact Cauchy surface, K may not be directedalthough it will be in cases of interest. However, when M is globally hyperbolic witha compact Cauchy surface, K will never be directed and we shall meet problems akinto those on the circle. The more complicated structures involved have been relegated,as far as possible, to the appendix to this chapter.The set of double cones in M whose causal complement has non-empty interioris even less likely to be directed. Both sets have in common that they form a basefor the topology of M and we will consider our nets of observables as being de�nedover K with the general philosophy that they can be extended to other regions, ifnecessary. In fact, we will consider a wider class of regions in subsequent chapters.Now, the geometry of spacetime enters the analysis only through the partially orderedset K and its relation of causal disjointness, introduced below. In view of furtherapplications and despite the degree of abstraction involved, we have emphasised therelevant properties of K.The selection criterion for localized charges in Minkowski space uses the vacuumrepresentation as a reference. Although there is no such preferred representation incurved spacetime, one expects there to be a preferred collection of representationssatisfying the conditions listed in Sec. 1.1. In the case of the Klein-Gordon �eld ona four dimensional globally hyperbolic spacetime, we may take the representationsinduced by the pure quasifree Hadamard states[59]. We shall choose one of theserepresentations as our reference representation and, whilst our sectors will depend onthis choice, the superselection structure will not since this depends only the net ofvon Neumann algebras. By 4) of Sec. 1.1, any two preferred representations generatethe same net of von Neumann algebras. We will denote our reference representationby �0 and its Hilbert space by H0.Once the reference representation has been �xed, it is just the causal structure ofMinkowski space that plays a role in the superselection criterion for localized charges.For this reason, it adapts well to curved spacetime. The causal structure enters inthe form of the relation ? of causal disjointness, de�ned in Ch. 2, and here to beconsidered as a relation on the ordered set K, satisfyinga) O1 ? O2 ) O2 ? O1.b) O1 � O2 and O2 ? O3 ) O1 ? O3. 15

c) Given O1 2 K, there exists an O2 2 K such that O1 ? O2.We write O? := fO1 2 K : O1 ? Og.As explained above, the geometry of spacetime enters through the partially or-dered set K together with the relation ? of causal disjointness. Hence we have topass from geometric or topological properties of (M;g) to properties of (K;?). Wewill need to know whether certain partially ordered sets are connected, a notion de-�ned in the appendix. But the basic idea is to move from one element O1 of K to anearby element O2, where nearby means that there is a third element O3 containingO1 and O2. A �nite series of such moves constitutes a path. K is connected if anytwo elements can be connected by a path. By virtue of Lemma 3A.1, we know thatK is connected and, see Lemma 2.2, that O? is connected except when M is twodimensional with a non{compact Cauchy surface.Lemma 2.2, itself, asserts that the set XM;g of pairs of spacelike separated pointsis pathwise connected again unlessM is two dimensional with a non{compact Cauchysurface. Since pairs of elements of K form a base for the topology in the productspace, we can again conclude by Lemma 3A.1 that the graph G? of the relation ? isconnected, G? = fO1 � O2 : O1 ? O2g:In the exceptional case, XM;g has two pathwise connected components. Indeedthe causal complement of a point is no longer connected but decomposes into a `left'causal complement and a `right' causal complement.These are the basic geometric considerations determining the statistics. The re-maining condition used in Sec. 3.3, the surjectivity of the projection from G?c , aconnected component of G?, to K has no geometric relevance seeing that it is au-tomatically satis�ed in the context of globally hyperbolic spacetimes. Thus, as willfollow from the results of Sec. 3.3, in a globally hyperbolic spacetime of dimensiongreater than 2, we get a net of symmetric tensor W �{categories, (Tt; "c), whereasin a 2{dimensional spacetime we shall in general get a braided tensor W �{categorywith two di�erent braidings "` and "r corresponding to the left and right causalcomplements of a double cone. Obviously, "` = "r�, where "� is de�ned by"�(�; �) = "(�; �)�:The next basic step is to establish the properties of charge conjugation. Thebasic tool here is a left inverse. The physical idea behind constructing left inverses isthat of transferring charge to spacelike in�nity and a geometric property is obviouslyinvolved. Expressed as a property of our partially ordered set K we need to assumethe existence of a net On of elements of K such that given O 2 K there exists an n0with On ? O for n � n0. We will say that a net On tends spacelike to in�nity. Such anet obviously exists whenever K is directed but it continues to exist for an arbitraryglobally hyperbolic spacetime with a noncompact Cauchy surface. The question ofwhether one can �nd a suitable substitute for globally hyperbolic spacetimes withcompact Cauchy surfaces is still open, a defect mitigated by the circumstance that16

a left inverse exists as a consequence of the equality of local and global intertwiners,postulated in Ch. 5.In this way, we establish in Sec. 3.4, the classi�cation of statistics and the existenceof charge conjugation for �nite statistics for the case of a globally hyperbolic spacetimeof dimension greater than two.3.2 The Selection CriterionOur discussion of superselection theory in this and in subsequent sections is in terms ofa partially ordered set K together with a binary relation ?. The necessary propertieswill be introduced as needed and there will be no speci�c reference to spacetime.We have adopted this procedure for clarity and with future applications in mind.Thus the best choice of K in a curved spacetime is not altogether clear. We havealready, for example, thought �t to use regular diamonds in place of double cones.On the other hand, we might like to go beyond strictly localized charges and workwith spacelike cones or to replace causal disjointness by its Euclidean counterpart,disjointness, as when working on the circle. In fact, we shall need to use results onsuperselection structure on the circle in Ch. 4 and, although these results have beendeveloped previously [28], [32], the formalism presented here includes this case andallows a uniform approach to all such problems. We shall also simplify the expositionby making use of the freedom to modify the binary relation on K. Thus this degreeof abstraction is now called for even if we have not been able to derive all results inan adequate generality.5Two nets A and B of �{subalgebras of B(H0) over K are said to be relatively localif A(O1) � B(O2)0; whenever O1 ? O2:This relation ful�lls the analogues of a), b) and c) above. Furthermore, there is amaximal net, the dual net Ad, which is relatively local to A. It is given byAd(O) = \fA(O1)0 : O1 ? Og:Since Add is the largest net local relative to Ad, A � Add. However A � B impliesBd � Ad, so that Ad = Addd. A net A is said to be local if A � Ad and thenAdd � Ad = Addd so that Add is local, too. We now compute the double dual:Add(O) = \O1?OAd(O1)0 = \O?O1 _O?O1 A(O):De�nition. A representation � of the net A is said to satisfy the selection criterionif � � O? ' �0 � O?; O 2 K:5Baumg�artel and Wollenberg[2] treat nets over partially ordered sets with a relation of causaldisjointness. In their applications to superselection structure they assume among other propertiesthat the partially ordered set is directed. When the partially ordered set is not directed, their notionof representation depends on a choice of enveloping quasilocal algebra.17

When K is directed this means that for each O there is a unitary VO such thatVO�(A) = AVO; A 2 A(O1); O1 2 O?;where, to simplify notation in the sequel, we have omitted the symbol �0 for thereference representation. We write T 2 (�; �0) to mean that T intertwines the rep-resentations � and �0 and let Rep?A denote the W �{category whose objects are therepresentations of A satisfying the selection criterion and whose arrows are the in-tertwiners between these representations. As far as superselection theory goes, thefollowing result allows one to replace the original net by its bidual.The Extension Theorem If each O? is connected, every object � of Rep?A ad-mits a unique extension to an object of Rep?Add. Furthermore there is a canonicalisomorphism of the corresponding W �{categories.This result is proved as Theorem 3A.4 of the Appendix. How to proceed when O?is not connected is exempli�ed by the well known case of a two dimensional Minkowskispace and we will not attempt a general analysis here. The theory of superselectionstructure rests on two assumptions. The �rst is a property derived by Borchers inMinkowski space as a consequence of additivity, locality and the spectrum condition.Here it involves the dual net, Ad.De�nition A net Ad satis�es Property B if given O, O1 and O2 in K such thatO ? O2, and O; O2 � O1 and a projection E 6= 0 in Ad(O), there is an isometryW 2 Ad(O1) with WW � = E.Lemma 3.1 If Ad satis�es Property B, the set of representations satisfying the se-lection criterion is closed under direct sums and (non-trivial) subrepresentations. Inother words, the W �{category Rep?A has direct sums and (non{zero) subobjects.The proof of this lemma will be omitted as it in no way di�ers from its Minkowskicounterpart. The characteristic assumption of superselection theory is a duality as-sumption.De�nition A net A is said to satisfy duality if A = Ad and essential duality ifAdd = Ad.To simplify notation, we shall suppose here that our net satis�es duality but, asa consequence of the Extension Theorem, the results remain valid under the weakerassumption of essential duality, whenever each O? is connected.In the Appendix, we have adopted the cohomological approach to superselectionstructure as this provides the most natural expression of the selection criterion. Inthe main text, we shall pursue the alternative strategy of working in terms of localizedendomorphisms rather than 1{cocycles.3.3 Localized EndomorphismsWhen K is directed, the analysis of superselection structure rests on the followingsimple construction: let � be a representation satisfying the selection criterion, picka unitary VO as above and set�(A) := VO�(A)V �O; A 2 A:18

Obviously � is a representation of A on H0 unitarily equivalent to � but, in fact,�(A) � A. To see this, pick O1; O2 2 K, O1 � O, O1 ? O2 and B 2 A(O2) then,writing V for VO,�(A)B = V �(A)V �B = V �(AB)V � = V �(BA)V � = BV �(A)V � = B�(A);Hence �(A) 2 Ad(O1) = A(O1), as required. Furthermore, � is localized in O, i.e.�(AB) = �(A)B; B 2 A(O1); A 2 A; O1 ? Oand we refer to � as a localized endomorphism. Now if � and �0 are localized endo-morphisms, an intertwiner R for the corresponding representations is automaticallyin A. For suppose � and �0 are localized in O and A 2 A(O1), O1 ? O, thenRA = R�(A) = �0(A)R = ARso that R 2 Ad(O) = A(O).We can thus write R 2 (�; �0) without specifying whether we treat � as a repre-sentation or as an endomorphism and, when studying superselection sectors, Rep?Amay be replaced by the full subcategory Tt of EndA. EndA is a tensor C�{categoryand we use the tensor product notation. Thus if S 2 (�; �0), we write R S todenote the intertwiner R�(S) 2 (��; �0�0). We characterize Tt by characterizing thecorresponding set �t of endomorphisms. The representation corresponding to � 2 �tsatis�es the selection criterion precisely when, given O 2 K, there is an equivalentendomorphism � localized in O. We then call � transportable since, transporting �by a suitable unitary U 2 A, it can be localized in any given O 2 K. �t is thus theset of transportable localized endomorphisms and �t(O) shall denote the subset ofendomorphisms localized in O.Lemma 3.2 If �; �0 2 �t then ��0 2 �t.Proof. As the product of endomorphisms localized in O is again localized in O, itsu�ces to observe that if U 2 (�; �) and U 0 2 (�0; �0) are unitary then U U 0 2(��0; ��0) is unitary.Thus the unitary equivalence class of ��0 depends only on the unitary equivalenceclasses of � and �0 and, regarding charge as the quality distinguishing one sector fromanother, this de�nes a composition of charges.When K is not directed, this simple scheme must be modi�ed. The basic com-plication is that localized endomorphisms are now not de�ned on the whole net A.Instead, an endomorphism � localized in O is just de�ned on the net O1 7! A(O1)with O � O1 and has the property that �(A(O1)) � A(O1). As explained in detailin the Appendix, we have a net O 7! Tt(O) of tensor W �{categories, the objects ofTt(O) are the transportable endomorphisms localized in O.It is also shown in the Appendix how a representation � satisfying the selectioncriterion gives rise to objects of Tt(a), a 2 �0 and how an interwiner T 2 (�; �0) be-tween two such representations leads to arrows ta, a 2 �0, between the corresponding19

objects of Tt(a). We can no longer study superselection sectors replacing RepA? byTt(O), more precisely, we have a faithful �{functor from RepA? to Tt(O) but cannotassert that it is an equivalence ofW �{categories. Thus, when K is not directed, Tt(O)may not give a description of superselection sectors. Nevertheless, as we shall see, ananalysis of localized endomorphisms still provides useful information.The basic step in this analysis is to investigate the relation between causal dis-jointness and commutation of localized endomorphisms and their intertwiners. It isnatural to say that an intertwiner T 2 Tt(O) is localized in O, but we need a �nernotion because we may have T 2 (�1; �0) where �i 2 �t(Oi) with Oi � O. In thiscase, we refer to O1 as being an initial support and O0 as being a �nal support ofT . As explained in the Appendix, we consider the set �1 of 1{simplices in K as apartially ordered set and let �?1 denote the subset of 1{simplices b with @1b ? @0bwith the induced order.Lemma 3.3 Let �?1;c be a connected component of �?1 , and suppose that given O0 2 K,there is a b 2 �?1;c with @0b = O0. Let Ti 2 (�i; �0i) be arrows in some Tt(O) thenT0 T1 = T1 T0;if there are b; b0 2 �?1;c so that @0b and @1b are initial supports of T0 and T1 and @0b0and @1b0 are �nal supports of T0 and T1.Proof. We �rst show that T0�0(T1) = T1�1(T0). This relation is trivial if T0 and T1are causally disjoint in the sense that there is a b 2 �?1 such that @0b contains aninitial and �nal support of T0 and @1b an initial and �nal support of T1. The ideaof the proof is to reduce to this trivial case. Replace T0 and T1 by T2 = T0 � U0 andT3 = T1 � U1, where U0 2 (�2; �0) and U1 2 (�3; �1) are unitary. ThenT2 T3 = T0 T1 � U0 U1; T3 T2 = T1 T0 � U1 U0;to be understood as valid in some Tt(O) for O su�ciently large. Thus if U0 and U1are causally disjoint, the validity or not of our relation is una�ected by the passagefrom T0 , T1 to T2, T3. But b and b0 lie in a connected component �?1;c by hypothesis,so after a �nite number of steps we can arrange that the initial and �nal supports ofboth intertwiners coincide. This is again the trivial case so T0�0(T1) = T1�1(T0), asrequired. It only remains to show that�0�1 � �1�0 = 0:The above computations show that the kernel of the left hand side does not changeif we shift to �2 and �3. However, by hypothesis, given O � b0, we can �nd b 2 �?1;cwith @0b = O and we can take �3 2 �t(@1b), when�0�3(A) = �0(A) = �3�0(A); A 2 A(O);completing the proof. 20

After this one crucial lemma, the standard results on the existence of a braid-ing follow without further geometric input. Of course the braiding will, in general,continue to depend on the choice of connected component.Theorem 3.4 Let �?1;c be a connected component of �?1 . If the projection mappingb 7! @0b from �?1;c to K is surjective then there is a unique intertwiner-valued function(�0; �1) 7! "c(�0; �1) 2 (�0�1; �1�0) such thata) "c(�00; �01) � T1 T2 = T2 T1 � "c(�0; �1); Ti 2 (�i; �0i); i = 0; 1;b) "c(�0; �1) = 1�0�1 , if there is a b 2 �?1;c such that �i 2 �t(@ib) i = 0; 1.Proof. The uniqueness claim tells us how to go about de�ning "c: given �1; �2 pickb 2 �?1;c and unitaries Ui 2 (�i; �i) where �i 2 �t(@ib) and we have no option but toset "c(�1; �2) = U�2 U�1 � U1 U2:By Lemma 3.3, such a choice, however made, automatically satis�es b). We have"c(�01; �02) = U 02�U 01� �U 01U 02, where U 0i 2 (�0i; � 0i ) and the product of supports of � 01and � 02 is contained in Xc. Set Si = U 0i �Ti�U�i then, by Lemma 3.3, S1S2 = S2S1and rearranging this identity gives a) and completes the proof of the theorem.Corollary 3.5 Under the hypothesis of Theorem 3.4a) "c(�1�2; �3) = "c(�1; �3) 1�2 � 1�1 "c(�2; �3);b) "c(�1; �2�3) = 1�2 "c(�1; �3) � "c(�1; �2) 1�3 ;If b 2 �?1;c implies �b 2 �?1;c, where j�bj = jbj, @0�b = @1b and @1�b = @0b, thenc) "c(�2; �1) � "c(�1; �2) = 1�1�2 .Proof. These equalities follow easily from the formula"c(�1; �2) = U�2 U�1 � U1 U2used to de�ne "c in the proof of Theorem 3.4.As a consequence of a) and b) or by direct computation, we also have"c(�; �) = "c(�; �) = 1�:In virtue of a) and b), ifK is directed, the pair (Tt; "c) is a braided tensorW �{categoryand when c) holds, too, we get a symmetric tensor W �{category. In the general casewe get a net O 7! (Tt(O); "c) of braided or symmetric tensor W �{categories, wherethe terminology implies that the inclusion Tt(O1) � Tt(O2) for O1 � O2 is not only atensor �{functor but also preserves the braiding.21

