String Localized Fields in a Strongly Nonlocal Model

Transcription

1 String Localized Fields in a Strongly Nonlocal Model arxiv:math-ph/ v1 18 Dec 2005 Detlev Buchholz a and Stephen J. Summers b a Institut für heoretische Physik, Universität Göttingen, Göttingen, Germany b Department of Mathematics, University of Florida, Gainesville FL 32611, USA December 4, 2005 Abstract We study a weakly local, but nonlocal model in spacetime dimension d 2 which satisfies the remaining Wightman axioms. We prove that this model is (quantitatively) maximally nonlocal; nevertheless, it has string localized observables of the sort which enable two body scattering theory to be defined. In two spacetime dimensions, it even has a covariant and local subnet of observables localized in bounded subsets of Minkowski space which has a nontrivial scattering matrix. his model exemplifies the algebraic construction of local observables from algebras associated with strongly nonlocal fields. 1 Introduction Until recently, locality has been viewed as indispensable in quantum field theory. However, more recent interest in nonlocal theories arising from research into quantum gravity and string theory (see, e.g. [15 17]), as well as the appearance of physically motivated weakly local but nonlocal models on other space times [11] and the possibility of using nonlocal fields to construct local theories [10,11,23 25,29,30], throws a new light on this matter. In this paper we examine a particular quantum field model in d 2 spacetime dimensions from the point of view of these and other recent developments. It is a weakly local but nonlocal quantum field, which satisfies the remaining Wightman axioms and the superstability conditions studied in [11,14]. In particular, we are interested in investigating just how nonlocal this model is and to which extent there are remnants of locality which have physical significance. 1

2 We shall show that this model is maximally nonlocal in a specific, quantitative sense. Nonetheless, it contains string localized observables which are sufficiently well behaved to allow the definition of two body scattering theory. And, in two spacetime dimensions, it contains a local, covariant net for which a full scattering theory can be defined, yielding a scattering matrix not equal to the identity. his illustrates in a concrete model a recently emerged approach [10,11,23 25,29,30] to algebraically construct local observables from relatively easily constructible nonlocal fields. he essential advantage of this approach is that the often much more difficult direct construction of local fields is simply avoided by using algebraic techniques to pass directly to the local net of observables. After introducing the model in the next section, we compute its modular structure and establish some immediate consequences in Section 3. In Section 4 we prove that the model is maximally nonlocal. We then investigate the independence properties of spacelike separated algebras in Section 5. In Section 6 we show that the nonlocal net of wedge algebras generated by this field contains local string localized observables for which scattering theory can be well defined, while in Section 7 we demonstrate that in two spacetime dimensions this nonlocal net of wedge algebras contains a covariant, local net indexed by double cones, and that the vacuum is cyclic for these double cone algebras. he scattering theoretical results are presented in Section 8. In the final section, we discuss the significance of these findings from a number of vantage points and make some further comments. 2 he Model he model we are studying in this paper has been known at least since R. Jost s classic monograph on axiomatic quantum field theory [21], where it was used to establish that weak local commutativity is a strictly weaker condition than local commutativity in the context of Wightman s axioms. We shall consider this model in Minkowski space time of dimension d, with d 2. In particular, one begins with a classical solution φ(x) of the massive Klein Gordon equation ( + m 2 )φ(x) = 0, m > 0. It can therefore be expressed in a momentum space expansion φ(x) = (2π) (d 1)/2 d d p δ(p 2 m 2 )θ(p 0 ) ( a(p)e ip x + a (p)e ip x), where p = (p 0, p 1,...,p d 1 ). However, in passing to the quantized field, the hypothesis of the Spin Statistics heorem is directly contravened by positing anticommutation relations: {a(p), a (p )}. = a(p)a (p ) + a (p )a(p) = 2ω(p) δ( p p ) 1I, where ω(p) = p 2 + m 2, and {a(p), a(p )} = {a (p), a (p )} = 0, 2

3 which, with a(p)ω = 0, yields spin 0 fermions. We write φ(x) = φ + (x) + φ (x), 1 where φ (x) = (2π) (d 1)/2 d d p δ(p 2 m 2 )θ(p 0 )a(p)e ip x, and φ + (x) = (2π) (d 1)/2 d d p δ(p 2 m 2 )θ(p 0 )a (p)e ip x. hen {φ + (x), φ + (y)} = 0 = {φ (x), φ (y)} and {φ + (x), φ (y)} = (x y) 1I, {φ (x), φ + (y)} = + (x y) 1I, where (ξ) = + ( ξ), and + (ξ) = (2π) (d 1) d d p δ(p 2 m 2 )θ(p 0 )e ip ξ. (2.1). he field φ(x) is nonlocal, since the commutator [φ(x), φ(y)] = φ(x)φ(y) φ(y)φ(x) 0 for spacelike separated x, y, and the anticommutator {φ(x), φ(y)} = {φ + (x), φ (y)} + {φ (x), φ + (y)} = 1 (x y) 1I, (2.2) where 1 (ξ) = + (ξ)+ (ξ), which is even in ξ and does not vanish outside the light cone, though it does decay exponentially as ξ approaches spacelike infinity. However, for the vacuum expectation of the commutator one observes Ω, [φ(x), φ(y)]ω = Ω, (φ (x)φ + (y) φ (y)φ + (x))ω = Ω, ({φ (x), φ + (y)} {φ + (x), φ (y)})ω = i (x y), where (ξ) = + (ξ) (ξ). But (x y) vanishes for spacelike separated x, y, since it is an odd, P +-invariant distribution, and since P + acts transitively upon the set of spacelike vectors. Hence, for spacelike separated x, y. Ω, [φ(x), φ(y)]ω = 0 (2.3) Note, as well, that the two point function has the form Ω, φ(x)φ(y)ω = Ω, φ (x)φ + (y)ω = + (x y). (2.4) Since the distribution on the right-hand side of (2.4) is P +-invariant with support on the branch of the mass hyperboloid having positive energy, there exists a strongly continuous unitary representation U(P +) of the proper orthochronous Poincaré group acting on the fermionic Fock space H generated by the field φ(x) 1 We do not follow the usual convention here, for later convenience. So φ + (x) is the negative frequency part of φ(x). 3