In view of the above results, it is obviously important to be able to compute theconnected components of �?1 . We �rst localize and try to compute the connectedcomponents of �?1 (O) := fb 2 �?1 : jbj � Ogbefore trying to compute those of �?1 . Needleess to say, neither step can be carriedthrough at this level of generality but we shall carry them through when K is the setof regular diamonds in a globally hyperbolic spacetime.Note that �?1 (O) is closely related to the local graph of the relation ?,G?(O) := fO1 � O0 : O1;O0 � O; O1 ? O0g:There is an obvious order{preserving injection i : G?(O) ! �?(O). We simplyconsider Oi as @ib and O as jbj. Conversely, we have an order{preserving surjections : �?1 (O)! G?(O) mapping b to @1b� @0b. b lies in the same component of �?1 (O)as i � s(b). Hence if s(b) and s(b0) lie in the same component, so do b and b0, thus wehave computed the components of �?1 (O) in terms of those of G?(O). Now if O is aregular diamond in a globally hyperbolic spacetime, then O itself with the inducedmetric is a globally hyperbolic spacetime with a non{compact Cauchy surface andthe connected components have been computed in Lemma 2.2.For passing from the local to the global computation, the strategy is to look forcoherent choices of components for the �?1 (O), i.e. we want a component �?1;c(O) foreach O such that �?1;c(O1) = �?1;c(O2) \ �?1 (O1); O1 � O2:Lemma 3.6 Given a coherent choice of components O 7! �?1;c(O), then �?1;c := fb 2�1 : b 2 �?1;c(jbj)g is a component of �?1 .Proof. K being connected, the result will follow from Lemma 3A.3 once we show that�?1;c(O) = �?1;c \ �?1 (O):But if b 2 �?1;c(O), jbj � O and since we have a coherent choice of components,b 2 �?1;c(jbj) giving an inclusion. The reverse inclusion is trivial, completing theproof.Now when K denotes the set of regular diamonds in globally hyperbolic spacetimewith dimension � 2, then �?1 (O) has a single component so that �?1 is connectedby Lemma 3.6. It remains to consider the case of a globally hyperbolic spacetimeof dimension two. We know that each �?1 (O) now has two components and thatone passes from one component to the other by reversing the orientation of the 1{simplices. We need a way of specifying a coherent choice of components. If the Cauchysurfaces are non{compact, then G? also has two components and one passes from onecomponent to the other by interchanging the two double cones. Hence mapping bto @1b � @0b must map the two components of �?1 (O) into di�erent components ofG?. Denoting the two components of G? by G? and G?r , the inverse images under22

the above map give us a coherent choice of components. Lemma 3.6 then shows usthat �?1 has precisely two components and that one passes from one component tothe other by reversing the orientation of 1{simplices.On the other hand, in a globally hyperbolic spacetime (M;g) of dimension twowith compact Cauchy surfaces, we know from the discussion in Sec. 3.1 that G? isconnected. However, �?1 continues to have two components and we need a di�erentprocedure for making a coherent choice of local components. To this end, we picka nowhere vanishing timelike vector �eld and restricting this to a regular diamondO, we have, by the discussion following Lemma 2.2, a coherent way of distinguishingthe left and right components of the set of spacelike points in the regular diamondand hence left and right components of G?(O) and �?1 (O). Thus by Lemma 3.6, �?1has two connected components and one passes from one component to the other byreversing the orientation of 1{simplices.3.4 The Left Inverse and Charge TransferThe classi�cation of statistics makes essential use of left inverses. WhenK is directed,we may proceed as follows.De�nition A positive linear mapping � on B(H0) is called a left inverse of a repre-sentation � of A on H0 if�(A�(B)) = �(A)B; A 2 B(H0); B 2 A; and �(1) = 1:There are some simple facts to be noted: �rst, a positive mapping is automaticallyself-adjoint, �(A�) = �(A)� so that we have �(�(A)B) = A�(B), A;B 2 A. Secondly,if �(B) = B, then �(B) = B. Thus � inherits any localization properties of �. Inparticular, if � is localized in O�(A) = A for A 2 A(O2); O2 ? Oand, by duality, if O � O1 then �(A(O1)) � A(O1). Consequently � maps A into A.Furthermore one may show that �(A�A) � �(A)��(A) and k�k � 1.The complications involved when K is not directed are treated in the Appendixwhere the relations with the left inverse of a localized endomorphism and the leftinverse of a cocycle are also discussed.Once we have left inverses, we may proceed to the classi�cation of statistics. Wesuppose we have permutation statistics. The basic result, stated abstractly, is asfollows.Theorem 3.10 Let � be an object in a symmetric tensor C�{category (T; ") and � aleft inverse of � with ��;� = �1� for some scalar � then � 2 f0g [ f�d�1 : d 2 Ngand depends only on the equivalence class of �. The Young tableaux associated withthe representations of Pn on (�n; �n), n � 1 are all Young tableaux:a) whose columns have length � d, if � = d�1 (para-Bose statistics of order d);b) whose rows have length � d if � = �d�1 (para-Fermi statistics of order d);23

c) without restriction, if � = 0 (in�nite statistics).Note that when � is irreducible, ��;�("(�; �)) is automatically a scalar, called thestatistics parameter of �. d is referred to as the statistics dimension and the sign isthe statistics phase, �� and corresponds to the Bose-Fermi alternative. In general,we say that � has in�nite statistics if there is a left inverse � with ��;�("(�; �)) = 0.Otherwise � is said to have �nite statistics. Assuming our category T has subobjects,� has �nite statistics if and only if � is a �nite direct sum of irreducible objects with�nite statistics. In the cases where we can have braid statistics there is, of course, nocorrespondingly complete classi�cation, not even if we invoke the special setting of atwo dimensional Minkowski space. However, many partial results are known in thatcase and the proofs presumably generalize without essential modi�cation.As explained in Sec. 3.1, to deduce the existence of a left inverse, we assume thatK has an asymptotically causally disjoint net On. Thus, given O 2 K there is an n0with On ? O for n � n0. Under such a hypothesis, every representation � satisfyingthe selection criterion can be obtained as a limit of unitary transformations. Phys-ically, this would be interpreted as creating charge by transferring it from spacelikein�nity. We pick unitary intertwiners Un 2 (�n; �) where �n is localized in On. Thecorresponding unitary transformation �Un, �Un(A) := UnAU�n, may be interpreted asan operation which transfers charge from On to O. Now if A 2 A(O0) and n is su�-ciently large so that O0 ? On then �Un(A) = �(A) so that, as far as A is concerned,we have created a charge in O. In the limit as n !1 this holds for all A 2 A andwe haveLemma 3.11 limk!1 kUkAU�k � �(A)k = 0, A 2 A.The physical idea is now to create the conjugate charge in O by transferring chargeto spacelike in�nity. More prosaically, we would like to get a left inverse by replacingUk by U�k and taking a limit. This will indeed be the case although the limitingprocedure is more delicate and we cannot work in the strong topology (i.e. pointwisenorm topology) for linear mappings on A.We consider the space M of bounded linear mappings on B(H0) equipped withthe pointwise �{topology, i.e. a net �n from M converges to � if �n(A) converges to�(A) in the �{topology for each A 2 A. The important fact for our purposes is thatthe unit ball M1 of M is compact in this topology,M1 = f� 2M : k�k � 1g:Lemma 3.12 The net �U�n possesses at least one limit point in M. Every limit pointof this net is a left inverse of �. The set of all left inverses of � is a nonvoid compactconvex subset of M.We omit the proof as it is identical with that already given for Minkowski space.The existence of an asymptotically causally disjoint net On is also used in the analysisof left inverses but there are no new geometric properties involved.Another important aspect of superselection structure which does not involvespacetime symmetries is the existence of a complete �eld net with gauge symme-try describing the superselection sectors[25]. This clearly involves no further input24

of a geometric nature as it is based on Corollary 6.2 of [24] which refers to a singleC�{algebra rather than a net of von Neumann algebras. We leave to the reader thetask of formulating a precise result so as to avoid having to introduce the relevantde�nitions from [25].3.5 Sectors of a Fixed{Point NetAlthough we have now succeeded in adapting the main results of superselection the-ory to globally hyperbolic spacetimes with non{compact Cauchy surface, there isanother important aspect to be discussed. As we have seen the Selection Criterionhas a natural mathematical extension to curved spacetime. In Minkowski space, how-ever, it is further justi�ed by there being a simple mechanism producing examplesof such sectors. Under rather general conditions, it su�ces to begin with a �eld netF in its vacuum representation and a group of unitaries, a gauge group, compact inthe strong operator topology, and inducing automorphisms of the �eld net. Thende�ning an observable net A as the �xed{point net: A(O) := F(O)G, the resultingrepresentation decomposes as a direct sum of irreducible representations satisfyingthe selection criterion. The equivalence classes of these representations are in 1{1 cor-respondence with the set G of equivalence classes of irreducible, continuous, unitaryrepresentations of G and the irreducible representation corresponding to � 2 G hasmultiplicity d(�), the dimension of �. The question is whether these results continueto hold in curved spacetime.The original result in [23] does not, as it stands, apply to curved spacetime asit involves translations and the cluster property. However the variant given in [25]involves only structural elements and geometric properties compatible with curvedspacetime and therefore can be stated here as a result on superselection sectors incurved spacetime. In fact, the following result is valid for a directed set K with abinary relation ? such that given O 2 K, there exists O1;O2 2 K with O;O1 � O2and O ? O1. This condition is related to our use of the Borchers Property.Theorem 3.13 Let F be a �eld net over K acting irreducibly on a Hilbert space Hequipped with a strongly compact group G of unitaries inducing automorphisms of thenet F. We de�ne the observable net A to be the �xed{point net:A(O) := F(O)G; O 2 K:We assume that the subspace H0 of G{invariant vectors is separable and that Ais represented irreducibly on H0, satisfying duality there and having the BorchersProperty. Furthermore, H0 is supposed to be cyclic for each F(O) and F(O1) andA(O2) to commute whenever O1 ? O2. Then A0 = G00 and letting � denote thede�ning representation of A on H� =X� d(�)��; � 2 G;where the �� are inequivalent irreducible representations satisfying the selection cri-terion and G denotes the set of equivalence classes of continuous irreducible unitaryrepresentations of G. 25

Despite this positive result, we must examine the assumptions carefully to seewhether they remain reasonable in the context of curved spacetime. To test theassumptions we turn to the examples of scalar free �elds de�ned using quasifreeHadamard states[59]. It is known that duality holds for the Klein-Gordon �eld ona globally hyperbolic spacetime for regular diamonds and that the associated vonNeumann algebra is the hyper�nite type III1 factor and hence satis�es the Borchersproperty. However, at least in the context of Theorem 3.12, this must be regardedas a �eld net rather than an observable net. Furthermore, we actually use dualityfor the modi�ed relation ~? of causal disjointness to pass from cocycles to localizedendomorphisms in the next section. This strengthened form of duality is equivalentto the original form whenever the nets are inner regular, as is the case for the Klein{Gordon �eld. An even stronger form of duality, ?{duality, is used in the discussionof left inverses in the next section. However, our basic result on regular diamonds,Lemma 2.1, shows that it is in fact equivalent to ?{duality for additive nets.As is well known, a geometric property is involved in passing from duality for the�elds to duality for the observables. We give here a variant on the proof of Theorem4.3 of [52], not a priori requiring each irreducible representation of the gauge groupto be realized on Hilbert spaces in F. In view of the Z2{graded structure of a �eldnet, it is appropriate to de�ne its dual net byFd(O) = \fFt(O1)0 : O1 ? Og:Here Ft, the twisted �eld net, can be de�ned as the transform of F under the unitarytransformation 2�1=2(1+ iV ), where V is the gauge transformation changing the signof Fermi �elds, see e.g. [23].Theorem 3.14 Let F be a �eld net over K on a Hilbert space H satisfying twistedduality under a compact group of unitaries G inducing automorphisms of the netF. Let H 0, the subspace of G{invariant vectors, be cyclic for each F(O). Then the�xed{point net A sati�es duality for each O 2 K provided O? is connected.Proof. Let E denote the projection onto H 0 then the conditional expectation m of Fonto A may either be de�ned by integrating over the action of G or bym(F )E = EFE; F 2 F:Now (AE)d(O) = \O1?O(AE(O1)0) = \O1?O(EFt(O1)E �H0)0;Since E is cyclic and separating for each Ft(O) and A(O) = m(F(O)),(EFt(O1)E �H0)0 = (EFt(O1)0E) � H0:Now using the fact that E is separating for each Ft(O)0 and that O? is path{connected, we obtain(AE)d(O) = E \O1?O Ft(O1)0E � H0 = A(O);since F satis�es twisted duality.What is still missing is a result allowing one to pass from the Borchers Propertyfor the �eld net to the corresponding property of the observable net.26

3.6 Appendix to Chapter 3In this Appendix, we begin by introducing various notions we shall need in connectionwith the partially ordered set K. We recall[53] that an 0{simplex a of the partiallyordered set P is just an element of P and a 1{simplex b consists of two 0{simplicesdenoted @0b and @1b contained in a third element jbj of P called the support of b. Moregenerally, an n{simplex is an order{preserving map into P from the set of subsimplicesof the standard n{simplex, ordered under inclusion. �n(P) or just �n will denote thepartially ordered set of n{simplices of P with the pointwise ordering.A partially ordered set P is connected if given a; a0 2 �0(P), there is a path froma to a0 in P, i.e. if there exist b0; b1; : : : ; bn 2 �1(P) with @0b0 = a, @1bn = a0 and@0bi = @1bi�1, i = 1; 2; : : : ; n. Obviously, if P is not connected, it is a disjoint unionof its connected components. We will be taking for P not only subsets of K with theinduced order but also of K�K with the product ordering. These notions are relatedto topological notions in the following way.Lemma 3A.1 Let P be a base for the topology of a space M and ordered underinclusion and suppose the elements of P are open, (non-empty) and path{connected.Then an open subset X of M is path{connected if and only if PX:=fO 2 P : O � Xgis connected.Proof. Any two points of X are contained in elements of PX so if this is connectedand each of its elements are path{connected the two points can be joined by a path inX. Conversely, given O0;O1 2 PX, there is a path in X beginning in O1 and endingin O0, if X is pathwise connected. Since P is a base for the topology, it is easy toconstruct a path in PX joining O1 and O0.A subset S of P of the form PX has the property that O 2 S and O1 � O impliesO1 2 S. Such subsets are referred to as sieves. If P is a base for the topology of Mthen a sieve S is a base for the topology of the open subset XS := [fO : O 2 Sg. Theconnected components of a partially ordered set are sieves, the union or intersectionof sieves is again a sieve.Corollary 3A.2 Under the hypotheses of Lemma 3A.1, the connected components ofP are of the form PX, where X runs over the path{connected components of M .We let Open(M) denote the set of open subsets of M ordered under inclusion andSieve(K) the set of sieves of K, then de�ning for a open set X of M , �(X) to be theset of O 2 K contained in X, � is an injective order-preserving map from Open(M)to Sieve(K). If we de�ne �(S) := XS, then � is order-preserving and a left inverse for�. The following result will prove useful in calculating the connected components ofa partially ordered sets.Lemma 3A.3 Let i 7! Pi be an order{preserving map from a partially ordered set Ito the set of sieves of a partially ordered set P ordered under inclusion. Suppose thatP = [i2IPi. Let C � P and set Ci := C \ Pi then C is a union of components of P ifand only if Ci is a union of components of Pi for each i 2 I. If I is connected and Ciis either empty or a component of Pi, i 2 I, then C is a component of P.27

Proof. If C is a union of components and b 2 �1(Pi) with @1b 2 Ci then b 2 �1(P) so@0b 2 C\Pi = Ci and Ci is a union of components. Conversely, if each Ci is a union ofcomponents and b 2 �1(P) with @1b 2 C, then jbj 2 Pi for some i. But Pi is a sieve sob 2 �1(Pi) and @1b 2 C \ Pi. Since Ci is a union of components, @0b 2 Ci � C so C isa union of components. Now C is a component, if any given pair a 2 Ci and a0 2 Ci0can be joined by a path in C. But I being connected, we may as well suppose i andi0 have an upper bound j 2 I. If Cj is a component, a and a0 can even be joined bya path in Cj, completing the proof of the lemma.Now an automorphism g of a partially ordered set P such that given O 2 P thereis a b 2 �1(P) with @1b � O and @0b � gO obviously leaves each connected componentof P globally invariant. If G is a connected topological group acting continuously ona topological space M and P is a base for the topology of M , then it is easy to seethat given O 2 K there is a O1 2 K and a neighbourhood N of the unit in G suchthat NO1 � O. It follows that G leaves any path{component of P globally invariant.Of course, this may also be deduced from Corollary 3A.2.After these generalities on partially ordered sets, we turn to the theory of super-selection sectors and need a partially ordered set K equipped with a binary relation? satisfying a), b) and c) of Sec. 3.1. Note that b) just says that O? is a sieve of K.There are two derived binary relations ~? and ? de�ned by supplementing O1 ? O2by requiring that there exists an O3 2 K such thatO1 ? O3; O2 ? O3or such that O1; O2 � O3;respectively. These relations automatically satisfy a) and b) but c) remains to bechecked and will not prove to be a problem in our applications to curved spacetime.The operation of passing from ? to ~? or ? is idempotent and if K is directed, allthree relations coincide. Furthermore, by Lemma 5.6, the corresponding notions ofduality coincide for additive nets when K is the set of regular diamonds in a globallyhyperbolic spacetime.If O1 ? O2 and O3 has non-trivial causal complement in O, i.e. if there exists anO4 with O3 ? O4, O3;O4 � O2 then trivially O1 ~?O3. Now a regular diamond is aunion of a sequence of smaller regular diamonds with non-trivial causal complementin the original regular diamond. Thus when K is the set of regular diamonds, thedi�erence between the relations ? and ~? is, in this sense, a boundary e�ect.The di�erence between ~? and ? merely re ects the potential di�culty of �ndingsuitably large regular diamonds. If we replace the setK of regular diamonds by the set~K of sieves in K with non-trivial causal complement, de�ning the causal complementS? of a sieve S to be the sieve S? := \O2SO?, then ~? = ?. In fact, if S1 ~?S2, then(S1 [ S2)? = S?1 \ S?2 6= ; so that S1 ~?S2.If K is a base of open sets of a topological space M and the relation ? on K isinduced by a relation ? on Open(M) satisfying a) and b) of Sec. 3.1 and which islocal in the sense that if X 2Open(M) and X � [iOi, then Oi ? O for all i implies28

X ? O. This condition is obviously satis�ed by the relation of causal disjointnesson a globally hyperbolic spacetime. It implies that �(X?) = �(X)?. We also have�(S)? = �(S?) for any sieve S in K.Lemma 3.A4 When restricted to causally closed open sets and sieves, the maps �and � are inverses of one another.Proof. If S is a sieve and X := �(S), then �(X)? = S?. If S is causally closed,so is X since � is injective. On the other hand, if X is causally closed and we setS := �(X), then S?? = �(X??) = �(X) and S is causally closed. It remains to showthat S = ��(S) if S is causally closed. But, in this case,S � ��(S) � ��(S)?? = S?? = S;completing the proof.By a representation � of a net of von Neumann algebras A overK we mean normalrepresentations �O of A(O) on a Hilbert space H� such that �O1 is �O2 restricted toA(O1), whenever O1 � O2 in K.IfG is a group of automorphisms ofK and (A; �) is a covariant net then a covariantrepresentation is a pair (�;U) consisting of a representation � of A and a unitaryrepresentation of G on H� such that U(g)�O(A) = �gO(�g(A)U(g), A 2 A(O), g 2 G.We now provide a cohomological interpretation of superselection sectors leadingto a proof of the Extension Theorem of Sec. 3.1. To enter into the spirit of thecohomological interpretation, we regard O?, O 2 K as being a covering of K, thecausal covering. The selection criterion selects those representations that are trivialon the causal cover and these representations allow a cohomological description inanalogy with locally trivial bundles.For each a 2 �0 we pick a unitary Va such thatVa�O(A) = AVa; A 2 A(O); O ? aand set z(b) := V@0bV �@1b; b 2 �1. Obviously if O 2 jbj?, z(b) 2 A(O)0 thus z(b) 2Ad(jbj). Furthermore, z(@0c)z(@2c) = z(@1c); c 2 �2so that z is a unitary 1{cocycle with values in the dual net Ad. We consider such1{cocycles as objects of a category Z1(Ad), where an arrow t in this category from zto z0 is a ta 2 Ad(a), a 2 �0, such thatt@0bz(b) = z0(b)t@1b; b 2 �1:This makes Z1(Ad) into a W �{category. Note that ktak is independent of a.If we were to make a di�erent choice V 0a of unitaries Va, then setting z0(b) :=V 0@0bV 0�@1b and wa := V 0aV �a , we see that wa 2 Ad(a) and w@0bz(b) = z0(b)w@1b. Thusw 2 (z; z0) is a unitary and the 1{cocycle attached to � is de�ned up to unitaryequivalence in Z1(Ad). More generally, if T 2 (�; �0) and � and �0 are trivial on thecausal cover and z and z0 are associated cocycles de�ned by unitaries Va and V 0a, asabove, set ta := V 0aTV �a ; a 2 �0:29