4 from the vacuum vector Ω, which satisfies the spectrum condition, leaves Ω invariant and under which φ(x) transforms covariantly. It follows that Θ 0 U(Λ, a)θ 0 = U(Λ, a) for any λ = (Λ, a) L + R d = P +. he subspace of U(P +)-invariant vectors in H is one-dimensional. Hence, the field φ(x) satisfies all of the Wightman axioms except local commutativity, which is replaced by weak local commutativity. his observation was made by Jost [21], though it may have been previously known to workers in the field. For our purposes here, we must introduce some further objects. Let V = ( 1) N(N 1)/2, where N is the Fock space number operator, and note that V 2 = 1. Define the additional field φ(x) = V φ(x)v 1, so that φ(x) = (φ + (x) φ (x))( 1) N2 = (φ + (x) φ (x))( 1) N, since ( 1) N = ( 1) N2. Because we have V Ω = Ω and U(λ)V U(λ) 1 = V, for all λ P +, also φ(x) is a U(P +)-covariant field satisfying the Wightman axioms except local commutativity and sharing the same vacuum and two point function (2.4). Although both φ(x) and φ(x) are not local fields, they are relatively local. In fact, on the one hand, one has while, on the other hand, one observes Hence, one finds [φ(x), φ(y)] φ(x) φ(y) = φ(x)(φ + (y) φ (y))( 1) N, φ(y)φ(x) = (φ + (y) φ (y))( 1) N φ(x) = (φ + (y) φ (y))(φ + (x)( 1) N+1 + φ (x)( 1) N 1 ) = (φ + (y) φ (y))(φ + (x) + φ (x))( 1) N. = ((φ + (x) + φ (x))(φ + (y) φ (y)) + (φ + (y) φ (y))(φ + (x) + φ (x))) ( 1) N = ({φ (x), φ + (y)} {φ (y), φ + (x)})( 1) N = ( + (x y) + (y x))( 1) N = ( + (x y) (x y))( 1) N = (x y)( 1) N. As pointed out above, (x y) vanishes for spacelike separated x, y. Proceeding to the corresponding nonlocal nets, for any nonempty open subset O of Minkowski space, we define R(O), resp. R(O), to be the von Neumann algebra generated by the operators φ(f), 2 resp. φ(f), for test functions f with support in O. Expressed in this algebraic language, we therefore have the following starting point. 2 Equation (2.2) entails that φ(f) is a bounded operator for every test function f. 4

5 Lemma 2.1 he two nets {R(O)}, { R(O)} are U(P +)-covariant, nonlocal nets which are relatively local in the sense that one has for any open O R d. R(O) R(O ) = V R(O ) V, In this paper a distinguished class of spacetime regions called wedges plays a. special role. Let W R = {x = (x0, x 1,...,x d 1 ) R d x 1 > x 0 } denote the right wedge. For d > 2, the set W of wedges is given by {λw R λp+}. In two dimensions, the set W of wedges has two disconnected components, one consisting of the translates of W R, {W R + x x R 2 }, and the other of the translates of. W L = WR. In the next section, the modular objects of the wedge algebras in the vacuum state will be computed. 3 Passivity and Modular Structure Given the above definition of the model, standard arguments [1, 21, 31] entail that both nets {R(O)}, { R(O)} are irreducible and that Ω is cyclic for both R(O) and R(O), for any nonempty open O. Hence, Lemma 2.1 entails that Ω is cyclic and separating for all wedge algebras R(W), R(W), W W. he omita-akesaki modular theory (cf. [7,22]) is thus applicable to the pairs (R(W), Ω), ( R(W), Ω), for all wedges W. he distribution (2.1) can also be understood as the two point function of a local spin 0 boson field satisfying all of the Wightman axioms and sharing the same one particle space as our field φ. By [2], it must be the case that for all test functions f, g with support in W R, the function R t + (f, g t ) (which also coincides with Ω, φ(f)u(v 1 (2πt))φ(g)Ω ) satisfies the modular condition, where v 1 (t), t R, is the boost subgroup in P + which leaves the wedge W R invariant (see [2]) and g t (x) = g(v 1 ( 2πt)x). Using the functorial character of the modular objects of such CAR algebras in Fock space, this entails that the modular group for both (R(W R ), Ω) and ( R(W R ), Ω) 3 is implemented by U(v 1 (2πt)), t R. his property has come to be called modular covariance (see, e.g., the review paper [5]). herefore, the modular unitaries for both (R(W R ), Ω) and ( R(W R ), Ω) are given by U(v 1 ( 2πt)), t R. Moreover, the vacuum state restricted to the wedge algebra R(W R ) (or R(W R )) is passive with respect to the automorphism group U(v 1 (t)), t R [27, hm. 1.2]. Because the algebras are not abelian and the vacuum is unique, it thus follows that they are type III 1 factors [27, hm. 4.3]. From Lemma 2.1 we have R(W L ) R(W R ), and since R(W L ) is invariant under the modular automorphism group of (R(W R ), Ω) and Ω is cyclic for 3 Recall that V commutes with U(P +). 5

6 R(W L ), we must have R(W L ) = R(W R ). Covariance 4 and the uniqueness of the modular objects entail that similar assertions hold for arbitrary wedges W W. We summarize these observations in the following proposition. Proposition 3.1 Let d 2. For any W W, Ω is cyclic and separating for R(W) and R(W), both of which are type III 1 factors satisfying modular covariance. he restriction of the vacuum state to the algebra R(W R ) or R(W R ) is passive for the automorphism group induced by the adjoint action of U(v 1 (t)), t R. Moreover, R(W ) = R(W), for all W W. (3.1) he modular group for both (R(W L ), Ω) and ( R(W L ), Ω) is implemented by U(v 1 ( 2πt)), t R. he relation (3.1) replaces Haag duality in this nonlocal model. In light of these facts, the arguments given in [4, 11] can be carried over directly to yield the following group theoretic version of the PC heorem, where henceforth J W denotes the modular involution corresponding to (R(W), Ω). Proposition 3.2 If d 2, one has J WR U(λ)J WR = U(θ 01 λθ 01 ), for all λ P +, where θ 01 = diag( 1, 1, 1,...,1) is the reflection which changes the sign of the 0 1 coordinates of the points in R d (the reflection about the edge of the wedge W R ). Moreover, U(λ)J W U(λ) 1 = J λw, for all λ P + and W W. We see, therefore, that the representation of the Poincaré group and the modular involutions associated with wedge algebras and the vacuum state act upon each other in a geometric manner. However, since the nets are not local, the Condition of Geometric Modular Action (CGMA) cannot be satisfied [9, hm. 5.3] and therefore cannot have the form found by Bisognano and Wichmann for finitecomponent Wightman fields [2,3]. In fact, we identify the modular involutions in the following proposition. o this end we define the operator U(θ 01 ) as U(θ 01 )φ(f 1 ) φ(f n )Ω. = φ(f j 1 ) φ(f j n ) Ω, for all n N and f i S(R d ), where f j (x) = f(θ 01 x). It is easy to see that U(θ 01 ) is antiunitary and that by defining U(θ 01 λ) = U(θ 01 )U(λ), for all λ P +, the unitary representation U of P + extends to a representation of the proper Poincaré group P + under which the two nets {R(O)}, { R(O)} transform covariantly. We 4 In two dimensions the corresponding assertions for R(W L ) are verified directly, as just explained for A(W R ). hen covariance is used to obtain the assertion for the second connected component of W. 6