Then ta 2 Ad(a) andt@0bz(b) = V 0@0bTV �@0bV@0bV �@1b = V 0@0bTV �@1b = V 0@0bV 0�@1bV 0@1bTV �@0b = z0(b)t@1b;so that t 2 (z; z0). Conversely, if t 2 (z; z0) then T := V 0�ataVa is independent of a sothat T�O(A) = �0O(A)T; A 2 A(O); O 2 K:and we clearly have a close relation between Z1(Ad) and the W �{category Rep?A ofrepresentations of A trivial on the causal cover.However, any cocycle z arising from such a representation has two special prop-erties that may not be shared by a general 1{cocycle. First, z is trivial on B(H 0), i.e.there are unitaries Va, a 2 �0, on H 0 such that z(b) = V@0bV �@1b, b 2 �1.If K is directed then ��(K) admits a contracting homotopy[53]. In this case every1{cocycle of Ad is trivial in B(H0). In general, if we consider the graph with vertices�0 and arrows �1 then the category generated by this graph has as arrows the pathsin K. Thus every 1{cocycle extends to a functor from this category. When z is trivialon B(H0) then z(p) for a path p depends only on the endpoints @0p and @1p of thepath. Conversely, if z(p) just depends on the endpoints of p and K is connected,then z is trivial on B(H0). To see this we pick a base point a0 2 �0, then givena 2 �0 a path pa with @0pa = a and @1pa = a0 and �nally de�ne y(a) = z(pa).z(p)y(@1p) = y(@0p), so we have trivialized z in B(H0).Secondly, for any path p, z(p)Az(p)� = A whenever A 2 A(O) and @0p, @1p 2 O?.The full subcategory of Z1(Ad) whose objects satisfy these two conditions will bedenoted by Z1t (Ad).The following simple result shows that the second condition is automatically sat-is�ed in an important special case.Lemma 3A.5 If O? is connected, then any object z of Z1(Ad), trivial on B(H0)satis�es z(p)Az(p)� = A; @0p ; @1p 2 O?; A 2 A(O):Proof. Since O? is connected, it su�ces to prove the result when the path p is a1{simplex b with jbj 2 O?. But then, z(b) 2 Ad(jbj) � A(O)0.Having discussed these two conditions, we can give our cohomological character-ization of the selection criterion.Theorem 3A.6 The W �{categories Rep?A and Z1t (Ad) are equivalent.Proof. We pick unitaries V �a , a 2 �0, as above, for each object � of Rep?A. Givenan arrow T 2 (�; �0) in that category, we de�ne for b 2 �1, a 2 �0F (�)(b) = V �@0bV ��@1b; F (T )a := V �0a TV ��a :Then F is a faithful �-functor and our computations above show that it is full. Hence,it remains to show that each object z of Z1t (Ad), is equivalent to an object in the30

image of F . We show this by constructing a representation �z. We pick unitaries Va,a 2 �0, on H0 such that z(b) = V@0bV �@1b; b 2 �1; and de�ne�zO(A) = V �a AVa; a 2 O?; A 2 A(O):This is well de�ned since K is connected and for any path p with @0p, @1p 2 O? wehave z(p) 2 A(O)0. Furthermore, the de�nition respects the net structure since�zO1(A) = �zO2(A); A 2 A(O1); O1 � O2:Hence we get a representation of the net A, trivial on the covering by constructionand V@0bV �@1b = z(b) is an associated 1{cocycle. This completes the proof.We now consider the problem of extending representations of a net A, trivial onthe causal cover, to representations of the bidual net Add, again trivial on the causalcover.Theorem 3A.7 If each O? is connected, every object � of Rep?A admits a uniqueextension to an object of Rep?Add. Furthermore there is a canonical isomorphism ofW �{categories Rep?A and Rep?Add.Proof. Let Va, a 2 �0 be unitaries realizing the equivalence of � and �0 on a?. Thenz(b) := V@0bV �@1b, b 2 �1 is an associated object of Z1t (Ad). Since each O? is connected,z is at the same time an object of Z1(Addd) by Lemma 3.A.4. If we de�ne~�O(A) := V �aAVa; A 2 Add(O); a 2 O?;this gives a well de�ned element of Rep?Add just as in the proof of Theorem 3A.6.Furthermore, ~� obviously extends � by the choice of the Va. If we make another choiceV 0a of the Va then V 0aV �a 2 Ad(a) so that ~� remains unchanged and is consequentlythe unique extension of � to an object of Rep?Add. Passing to the extensions doesnot change the intertwiners by Theorem 3A.6.For the further development of superselection theory, we must assume dualityA = Ad, although essential duality would do whenever each O? is connected. We shalleven need to assume ~?{duality, but this coincides with duality in curved spacetimewhose status is commented on in Sec. 4.2.The next goal is to show that sectors have a tensor structure. More precisely,we shall show that Z1(A) has a canonical structure of a tensor W �{category arisingby adjoining endomorphisms. If A is a net of von Neumann algebras, then there isan associated net O 7! EndA(O) of tensor W �{categories. EndA(O) has as objectsthe normal endomorphisms of the net O1 7! A(O1), i.e. normal endomorphisms �O1of A(O1) compatible with the net structure. An arrow T 2 (�; �) in EndA(O) is aT 2 A(O) such that T�(A) = �(A)T; A 2 A(O1); O � O1:The tensor structure is de�ned on the lines of Sec. 3.3 and the net structure is givenby the obvious restriction mappings. 31

The construction of appropriate endomorphisms is just a variant on that alreadyused to pass from a 1{cocycle z 2 Z1t (Ad) to a representation �z. Given a 2 �0, andA 2 A(O), a � O pick a path p with @0p = a and @1p 2 O? and sety(a)(A) := z(p)Az(p)�:y(a)(A) is independent of the choice of p since z 2 Z1t (Ad). Given X 2 A(O1) withO1 ? O, O2 with O2 ? O and O2 ? O1 and choosing @1p = O2, we see that y(a)(A)and X commute so that y(a)(A) 2 A(O) by ~?{duality. Thus y(a) is an object ofEndA(a).But y(a) is not only localized in a in the sense of net automorphisms but also inthe sense of superselection theory in that y(a)(A) = A whenever A 2 A(O1) whereO1 2 a? and O1; a � O, since the endpoints of p lie in O?1 . We write �(a) to denotethe objects of EndA(a) satisfying this second localization condition and denote byT(a) the corresponding full tensor C�{subcategory of EndA(a).Lemma 3A.8 Let p be a path with @1p; @0p � O thenz(p)y(@1p)(A) = y(@0p)(A)z(p); A 2 A(O):Proof. Given A 2 A(O) and a path p with @1p; @0p � O, pick paths p0; p00 with@0p0 = @1p, @0p00 = @0p and @1p0; @1p00 2 O?, thenz(p)y(@1p)(A) = z(p)z(p0)Az(p0)� = z(p00)Az(p00)�z(p) = y(@0p)(A)z(p);as required.Furthermore if t 2 (z; z), A 2 A(O) and p is a path with @0p = a � O and@1p � O? thentay(a)(A) = taz(p)Xz(p)� = z(p)t@1pAz(p)� = z(p)Az(p)�ta = y(a)(A)ta:In other words ta 2 (y(a); y(a)).These results admit the following interpretation.Theorem 3A.9 Let A be a net over (K;?) satisfying ~?{duality. If z is a 1{cocycleof A trivial in B(H0) then (y; z) is a 1{cocycle in the net T of tensor W �{categoriesand the map z 7! (y; z) together with the identity map on arrows is an isomorphismof Z1t (A) and Z1t (T).Now, T being a net of tensor W �{categories, Z1(T) is itself a tensor W �{category.Given 1{cocycles (y1; z1) and (y2; z2), their tensor product is the 1{cocycle (y; z),where y(a) = y1(a)y2(a); z(b) = z1(b)y1(@1b)(z2(b)):If both (y1; z1) and (y2; z2) are trivial in B(H0) then so is their tensor product. Thetensor product on arrows is de�ned as follows: if ti maps from (yi; zi) to (y0i; z0i) fori = 1; 2, then the tensor product t1 t2 is given by(t1 t2)a = t1;ay1(a)(t2;a):32

This completes our goal of describing superselection structure in terms of a tensorW �{category. Note that we could have used the subnet Tt in place of T de�ned byrequiring an object � of T(O) to be transportable, i.e. there exists a map a 7! �a,where �a is an object of T(a) and �a = � when a = O and a map �1 3 b 7! u(b),where u(b) is an arrow from �@1b to �@0b in T(jbj). In fact the tensor W �{categoriesZ1t (T) and Z1t (Tt) are canonically isomorphic. In Sec. 3.3, we show how to get a net(Tt; "c) of braided tensor W �{categories and it is a simple general fact that this leadsto a braided tensor W �{category, (Z1t (Tt); "c). We need only set"c(z; z0)a := "(y(a); y0(a)):Since this expression obviously acts correctly on the arrows evaluated in a and thelaws for a braiding hold for each a, the only point that has to be checked is that"(z; z0) is an arrow from z � z0 to z0 � z. However, if b 2 �1, z(b) 2 (�@1b; �@0b) inTt(jbj) and similarly for z0(b). Thusz0(b)� z(b) � "(�@1b; �0@1b) = "(�@0b; �0@0b) � z(b)� z0(b);as required.Thus the cohomological approach leads to a braided tensor W �{category (Z1t (Tt),�c) describing superselection structure and in the context of globally hyperbolic space-times this is even a symmetric tensor W �{category for spacetime dimensions � 2. Itshould be noted that except whenK is directed, we have not given a direct descriptionof this structure in terms of transportable localized endomorphisms. In particular, itnot clear that every transportable localized endomorphism arises from a 1{cocycle.Furthermore, if � and � are in �t(O) and T is a bounded operator on the ambientHilbert space, such thatT�(A) = �(A)T; A 2 A(O1); O � O1;then T commutes with A(O2) for O2 ? O provided there is a O1 with O;O2 �O1. This means, we would need duality with respect to the modi�ed relation ? tobe able to conclude that T 2 A(O) and hence that T is an arrow from � to � inTt(O). Conversely, if � and �0 are representations satisfying the selection criterionand restricting to endomorphisms � and �0 in �t(O) then it is not clear that an arrowT 2 (�; �0) in Tt(O) will at the same time intertwine � and �0.These points should be bourne in mind, when, in the main body of the text, weavoid the cohomological description and put the emphasis on transportable localizedendomorphisms.To proceed with the analysis of statistics, we need to use left inverses and weexamine, at this point, the notions involved and the relations between them. If �is a representation of A on H0 then we de�ne a left inverse � of � to be given byunital positive linear mappings �O on B(H0) compatible with the net inclusions andsatisfying �O(A�O(B)) = �O(A)B; A;B 2 A(O):33

Note that if �O(B) = B then �O(B) = B. If � is localized in O in the sense that�O1(A) = A; O ? O1; A 2 A(O1);then � is localized in O in the same sense. Furthermore, if O1?O2 and O � O2,then �O2(A)B = B�O2(A) for A 2 A(O2) and B 2 A(O1). In fact, picking O3 withO1;O2 � O3 we have �O2(A)B = �O3(A)B = �O3(A�O3(B))= �O3(A�O1(B)) = �O3(AB):Since A and B commute, we interchange them and reverse the steps to conclude that�O2(A) and B commute. This proves the following result.Lemma 3A.10 If � is a left inverse for a representation � localized in O then � islocalized in O and if duality holds for the relation ?, �O1A(O1) � A(O1) for O � O1.The restriction of � to the net O1 7! A(O1), O1 � O is a localized endomorphism� and a object of the tensor W �{category EndA(O). The above notion of left inverseadapts easily to localized endomorphisms. If � is localized in O, a left inverse of � isa family O1 � O 7! �O1 of unital positive linear mappings on the A(O1), compatiblewith the net inclusions and satisfying�O1(A�O1(B)) = �O1(A)B; A; B 2 A(O1):Obviously, a left inverse for � considered as a representation yields a left inverse forthe endomorphism � on restriction. If �� is a conjugate for � then we get a left inverse� for � by setting �O1(A) := V ���O1(A)V; A 2 A(O1); O1 � O;where V 2 (id; ���) is an isometry.The restriction of � to the net O1 7! A(O1), O1 � O is a localized endomorphism� and a object of the tensor W �{category EndA(O). We now show that a left inverse� for � induces a left inverse of � in the categorical sense [47]. In other words, weneed a set ��;� : (��; �� )! (�; � );of linear mappings where �, � are objects of the category. These have to be naturalin � and �; i.e. given S 2 (�; �0) and T 2 (�; � 0) we have��0 ;� 0(1� T �X � 1� S�) = T � ��;�(X) � S�; X 2 (��; �� );and furthermore to satisfy���;��(X 1�) = ��;� (X) 1� ; X 2 (��; �� )for each object �. We will require that � is positive in the sense that ��;� is positivefor each � and normalized in the sense that��;�(1�) = 1�:34

We say that � is faithful if ��;� is faithful for each object �.Now, given T 2 (��; �� ), we recall that T 2 A(O). Hence we set��;� (T ) = �O(T )and since �O(T ) 2 A(O) by Lemma 3A.9, we conclude without di�culty that we geta left inverse for � in this way.On the other hand, if we are dealing with a representation satisfying the selectioncriterion then we know that, by passing to an associated 1{cocycle, we get a �elda 7! y(a) of localized endomorphisms under the weaker assumption that dualityholds for the relation ~?. In this case, we would actually like a left inverse for the1{cocycle considered as an object of the tensor W �{category Z1t (A). To this end, wepick, for each of the associated endomorphisms y(a) a left inverse �a and ask whethera 7! �a(ta) is an arrow from z0 to z00, whenever a 7! ta is an arrow from z � z0 toz � z00. Thus ta 2 A(a) and(z � z00)(b)t@1b = t@0b(z � z0)(b):It follows thatz00(b)�@1b(t@1b) = �@1b(y(@1b)(z00(b))t@1b) = �@1b(z(b)�t@0bz(b))z0(b)and we deduce the following lemma.Lemma 3A.11 If z 2 Z1t (A) and a 7! y(a) is the associated �eld of endomorphisms.Then a �eld a 7! �a of left inverses of the y(a) de�nes a left inverse for z by theformula �z0 ;z00(t)a := �a(ta)provided �@0b = �@1bAdz(b)� for b 2 �1.There is no a priori reason to suppose that every left inverse for a 1-cocycle arisesfrom such a �eld of left inverses. In particular a map t 2 (z; z0) 7! ta 2 (y(a); y0(a))might not be surjective. We can also not just begin with a left inverse �a for y(a)since it is not clear that we get a �eld of left inverses using the cocycle. However,if we assume, as in Sec. 3.4, that K has an asymptotically causally disjoint net On,then we can construct left inverses for 1-cocycles. If z is an object of Z1t (A), wedenote by z(a; n), the evaluation of z on a path p with @0b = a and @1b = On. Thisis independent of the chosen path. We now de�ne �a(X) to be a Banach{limit overn of z(a; n)�Xz(a; n). Then �a is a positive linear map satisfying�a(X)A = �a(X�aO(A)); A 2 A(O):Furthermore, from the cocycle identity we have�@0b = �@1bAd(z(b)):Since each �a de�nes a left inverse for y(a), we have constructed a left inverse for zby Lemma 3A.11. 35

One sometimes wishes to consider nets de�ned over a wider class of regions thansay just the set of regular diamonds. Thus in Sections 4 and 5, we are interested inde�ning the von Neumann algebras of wedge regions. Furthermore, another reasonfor wanting von Neumann algebras associated with large rather than small regionsis that we can only compose endomorphisms if we �nd a joint localization region forthe endomorphisms involved. We consider here the task of extending the domain ofde�nition of the net in the context of the present formalism where K is a partiallyordered set commenting on the relation with regions of spacetime afterwards. Thusinstead of a region, we use the notion of a sieve S, see above, and consider the set ~Kof sieves S of K such that neither S nor S? are the empty set, ordered under inclusion.To each such sieve S, we associate the von Neumann algebra A(S) generated by theA(O) with O 2 S in the de�ning representation.We now show that a representation � of A satisfying the selection criterion has anatural extension to a representation of the net S 7! A(S). We pick for each a 2 �0a unitary Va such that �O(A) = V �a AVa; A 2 A(O); O 2 a?;and then de�ne �S(A) := V �a AVa; A 2 A(S); a 2 S?:Note that this expression is well de�ned being independent of the choice of a 2 S?since if a0 2 S? then Va0V �a 2 \O2SA(O)0 = A(S)0:In the same way, we see that �S is independent of the choice of a 7! Va. Note, toothat we get a representation of the extended net in that if S1 � S2 then �S1 is therestriction of �S2 to A(S1). Obviously, an intertwiner T 2 (�; �0) over K remains andintertwiner over ~K so that e�ectively Rep?A remains unchanged when we extend thenet.That part of the formalism related to the concept of localized endomorphism ishowever sensitive to exteding the net. Although localized endomorphisms do not playthe same fundamental role as 1{cocycles, we have found it convenient to use themin developing the theory. The problems involved in using them are two: they arenot de�ned on the whole net and the natural map (z; z0) 7! (y(a); y0(a)) may not besurjective. Extending the net improves matters in that localized endomorphisms arethen de�ned on more operators and hence have fewer intertwiners. Since localizedendomorphisms require subsets satisfying ?{duality, we bene�t from the equality~? = ? on ~K.Supposing we have as usual a �eld a 7! y(a), a 2 �0, of localized endomorphismsderived from a 1{cocycle, then we know that if O~?O1 and a ? O1, y(a)(A(O)) �A(O1)0. Hence y(a)(A(S)) � \O12S~?A(O1)0. We conclude that if ~?{duality holds forS in the de�ning representation in the sense thatA(S) = A(S~?)0;36

then y(a) acts as an endomorphism of A(S), y(a)(A(S)) � A(S). Now if S satis�es~?{duality then so does S~?. Furthermore, A(S) = A(S?)0 � A(S??). Thus A(S) =A(S??). Hence, we may as well restrict attention to causally closed sieves and chooseas our index set the set L of non-trivial causally closed sieves S for which ~?{dualityholds either for S or for S?. This choice has the disadvantage of depending on thetheory under consideration but it allows a smooth treatment of endomorphisms. Inparticular, if ~? duality holds for S and a 2 S? then the endomorphism y(a) associatedwith a 1{cocycle satis�es y(a)(A(S?)) � A(S?), because, as we have seen above,duality holds for S~? and A(S?) = A(S~?).We shall be assuming ~?{duality for the elements of K. Thus K � L and fO? :O 2 Kg � L. Thus L is both coinitial and co�nal in ~K. Let us call two localizedendomorphisms comparable if they are both localized in a common sieve in ~K andhence in some element of L. In this case, it makes sense to talk about intertwiningoperators between the two localized endomorphisms. If �i is localized in Si, i = 1; 2,then �1 and �2 are comparable, if and only if S1 \ S2 6= ;.We turn now to the notion of left inverse. If we consider � as a representation ofthe extended net S 7! A(S), then there is an obvious modi�cation of the notion ofleft inverse as we just need to replace O everywhere by S. Suppose � is localized inS and � is a left inverse for �, then given S1 � S and O 2 S?1 , we remark that thereis a sieve S2 with O 2 S2 and S1 � S2. Given A 2 A(S1) and B 2 A(O) we have�S1(A)B = �S2(A)B = �S2(A�S2(B))= �S2(A�O(B)) = �S2(AB):Since A and B commute, we interchange them and reverse the steps to conclude that�S1(A) and B commute. Recalling that ~? = ? on ~K, this proves the following result.Lemma 3A.12 Let � be a left inverse for a representation � of the extended netS 7! A(S) localized in S. Then, if S � S1 and ~?{duality holds for S1, �S1A(S1) �A(S1).4 The Conformal Spin and Statistics Relation forSpacetimes With Bifurcate Killing HorizonIn the present chapter, we shall specialize our considerations to the class of spacetimeswith a bifurcate Killing horizon (bKh), whose de�nition we now summarize, followingKay and Wald [40]. The interested reader is strongly recommended to consult thisreference for further details not spelled out here. The main purpose here is to showthat, from the original theory, we can construct a family of local algebras localizedon the horizon, which possesses a conformal symmetry. Therefore horizon localizedsuperselection sectors have a conformal spin and we prove that this coincides withtheir statistics phase. 37