7 recall that in even spacetime dimensions, the field φ(x) admits a PC-operator Θ 0 [21] such that Θ 0 Ω = Ω and Θ 0 φ(x) Θ 0 = φ( x) (3.2) Proposition 3.3 Let d 2. For any wedge region W, the modular involution J W corresponding to (R(W), Ω) is given by J W = U(λ)V U(θ 01 )U(λ) 1, where W = λw R. When d = 2, in which case U(θ 01 ) = Θ 0, one has J W = U(λ)V Θ 0 U(λ) 1, whether W = λw R is from the connected component in W containing W R or W = λw L is from the other connected component in W. For d 2, J WL = J WR, and V J W V = J W is the modular involution corresponding to ( R(W), Ω). Note that for d = 4, this entails J WR = V U(R 1 (π))θ 0, where R 1 (π) L + is the rotation through the angle π about the x 1 -axis (compare with [2,3]). Proof. Consider first W = W R. Clearly, V U(θ 01 )Ω = Ω and J WR R(W R ) J WR = V U(θ 01 ) R(W R ) (V U(θ 01 )) 1 = V R(W R )V = R(W R ) = R(W R ), which are necessary conditions. But what must be shown is that V U(θ 01 ) 1/2 W R AΩ = A Ω, (3.3) for all A R(W R ), where WR = e 2πK 1 (e itk 1 = U(v 1 (t)), t R) is the modular operator for the pair (R(W R ), Ω) by Prop It clearly suffices to verify (3.3) for the special choice A = φ(f 1 ) φ(f n ), n N, and f 1,...,f n real-valued test functions with support in W R. Since the difference between φ(f 1 ) φ(f n ) and the normal ordered product : φ(f 1 ) φ(f n ) : is an element of R(W R ), it suffices to take A =: φ(f 1 ) φ(f n ) :. But then AΩ is an n-particle vector with wave function f 1 + f n +, where the superscript + indicates that the function is to be restricted to the positive energy mass hyperboloid and the wedge products indicate that the tensor product is totally antisymmetrized. Note that U(θ 01 ) : φ(f 1 ) φ(f n ) : Ω =: φ(f j 1 ) φ(f j n ) : Ω 7

8 and it W R : φ(f 1 ) φ(f n ) : Ω =: φ(f 1,t ) φ(f n,t ) : Ω (recall that f t (x) = f(v 1 ( 2πt)x)). So, since v 1 (iπ) = θ 01 [2] and θ 2 01 = 1, one sees that the wave function for the vector U(θ 01 ) 1/2 W R : φ(f 1 ) φ(f n ) : Ω is f 1 f n, where the superscript indicates that the function is to be restricted to the negative energy mass hyperboloid. But f 1 f n = ( 1) n(n 1)/2 f n f 1, so that (3.3) is proven, since f n f 1 is the wave function corresponding to the vector (: φ(f 1 ) φ(f n ) :) Ω. U(P+) covariance and the uniqueness of the modular objects yield the assertion for all W in the connected component of W containing W R. For d > 2, that is already all of W. For d = 2 a similar argument yields J WL = V Θ 0 = J WR. For d > 2, observe further that since W R = R(π)W R, where R(π) is the rotation diag(1, 1, 1, 1,..., 1), and R(π)θ 01 = θ 01 R(π), it follows from Prop. 3.3 that J WL = J WR. Finally, observe that V U(θ 01 ) = U(θ 01 )V. ogether with the fact that the modular group for (R(W L ), Ω) is U(v 1 ( 2πt)), t R, the relation J WR = J WL entails that one can use the proof of Prop. 2.6 in [11] to conclude the following weak locality property of the model, which is prima facie stronger than the earlier observation (2.3). Proposition 3.4 Let d 2. For any wedge region W, one has Ω, ABΩ = Ω, BAΩ, for all A R(W) and B R(W ). Nonetheless, the model is extremely nonlocal, as shall be seen next. 4 How Nonlocal is the Model? In order to gauge the nonlocality of the model under investigation, we employ a quantitative measure of nonlocality recently introduced in [12]. Let A, B be two von Neumann algebras acting on a Hilbert space H (which may arise as the algebras of observables of two quantum subsystems in, e.g. coincidence experiments). In [12] we proposed as a natural measure of the degree of commutativity of these algebras, i.e. the commensurability of the observables of the two subsystems, in a normal state ω on the von Neumann algebra A B generated by A and B (equivalently on B(H)) the quantity C ω (A, B) =. sup ω(e F) ω(e)ω(f), E,F 8

9 where the supremum extends over all pairs of projections E A, F B. hus, a quantitative measure of the commutativity of these two algebras is the quantity C(A, B). = inf ω C ω(a, B), the infimum being taken over all normal states on A B. he values of both C(A, B) and C ω (A, B) lie in the interval [0, 1]. For the quantity C(A, B), the extreme value 0 corresponds to the commensurability of the respective observables and the statistical independence of the given algebras (see [12]), and the value 1 corresponds to maximal incommensurability of the underlying observables. he latter case is realized, for example, in quantum mechanics by the algebras generated by the position and momentum operator, respectively. It is straightforward to provide examples of noncommuting matrix algebras for which 0 < C(A, B) < 1. herefore, if for spacelike separated regions O 1, O 2, one has C(R(O 1 ), R(O 2 )) = 1, we say that the model is maximally nonlocal. We shall compute this invariant in the model under investigation. As a convenient shorthand in the following computations, we write φ(f) φ(g) + φ(g)φ(f) = 2 f g 1I (cf. (2.2)) and recall that φ(f) = φ(f). Let f 1, f 2 be real test functions which are orthonormal in the sense f 1 f 1 = f 2 f 2 = 1, f 1 f 2 = 0, so that φ(f 1 ), φ(f 2 ) are self-adjoint involutions and φ(f 1 )φ(f 2 ) = φ(f 2 )φ(f 1 ). For such test functions, the operator iφ(f 1 )φ(f 2 ) is self-adjoint and unitary; hence it is an involution. It follows that P ± = 1 2 (1I ± iφ(f 1)φ(f 2 )) are orthogonal projections in H and P + + P = 1I. Similarly, let g 1, g 2 be another pair of such test functions and set Q ± = 1 2 (1I ± iφ(g 1)φ(g 2 )). We begin the analysis of the relation between these operators with the following lemma. Lemma 4.1 P Q < 1 if and only if P Q = 0, where stands for an independent choice of + or. Proof. Since P Q is the largest projection contained in P and Q, P Q 0 entails P Q = 1. o prove the converse, consider the C algebra C generated by {φ(f 1 ), φ(f 2 ), φ(g 1 ), φ(g 2 )}. In view of the algebraic structure induced by the stated assumptions, C is finite dimensional. Moreover, if the test functions f 1, f 2, g 1, g 2 have compact support, Ω is separating for C (since C would contain R(W) for a suitable wedge W). Hence, C is faithfully represented on H. Let U = 1 2 φ(f 1 if 2 ), V = 1 2 φ(g 1 ig 2 ). It is straightforward to verify the equalities UU = 1 2 (1I + iφ(f 1)φ(f 2 )) = P +, U U = P, 9

10 and V V = Q +, V V = Q. Since 0 = P + P = UU U U, one has UU U 2 = UU U UUU = 0 (using X X = X 2 = X 2 ), so that UU U = 0. Similarly, U U 2 = UUU U = 0, entailing (U ) 2 = 0 = U 2. 5 One therefore has so that P + Q + = U V. U V 2 = V UU V = P + V 2 = V P + 2 = P + V V P + = Q + P + 2 = P + Q + 2, Now assume U V = P + Q + = 1. Since U V C, there exist unit vectors Φ, Ψ H such that 1 = (Φ, U V Ψ) = (UΦ, V Ψ). Since both UΦ and V Ψ are bounded in norm by 1, this equality entails that 0 UΦ = V Ψ, so that UH V H {0}. But UH = P + H, since UH UU H = P + H and P UH = U U 2 H = {0}, which implies UH P + H. Similarly, V H = Q + H. hus, P + H Q + H = UH V H {0}, and consequently P + Q + 0. One establishes the same assertion for the remaining cases in a similar manner. Since UV + V U = 1 2 f 1 + if 2, g 1 ig 2 1I, the following is an immediate consequence. Proposition 4.2 Let f 1, f 2, g 1, g 2 be functions as described above such that f 1 + if 2, g 1 ig 2 0. hen one has P + Q + < 1. If one of the other three expressions of the form f 1 ±if 2 g 1 ±ig 2 (the ± are understood to be taken independently of each other) is different from 0, then for the corresponding projections one has P Q < 1. Proof. Assume, for example, that P + Q + = 1. hen, as was shown in the preceding proof, UH V H {0}. For any nonzero Φ UH V H, one would then have 1 2 f 1 + if 2 g 1 ig 2 Φ = (UV + V U)Φ = 0, since, as shown in the preceding proof, U 2 = 0 = V 2. he arguments for the remaining cases are similar. As is evident from (2.2), since the corresponding subspaces of test functions are infinite dimensional, there exist functions f 1, f 2 localized in a given spacetime region and functions g 1, g 2 localized in any other given, spacelike separated region such that at least one of the four expressions f 1 ± if 2 g 1 ± ig 2 (again, the ± are understood to be taken independently of each other) is different from 0. his yields the following corollary, which we shall employ in the next section. 5 Note for use in the proof of the following proposition that the same argument yields V 2 = 0. 10

11 Corollary 4.3 Let O 1 and O 2 be spacelike separated nonempty open regions in d-dimensional Minkowski space time with d 2. here exist nonzero projections P R(O 1 ), Q R(O 2 ) such that PQ < 1. We continue now with the computation of C ω0 (R(W 1 ), R(W 2 )), where ω 0 is the vector state on B(H) induced by Ω. Let > 0 and u, v be test functions of the form u (x) = s (x 0 )h( x), v (x) = a (x 0 )h( x), where h S(R d 1 ) is a fixed real-valued test function with support in a ball of radius R > 0, and s, resp. a, is a real-valued even, resp. odd, element of S(R) with support in [, ]. hen φ(u ) and φ(v ) are self-adjoint and ω 0 (φ(u ) 2 ) = (2π) (d 1) d d p δ(p 2 m 2 )θ(p 0 ) s (p 0 ) 2 h( p) 2, ω 0 (φ(v ) 2 ) = (2π) (d 1) ω 0 (φ(u )φ(v )) = (2π) (d 1) ω 0 (φ(v )φ(u )) = (2π) (d 1) d d p δ(p 2 m 2 )θ(p 0 ) a (p 0 ) 2 h( p) 2, d d p δ(p 2 m 2 )θ(p 0 ) s (p 0 ) a (p 0 ) h( p) 2, d d p δ(p 2 m 2 )θ(p 0 ) a (p 0 ) s (p 0 ) h( p) 2. In view of the reality and symmetry properties of a, s, we have s (p 0 ) = s (p 0 ) and a (p 0 ) = a (p 0 ). Consequently ω 0 (φ(u )φ(v ) + φ(v )φ(u )) = 0, which yields u v = 0. We now further constrain such u, v to satisfy u u = 1 = v v, i.e. (2π) (d 1) d d p δ(p 2 m 2 )θ(p 0 ) α (p 0 ) 2 h( p) 2 = 1, where α equals s, resp. a. As above, we define the projections E ± = 1 2 (1 ± iφ(u )φ(v )) and note that ω 0 (E ±) = 1 2 ± i 2 (2π) (d 1) d d p δ(p 2 m 2 )θ(p 0 ) s (p 0 ) a (p 0 ) h( p) 2. (4.1) We show in the Appendix that there exist choices of a, s, h satisfying all of these constraints, which make the quantity ω 0 (E ±) arbitrarily close to 1 for sufficiently large. With these preparations, we can prove the following theorem. heorem 4.4 With d 2 and for arbitrary wedges W 1, W 2, one has C(R(W 1 ), R(W 2 )) = 1. 11