4.1 Spacetimes with bKhA spacetime with a bKh is a triple (M;g; �t) where (M;g) is a four-dimensional,globally hyperbolic spacetime, although spacetimes with a bKh generalize to otherspacetime dimensions. (�t)t2R is a non-trivial one-parameter group of isometries of(M;g), assumed to be C1, and hence the ow of a Killing vector �eld � onM for themetric g. We often refer to (�t)t2R as the Killing ow (of the spacetime with bKh).We shall assume that (M;g) is orientable and that the set � �M of �xed points of(�t)t2R is a two-dimensional smooth, acausal, orientable, connected submanifold ofM .It is worth noting that �, when compact, automatically lies in some Cauchy-surface,see [40] for a proof.From this data we can construct the bKh, h, as follows: at each point p 2 � wechoose a pair of linearly independent, lightlike, future-oriented vectors �A(p); �B(p) 2TpM , normal to �. They are unique up to scalars and they may be chosen so that� 3 p 7! �A(p) and � 3 p 7! �B(p) are smooth vector �elds along � since M and �are orientable. Now let Ap and Bp be the maximal geodesics with tangents �A(p)and �B(p) at p 2 �, respectively. Since (�t) leaves each p 2 � �xed, it maps each ofthe curves Ap and Bp into itself. Moreover, Ap and Ap0 do not intersect for p 6= p0,and the same holds with B in place of A. Now one de�nes sets hA and hB to be thelightlike hypersurfaces inM formed by the Ap and Bp, respectively, as p ranges over�. Then h := hA [ hB is the bKh, and one distinguishes the following subsets:hRA := (hAn�) \ J+(�) ; hLA := (hAn�) \ J�(�) ;hRB := (hBn�) \ J�(�) ; hLB := (hBn�) \ J+(�) :The Killing vector �eld � is conventionally assumed to be future oriented on hRA.The bKh divides the spacetime M locally into four disjoint parts, F := J+(�),P := J�(�), R := (J�(hRA)nhRA)\(J+(hRB)nhRB) and L := (J�(hLB)nhLB)\(J+(hLA)nhLA),the future, past, right and left parts of the bKh, respectively.To give a rather simple illustration, consider (M;g) as Minkowski spacetime (ofdimension 4). Then choose an inertial coordinate system and de�ne � as the two-dimensional hyperplane f(x0; x1; x2; x3) 2 R4 : x0 = x1 = 0g. There is a smooth,one-parameter group �t = �t, t 2 R, of pure Lorentz transformations leaving � �xed;they are de�ned by�t(x0; x1; x2; x3) := (cosh(t)x0 + sinh(t)x1; sinh(t)x0 + cosh(t)x1; x2; x3) : (4.1)Then h = hA [ hB is a bKh, where hA = f(u; u; x2; x3) : u 2 R; (x2; x3) 2 R2g andhB = f(v;�v; x2; x3) : v 2 R; (x2; x3) 2 R2g. Here, the regions R and L correspondto the usual \right wedge" and \left wedge" regions in Minkowski spacetime. Otherimportant examples of spacetimes with a bKh include e.g. deSitter and Schwarzschild-Kruskal spacetimes, as well as the Schwarzschild-deSitter spacetimes and (certainregions of the) Kerr-Newman spacetimes. (The latter have at least two bKhs withdi�erent surfaces gravities, see below. This leads [40] to conclude that there are noregular, Killing- ow invariant states of the free scalar �eld on such spacetimes.)Let us now look at how the Killing ow acts on the bKh in greater detail. Eachof the geodesic generators Ap of the hA-part of the bKh is de�ned on some interval38

Ip. We may choose an a�ne parametrization of Ap, with a�ne parameter U , suchthat Ap(U = 0) = p and ddU Ap��U=0 = �A(p) for all p 2 �. This parametrizes all thegeodesics and, since the vector �eld �A(p) depends smoothly on p 2 �, by assumption,the a�ne parametrization of the curves Ap depends smoothly on p 2 �. Since Apis left invariant under the Killing ow, Ip must be invariant under a (non-trivial)smooth representation of the additive group R (with 0 as the only �xed point), andthus Ip = R. A similar result holds for the domains of the geodesic generators Bpof hB. Therefore, each point q 2 hA is uniquely determined by the pair (U; p), whereq = Ap(U). Hence we have a di�eomorphism A : hA ! R� � assigning to q 2 hAthe pair (U; p) 2 R�� with q = Ap(U). 6 As explained below, certain choices of �Aand �B turn out to be particularly useful for our purposes and lead to the followingrelation (cf. [40], see also [57]):�t� A�1(U; p) = A�1(e�tU; p) ; t; U 2 R; p 2 � ; (4.2)where the number � > 0, called the surface gravity, is an invariant of the bKh underconsideration. (For the Schwarzschild-Kruskal spacetime of a black hole with massmbh > 0, � is proportional to mbh. The reader is referred to [40],[61] for moreinformation about the notion of surface gravity.) Constructing a di�eomorphism B : hB ! R� �, similarly, where B(q) = (V; q) i� q = Bp(V ), the a�ne geodesicparameter being now denoted by V , one can show that�t� B�1(V; p) = B�1(e��tV; p) ; t; V 2 R; p 2 � ; (4.3)with the same � > 0 as in the previous equation.There are a few other geometric actions on hA and hB, induced by identifying theseparts of the bKh with R� � via the maps A and B. First, there are the a�netranslations `a� A�1(U; p) := A�1(U + a; p) ; (4.4)`a� B�1(V; p) := B�1(V + a; p) ; a; U; V 2 R; p 2 � : (4.5)In contrast to the dilations on hA and hB, induced by restricting the Killing ow to thebKh, the translations will not, in general, extend to isometries of the full spacetime.Another action is the (a�ne) re ection, 7�� A�1(U; p) := A�1(�U; p) ; (4.6)�� B�1(V; p) := B�1(�V; p) ; U; V 2 R; p 2 � : (4.7)Again, � need not extend to an isometry of the full spacetime to the bKh. However,Kay and Wald [40] have shown that, if the spacetime with bKh is analytic, there is6Notice that A depends on the choice of the vector �eld � 3 p 7! �A(p) along �. It may berescaled at each point: ~�A(p) = �(p)�A(p), with � : � ! R a smooth, strictly positive function,would serve just as well when constructing hA. A similar remark applies to the hB-horizon.7The de�nitions of `a and � involve A (or B) so these quantities, cf. the previous footnote,depend on the scaling freedom when choosing A (or B).39

a neighbourhood N of h and an orientation and chronology-reversing isometry j ofN (\horizon re ection") commuting with the action of (�t) which re ects the a�neparameter of geodesics passing orthogonally through �.In the next step, we shall specify some families of regions analogous in some respects tothe \shifted wedges" in Minkowski spacetime. With their help, we can then formulatea version of geometric modular action for quantum �eld theories on spacetimes witha bKh in the operator-algebraic framework. To begin with, we note (cf. [40]) thatthe parts F , P , R and L of a spacetime with bKh (see above) satisfyF \ P = � ; F \R = ; ; P \ R = ; ; (4.8)F \ L = ; ; P \ L = ; :Thus, as we have already seen from the example above, R and L may be viewedas playing the role of the right and left wedge regions in Minkowski spacetime. If~M := F [ P [ L [ R, then ~M , L and R, with the appropriate restrictions of g asLorentzian metric, are globally hyperbolic spacetimes. It may, however, happen that~M 6=M , see [40] for examples. As we shall later assume thatM = ~M , this possibilityneed not concern us. One can see from (2.3) that the regions F , P , R and L areinvariant under the Killing ow (�t). This implies that ~M is also invariant under (�t).For open intervals (a; b) with a < b and a; b 2 R[ f�1g, we now de�nehA(a; b) := f A�1(U; p) : a < U < b; p 2 �g ; (4.9)with an analogous de�nition of hB(a; b). Notice that with this notation,hRA = hA(0;1) ; hLA = hA(�1; 0) : (4.10)The \shifted right wedge" can then be de�ned asRa := R n clJ�(hA(�1; a)) (4.11)for a > 0, where cl means \closure".Lemma 4.1. �t(Ra) = Re�t�a for all t 2 R; a � 0 : (4.12)Proof. Since (�t) is a group of isometries leaving R invariant,�t(Ra) = �t �R n clJ�(hA(�1; a))� (4.13)= �t(R) n �t(clJ�(hA(�1; a)))= R n clJ�(ha(�1; e�t � a))= Re�t�a : 40

Similarly, setting L�a := L n clJ+(hA(�1;�a)) (4.14)for a > 0 (!), we �nd as before that�t(L�a) = L�e��t�a ; t 2 R; a > 0 : (4.15)In this section, a non-void open O � M is called a diamond if it is of the formO = intD(G) where G is an open subset of a Cauchy surface C (not necessarilyacausal) such that @G is continuous and O? non-void; moreover O or O? is requiredto be connected.Below we study nets of von Neumann algebras indexed by the diamond regionsin a given spacetime with bKh. Hence we would like the regions Ra and L�a tobe diamonds. Our task is thus to verify this if �A and �B are chosen suitably.By assumption, there is an acausal Cauchy surface C passing through �. Let C1 beanother acausal Cauchy surface lying strictly in the future of C, i.e. C1 � intJ+(C) =J+(C)nC. Then we suppose that �A has been chosen such that each point q 2 hA\C1has a�ne parameter U = 1, which means that q = �1(1; p) for some p 2 �. Clearlysuch a choice is always possible (it amounts to a suitable choice of the smooth rescalingfunction � : �! R). Under the Killing ow �t we get a familyCe�t := �t(C1), t 2 R, ofacausal Cauchy surfaces (not necessarily forming a foliation) having the property thateach q 2 hA \ Ce�t is represented as q = �1A (e�t; p) with suitable p 2 �. Obviously,a similar construction can be carried out with a Cauchy surface C�1 lying strictlyin the past of C and leads to family of acausal Cauchy surfaces C�e�t = �t(C�1).(Moreover, similar constructions can be made for �B, hB.) As we �rst chose C1 andthen adjusted �A to give all points of C1\hA a�ne parameter U = 1 it is not obviousthat we can choose C�1 to give all points of C�1 \ hA a�ne parameter U = �1.It would su�ce if there were a global isometry of M acting as a horizon-re ectionsymmetry j since then one may simply choose C�1 = j(C1). The existence of such anisometry will be required later, but not for the next lemma, where an arbitrary pairof Cauchy surfaces C1 and C�1 with the indicated properties is assumed given, andthe corresponding vector �elds �A and �(�)A assumed chosen so that each point onC1 \ hA has a�ne parameter U = 1 with respect to �A and each point on C�1 \ hAa�ne parameter U = �1 with respect to �(�)A .Lemma 4.2. If M = ~M , then R? = L, L? = R and R;L and Ra; L�a, a > 0, arediamonds.Proof. By assumption, we haveM = F [P [R[L, and F[P = J(�). Since � is partof a Cauchy surface, it follows that �? = intD(Cn�). Hence R [ L = intD(Cn�).Now de�ne CR := C \ R, CL := C \ L. Then CR \ CL = ; since L \ R = ; (see[40]), and CL [ CR = Cn�. Therefore we obtain intD(Cn�) = intD(CR [ CL) =intD(CL) [ intD(CR) where the last equality is a consequence of the fact that CLand CR are disjoint open subsets of a Cauchy surface. The boundary of CL and CRis in both cases the smooth manifold �. Hence L = intD(CL) and R = intD(CR)41

are diamonds, and since CL and CR are disjoint and their union yields C up to thecommon boundary � of CL and CR, this entails R? = L and L? = R.Now we de�ne the following sets: �a := Ca \ hA, CaR := Ca \ R, CaL := Ca \ L,CaF = Ca \ F . One can see that Ca \ P = ;, for there would otherwise be causalcurves joining pairs of points on Ca and this is excluded. It follows that Ca =CaL [ CaR [ CaF is the union of three disjoint parts, and intD(CaR) = (CaF [CaL)?. The common boundary of CaR and CaL [ CaF is the smooth manifold �a,implying that intD(CaR) is a diamond. Moreover, it is obvious that hA(a;1) �J+(�a), hA(�1; a) � J�(�a), and by standard arguments it follows that J+(�a) =clJ+(hA(a;1)) and J�(�a) = clJ�(hA(�1; a)). Let us check that Ra = intD(CaR).First we notice that intD(CaR) � R is fairly obvious (R is causally closed, i.e.R? ? = R, and CaR is an acausal hypersurface in R), and so is intD(CaR) = (CaF [CaL)? � �?a = MnJ(�a), implying intD(CaR) � Ra. To show the reverse inclusionit is su�cient to prove that Ra \ clJ(CaF [ CaL) = ;. We have clJ(CaF [ CaL) =clJ(CaF ) [ clJ(CaL) and CaL � L and R = L? imply that R \ clJ(CaL) = ;.Now consider an arbitrary past-directed causal curve starting at some point onCaF . For to meet Ra, it must intersect hA. However, any intersection of withhA must be contained in hA(�1; a] since is past-directed and we have seen thathA(a;1) � J+(�a) � J+(CaF ). Thus, since only the part of lying in the causalpast of its intersection with hA can enter R, never meetsRa = RnclJ�(hA(�1; a)),showing that clJ(CaF ) \ Ra = ;. Therefore Ra = intD(CaR) is a diamond. Ananalogous argument works for L�a.4.2 Conformal Spin-Statistics RelationOur aim in this subsection will be to show that the net O 7! A(O) on a spacetime withbKh induces a net of von Neumann algebras (a; b) 7! C(a; b), indexed by the openintervals (a; b) of the real line and allowing an extension to a conformally covarianttheory on the circle S1. Moreover, we shall see that this net is to be viewed ascontaining precisely the observables localized arbitrarily closely to the hA-horizon.(A similar construction works for the hB-horizon). The variant of Wiesbrock's resultson modular inclusion [64] which is needed to show this may be familiar to experts,but for the reader's convenience we present the arguments in an appendix to thischapter (Sec. 4.3).Earlier results [32, 33] on the spin-statistics connection for conformally covarianttheories on S1 then apply, yielding a conformal spin-statistics theorem for the subnetsof the initial theory consisting of observables concentrated on the parts hA and hB ofthe horizon.We begin with a spacetime with a bKh, (M;g; �t;�; h), where we assume hence-forth thatM = ~M (cf. Sec. 4.1). Furthermore, we assume given a net K 3 O 7! A(O)assigning to each member O in the collection K of regions in M a von Neumann al-gebra A(O) on a Hilbert space HA. For convenience, we shall work not with K, thecollection of regular diamonds ordered under inclusion, but extend the domain ofour observable net A in the canonical way to include a larger collection L of opensubsets of our spacetime. As discussed in the appendix to Sec. 3, this choice does not42

change the superselection structure in that each representation satisfying the selec-tion criterion based on K extends uniquely to a representation satisfying the selectioncriterion based on L, the intertwining operators thereby remaining unchanged. Againas discussed in the Appendix to Sec. 3, the formalism changes only in so far as thelocalized endomorphisms are now de�ned on larger algebras and this proves to be anadvantage. We choose L to be the set of non-empty causally closed subsets S of Mwith non-empty causal complements such that for the given net A ~?{duality holdseither for S or for S?. By virtue of Lemma 3A.4, this is the same as the partiallyordered set L de�ned in the Appendix to Sec. 3 in terms of sieves. We recall, too,that if A is additive, or even inner regular, as a net over K, then ~?{duality coincideswith ?{duality.Indeed, even though we assume duality for all diamonds, such assumption isactually used only for two kinds of regions, the translated wedges La and Ra, andsome tubular neighborhoods of the horizon intervals hA(a; b) or hB(a; b), which are inturn tubular neighborhoods in hA or hB of a suitable translation of �. We observethat the obstructions to duality are usually homological in nature, and that is whyduality is generally assumed to hold for regular diamonds. On the other hand thesurface �, even though not necessarily homologically trivial, is often relatively trivial,meaning that k-cycles in � which are trivial in M are trivial in � too.In the following we shall consider the subnet of O 7! A(O) generated by theobservables located arbitrarily closely to the (half) horizon hA. Let us adopt thesetting of Lemma 4.2 and start with a given acausal Cauchy surface C containing �and choose an acausal Cauchy surface C1 lying strictly in the future of C and thevector �eld �A so that each point on C1 \ hA has a�ne parameter U = 1. Then wede�ne for 0 < a < b <1,BRA(a; b) :=\O fA(O) : O � hA(a; b)g (4.16)where the intersection is taken over diamonds O. Likewise, one may also assumethat another acausal Cauchy surface C�1, lying strictly in the past of C, has beenselected and that another (possibly identical) copy �(�)A of �A has been chosen togive each point of C�1 \ hA an a�ne parameter U = �1. Correspondingly, we set for�1 < �b < �a < 0 ;BLA(�b;�a) :=\O fA(O) : O � hA(�b;�a)g: (4.17)Finally, with these assumptions, one may also de�neBA(a0; b0) :=\O fA(O) : O � hA(a0; b0)g; (4.18)for �1 < a0 < b0 < 1. Substituting B for A in the above, algebras BR=LB (a; b),BB(a0; b0) can be de�ned and all results formulated in the sequel for the algebras BAhold with obvious modi�cations for the algebras BB too.43

Lemma 4.3. Suppose that the net O 7! A(O) satis�es the following assumptions:� (I) Irreducibility: _O2KA(O) = B(H):� (II) Additivity: O � [i2IOi, Oi; O 2 K ) A(O) � _i2IA(Oi).� (III) Haag duality: A(O?) = A(O)0, O 2 K (implying locality).Then BRA(a; b) = A(Ra) \ A(Rb)0 (4.19)BLA(�b;�a) = A(L�a) \A(L�b)0 (4.20)BA(�a0; b) = A(L�a0)0 \A(Rb)0 (4.21)for all 0 < a < b <1, �a0 < 0.Proof. We shall only give the proof of the �rst equality, since the remaining cases arecompletely analogous, requiring some largely obvious notational changes.We recall that Ca = �ln a=�(C1) for any a > 0, and also the notation �a = Ca\hA,CaR = Ca \ R, CaF = Ca \ F and CaL = Ca \ L used in the proof of Lemma4.2. Then we de�ne the subsets eLa := (Ra)? = intD(CaL [ CaF ), Fa := J+(�a)and Pa := J�(�a), and analogous sets with a replaced by b. Next, we de�ne C_ :=CaL[CaF [hA(a; b)[CaR, and aim at demonstrating that this set is a Cauchy surface.It is fairly obvious that C_ is achronal, i.e. C_\intJ�(C_) = ;. It is also not di�cultto check that M = eLa [Rb [ Fa [ Pb where the sets forming the union are pairwisedisjoint except for the intersection Fa \ Pb = hA(a; b). Now let be an arbitraryendpointless causal curve in M . If enters eLa or Rb, it must intersect CaL [ CaF orCbR, hence C_. Suppose that enters Fa. Since Fa is past-compact, must intersectone of the regions Rb, eLa or Pb, as would otherwise have a past-endpoint. On theother hand, a causal curve without endpoint intersecting Fa can only meet Pb if itintersects hA(a; b), too. Hence, if enters Fa, it must also intersect C_. Using thesame argument with obvious modi�cations for the case that enters Pb, one arrivesat the same conclusion. This shows that every causal curve without endpoints in Mintersects C_, implyingM = D(C_), and therefore C_ is a Cauchy surface.Now we note that intD(U) � hA(a; b) for each open neighbourhood U of hA(a; b)in C_ since J(hA(a; b)) = Pb[Fa has empty intersection with cl(C_nU). Thus hA(a; b)is an intersection of diamonds. Moreover, whenever O � hA(a; b) is any diamond, it isobvious that we can �nd some open subset U of C_ with piecewise smooth boundaryhA(a; b) � U � O \ C_, implying hA(a; b) � intD(U) � O. Hence, to establishthe lemma, it su�ces to consider diamonds of the form O = intD(U). Obviously,the causal complement O? of each such O may be written as O? = O?R [ O?L whereO?R = O? \Ra = intD(CaRnU) and O?L = O? \ eLa = intD((CaL [CaF )nU) are bothdiamonds. Notice that the union of O?L and O?R over all O = intD(U) yield eLa and44

Rb, respectively. Consequently we have\O A(O) = (_O A(O)0 )0 = (_O A(O?) )0= (_O A(O?L [ O?R) )0 = (_O A(O?L ) _ A(O?R) )0= (A(eLa) _A(Rb) )0 = (A(Ra)0 _A(Rb) )0= A(Ra) \ A(Rb)0 ;where the second equality follows from Haag duality, the third has been justi�edabove, the fourth and �fth equalities use additivity and the last but one again followsfrom Haag duality.The formulation of the subsequent result necessitates introducing further assump-tions and related notation.We shall write BRA(a;1) := Wb>aBRA(a; b), and de�ne the other horizon-algebrasassociated with unbounded intervals in a similar manner by additivity. Let 2H be a unit vector vector, then we denote by HRA() := BRA(0;1), HLA() :=BLA(�1; 0) and HA() := BA(�1;1) the Hilbert subspaces generated by ap-plying the various algebras of observables concentrated on the hA-horizon on thatvector. We say that (BR(0;1);) is a standard pair if is separating for BR(0;1).It is by de�nition cyclic with respect to the Hilbert subspace HRA(). The modularobjects (with respect to HRA()) of such a standard pair will be denoted by JR;,�R;. The like objects for L in place of R are de�ned similarly.In the following, we shall focus attention on the next two assumptions:(IV) Geometric modular group on the horizon: There is a unit vector 2 H sothat(i) (BRA(0;1);) is a standard pair, and�itR;BRA(a;1)��itR; = BRA(e�2�t=�a;1) ; (4.22)(ii) (BLA(0;1);) is a standard pair, and�itL;BLA(�1;�a)��itL; = BLA(�1;�e2�t=�a) ; (4.23)for all a > 0, t 2 R, where � > 0 is the surface gravity of the bKh.(V) Geometric modular conjugation on the horizon: For the as in (IV), we haveHA() = HRA() = HLA() and moreoverJR;BRA(a;1)JR; = BLA(�1;�a) ; a � 0 : (4.24)Let us now assume that the net O 7! A(O) satis�es assumptions (I{IV). Thuswe see that (BRA(1;1) � BRA(0;1);) is a +hsm inclusion and (BLA(�1;�1) �BLA(�1; 0);) is a {hsm inclusion. Then the results of [64, 1] yield two continuousunitary groups UR=L(a), a 2 R, having positive/negative spectrum and satisfying the45

following relations for a > 0:��itR UR(a)�itR = UR(e2�ta) ; �itLUL(a)��itL = UR(e2�ta) ;JRUR(a)JR = UR(�a) , JLUL(a)JL = UL(�a) ,UR(a)BRA(0;1)UR(�a) = BRA(a;1) ;UL(a)BLA(�1; 0)UL(�a) = BLA(�1;�a)where we have dropped the index on the modular objects to simplify notation.Without further assumptions, UR and UL are unrelated and so are the nets BRA andBLA. However, if we suppose that (V) holds, too, then it follows from the way theseunitaries are constructed (cf. [64]), that JRUR(a)JR = UL(a), a 2 R. Therefore weobtain the following:Corollary 4.4. Under assumptions (I{IV) the nets of horizon-algebras indexed bythe intervals of the half real lines,(a; b) 7! BRA(a; b) ; 0 < a < b <1 ;(�b;�a) 7! BLA(�b;�a) ; �1 < �b < �a < 0 ;extend to local conformal nets I 7!MR(I) and I 7!ML(I) of von Neumann algebrason S1 on the Hilbert spaces HR0 = BRA(a; b) and HL0 = BL0 (�b;�a), respectively(where the 0 < a < b <1 are arbitrary).If (V) is assumed, too, then the net (a0; b0) 7! BA(a0; b0) on the full real line extendsto a local conformal net I 7!M(I) on HA().Proof. The �rst part is a variant of Wiesbrocks's result [64, 65], cf. also [33]. Wesupply the relevant argument as Proposition 4A.2 in Sec. 4.3.If assumption (V) is added so that JR intertwines UR and UL, the adjoint action ofUR(a) on the net BA is geometrically correct, i.e. UR(a)BA(a0; b0)UR(�a) = BA(a0+a; b0 + a), a 2 R, a0 < b0. Thus the net BA together with its dilation and translationsymmetries coincides with both CR and CL (derived from the nets BRA and BLA as inProposition 4A.2) and their respective translation and dilation symmetries. Thus thecorresponding extensions to conformally covariant theories coincide.Condition (IV) may be viewed as a weak form of the Hawking-Unruh e�ect: anobserver moving with the Killing ow of the bKh registers a thermal ensemble inthe \vacuum" state (see [56, 62]). The term \vacuum" here means a state invariantunder the space-time isometries and ful�lling additional stability conditions, in fact(IV) and (V) may be viewed as a weak form of such conditions, namely applyingto the subsystem of observables concentrated on the horizon. As the group of a�netranslations along the geodesic generators of the horizon has positive generator de-rived from the modular inclusion of horizon-algebras, can be justly interpreted as avacuum vector for the horizon-algebras (cf. the principle of geometric modular action[17] or modular covariance [15]). Clearly, if induces a KMS-state for the Killing ow at the Hawking temperature on A(R), then (IV,i) follows by Lemma 4.3. Like-wise, if induces a KMS-state for the Killing ow at negative Hawking temperatureon A(L), then (IV,ii) follows by Lemma 4.3.46

The motivation for Condition (V) is that, a horizon (or wedge) re ection sym-metry should be implemented in a \vacuum" representation by the modular conju-gations JR, in analogy with the Bisognano-Wichmann result for quantum �elds inMinkowski space [4, 5, 6, 56]. Our condition is actually a bit weaker in that JRneed not implement a point-transformation of the underlying spacetime manifold.However, Condition (V) implicitly imposes a relation between the horizon segmentshA(�1;�a) and hA(a;1).We �nally comment on whether these assumptions are realistic. For the freescalar �eld, conditions (I,II) hold generally in representations induced by quasifreeHadamard states (for O 2 K based on relatively compact subsets of Cauchy surfaces,and, in more special cases, even when the base is unbounded), see [59]. The Hartle-Hawking state, i.e. the candidate for the \vacuum" state of the free scalar �eld on theSchwarzschild-Kruskal spacetime, should also satisfy all the assumptions [37, 40] ((III)has not been checked in the generality formulated here, but a version of (III) su�cientto imply the spin-statistics theorem in the sequel does hold). As is known from theBisognano-Wichmann result [4], the assumptions are ful�lled for local von Neumannalgebras generated by (�nite-component) Wightman �elds in Minkowski spacetime((III) then holds for wedge-regions and this su�ces to establish the spin-statisticsrelations [31; 42]). Results of Borchers [5, 6] yield (III{V) generally for algebraicquantum �eld theories in two spacetime dimensions. With additional conditionsthese generalize to higher dimensions[7, 8, 66].Now we can formulate the conformal spin and statistics theorem. Our aim is tode�ne the spin of a sector as the conformal spin on the horizon. To this end weneed to restrict to considering sectors that are horizon localizable, namely having arepresentative which acts trivially on the algebras BA(a; b)0 for some a; b 2 R (orthe same for the B horizon). However this is not su�cient in general because thesector on the horizon may not be covariant. As shown in [33] covariance of localizedendomorphisms with �nite statistics is automatic when the net is strongly additive,which is always the case for the dual net. Unfortunately extending a sector on aconformal net to a sector on the dual net may produce soliton sectors. Thereforewe shall only consider those sectors which are not only horizon localizable, but alsodual localizable, namely which give rise to a localized sector on the dual net of thehorizon conformal net. Clearly if we have a dual localizable sector on the net O 7!A(O) satisfying assumptions (I{V) with non-zero statistical parameter �, we obtaina covariant sector on the dual net on the horizon with the same statistical parameter,since this is determined by the intertwiners. The following theorem is now a simpleconsequence of the conformal spin and statistics theorem in [32].Theorem 4.5. Let O 7! A(O) be a theory on a spacetime with bKh satisfying as-sumptions (I{V) and � a dual localizable sector with �nite statistics. Then � givesrise to a covariant sector on the dual net on the horizon, therefore a conformal spins� is de�ned, and the conformal spin and statistics relation holds, namely s� = ��.47

Remarks Concluding Section 4.2The idea of passing from a quantum �eld theory initially formulated over a four-dimensional spacetime to observables concentrated on a lightlike hypersurface (i.e.pieces of a bKh) is not a new one and once was popular in quantum �eld theoryunder the keyword \in�nite momentum frame". [43, 55] are just two references inthis direction. This matter is studied for the �rst time in the operator algebraicframework in [26]. One motivation is that symmetries may be enhanced by restrictingto a subtheory concentrated on a lightlike hypersurface, a particularly attractivepossibility for quantum �eld theory in curved spacetimes where symmetries of theunderlying four-dimensional spacetime are rather limited. As proved in this section,for bKh spacetimes restricting to the horizon does indeed give conformally covariantnets.Sewell [56] was the �rst to observe that this allows one to formulate a Bisognano-Wichmann theorem relating to the Hawking e�ect for quantum �elds on blackholespacetimes, in the setting of a Wightman �eld theory (see e.g. [62] for further dis-cussion). In this context, two papers rigorously establish related results for free �eldmodels [37, 22]. Kay and Wald [40] realized that such results may be generalized tospacetimes with a bKh and obtained strong theorems for free �elds in this setting.An operator-algebraic version of aspects of Sewell's work appears in [57] where thenets BA(a; b) are used.We ought to mention that in general it is not very clear how \big" the algebrasBA(a; b) (or BB(a; b)) are in the original algebras A(O).If is cyclic for BA(a; b) then it is resonable to expect that sectors are horizonlocalizable (on the A-horizon). Moreover in this case the conformal net on the horizonis strongly additive by de�nition, therefore it coincides with its dual net (cf. [33]),and then horizon-localizability and dual localizability are equivalent.It is known that is cyclic for BA(a; b) when free �elds on the n-dimensionalMinkowski space are considered, n 6= 2. We give here a simple argument based on[16].By a \free �eld" on Minkowski space we here mean a local net A of von Neumannalgebras indexed by regions of Minkowski space which can be constructed by secondquantization from a net K of real vector spaces in a complex Hilbert space H, plusthe usual assumptions of Poincar�e covariance, positive energy, and in particular theBisognano-Wichmann property and irreducibility: \WA(W ) = C I.Working in the �rst quantization space H from now on, we �rst observe thatirreducibility means \WK(W ) = f0g and, by the Bisognano-Wichmann property,this is equivalent to there being no �xed vectors for the action of the Poincar�e groupon H.Then, by a Theorem of Mackey (cf. e.g. [68], Proposition 2.3.5), the absenceof invariant vectors for the whole Poincar�e group is equivalent, when n 6= 2, tothe absence of invariant vectors for any given translation, hence the spectrum of thegenerator of any light-like translation is strictly positive, i.e. zero is not an eigenvalue.Now, given two wedges W1, W2, the cyclicity of the vacuum vector for A(W1)\A(W2) is equivalent to (K(W1) \ K(W2)) + i(K(W1) \ K(W2)) being dense in H,48

this being in turn equivalent to having fv 2 dom(sW1) \ dom(sW2) : sW1v = sW2vgdense in H, where sWj denotes the \�rst quantized" Tomita operator de�ned bysWj(� + i�) := � � i�, �; � 2 K(Wj). When W1 = f(t; x) : x1 > jx0jg and W2 is atranslation of the causal complement of W1, W2 = f(t; x) : x1 � c < �jt� cjg, c > 0,the situation met when considering the vector space associated with the interval(0; c) on the A-horizon, this is in turn equivalent, again by the Bisognano-Wichmannproperty, to the density of the spacefv 2 dom(�1=21 T (c)�1=21 ) : T (c)�1=21 T (c)�1=21 v = vg; (4.25)where a! T (a) denotes the representation of the light-like translations along the A-horizon and �1 denotes the \�rst quantized" modular operator for the space K(W1).This property clearly depends only on the restriction of the representation of thePoincar�e group to the subgroup P1 generated by boosts and light-like translationswith strictly positive generator (relative to the wedge W1). As the logarithm ofthe generator of translations and the generator of the boosts give rise to (and aredetermined by) a representation of the CCR in one dimension, the strictly positiveenergy representations of P1 have a simple structure: they are always a multiple of theunique irreducible representation. Therefore the density of the space in eqn. (4.25)holds either always or never, and hence can be checked in the irreducible case. Butthis is the case of the current algebra on the circle, where cyclicity holds by conformalcovariance.Of course, the vector is not expected to be cyclic in general for the algebragenerated by the BA(a; b), and it might even happen that BA(a; b) contains onlymultiples of the identity. Field nets giving rise to non-trivial superselection sectorsof the observable net localizable on the horizon can easily be constructed just byrequiring the vacuum to be cyclic for the horizon �eld algebras. However it is notclear, in general, how strong the requirement of dual localizability is.4.3 Appendix to Chapter 4For the bene�t of the non-expert reader, we present in detail in this Appendix thearguments leading from the results in [64, 65, 33] to Corollary 4.5. To begin with, westate a result about modular inclusions needed in the following.Lemma 4A.1. Let (N � M;) be a pair of von Neumann algebras with a unitvector cyclic and separating for M and such that �itN��it � N for all �t � 0 (ort � 0), where �it, t 2 R, is the modular group of M;. Then M = _t2R�itN��it ifand only if is cyclic for �itN��it for some (hence for any) t 2 R.Proof. If is cyclic for �itN��it for a given t, then it is cyclic for _t2R�itN��it,too. However this von Neumann algebra is invariant under the modular group of M,and hence coincides with M by Takesaki's theorem. Conversely, let � be orthogonalto �itN��it. Then for any x 2 �itN��it we have x 2 dom(�1=2), hence thefunction z 7! (�izx; �) is analytic on the strip �i=2 < =z < 0 and continuous onthe boundary. But as we have a +hsm inclusion, it vanishes for negative real z and49

hence everywhere. Thus � is orthogonal to _t2R�itN��it = M, completing theproof.Proposition 4A.2. Let (N � M;) be a a pair of von Neumann algebras with aunit vector which is cyclic and separating for M and such that _t2R�itN��it =Mand �itN��it � N for all �t � 0, where �it, t 2 R, is the modular group of M;.Then, settingH0 = (N \ (��iN�i)0) (4.26)C(a; b) = (��i log a2� N�i log a2� ) \ (��i log b2� N�i log b2� )0 �H0; 0 < a < b; (4.27)the family (a; b) 7! C(a; b) extends to a local conformal net of von Neumann algebrasacting on the Hilbert space H0.Proof. Set Na = ��i log a2� N�i loga2� ; a > 0By the previous lemma is cyclic for Na, a > 0, therefore we may apply a resultof Wiesbrock and Araki-Zsido ([64, 1]) to the +hsm inclusion (N �M;) and get aone parameter group of unitaries U(a) on H with positive generator satisfying��itU(a)�it = U(e2�ta)JU(a)J = U(�a)Hence we have Na = U(a)MU(a)� ; a � 0 ;and this equation is used to de�ne Na for negative a.We now setC(a; b) = Na \N0b �H0 ; �1 < a < b < +1C(�1; b) = _a<bC(a; b) ; �1 < b < +1C(a;+1) = _b>aC(a; b) ; �1 < a < +1and the de�nition of C(a; b) clearly agrees with (4.26) when 0 < a < b <1. Further-more, H0 = N \ N0e2� = C(1; e2�):Moreover, the operators J;� restricted to H0 give the modular conjugation andoperator of (C(0;1);). Similarly, using the results of [64, 1] anew, the restrictionof U(a) to H0 (again denoted by U(a)) coincides with the unitary group derivedfrom the +hsm inclusion (C(1;1) � C(0;1);). Now a standard Reeh-Schliederargument, based on the positivity of the generator of U(a), shows that C(�1; b)is independent of b, while the \modular" Reeh-Schlieder argument in Lemma 4A.1shows that C(a; b) is independent of a 2 (�1; b). Thus the inclusion (C(1;1) �C(0;1);) is standard. We have proved that H0 = C(a; b) for any �1 � a < b �+1, and that C gives a translation-dilation covariant net of von Neumann algebrason H0. Then we get a conformally covariant net by a result of Wiesbrock ([65], seealso [33]). 50