12 Proof. Given the wedges W 1, W 2, choose translations x (1), x(2) Rd such that the functions u (i) v (i). = s (x 0 x (i),0 )h( x x(i) ). = a (x 0 x (i),0 )h( x x(i) ), have support in W i, i = 1, 2. Since s, a and h all have compact support, this is always possible. u (2) Next, consider the algebra C generated by U(λ)φ(u (2) + iv(2) )U(λ) 1, for fixed and arbitrary Poincaré transformation λ in the endomorphism semi- + iv(2) group 6 of W 2. Since u (2), v(2) have compact support, the energy-momentum spectral support of φ(u (2) + iv(2) )Ω is the full mass hyperboloid. Hence, as λ runs through the said semigroup, U(λ)φ(u (2) + iv(2) )Ω runs through a total set in the one-particle subspace of H. hus, due to the Fock space structure of the model, Ω is cyclic for C. Since the boosts leaving W 2 invariant are contained in this semigroup, and since the representation of this boost subgroup in U(P+) coincides (after reparametrization) with the modular group of the pair (R(W 2 ), Ω), one must have C = R(W 2 ). Hence, if it were not possible, for all sufficiently large > 0, to choose u (i), v(i), λ so that u (1) iv(1) λ(u (2) + iv(2) ) 0, where λ denotes the natural action on the test function space of the said endomorphism semigroup, then φ(u (1) iv (1) ) would have to anticommute with all field operators φ(f) R(W 2 ). But this would have to be true for all admissible choices of u (1) iv(1), and the set of all such functions is total in the subspace {g S(R d ) supp(g) W 1 }. Hence, it would have to be the case that {φ(g), φ(f)} = 0, for all φ(g) R(W 1 ) and φ(f) R(W 2 ), which is excluded by (2.2). he choice of such test functions and λ as indicated is therefore possible. Consider then the corresponding projections P = 1 2 ( ) 1 + iφ(u (1) )φ(v(1) ) Q = 1 ( ) 1 iφ(λu (2) 2 )φ(λ 2v (2) ). By the above preparations it follows that P R(W 1 ), Q R(W 2 ) and P Q = 0. Moreover, in view of the invariance of ω 0 under Poincaré transformations, one has ω 0 (P ) = ω 0 (Q ) = ω 0 (E+ ) 1 in the limit as. But then ω 0 (P Q ) ω 0 (P )ω(q ) = ω 0 (E + )2 1, 6 i.e. the set of all λ P + such that λw 2 W 2 12

13 as. hus C ω0 (R(W 1 ), R(W 2 )) = 1. Since Ω 0 is faithful on R(W 1 ) R(W 2 ) as long as W 1 W 2 has nonempty interior, it follows that C(R(W 1 ), R(W 2 )) = 1 in such cases [13]. However, by choosing sufficiently large spacelike translations x (1) (which is admissible, since W 1 and W 2 are wedges), one has P, Q, x(2) R(O) for any fixed relatively compact O. Hence, by the irreducibility of φ, it follows that P and Q converge weakly to the identity operator. his yields the desired conclusion in the generality asserted. he wedge algebras in this model therefore manifest maximal nonlocality. It should be noted one can also prove that the same is true of the subalgebras R + (W 1 ), R + (W 2 ) generated by even products of the corresponding field operators, but it is not clear if C(R(O 1 ), R(O 2 )) = 1 holds when O 1, O 2 are bounded regions. 5 Independence Properties In [12] we proved in a general setting that under certain operationally motivated conditions on the state space of two von Neumann algebras A, B acting upon a common Hilbert space (which entail C(A, B) = 0), the algebras A and B must commute and satisfy a strong independence condition. Since the other extreme value of C(A, B) is attained in this model, it is of interest to understand which independence properties are satisfied, resp. violated, by the algebras R(W). In [19] a notion of the independence of two spacelike separated subsystems was introduced, the basic idea of which is that if each of the two subsystems can be prepared in any state, independently of the preparation of the other subsystem, then the two subsystems manifest a strong mutual independence. In algebraic quantum theory, there are two natural formulations [32] of Haag and Kastler s original suggestion, one in the category of C algebras and the other in the category of W algebras. Definition 5.1 Let A and B be subalgebras of a C algebra C. he pair (A, B) is (or A and B are) said to be C -independent if for every state φ 1 on A and every state φ 2 on B there exists a state φ on C such that φ A = φ 1 and φ B = φ 2. Replacing C algebra by W algebra and restricting all states to be normal states on their respective algebras yields the notion of W -independence. hese notions are physically meaningful also when the algebras (A, B) do not commute the same operational interpretation is valid. W -independence always implies C - independence [18], but when the algebras do not commute, then the converse can fail [20]. Note, however, that C -independence and W -independence are equivalent for commuting σ-finite von Neumann algebras [18]. We show that spacelike separated algebras from the same net in this model are not C -independent (and hence, not W -independent). his fact is not due simply to the lack of commutativity: there exist C -independent pairs of algebras which do not mutually commute cf. [32]. 13