5 The Spin and Statistics Relation for Spacetimeswith Rotation SymmetryIn this section, we present a proof of the spin and statistics relation for superselectionsectors on a globally hyperbolic spacetime with some rotational symmetry.The main assumption here is the existence of a suitable family of regions, calledwedges, each being equipped with a re ection mapping it to its causal complementand of a net of von Neumann algebras with a common cyclic vector whose modularconjugations implement the said re ections, in the spirit of [18] and [45].Moreover we assume the existence of rotational spacetime symmetries, rotatinga wedge to its causal complement and belonging to the commutator of the space-time symmetry group. As we shall see, our geometric assumptions are satis�ed inmany interesting spacetimes and form the geometric basis for the rotational spin andstatistics theorem, explained in more detail below.5.1 Geometric AssumptionsA spacetime with rotation and re ection symmetry is a quadruple (M;W; G+; j),where M is a globally hyperbolic spacetime,W is a family of open subregions calledwedges, G+ is a Lie group of proper (i.e. orientation preserving) transformations ofM and j is a map fromW to the antichronous (i.e. time reversing) re ections in G+;we write it as W 7! jW . We denote the orthochronous subgroup of G+ by G"+ andthe identity component of G+ by G0. The universal covering of G0 is denoted by eG.The Z2 action implemented by any jW on G0 lifts to an action on eG. The quadruplehas to satisfy the following properties:(a) j leaves W globally invariant and veri�es jW (W ) = W? and jgW = gjW g�1,W 2W, g 2 G+.(b) There is W 2W and an element h in the Lie algebra of G0 such that(1) exp(2�h) is the identity in G0,(2) jW exp(th)jW = exp(�th),(3) exp(�h)W =W?,(4) \0�t��=2 exp(th)W is non{empty.(c) h belongs to the commutator of the Lie algebra of G0.Remark 5.1. Two wedges W , fW are called orthogonal if jWfW = fW and jfWW =W .It is easy to see that W and exp(�2h)W are orthogonal. Indeed, making use ofassumptions (b 2) and (b 3), we getjW exp(�2 h)W = exp(��2h)jWW= exp(��2h)W? = exp(��2h) exp(�h)W= exp(�2h)W:51

The second equation is proved analogously.We shall also consider spaces where property (c) is replaced by the following prop-erty:(c0) There exists a wedge fW , orthogonal to W , such that jfW commutes with exp(th).Remark 5.2. (i) Assumption (a) has to be seen as a part of the de�nition of a wedge.The �rst part of property (a) says that any wedge is G+{equivalent to its causalcomplement, hence a wedge is in some sense \a half" of M or, more precisely, is thecausal completion of \a half" of a Cauchy surface. The second part means that jWcommutes with the stabilizer of W and, when G+ acts transitively on W, says that jis determined by its value on one wedge.(ii) Properties (b) describe the rotation symmetry. In view of property (b 1) we callthe group elements exp(th) rotations. Property (c) ensures that all characters of eGare trivial on the cycle fexp(th); t 2 [0; 2�]g, since the latter belongs to the commu-tator subgroup of eG where all characters are trivial. As we shall see, this makes thespin well de�ned.(iii) The element jW , seen as an automorphism of the Lie algebra of G0, has eigenval-ues 1 and �1 and by (a) the eigenspace corresponding to 1 consists of generators oftransformations preserving W . Therefore (b 2) essentially says that not all rotationspreserveW . More precisely,W may be rotated to its spacelike complement by (b 3).(iv) Property (b 3) mainly expresses the fact that 2� is the minimal period of theone-parameter group exp(th).(v) Property (b) is stated for one wedge W , but then holds for any wedge in thefamilyW0 := fgW : g 2 G+g. We are of course interested in the case where the cyclefexp(th); t 2 [0; 2�]g is not homotopy trivial and hence gives rise to a non-trivialnotion of spin. However this is not needed for the proof of the spin and statisticstheorem nor do we require that the exp(2�h) generate the homotopy group of G0.(vi) Property (c0) implies that W , e�=2hW and fW are mutually orthogonal. It alsoimplies that rW := jW jfW is an involution in G"+ and that exp(2th) = [exp(th); rW ],where the square brackets here denote the multiplicative commutator. As a conse-quence, the rotations exp(th) belong to the commutator subgroup of G"+. In this sense(c0) is a weak form of (c). G"+ and G0 do not always coincide. Of course rW 2 G"+,but we do not require that rW belongs to G0.(vii) Property (b) �xes the the generator h up to a sign, indeed (b 1) �xes the genera-tor up to an integer, (b 3) implies this integer to be odd, and (b 5) requires this integerto be 1 or �1. When the spacetime is two-dimensional, i.e. when the Cauchy surfaceis 1-dimensional, the orientation �xes a direction on any spacelike curve (from left toright). In this case we choose the sign in such a way that the element h generates arotation in the prescribed direction.Assumptions (a), (b), (c) and (c0) in some spacetimesIn the case of the n-dimensional Minkowski spacetime Mn, a wedge is any G+{transform of the region W = fjx0j < x1g if n > 2, and of the region fx > 0g if n = 1.52

Taking jW to be the re ection w.r.t. the edge of the wedge, the map j turns out tobe uniquely de�ned by property (a).When G+ is the proper Poincar�e group and n � 3, property (b) holds with Was above and h as the generator of rotations in the (x0; x1)-plane. Indeed the properorthochronous Poincar�e group is perfect, hence property (c) is obviously satis�ed. Ifn � 4 then (c0) is satis�ed too, with fW = fjx0j < x2g.WhenG+ is the proper conformal group, properties (b) and (c) are satis�ed for anyn � 1, h being the generator of a suitable group of (conformal) rotations. Property(c0) is satis�ed when n � 3, W being as before, h being the generator of rotations inthe (x0; x1)-plane and fW a double cone with spherical basis centred on the origin.Since the n-dimensional de Sitter spacetimeDn may be de�ned as the hyperboloidx20+1 = jxj2 inMn+1, the wedges can be de�ned as the intersection of this hyperboloidwith the wedges in Mn+1 whose edge contains the origin. Then properties (b), (c) or(c0) hold for Dn if and only if properties (b), (c) or (c0) hold forMn+1 (with Poincar�esymmetry), respectively.Note that the Cauchy surface of Dn is compact and the same is true for Mn withconformal symmetry, since in this case the quantum �eld theories actually live on (acovering of) the Dirac-Weyl compacti�cation of Mn (cf. [14]).Whenever the spin makes sense in the above examples, i.e. whenever (c) or (c0)holds, the group G"+ has no non-trivial �nite dimensional representations, a muchstronger requirement than (c) or (c0). In this case the spin and statisitcs relation maybe proved as in [45].Moreover, in these examples, modular covariance makes sense, i.e. there is a nat-ural de�nition of the geometric action of �it, furthermore, the Bisognano-Wichmannproperty has been proved for Wightman �elds ([4, 10]), wedges separate spacelikepoints and every double cone is an intersection of wedges. Therefore geometric mod-ular conjugation follows from modular covariance (as in [31], cf. [18]) and modularcovariant free �elds may be constructed canonically as in [16] by second quantizing(anti-)unitary representations of G+.We now describe a class of spacetimes where these additional features do nothold, namely where the group admits one-dimensional representations and the wedgesdo not separate points. Nevertheless, these cases are still covered by the spin andstatistics theorem we are going to present below.53

Spherically symmetric black holesWe call spherically symmetric black holes those spacetimes (K; gK) whose structure isvery similar to the Schwarzschild-Kruskal spacetime, i.e. they are isometric toX�Sn,X being the set of points (x0; x1) 2 R2 with x20 � x21 < �2, � 2 R [ f1g, with themetric8 ds2K = a(x20 � x21)(dx20 � dx21)� b(x20 � x21)d2� ; (5.1)where d2� is the usual Riemannian metric on the sphere Sn and a and b are smooth,strictly positive functions. Then the hypersurface x0 = 0 is a Cauchy surface and(K; gK) is globally hyperbolic. The structure of such spacetimes is in some respectssimilar to that of Minkowski spacetime. For instance, if points in X � Sn are rep-resented as (x0; x1; �), then one may de�ne a one-parametric group of isometries �t,t 2 R, by replacing the pair (x2; x3) 2 R2 by � 2 Sn in de�nition (4.2) and then de�ne� and hA and hB, correspondingly. Hence (K; gK) has the structure of a spacetimewith bKh, where the Killing ow is �t = �t, t 2 R. Moreover, there is an horizonre ection j(x0; x1; �) = (�x0;�x1; �) which is a PT symmetry, i.e. an orientation andchronology-reversing isometry.Let us investigate further the isometries of such spacetimes. To simplify thematter a bit, we assume that (K; gK) does not admit translations in the X-part ofK = X � Sn as symmetries. (This is not really a restriction; our �ndings can bemodi�ed by taking the semidirect product of the translational symmetry group TXwith the non-translational symmetry group G in the presence of such symmetries.For our treatment of the connection between rotational spin and statistics, transla-tional symmetries are irrelevant.) Since (K; gK) is orientable and time-orientable, weconsider the groups G+ and G"+ of proper (i.e. orientation preserving) and properorthochronous (i.e. time-orientation preserving) isometries, respectively. In the fol-lowing, we describe the proper orthochronous subgroup G"+.The form of the metric tensor gK and the assumed triviality of TK imply that allisometries leave � globally �xed and that an element of G"+ acting trivially on � hasto be an element of the Killing ow. Conversely, orientation preserving isometriesof �, i.e. elements of SO(n + 1), naturally give rise to symmetries in G"+. Indeed,R(x0; x1; �) = (x0; x1; R�), R 2 SO(n + 1), gives an isometry of (K; gK). To extendorientation reversing isometries of � to orientation preserving isometries of K, weobviously need a di�erent procedure. To this end we note that each orientationreversing isometry of � � Sn can be written as a product of a rotation in SO(n+1)and an equatorial re ection rQ, where Q denotes the Sn�1 equator of �xed points ofsuch a re ection. More precisely, rQ re ects points on Sn about Q along the greatcircles orthogonal to the equator Q. In other words, rQ acts as a re ection of thenormal geodesic spray of Q in Sn. Note that such equatorial re ections generate theaction of O(n+1) on Sn. In fact, if an equator Q1 is inclined at angle � to an equatorQ2, then rQ1rQ2 is a rotation by 2� about the axis de�ned by the intersection of Q1and Q2.8It is customary to write coordinate indices as upper indices, but our deviating from this con-vention is unlikely to cause confusion. 54

Now choosing a normalized, timelike, future-oriented, rotation-invariant normalvector �eld �0 along � there is a unique normalized, spacelike, outward-oriented,rotation invariant normal vector �eld �1 along � such that �0 + �1 is parallel to thevector �eld �A. It is therefore equivalent to choosing an orthonormal frame on theX{component of K = X �Sn. Moreover the Killing ow acts transitively on the setof possible such choices.An equatorial re ection rQ extends to an orientation and chronology-preservingisometry rQ;0 2 G"+ by setting rQ;0 := (x0;�x1; rQ�) and we de�ne rQ;t := �trQ;0��1t ,where �t, t 2 R, is the Killing ow. Each rQ;t 2 G"+ is an involution. On the otherhand, by the above observation, each involution in G"+ restricting to some rQ on �must be of the form rQ;t for some t 2 R. Clearly, rQ;t determines a unique normalized,spacelike, outward-oriented, rotation invariant normal vector �eld �1 along � whichis anti-invariant under rQ;t.Thus G"+ is generated by the Killing ow, the (extensions of the) orientationpreserving isometries of � and the re ections rQ;t so that G"+ � (R�SO(n+1))��Z2,where � denotes the conjugation by rQ;0, for some given equator Q. ConsequentlyG0 � (R� SO(n + 1)) and G+, being generated by G"+ and the horizon re ection j,is isomorphic to (R� SO(n+ 1))��Z2�Z2. The following lemma obviously holds.Lemma 5.3. On a spherically symmetric black hole, the commutator subalgebra ofthe Lie algebra of the identity component G0 of the group of proper isometries isisomorphic to so(n+1). The commutator subgroup of G0 is isomorphic to SO(n+1).We now show that the re ection symmetries rQ;t are naturally associated withwedge-like subregions of K. Indeed, given a normalized, spacelike, outward-oriented,rotation-invariant normal vector �eld �1 along �, its (two-sided, maximally extended)geodesic spray gives a geodesic-foliated Cauchy surface containing �, and it is easyto see that all such Cauchy surfaces arise in this way. Therefore, given a re ectionrQ;t and an open hemisphere E in � � Sn with @E = Q, we may consider theopen causal completion W (E; t) of the part of the Cauchy surface generated by thespacelike vectors determined by rQ;t and based on E. Put di�erently, de�ning E0 :=f(0; x1; �) : x1 2 R ; � 2 Eg and Et := �tE0��1t , t 2 R, then W (E; t) = intD(Et)where D(Et) is the domain of dependence of Et. We also mention that the edge ofthe wedge W (E; t) is the spacelike cylinder generated by the geodesic spray of thevectors of �1 based on @E, i.e. the set �tf(0; x1; �) : x1 2 R ; � 2 @Eg. Hence eachW (E; t) is a diamond. The set of such wedge{regions will be denoted by W0. Thefollowing proposition immediately follows.Proposition 5.4. (i) W (E; t)? = r@E;tW (E; t) = W (E 0; t), where E 0 denotes theinterior of the complement of E.(ii) R W (E; t) = W (RE; t), for any R 2 SO(n + 1).(iii) �sW (E; t) = W (E; s+ t).(iv) The group G"+ acts transitively on the family W of wedges W (E; s).(v) The group G"+ is generated by the re ections rQ;t.55

Now we show that these spacetimes �t in the scheme proposed at the beginningof this section. Let us de�ne W as W0 [ fRg [ fLg, jR = jL as the horizon re ectionj and jW (E;t) = jR r@E;t.Proposition 5.5. If n � 2 then properties (a), (b), (c) and (c0) hold. If n = 1 thenproperties (a), (b) and (c0) hold.Proof. Proposition 5.4 immediately gives (a). Then let W = W (E; t) and chooseh 2 so(n + 1) as an eigenvector with eigenvalue �1 of jW (E;t), normalized in such away that exp(#h) is a rotation through an angle #. Then property (b) is obviouslysatis�ed and choosing fW = R we get property (c0). When n � 2, (c) follows byLemma 5.3.5.2 Quantum Field Theories on Spacetimes with RotationSymmetryNow we consider a net O 7! A(O) of von Neumann algebras indexed by elementsO 2 K [W where K is the set of regular diamonds and W is a set of wedges withthe properties discussed in the previous section; this net describes the observables ofa local quantum theory on M . We require irreducibility, additivity and Haag dualityas in assumptions (I-III) of Sect. 4.2 and, moreover,(VI) Reeh-Schlieder property: There exists a unit vector (vacuum) cyclic for thevon Neumann algebras associated with all wedge regions.(VII) Geometric modular conjugation:JWA(O)JW = A(jWO);where O is any regular diamond and JW denotes the modular conjugation as-sociated with the algebra A(W ) and the vector , cf. Sect. 4.2.(VIII) Covariance: There exists a unitary representation U of the group G"+ such thatU(g) = for any g 2 G"+, U(g)A(O)U(g)� = A(gO) for any g 2 G"+ and anyregular diamond O and JWU(g)JW = U(jW gjW ) for any wedge W .Let us note that, under the previous hypotheses, the representation U extends toan (anti)-unitary representation of G+ with a geometric action on the net verifyingU(jW ) = JW .Proposition 5.6. Under the above assumptions, the net satis�es duality for the re-lation ?, namely A(O) = \O1?OA(O1)0where (cf. Appendix to Section 3) O1?O if O1 ? O and 9O2 2 K : O1, O � O2.56

Proof. Let O1 ? O. By Lemma 2.1, for any x 2 O1 there exist Ox, ~Ox 2 K such thatO ? Ox and O, Ox � ~Ox, in particular O?Ox. ThenA(O) � \O1?O \x2O1 A(Ox)0 = \O1?OA(O1)0 = A(O)where the �rst equality follows by additivity and the second by duality.(IX) equivalence of local and global intertwiners: Given a representation � satisfyingthe selection criterion and localized in a wedgeW , let �W denote the associatedendomorphism of A(W ), then(�; �) = (�W ; �W ):Remark 5.7. (i) This assumption implies factoriality for the algebras associated withwedge regions, that irreducibility of representations coincides with irreducibility on awedge and that the equality (�; �0) = (�W ; �0W ) holds for pair of representations (see[32]).(ii) Assumption (IX) has been shown to follow from dilation invariance [54], and itis conjectured that it already follows from the existence of a non-trivial scaling limit.We give an explicit proof of its validity for Minkowski space of any dimension in theAppendix to this section.(iii) If we assume G"+ to be continuously represented by automorphisms �g, G+ to begenerated by fjW ;W 2Wg and AdJW1JW2 = �g, with g = jW1jW2 , we get covariance(VIII). Moreover we obtain algebraic covariance for any sector with �nite statistics,namely � ' �g���1g , g 2 G"+. By an argument of M�uger [48], this implies that anysector is covariant w.r.t. a continuous representation of a central extension of G"+.When the wedges separate spacelike points, i.e. regular diamonds are intersections ofwedges, geometric modular conjugation (VII) also follows (cf. [18]).Spin and Statistics under property (c)Theorem 5.8. Let � be a representation satisfying the selection criterion and lo-calized in O � W . Suppose the associated endomorphism �W of the von Neumannalgebra of the wedge W has �nite index. Let j be the antilinear morphism imple-mented by the modular conjugation of (A(W );). Then j � � � j is a conjugate of �and � has �nite statistics.Remark 5.9. To inclusions of von Neumann algebras one can assign an invariant, apositive number called the index (cf. [44] and refs. cited there). The index of theendomorphism �W is that assigned to the inclusion �W (A(W )) � A(W ). For dis-cussion of the relation between the statistical dimension of a superselection sector inquantum �eld theory in Minkowski spacetime and the index of its associated localizedendomorphisms, the reader is again referred to [44].57

Proof. Pick a representation �0 equivalent to � and localized inW?. Then arguing asin [30], we see that j�j and �0 yield conjugate endomorphisms of the von Neumannalgebra of the wedge W?. The next step is to deduce from Assumption IX that j�jand � are conjugate representations. This circumstance is obscured by the fact thatthe product even of localized representations is de�ned only up to equivalence. Forthis reason, we use cocycles from Z1t (A) instead of representations, recalling Theorem3A.5. We have a faithful tensor �-functor Z1t (A) ! T(a) taking a cocycle z into theassociated endomorphism y(a) in a and an arrow t into ta. If a � W , then there is atensor �-functor from T(a) into the category of endomorphisms of the von Neumannalgebra of the wedge W , mapping an object � onto its restriction to the algebra ofthe wedge �W and acting as the identity on arrows. Assumption IX means that thecomposition of these functors is even full. Thus if y(a)W and �y(a)W are the images ofz and �z and are conjugates, z and �z are conjugates. If z is a cocycle associated with �0and �z is a cocycle associated with j�j, then the endomorphisms of A(W?) obtainedby restriction are conjugates and so are the equivalent endomorphisms y(a)W and�y(a)W . Hence z has a left inverse and �nite statistics.By assumption, JW implements a spacetime re ection consisting of a time reversing(since JW is anti-unitary) and a space reversing transformation since, preserving theoverall orientation, it has to reverse the orientation of any globally invariant Cauchysurface. Therefore the previous theorem is indeed a PCT theorem.In the following we choose a rotationally symmetric spacetime (M;W; G+; j) sat-isfying properties (a), (b) and (c), a local net O 7! A(O) verifying the above assump-tions and an irreducible, eG-covariant, superselection sector with �nite statistics.If � is a representation obeying the selection criterion with �nite statistics, asabove, let � be a localized endomorphism de�ned using an associated cocycle. Thestandard left inverse for the cocycle gives us a left inverse � for �, cf. Lemma 3A.10.When the statistics operator "(�; �) is uniquely de�ned, namely when the space-timedimension is greater than or equal to 3, ��;�("(�; �)) is an intertwiner between � anditself. Therefore, when � is irreducible, it is a complex number, cf. Sec. 3.4.When the dimension of a Cauchy surface is one, there are two choices for thestatistics and correspondingly two choices for the statistics parameter. In this case,we choose the statistics operator " associated with the connected component of G?where the 1-simplices have the chosen orientation (cf. Remark 5.2 (vii)).Let us note that, by Assumption IX, a left inverse exists even when a Cauchysurface is compact.The preceding theorem shows that the statistics phase is well de�ned. In fact, thesame is true for the spin, as the following proposition shows.Proposition 5.10. Let � be a representative of the given sector and (�;U�) a co-variant representation. Then:(i) The quantity s := U�(exp(2�h)) is a complex number of modulus one dependingonly on the equivalence class of � and not on the representation U�. It is calledthe spin of the sector. 58

(ii) Given U�, let � := AdV � � be an equivalent representation, then (�; U�) is acovariant representation, where U� := V U�V �does not depend on the intertwiner V .Proof. (i) Since � is irreducible, U� is �xed up to a one-dimensional representa-tion. By Assumption (c), one-dimensional representations are trivial on exp(th),hence U�(exp(2�h)) does not depend on the chosen representation. Since exp(2�h)is the identity element in G0, the corresponding element in ~G is a central element, soU�(exp(2�h) is a scalar by irreducibility. Equation 5.2 shows that s does not dependon the representative �. (ii) is obvious.A priori s depends on the Lie algebra element h. However, this possibility isruled out a posteriori by the spin and statistics relation. In the following, we �x theassignment � 7! U� for any representative �, as described in the above proposition.Now we may state the main theorem of this section. The proof will require somelemmas.Theorem 5.11. Let us consider a local net O 7! A(O) on a rotationally symmetricspacetime (M;W; G+; j), satisfying the above assumptions (I-III), (VI-IX), and anirreducible ~G-covariant superselection sector with �nite statistics on such a net. Thenthe spin of the sector agrees with its statistics phase.Let � be a representative of a sector with �nite statistics, let O be contained in awedge W and let � be an object of EndA(O) associated with � and set�� := j � � � j;where j is the modular antilinear morphism associated with A(W ) and . �� is anobject of EndA(jWO). Let V denote the Araki-Connes-Haagerup standard imple-mentation (cf. e.g. [32]) of the restriction of � to A(W ).Lemma 5.12. (cf. Lemma 3.1 of [32]) Let ~W be a wedge orthogonal to W . Let� ~W and �� ~W denote the restrictions of � and �� to A( ~W ), then (id; � ~W �� ~W ) is onedimensional and V 2 (id; � ~W �� ~W ) \ A(~O), where ~O is any element of L 9 containingO and jWO with ~O � ~W .Proof. We remark that the existence of conjugates for �nite statistics depends onAssumption IX and was discussed in the proof of 5.11. Since we are dealing with asector, Assumption IX implies that (id; � ~W �� ~W ) is one dimensional and contained inA(~O). In fact, let z yield � in O, i.e. y(a) = � for a = O, then the cocycle �z, de�nedby �z(b) = j(z(jW b)); b 2 �1;9Recall that ~O is in L if ~?-duality holds either for ~O or for ~O?, cf. the discussion at beginningof Sect. 4.2. 59

yields j � � � j in �a = jWO. Let b 2 �1 be de�ned by @ob = a, @1b = �a and jbj = ~O. Asimple computation shows thaty(a)(�z(b))V A = y(a)�y(a)(A)y(a)(�z(b))V; a 2 A( ~W ):Thus by Assumption IX, y(a)(�z(b))V 2 A(O). But �z(b) 2 A(jbj) = A(~O). HenceV 2 A(~O) as claimed. Obviously, an isometry V in (id; � ~W �� ~W ) will implement �W .Now a simple computation shows that j(V ) 2 (id; ��jW ~W �jW ~W ). But jW ~W = ~W sinceW and ~W are orthogonal. Hence, we may suppose that V = j(V ) and di�ers at mostby a sign from the standard implementation of the restriction of � to A(W ).Let � be a representative of a sector with �nite statistics and let z be an associatedcocycle. Let O be a diamond contained in the intersection of two wedges W1 and W2and � the object of EndA(O) associated with z. Write ji for the modular antilinearmorphism associated with A(Wi) and ��i for ji � � � ji, i = 1; 2.Lemma 5.13. Let �, ��i and Wi, i = 1; 2, be as above and suppose there exists ag 2 ~G with W2 = gW1. The following identity between representations of the netO1 7! A(O1), O1 � O, holds:���1 = AdU�(j1gj1g�1)���2AdU(j1gj1g�1)�;where g 7! j1gj1 denotes by abuse of notation the action of j1 lifted to ~G and j1 := jW1 .Proof. We have J2 = U(g)J1U(g)�, hence J1J2 = U(j1gj1g�1) and j1j2 = AdU(j1gj1g�1),therefore ��1 = AdU(j1gj1g�1)��2AdU(j1gj1g�1)�:Thus by covariance ���1 = �AdU(j1gj1g�1)��2AdU(j1gj1g�1)�= AdU�(j1gj1g�1)���2AdU(j1gj1g�1)�:Lemma 5.14. Let �, W1 and W2 and g be as in the previous lemma. Then there isa (unique) complex number c(�;W1; g) of modulus one such thatU�(j1gj1g�1)V2U(j1gj1g�1)� = c(�;W1; g)V1: (5.2)Proof. By Lemma 5.12, V1 2 (id; � ~W1 �� ~W1). Furthermore, by the previous lemma,U�(j1gj1g�1)V2U(j1gj1g�1)� belongs to the same one dimensional space of intertwin-ers.Lemma 5.15. Let � and � be two endomorphisms associated with a given sector asabove, the �rst localized in W1\W2, W2 = gW1, the second in hW1 \hW2, g; h 2 ~G.Then c(�;W1; g) = c(�; hW1; hgh�1). 60

Proof. We �rst observe that if � = AdW �� for some unitary W 2 A(W1 \W2), thenV �i = W �JiW �JiV �i and this implies that c(�;W1; g) = c(�;W1; g). Then we note thatc(�;W1; g) = c(��1h ��h; hW1; hgh�1), where �h = AdU(h), because U(h) establishesan isomorphism between the original structure and the structure transformed byh. Since ��1h ��h and � are associated with the same sector and both localized inhW1 \ hW2 and hgh�1hW1 = hW2, the result now follows.The previous lemma shows that for the given sector there is a well de�ned functionc(W; g) satisfying c(W; g) = c(hW;hgh�1)whenever W \ gW 6= ;.Lemma 5.16. Let W 2 W. Then the function g 7! c(W; g) is a local group charac-ter, namely for any g; h 2 ~G such that W \ gW \ ghW 6= ;, we havec(W; g)c(W;h) = c(W; gh) :Proof. Choose associated endomorphisms localized in W \ gW \ ghW and denotethe involutions associated with W and gW by j1 and j2, respectively. Then from thede�nition of c for the pairs (W; g) and (gW; hgW ) and the equality(j1gj1g�1)(j2hj2h�1) = j1gj1g�1(gj1g�1)h(gj1g�1)h�1 = j1hgj1(hg)�1one obtains the relation c(W; g)c(gW; h) = c(W; gh)which means that the function c is a local groupoid character. Then, making use ofLemma 5.15 we getc(W; g)c(W;h) = c(W; ghg�1) = c(W; (ghg�1)g) = c(W; gh):In Proposition 5.10, we only used properties (b 1), (b 2). The rest of the argumentmakes essential use of further properties, more precisely, (b 2) and (b 3) are usedin the following proposition, whilst (b 3) and (b 4), or rather, the orthogonality ofRemark 5.1, are used to conclude the proof of Theorem 5.11.Proposition 5.17. Under the given assumptions, we have c(W; exp �2h) = 1.Proof. Since g 7! c(W; g) is a local representation, it is locally trivial on the commu-tator of eG, hence, by assumption (c), there exists " > 0 such that c(W; exp th) = 1for jtj � ". Because of assumption (b 4) the result follows by applying Lemma 5.16su�ciently often.Lemma 5.18. Let � be an endomorphism associated with the sector and localizedin O � W1 \W2, where W1 and W2 are orthogonal wedges, W2 := exp(�2h)W1 (cf.Remark 5.1). Let the standard implementations of its restriction to the algebrasA(W1), A(W2) be denoted by V1, V2 as before. Then the statistics parameter �� canbe written as �� = V �1 V �2 V1V2. 61

Proof. As in [32], Lemma 3.5, we �rst show �� = �(V �1 )V1; indeed if �0 is localizedin W1 \ W?2 and u is a unitary in (�; �0) in EndA(W1), then u 2 A(W1). Since(W1 [ W2)? 6= ;, u�A = u��0(A) = �(A)u�, for A 2 A(W2). But V1 2 A(W2) byLemma 5.12. Thus �(V �1 )V1 = u�V �1 uV1. Now ��1 := j1 � � � j1 is localized in W?1 \W2and, again since (W1 [ W2)? 6= ;, �, �0 and ��1 are comparable and �1(u) = u.Thus V �1 uV1 = �(u), where � is the left inverse of �. Hence �(V �1 )V1 = u��(u) =�("(�; �)) = ��. Now V2 2 A(W1) and implements � on A(W2). ��2 := j2 � � � j2 islocalized in W1 \W?2 and since (W1 [W2)? 6= ;, ��2(V1) = V1 so we haveV �1 V �2 V1V2 = V �1 �(V1) = �(�(V �1 )V1) = �(��) = ��: (5.3)Before proceeding to the proof of Theorem 5.11, we prove a result about orthog-onal wedges.Lemma 5.19. Given two orthogonal wedges W1, W2 with re ections j1 and j2, thereis a region O with non{empty causal complement which is invariant under j1 and j2.Proof. Take O1 and O2 orthogonal to each other and contained in W1 \W2, and setO = O1 [ j1O1 [ j2O1 [ j1j2O1. Clearly O is causally disjoint from O2 and invariantunder j1 and j2.Proof of Theorem 5.11. We follow the reasoning of [32]. Consider the two orthogonalwedges W1, and W2 = exp(�2h)W1 as in the preceding lemma and choose a represen-tative endomorphism localized in a regular diamond O 2 W1 \W2 and chosen suchthat there is an ~O containing O, j1O, j2O and j1j2O. Then ���1j2���1j2 = ���2j1���2j1and are objects of EndA(~O). V1J2V1J2 and V2J1V2J1 intertwine from the identityto this object in EndA(~O). Thus �� := (V1J2V1J2)�V2J1V2J1 is a scalar and we �rstshow that it belongs to (0; 1], as in Lemma 3.4 in [32], observing that�� = V �1 U(exp�h)V �1 V2U(exp�h)V2: (5.4)Then, by Equation 5.2 with g = exp �2h and its adjoint and using the equationc(W; exp �2h) = 1, proved in Proposition 5.17, we getV �2 V1 = s�U(exp�h)V �1 V2U(exp�h): (5.5)Inserting this equation into the expression for the statistics parameter given byLemma 5.18 and comparing with Equation 5.4 we obtain�� = V �1 V �2 V1V2 = s�V �1 U(exp�h)V �1 V2U(exp�h)V2 = s���and the result follows easily. 2We conclude this subsection showing that the Spin and Statistics relation makessense and is true for reducible covariant representations, too. Clearly the result followsfrom the irreducible case once we can show that the irreducible subrepresentationsare still covariant. 62

Proposition 5.20. Let � be a representation satisfying the selection criterion andwith �nite statistics and covariant under the group eG. Then there exists a covariantrepresentation (�;U�), where U� acts trivially on �(A)0. U� is unique up to a onedimensional representation and any other choice of U� may be written as a productof U� and a representation Uo� contained in �(A)0. In particular, each irreduciblecomponent of � is eG-covariant.Proof. Since � has �nite statistics, �(A)0 and hence the centre of �(A) are �nitedimensional. Therefore if (�; eU�) is a covariant representation, eU� is trivial on suchcentre. Then, since eU� implements automorphisms of �(A), it implements an action ofeG by automorphisms of �(A)0, preserving any factorial component. Thus this actionis implemented by a unitary representation Uo in �(A)0. Then g 2 eG! eUr(g)Uo(g)�is a representation of eG acting trivially on �(A)0. Clearly such a representationdecomposes into representations of the irreducible components of �, so these areeG-covariant.Remark 5.21. The given proof of the spin and statistics relation does not rely on thecontinuity of the representations U or U�. Even Proposition 5.20 does not requirecontinuity, because it relies on the fact that a connected Lie group acts trivially on a�nite set and this is true without assuming continuity.Spin and Statistics under property (c0)Now we give a proof of the Spin and Statisitics Theorem for rotationally symmetricspacetimes satisfying (c0) rather than (c), such as the 3-dimensional Schwarzschild-Kruskal spacetime, for example.Recall that in this case there is an involution rW := jW jW0 2 G"+ anticommutingwith h (cf. Remark 5.2 (vi)).Let us denote by the subgroup of G"+ generated by G0 and rW by G1. If rW doesnot belong to G0, G1 is isomorphic to G0 ��Z2, where � = AdrW . In the same waywe can consider the group eG1 � eG �� Z2, where, by an abuse of notation, � liftedto eG is still denoted by �. We shall also denote the corresponding element in eG1 byrW . Clearly the covering map extends to an epimorphism from eG1 to G1. We wantto show that any eG-covariant sector with �nite statistics is eG1-covariant, too.Proposition 5.22. Let � be an irreducible representation of A obeying the selectioncriterion, with �nite statistics, and covariant under the group eG. Then it is covariantunder eG1.Proof. Of course we may restrict to the case rW =2 G0. Since r � rW is the product ofjW with jW0, such re ections are implemented by the corresponding modular involu-tions J , J0, and j�j is equivalent to j0�j0, both being conjugate endomorphisms of �,there exists a unitary Ur intertwining � and �(r)��(r). Since r2 = 1, U2r implementsthe trivial action on A, hence, � being irreducibile, U2r is a constant and we maychoose Ur selfadjoint. Then UrU�(rgr)Ur is another representation of eG realizing thecovariance of �. Since � is irreducible, we get UrU�(g)Ur = �(g)U�(rgr), where �(g)63

is a character of eG. Applying this relation twice, we get U�(g) = �(g)�(rgr)U�(g),namely �(g)�(rgr) = 1. Now observe that, since � is a Lie group representation, it isthe exponential of a Lie algebra morphism � from the Lie algebra of G0 to R. SinceeG is simply connected, �=2 exponentiates to a character, which we denote by p�,whose square gives �, and we getUrp�(g�1U�(g))Ur = p�(rg�1r)U�(rgr);namely p�(g�1)U�(g) and Ur yield the required representation of eG1.Theorem 5.23. Let (M;G+; j;W) be a rotationally symmetric spacetime satisfyingproperties (a) and (b) and (c0), (A; U;) a covariant net verifying the mentionedaxioms (I-III), (VI-IX), and let � be a eG-covariant sector with �nite statistics. Thenthe spin and statistics relation holds.Proof. By property (c0), s� does not depend on U�, as observed in Remark 5.2 (vi).Concerning the relation between spin and statistics, we may de�ne a function c(W; g),g 2 eG1, as in the proof of Theorem 5.11, which is indeed a local group representationnamely, if g; h 2 eG1 verify W \ gW \ ghW 6= ;, we have c(W; g)c(W;h) = c(W; gh).Setting ~r := exp(�2h)rW exp(��2h), we get~rW = exp(�2 h)rW exp(��2h)W = exp(�h)rWW = Wand ~r exp(th)~r = exp(�th). Hence, for su�ciently small t,c(W; exp(2th)) = c(W; ~r)c(W; ~r exp(th))c(W; exp(th))= c(W; ~r exp(th)~r)c(W; exp(th))= c(W; exp(�th))c(W; exp(th) = 1:The proof now continues as in Theorem 5.11.As before, the Spin and Statistics relation for reducible representations follows assoon as we prove that the irreducible representations are ~G1-covariant, and this is aconsequence of Proposition 5.20 and Proposition 5.22.Remark 5.24. Generally speaking, asking for an irreducible endomorphism to be co-variant corresponds to asking for a projective representation of the group G0, namelya representation of a central extension of G0 by some subgroup of T1 implementingthe action of G0 on �(A). This means that there may be an extension at the Liealgebra level, not only a covering. However, we are not aware of any physical ex-ample where non-trivial Lie algebra central extensions exist (for the Poincar�e groupon the two-dimensional Minkowski space, such non-trivial extensions exist, but areincompatible with the positive energy requirement). As a consequence, we have onlytreated the case of the universal covering.64