14 heorem 5.2 Let O 1, O 2 be spacelike separated open regions in d-dimensional Minkowski space time with d 2. hen (R(O 1 ), R(O 2 )) is not C -independent. he same is true of ( R(O 1 ), R(O 2 )). Proof. Corollary 4.3 entails the existence of nonzero projections P R(O 1 ), Q R(O 2 ) such that PQ < 1 = P Q. he first assertion thus follows from Prop. 3 in [18]. For the second, just consider the projections V PV, V QV. On the other hand, we observe that W -independence holds for pairs (R(O 1 ), R(O 2 )). heorem 5.3 Let O 1, O 2 be spacelike separated double cones or wedges 7 in d- dimensional Minkowski space time with d 2. hen (R(O 1 ), R(O 2 )) is W - independent. Proof. For regions O 1, O 2 as described, there exists a wedge W W such that O 1 W and O 2 W [33, Prop. 3.7]. Moreover, since (R(W), R(W )) is a pair of commuting factors, it satisfies the Schlieder property 8 and thus is C - independent [28]. Hence, by [18, Prop. 8] the pair is W -independent. Since W -independence is inherited by subpairs [32], the assertion follows. Although the pair (R(W), R(W )) is not C -independent, it still manifests some of the weaker independence properties. Recalling that J WL = J WR and that the modular group for (R(W L ), Ω) is U(v 1 ( 2πt)), t R, one can employ the proof of Prop. 2.7 in [11] to conclude the following version of extended locality. Proposition 5.4 For d 2 and any wedge W W, one has R(W) R(W ) = C 1I. Hence, although R(W) and R(W ) do not commute, they only have trivial elements in common. Next, arguments presented in [11] can be modified to show that (R(W 1 ), R(W 2 )) satisfies an extended Schlieder property, as long as W 2 W 1. Proposition 5.5 Let d 2 and W 1, W 2 W be strictly spacelike separated, i.e. W 2 W 1. For any n N, A 1,k R(W 1 ) and A 2,k R(W 2 ), k = 1,...,n, such that n k=1 A 1,kA 2,k = 0, one must have n n ω(a 1,k )A 2,k = 0 = A 1,k ω(a 2,k ), k=1 for all normal states ω on B(H). In particular, if A 1 A 2 = 0, then either A 1 = 0 or A 2 = 0. 7 he proposition is valid for more general spacelike separated open regions see the hypothesis of [33, Prop. 3.7]. 8 he pair (A, B) satisfies the Schlieder property if A A, B B and AB = 0 imply either A = 0 or B = k=1

15 Proof. Since the proof follows in the steps of the arguments given in Section III in [11], only the necessary changes to be made will be indicated. For a given wedge W W, let G W P + be the subgroup consisting of the boost group leaving W invariant and the group of translations in the (Euclidean) orthogonal complement of W s edge in R d (in two dimensions, G W = P +). he assumption W 2 W 1 entails that there exists a wedge W W and a neighborhood N of the identity in G W such that W 1 λ 0 λ 1 W and W 2 λ 0 λ 1 W for all λ 0 λ 1 N. he group G W replaces the group SO 0 (2, n 1) in the arguments in [11]. Due to covariance, it is no loss of generality to take W = W R. hen G W consists of v 1 (t), t R, and the translations in the 0 1-plane. It is crucial to note that if O is any bounded open region in Minkowski space, then there exists a translation a G W such that O + a W. It therefore follows that λ G W U(λ)R(W))U(λ) 1 = B(H). (5.1) It is also important to observe that the subgroup of P + generated by the elements λv 1 (t)λ 1, t R, λ N, coincides with G W, which can be verified by a simple calculation. he necessary ingredients for the arguments given in Section III of [11] are therefore in place here, as well. 6 String Localized Observables As nonlocal as we have seen the net {R(W)} W W to be, nonetheless it accomodates quantities localized in unbounded regions which commute when spacelike separated and for which two body scattering theory can be defined. Let W 1, W 2 be wedges such that W 1 W 2 and W 1 W R. Consider the distributions h ± (x) 1I = {φ(h), φ ± (x)} = d d p θ(±p 0 )δ(p 2 m 2 ) h(p)e ipx 1I in the special case where h(x) = δ(x 0 )k( x). 9 hen d p h ± (x) = 2ω( p) k( p)e ±iω( p)x 0 i p x. We pose the question: does there exist such an h so that h + (f) + h (f) = 0 for all test functions f such that supp(f) W 1 and so that h + (g) h (g) = 0 for all g such that supp(g) W 2? If so, then we would have {φ(h), φ(f)} = 0, supp(f) W 1 (6.1) 9 Note that the bounded operator valued distribution φ on S(R d ) can be extended to such test functions. 15

16 and [φ(h), φ(g)] = {φ(h), φ + (g) φ (g)}v = 0, supp(g) W 2. (6.2) Since h ± are solutions of the Klein Gordon equation, these conditions can be stated as the following four conditions (understood as distributional equations) for all (0, x) W 1 and h + (0, x) + h (0, x) = 0 = h + (0, x) + h (0, x), ĥ + (0, x) ĥ (0, x) = 0 = ĥ+(0, x) ĥ (0, x), for all (0, x) W 2, where ĥ±(0, x) = d dx 0 h ± (x) x0 =0. For the special choice of h made above, these conditions are equivalent to d p 2ω( p) k( p)e i p x = 0, for all (0, x) W 1, and d p k( p)e i p x = 0, for all (0, x) W 2. wo of the four conditions are identically satisfied due to the special choice of h. We shall show that the latter two conditions can be satisfied. Let l 1 S(R) be any test function with support in the complement of the intersection of W 1 W 2 with the x 1 -axis. Further, let p = (p 1, p ), x = (x 1, x ), with p, x in the obvious d 2-dimensional subspace of R d. Choose an arbitrary test function l on this subspace and define l( x) = l 1 (x 1 )l (x ). Note that p 2 + m 2 = p 1 + i m 2 + (p ) 2 p 1 i m 2 + (p ) 2, and choose k S(R d 1 ) so that k( p) = p 1 + i m 2 + (p ) 2 l( p). In order to see that k and 1 k have the desired support properties, consider the ω distributions 1 χ ± (u) = dq e iqu q ± iκ on R, taking the principal branch of z 1/2 in the denominator. he analyticity properties of the integrand entail that χ ± is independent of the choice of κ > 0 and that supp(χ ± ) ±[0, ). It thus follows that supp(χ l 1 ) does not intersect (the intersection of the x 1 -axis with) W 1. he pseudodifferential operator i x 1 +ω, where ω (p ) = m 2 + (p ) 2, is a local differential operator in x 1 (it is not local in x ), so that (i x 1 +ω )l( x) is still localized in the complement of the intersection of W 1 W 2 with the x 0 = 0 hyperplane, entailing supp(χ + (i x 1 + ω )l( x)) does 16

17 not intersect (the intersection of the x 0 = 0 hyperplane with) W 2. But then d p 2ω( p) k( p)e i p x = dp e ip x l(p1, p ) dp 1 p 1 i e ip 1x 1 p 2 + m 2 = (χ l)(x 1 ) dp e ip x l (p ) vanishes for all (0, x) W 1. Similarly, d p k( p)e i p x = dp e ip x dp 1 (p 1 + i p 2 + m 2 ) l(p 1, p ) p 1 + i p 2 + m 2 e ip 1x 1 = (χ + (i x 1 + ω )l)( x) is zero for all (0, x) W 2. Observe that the choice of l S(R d 2 ) is arbitrary. We can now prove the following result, employing the notation γ(a) =. ZAZ, where Z = ( 1) N and A B(H). Note that γ(φ(x)) = φ(x). We also write A ± = 1(A ± γ(a)) for any A B(H) and set A. 2 ± = {A ± A A} for all A B(H). Lemma 6.1 Let d 2 and W 1, W 0 be wedges such that W 1 W 0. hen there exist nonzero elements φ(h) R(W 0 ) such that [φ(h), A + ] = 0, A + R + (W 1 ), and {φ(h), A } = 0, A R (W 1 ). (6.3) Indeed, there are so many such elements that Ω is a cyclic vector for the algebra they generate. Proof. After a suitable Poincaré transformation and choosing W 2 = W 0, the immediately preceding discussion is applicable here. Hence, there exist φ(h) R(W 0 ) such that (6.1) and (6.2) both hold. But (6.2) entails φ(h) R(W 0 ) = R(W 0 ), by Prop And since R ± (W 1 ) is generated by even, resp. odd, products of field operators φ(f), with supp(f) W 1, assertion (6.3) follows from (6.1). Since l 1 has compact support, its Fourier transform does not vanish in any nonempty open set. Moreover, one can choose l so that the same is true of l 1 l. and thus also of k. Moreover, one can choose l 1 so that its support is in the interior of the complement of the intersection of W 1 W 2 with the x 1 -axis, which has as a consequence that the desired support properties of k are still satisfied by all translates of k by elements of some neighborhood of the origin in R d 1. It then follows that Ω is cyclic for the algebra generated by all field operators φ(h) with test function h as specified in the construction above. he preceding result makes it natural to consider another net of algebras. For any A B(H), let A t. = A+ + A Z. Note that φ(x) φ t (x). For any wedge W W we define R t (W) =. {A t A R(W)}. Due to the nonlocal nature of the field φ(x), one has R t (W) R(W ) = R(W). 17

18 Corollary 6.2 Let d 2 and W 1, W 0 be wedges such that W 1 W 0. hen the algebra R(W 0 ) R(W 1 ) is nontrivial, and Ω is cyclic for the algebra R(W 0 ) R t (W 1 ), which contains all of the operators φ(h) from Lemma 6.1. Proof. he algebra generated by products of even numbers of operators φ(h) from the preceding lemma is contained in R(W 0 ) R(W 1 ), and all such operators φ(h) are contained in R(W 0 ) R t (W 1 ). In light of this result, let us fix a wedge W 0 R d. We say that the wedges W 1, W 0 are coherent if there exists a translation x R d such that W 1 + x W 0. (his is an equivalence relation.) Note that the edges of coherent wedges are parallel d 2-dimensional spacelike surfaces. We define the algebras F(W 2 W 1 ). = R(W 1 ) R t (W 2 ), where W 0, W 1, W 2 W are coherent and W 2 W 1. By the preceding corollary, Ω is a cyclic vector for F(W 2 W 1 ). Let L 0 L + be the stability subgroup of W 0. It is clear that U(L 0 R d ) acts covariantly on the family of such algebras F(W 2 W 1 ), W 1, W 2 as specified, and that L 0 R d contains the boost subgroup leaving W 0 invariant. Hence, U(L 0 R d ) contains the modular group of (R(W 0 ), Ω). It follows that F(W 2 W 1 ) = R(W 0 ) W 2 W 1 W 0 and F(W 2 W 1 ) = R t (W 0 ), W 0 W 2 W 1 (similarly for the translations of W 0 ) since the algebra on the left hand side of both equations has Ω as a cyclic vector and is invariant under the modular automorphism group of (R(W 0 ), Ω), resp. (R t (W 0 ), Ω) (cf. Prop. 3.1 and recall that Z commutes with U(P +)). We next consider the other class of algebras which arises in Corollary 6.2, namely A(W 2 W 1 ). = R(W 1 ) R(W 2 ), for W 2 W 1 and W 0, W 1, W 2 W coherent. We shall show that in more than two spacetime dimensions this algebra contains no odd elements, which immediately entails that Ω cannot be cyclic for this algebra, in contast to the algebra F(W 2 W 1 ). Proposition 6.3 Let d > 2 and W 0, W 1, W 2 W be coherent for fixed W 0 with W 2 W 1. hen R(W 1 ) R(W 2 ) contains no nonzero odd element, so that Ω is not cyclic for A(W 2 W 1 ). Proof. Choose coherent wedges such that W 6 W 5 W 2 and W 1 W 4 W 3 and let X R(W 1 ). In light of F(W 4 W 3 ) R t (W 1 ), so that ZF(W 4 18

19 W 3 )Z R(W 1 ), X must anticommute with the elements of F (W 4 W 3 ). If also X R(W 2 ) and γ(x) = X, then the operator Y. = XZ anticommutes with the elements of R (W 2 ) F (W 6 W 5 ). Let a be an arbitrary translation along the edge 10 of W 0, and set A(a) = U(a)AU(a) 1, for any A B(H). Since Z(a) = Z, this entails Y (a) = X(a)Z, so that (X Y )(a) = (X X)(a) Z. Letting a tend to spacelike infinity, the vector (X X)(a)Ω converges weakly to ω 0 (X X)Ω. But (X X)(a) R(W 1 ) for all such a, and since Ω is separating for R(W 1 ), the operator (X X)(a) converges in the weak operator topology to ω 0 (X X) 1I. his limit is different from 0 if and only if X 0. o show that the weak limit of (X Y )(a) is also a multiple of the identity, choose φ(f 1 ) φ(f m ) F (W 4 W 3 ) and φ(g 1 ) φ(g n ) F (W 6 W 5 ) (so m and n are odd) and compute: φ(f 1 ) φ(f m )Ω, (X Y )(a) φ(g 1 ) φ(g n )Ω = = ( 1) m+n XΩ, (φ(f m ) φ(f 1 ) φ(g 1 ) φ(g n )) ( a) Y Ω. Since X Y is even, this expression vanishes if n + m is odd. If n + m is even, then for any test function h S(R d ), one has [(φ(f m ) φ(f 1 ) φ(g 1 ) φ(g n )) (a), φ(h)] 0, as a tends to spacelike infinity, since this commutator can be expanded into a finite sum of terms involving bounded operators multiplied by anticommutators of φ(h) with φ(f )(a) or φ(g )(a), which tend to 0 by the Riemann-Lebesgue Lemma. Since the field φ is irreducible and h was arbitrary, it follows that (φ(f m ) φ(f 1 ) φ(g 1 ) φ(g n )) (a) weakly Hence the quantity converges to Ω, φ(f m ) φ(f 1 ) φ(g 1 ) φ(g n )Ω 1I. φ(f 1 ) φ(f m )Ω, (X Y )(a) φ(g 1 ) φ(g n )Ω Ω, X Y Ω φ(f 1 ) φ(f m )Ω, φ(g 1 ) φ(g n )Ω as a tends to spacelike infinity. But the span of the set of vectors of the form φ(f 1 ) φ(f m )Ω with φ(f ) F (W 4 W 3 ), m N, together with Ω is dense in H, by Corollary 6.2 (similarly for φ(g 1 ) φ(g n )Ω with φ(g ) F (W 6 W 5 ), n N, together with Ω). herefore, the operator (X Y )(a) converges weakly to ω 0 (X Y ) 1I, so that the equality (X X)(a) Z = (X Y )(a), for all a as described, entails ω 0 (X X) Z = ω 0 (X Y ) 1I = ω 0 (X XZ) 1I = ω 0 (X X) 1I, 10 Note that by the coherence of the wedges, translation by a leaves invariant all the algebras arising in this proof. 19