5.3 Appendix. Equivalence between local and global inter-twiners in Minkowski spacetimeIn this appendix we prove that Assumption IX concerning the equivalence of localand global intertwiners holds for sectors localized in a wedge region of a Minkowskispace of arbitrary dimension. The argument is a straightforward adaptation of thatgiven in [32] for a conformal net on S1.In the following, A is a net of von Neumann algebras on the d + 1-dimensionalMinkowski spacetime. We assume Poincar�e covariance with positive energy anduniqueness of the vacuum, additivity and Haag dualityA(O) = A(O0)0if O is either a double cone or a wedge region.If �; � are endomorphisms of A localized in the wedge region W , we consider theirintertwiner space (�W ; �W ) := fT 2 A(W ) : �(A)T = T�(A); 8A 2 A(W )g. Byduality we always have (�; �) � (�W ; �W ).Theorem A5.1 Let W be a wedge region and �, � be endomorphisms with �nitedimension localized in a double cone O � W . Then(�W ; �W ) = (�; �):Namely, if T 2 (�W ; �W ) then T intertwines the representations � and � of A.In the following � denotes an endomorphism with �nite dimension of the quasi-local observable C�-algebra A localized in a double cone O contained in the wedgeW . We may assume that W = fx 2 Rd+1 : �x1 > jx0jg.We shall denote byR2 the 2-dimensional x0�x1-plane and byP the corresponding2-dimensional Poincar�e group, namely the semidirect product of the 2-dimensionaltranslations fT (x)gx2R2 and boosts f�(s)gs2R associated to W : each g 2 P can bewritten uniquely as a product g = T (x)�(s).The endomorphism � restricts to an endomorphism of the C�-algebra associatedwith W + x and then extends to the von Neumann algebra A(W + x), for x1 >0, hence giving rise to an endomorphism the C�-algebra A1, the norm closure of[x2R2A(W + x). We will still use � to denote this endomorphism. Since P is simplyconnected, there is a unitary representation U� of P expressing the covariance of �with respect to P�g(A) = U�(g)AU�(g)� = z�(g)U(g)AU(g)�z�(g)�; A 2 A1; g 2 P: (5.6)As the cocycle z� is a local operator by Haag duality (this is the essential point aboutthe 2-dimensional x0 � x1-net inherited from the higher dimensional original net) �is an action of P by automorphisms of A1.We consider now the semigroupP0, the semidirect product of the boosts �(s) withthe positive translations, where we say that T (x) is positive if x 2 R2 with x1 > jx0j.P0 is an amenable semigroup and we need an invariant mean m constructed asfollows: �rst we average (with an invariant mean) over positive translations and then65

over boosts. Observe that f ! RP0 f(g)dm(g) gives an invariant mean on all Pvanishing on f if, for any given s 2 R, the map x 2 R2 ! f(T (x)�(s)) vanishes ona right wedge.Then we associate to m the completely positive map � of A1 to B(H) given by�(A) := ZP0 z�(g)�Az�(g)dm(g); A 2 A1:Lemma A5.2 � is a left inverse of � on A1. Moreover � is locally normal, i.e. hasnormal restriction to A(W + x), x 2 R2, and P-invariant, namely� = ��1g ��g; g 2 P:We have set �g � AdU(g).Proof. Let A belong to A(W + x), x 2 R2. By formula 5.6�(�(A)) = ZP0 �g(�(�g�1(A)))dm(g) = Abecause of the above property of m since the integrand is constantly equal to A onthe set g 2 P0 : g�1W \ O = ;. Then the localization of � and Haag duality implythat the range of � is contained in A1.Setting E = � � � gives a conditional expectation of A1 onto the range of �that restricts to a conditional expectation Ex of A(W + x) onto �(A(W + x)) ifW + x � O. Since �W+x is assumed to have �nite index, Ex is automatically normal[44]. Therefore � � A(W + x) = ��1W+xEx is normal for x = (0; x1) with x1 > 0, hencefor any x.Concerning the P-invariance of � we have, making use of the cocycle condition,��1g ��g(A) = ��1g (ZP0 z�(h)��g(A)z�(h)dm(h))= ��1g (ZP0 z�(h)�z�(g)�g(A)z�(g)�z�(h)dm(h))= ZP0 z�(hg�1)�Az�(hg�1)dm(h) = �(A)Corollary A5.3 ' = !� is a locally normal �-invariant state on A1, where ! =( � ;).Proof. We have '�g = !��g = !�g� = !� = ' and ' is locally normal becauseboth ! and � are locally normal. 66

Let f�'; �';H'g be the GNS triple associated with the above state ' and V bethe unitary representation of P onH' given by VgA�' = �g(A)�' for A 2 A1. Noticethat V is strongly continuous because ' is locally normal.Lemma A5.4 If � is irreducible then'(x) = ZP0 �g(x)dm(g); x 2 A1Proof. If A 2 A(W + x) and B 2 A1 is localized in a double cone, the commutatorfunction R2 3 x 7! [�T (x)�(s)(A); �(B)] = �T (x)�(s)([A; �(��1T (x)�(s)(B)]) vanishes on aright wedge, hence [RP0 �g(A); �(B)dm(g)] = RP0[�g(A); �(B)]dm(g) = 0.Since � is locally normal, RP0 �g(A)dm(g) commutes with every �(A(W + x)),thus with �(A1); but � being irreducible, it is therefore a scalar equal to its vacuumexpectation value:ZP0 �g(A)dm(g) = ZP0 !(�g(A))dm(g) = ZP0 !(z�gAzg)dm(g) = !�(A) = '(A);as ! is normal and �-invariant.Corollary A5.5 If � is irreducible, the two-parameter unitary translation groupV (T (x)) satis�es the spectrum condition.Proof. One may repeat the proof of Corollary 2.7 of [32] for each of the one-parameterlight-like unitary translation groups.Corollary A5.6 If � is irreducible, ' is faithful on [�(A(W + x)).Proof. A1 is a simple C�-algebra since it is the inductive limit of type III factors(that are simple C�-algebras). Therefore �' is one-to-one and the statement willfollow if we show that �' if cyclic for Bx � �(A(W + x))0; x1 > 0. To this end wemay use a classical Reeh-Schlieder argument. If 2 H is orthogonal to Bx�', andx�y 2 W , then for all A 2 By we have (A�'; V (T (x)) ) = 0 for x in a neighborhoodof 0, thus for all x 2 R2 by the spectrum condition shown by Corollary A5.5. Hence,setting �x � �T (x) and �x � �T (x), is orthogonal to ([x�x(By))�', thus = 0because [x�x(By) is irreducible since([x �x(By))0 =\x �x(�(A(W + y)) =\x �(�x(A(W + y)))= �(\x �x(A(W + y))) =\x A(W + x) = Cby the local normality of �.Proposition A5.7 (�W+x; �W+x) does not depend on the wedge W + x � O.67

Proof. We begin with the case where � is irreducible and assume for convenience that�O �W . Notice then that (�W ; �W ) is �nite-dimensional and, by covariance, globally�g-invariant with g in the subgroup of boosts because these transformations preserveW . Therefore (�W ; �W )�' is a �nite-dimensional subspace of H' globally invariantfor V (�(s)), s 2 R. By Proposition B.3 of [32] we thus have V (T (x))A�' = A�' forevery element A 2 (�W ; �W ), thus �T (x)(A) = A because �' is separating. It followsthat if A 2 (�W ; �W ) and B 2 A(W )[A; �(�g(B))] = �g([��1g (A); �(B)]) = �g([A; �(B)]) = 0namely A 2 (�W ; �W )) A 2 (�; �) = C :Since the converse implication is obvious by wedge duality we have the equality ofthe two intertwiner spaces.Now if � is any endomorphismwith �nite index, (�; �) is �nite-dimensional because(�; �) � (�W ; �W ) and � decomposes into a direct sum of irreducible endomorphisms ofA1 which are covariant, therefore the preceding analysis shows that (�W ; �W ) = (�; �)in this case, too. Since (�; �) is translation invariant, we get (�W+x; �W+x) = (�; �)whenever O �W + x and, since x was arbitrary, the result follows.Proof of Theorem A5.1. The case � = � follows immediately by Proposition A5.6:if T 2 (�W ; �W ) then T also belongs to (� ~W ; � ~W ) for any wedge ~W � W hence byadditivity T is a self-intertwiner of � on the whole algebra A.To handle the general case, consider a direct sum endomorphism � := � � �localized in W , thendim(�W ; �W ) = dim(�W ; �W ) + dim(�W ; �W ) + 2dim(�W ; �W )while dim(�; �) = dim(�; �) + dim(�; �) + 2dim(�; �)therefore dim(�W ; �W ) = dim(�; �) and since we always have (�; �) � (�W ; �W ) thesetwo intertwiner spaces coincide. 2AcknowledgmentsR.V. has been in part supported through the Operator Algebras Network funded bythe EU under contract CHRX-CT94-0566. R.V. also wishes to thank all the membersof the operator algebra group at the Dipartimento di Matematica, Universit�a di Roma\Tor Vergata", for their kind hospitality in 1996.Three of the authors (D.G., J.R., R.V.) would like to thank the Erwin Schr�odingerInstitute, Vienna, as well as the Organizers of the Workshop on Quantum FieldTheory in September 1997, D. Buchholz and J. Yngvason, for the opportunity ofparticipating the workshop. The excellent working conditions provided a basis fordiscussions relevant to the present paper.We would also like to thank K.-H. Rehren for pointing out a gap in an earlierversion of Section 4. 68

References[1] H. Araki, L. Zsido, \Extension of the structure theorem of Borchers and itsapplication to half-sided modular inclusions" manuscript, preliminary version(1995), to appear[2] H. Baumg�artel, M. Wollenberg, Causal nets of operator algebras, AkademieVerlag, Berlin, 1992[3] J.K. Beem, P.E. Ehrlich, Global Lorentzian geometry, Marcel Dekker, NewYork, 1981[4] J.J. Bisognano, E.H. Wichmann, \On the duality condition for quantum �elds",J. Math. Phys. 17, 303 (1976)[5] H.-J. Borchers, \The CPT-theorem in two-dimensional theories of local observ-ables", Commun. Math. Phys. 143, 315 (1992)[6] H.-J. Borchers, \On modular inclusion and spectrum condition", Lett. Math.Phys. 27, 311 (1993)[7] H.-J. Borchers, \When does Lorentz invariance imply wedge duality?", Lett.Math. Phys. 35, 39 (1995)[8] H.-J. Borchers, \Half-sided modular inclusions and the construction of thePoincar�e Group", Commun. Math. Phys. 179, 703 (1996)[9] H.-J. Borchers, D. Buchholz, \Global properties of vacuum states in de Sitterspace", Ann. Inst. H. Poincar�e 70, 23 (1999)[10] J. Bros, H. Epstein, U. Moschella, \Analyticity properties and thermal e�ectsfor general quantum �eld theory on de Sitter space-time", Commun. Math.Phys. 196, 535 (1998)[11] M. Brown, \Locally at imbeddings of topological manifolds", Annals of Math.75, 331 (1962)[12] R. Brunetti, K. Fredenhagen, \Interacting quantum �elds in curved space:Renormalizability of '4", in the Proceedings of the Conference \Operator al-gebras and quantum �eld theory" held in Rome, July 1996, S. Doplicher, R.Longo, J. Roberts, L. Zsido eds, International Press, 1997;|, \Microlocal analysis and interacting quantum �eld theories: Renormaliza-tion on physical backgrounds", preprint math-ph/9903028[13] R. Brunetti, K. Fredenhagen, M. K�ohler, \The microlocal spectrum conditionand Wick polynomials of free �elds in curved spacetimes", Commun. Math.Phys. 180, 633 (1996)[14] R. Brunetti, D. Guido, R. Longo, \Modular structure and duality in conformalQuantum Field Theory", Commun. Math. Phys. 156, 201 (1993)69

[15] R. Brunetti, D. Guido, R. Longo, \Group cohomology, modular theory andspace-time symmetries", Rev. Math. Phys., 7, 57 (1994)[16] R. Brunetti, D. Guido, R. Longo, \First quantization via BW property", inprogress.[17] D. Buchholz, S.J. Summers, \An algebraic characterization of vacuum statesin Minkowski space", Commun. Math. Phys. 155, 442 (1993)[18] D. Buchholz, O. Dreyer, M. Florig, S.J. Summers, \Geometric modular actionand spacetime symmetry groups", preprint math-ph/9805026[19] D. Buchholz, M. Florig, S.J. Summers, \Hawking-Unruh temperature and Ein-stein causality in anti-de Sitter space-time", hep-th/9905178[20] C.J.S. Clarke, \A title of cosmic censorship", Class. Quantum Grav. 11, 1375(1994)[21] J. Dieckmann, \Cauchy surfaces in globally hyperbolic spacetimes", J. Math.Phys. 29, 578 (1988)[22] J. Dimock, B.S. Kay, \Classical and quantum scattering theory for linear scalar�elds on the Schwarzschild metric. I." Ann. Phys. (N.Y.) 175, 366 (1987)[23] S. Doplicher, R. Haag, J.E. Roberts: \Fields, observables and gauge transfor-mations I", Commun. Math. Phys. 13, 1 (1969)[24] S. Doplicher, J.E. Roberts: \Endomorphisms of C�{algebras, cross productsand duality for compact groups", Ann. Math. 130, 75 (1989)[25] S. Doplicher, J.E. Roberts: \Why there is a �eld algebra with a compact gaugegroup describing the superselection structure in particle physics", Commun.Math. Phys. 131, 51 (1990)[26] W. Driessler, \On the structure of �elds and algebras on null planes, I", ActaPhys. Austriaca 46, 63 (1977)[27] K. Fredenhagen, R. Haag, \On the derivation of Hawking radiation associatedwith the formation of a black hole", Commun. Math. Phys. 127, 273 (1990)[28] K. Fredenhagen, K.-H. Rehren, B. Schroer, \Superselection sectors with braidgroup statistics and exchange algebras". 1, Commun. Math. Phys. 125, 201(1989); 2, Rev. Math. Phys. Special Issue, 111 (Dec. 1992)[29] R. Geroch, \Domain of dependence", J. Math. Phys. 11, 437 (1970)[30] D. Guido, R. Longo, \Relativistic invariance and charge conjugation in quan-tum �eld theory", Commun. Math. Phys. 148, 521 (1992)[31] D. Guido, R. Longo, \An algebraic spin and statistics theorem", Commun.Math. Phys. 172, 517 (1995) 70

[32] D. Guido, R. Longo, \The conformal spin and statistics theorem", Commun.Math. Phys. 181, 11 (1996)[33] D. Guido, R. Longo, H.-W. Wiesbrock, \Extensions of conformal nets andsuperselection structures", Commun. Math. Phys. 192, 217 (1998)[34] R. Haag, Local quantum physics, 2nd ed., Springer, Berlin, Heidelberg, NewYork, 1996[35] S.W. Hawking, \Particle creation by black holes", Commun. Math. Phys. 43,199 (1975)[36] S.W. Hawking, G.F.R. Ellis, The large scale structure of space-time, CambridgeUniversity Press, 1973[37] B.S. Kay, \The double-wedge algebra for quantum �elds on Schwarzschild andMinkowski spacetimes", Commun. Math. Phys. 100, 57 (1985)[38] B.S. Kay, \Quantum �elds in curved spacetime: Non global hyperbolicity andlocality", in the Proceedings of the Conference \Operator algebras and quantum�eld theory" held in Rome, July 1996, S. Doplicher, R. Longo, J. Roberts, L.Zsido eds, International Press, 1997[39] B.S. Kay, M.J. Radzikowski, R.M. Wald, \Quantum �eld theory on spacetimeswith a compactly generated Cauchy-horizon", Commun. Math. Phys. 183, 533(1997)[40] B.S. Kay, R.M. Wald, \Theorems on the uniqueness and thermal properties ofstationary, nonsingular, quasifree states on spacetimes with a bifurcate Killinghorizon", Phys.Rep. 207, 49 (1991)[41] M. Keyl, \Causal spaces, causal complements and their relations to quantum�eld theory", Rev. Math. Phys. 8, 229 (1996)[42] B. Kuckert, \A new approach to spin and statistics", Lett. Math. Phys. 35,319 (1995)[43] H. Leutwyler, J.R. Klauder, L. Streit, \Quantum �eld theory on lightlike slabs",Nouvo Cimento 66A, 536 (1970)[44] R. Longo, \Index of subfactors and statistics of quantum �elds. I", Commun.Math. Phys. 126, 217 (1989)[45] R. Longo, \On the spin-statistics relation for topological charges", in the Pro-ceedings of the Conference \Operator algebras and quantum �eld theory" heldin Rome, July 1996, S. Doplicher, R. Longo, J. Roberts, L. Zsido eds, Interna-tional Press, 1997[46] R. Longo, \An analogue of the Kac{Wakimoto formula and black hole condi-tional entropy", Commun. Math. Phys. 186, 451 (1997)71

[47] R. Longo, J.E. Roberts, \A theory of dimension", K-Theory 11, 103 (1997)[48] M. M�uger, \On soliton automorphisms in massive and conformal theories",Rev. Math. Phys. 11, 337 (1999)[49] B. O'Neill, Semi-Riemannian geometry, Academic Press, New York, 1983[50] M.J. Radzikowski, \Micro-local appraoch to the Hadamard condition in quan-tum �eld theory in curved space-time", Commun.Math. Phys. 179, 529 (1996)[51] K.-H. Rehren, \Algebraic holography", preprint hep-th/9905179[52] J.E. Roberts: Net cohomology and its applications to �eld theory. In: Quantum�elds { algebras, processes, ed. L. Streit, pp. 239-268. Springer, Wien, NewYork, 1980[53] J.E. Roberts: Lectures on algebraic quantum �eld theory. In: The algebraictheory of superselection sectors. Introduction and recent results, ed. D. Kastler,pp. 1-112. World Scienti�c, Singapore, New Jersey, London, Hong Kong 1990[54] J.E. Roberts: \Some applications of dilation invariance to structural questionsin the theory of local observables", Commun. Math. Phys. 37, 273 (1974)[55] F. Rohrlich, \Null plane �eld theory", Acta Phys. Austriaca, Suppl. 8 (Con-ference Proc., Schladming 1971), 227 (1971)[56] G.L. Sewell, \Quantum �elds on manifolds: PCT and gravitationally inducedthermal states", Ann. Phys. (N.Y.) 141, 201 (1982)[57] S.J. Summers, R. Verch, \Modular inclusion, the Hawking temperature, andquantum �eld theory in curved spacetime", Lett. Math. Phys. 37, 145 (1996)[58] M. Takesaki, Tomita's theory of modular Hilbert-algebras and its applications.Lecture Notes in Mathematics Vol. 128 Springer, Berlin-Heidelberg-New York1970[59] R. Verch, \Continuity of symplectically adjoint maps and the algebraic struc-ture of Hadamard vacuum representations for quantum �elds in curved space-time", Rev. Math. Phys. 9, 635 (1997)[60] R. Verch, \Notes on regular diamonds", preprint, available as ps-�le athttp://www.lqp.uni-goettingen.de/lqp/papers/[61] R.M. Wald, General relativity, University of Chicago Press, 1984[62] R.M. Wald, Quantum �eld theory in curved spacetime and black hole thermo-dynamics, University of Chicago Press, 1994[63] R.M. Wald, \Gravitational collapse and cosmic censorship", gr-qc/9710068, toappear in \The Black Hole Trail", ed. by B. Iyer.72

[64] H.-W. Wiesbrock, \Half-sided modular inclusions of von Neumann algebras"Commun. Math. Phys. 157, 83 (1993)[65] H.-W. Wiesbrock, \Conformal quantum �eld theory and half-sided modularinclusions of von Neumann algebras " Commun. Math. Phys. 158, 537 (1993)[66] H.-W. Wiesbrock, \Symmetries and modular intersections of von Neumannalgebras", Lett. Math. Phys. 39, 203 (1997)[67] H.-W. Wiesbrock, \Modular intersections of von Neumann algebras is quantum�eld theory", Commun. Math. Phys. 193, 269 (1998)[68] R. Zimmer, \Ergodic Theory of Semisimple Groups", Boston-Basel-Stuttgart:Birkh�auser, 1984

73