20 since ZΩ = Ω. his is a contradiction, unless X = Y = 0. he argument fails for d = 2, since the required translations a do not exist. Indeed, the conclusion of this proposition is false when d = 2, as we shall see in the next section. We remark that the algebras F(W 2 W 1 ) and A(W 2 W 1 ) are not determined merely by prescribing the localization region W 2 W 1 but encode additional information to specify that W 1 W 2 actually comes from the pair W 1, W 2 with W 2 W 1 instead of the pair W 2, W 1 with W 1 W 2. It would be interesting to determine a mathematically natural setting for this structure. Note further that the algebras A(W 2 W 1 ) are local, in the sense that if W 0, W 1,...,W 4 W are coherent with W 2 W 1 and W 4 W 3 such that W 4 W 3 is spacelike separated from W 2 W 1, then A(W 4 W 3 ) A(W 2 W 1 ). 7 Compactly Localized Observables In this section we investigate the existence of compactly localized observables in this model. We begin by considering d = 2. In two dimensions, the model under study here coincides with the S 2 = 1 factorizing S matrix model discussed in [23,24]. It was proven in [23] that the net this field generates satisfies a certain nuclearity condition, which entails that if W 2 W 1, then the inclusion R(W 2 ) R(W 1 ) is split, i.e. there exists a type I factor N such that R(W 2 ) N R(W 1 ). Using an argument from [10,11] (cf. Prop. 5.1 in [11]) and Prop. 3.1, it then follows that the relative commutant A(W 2 W 1 ) is a type III 1 factor, which therefore has a dense G δ set of cyclic vectors. We shall show that, in fact, the vacuum vector Ω is cyclic for these algebras. Indeed, since the inclusion R(W 2 ) R(W 1 ) is split, one can make use of the universal localizing map precisely as in [8] to find a self-adjoint and unitary operator Z 1 R + (W 1 ) (since the inclusion is invariant under γ, ZΩ = Ω, and γ(z) = Z, Z 1 can also be chosen to be left fixed under γ) such that the adjoint action of Z 1 on R(W 2 ) coincides with that of Z. Hence, choosing any operator φ(h) F(W 2 W 1 ) as in Corollary 6.2 and setting X(h) = φ(h)z 1, we have X(h) R (W 1 ) and, for any φ(f) R(W 2 ), X(h)φ(f) = φ(h)z 1 φ(f) = φ(h)φ(f)z 1 = {φ(h), φ(f)}z 1 + φ(f)φ(h)z 1 = φ(f)x(h), since {φ(h), φ(f)} = 0, because of φ(h) R t (W 2 ) and φ(f) t = φ(f)z. hus, one has X(h) R(W 2 ), so that X(h) A(W 2 W 1 ) = R(W 1 ) R(W 2 ). Now, consider any third wedge W 3 satisfying W 2 W 3 W 3 W 1. For any n N, choose φ(h 1 ),...,φ(h n ) F(W 2 W 3 ) ( F(W 2 W 1 )) as in Corollary 6.2. Since also the inclusion R(W 3 ) R(W 1 ) is split, one can choose Z 1 R(W 1 ) as above such that the adjoint action of Z 1 on R(W 3 ) (and therefore also on R(W 2 )) 20

21 coincides with that of Z and γ(z 1 ) = Z 1. hen X(h i ) = φ(h i )Z 1 A(W 2 W 1 ), for all i = 1,...,n, and ( n ) ( n ) X(h i ) Ω = Z1 n ( 1) n(n+1)/2 φ(h i ) Ω. i=1 If Ω were not cyclic for A(W 2 W 1 ), there would therefore have to exist a nonzero Φ H orthogonal to all such vectors. Since Z 1 is an involution, Lemma 6.1 would entail P n Φ = 0, if n is even, and P n Z 1 Φ = 0, if n is odd, where P n is the projection onto the eigenspace of N corresponding to the eigenvalue n (the n- particle subspace). But P n Φ = 0, if n is even, implies that Φ is in the eigenspace H odd of Z corresponding to the eigenvalue 1, and γ(z 1 ) = Z 1 entails therefore that Z 1 Φ H odd. Hence, the condition P n Z 1 Φ = 0, if n is odd, yields Z 1 Φ = 0. Multiplying both sides of this equality by Z 1 yields Φ = 0. Hence, Ω is cyclic for A(W 2 W 1 ). Note that in two dimensions, W 2 W 1 is a nontrivial double cone, and every nontrivial double cone is an intersection of such a pair of wedges. In two dimensions, there are two coherent families of wedges, one based on W R and the other based on W L. Given a double cone O and a coherent family of wedges determined by a fixed W 0 W, we define B(O). = A(W 2 W 1 ), where W 1, W 2 W are the unique members of the specified coherent family satisfying the conditions W 2 W 1 and O = W 2 W 1. From above, B(O) is a type III 1 factor with Ω as a cyclic vector. hus, one can find a nontrivial covariant, local net {B(O)} inside the nonlocal net {R(W)} W W of wedge algebras. he adjoint action of V upon the net {B(O)} yields a corresponding subnet in { R(O)} with the same properties. We have therefore demonstrated the following result. heorem 7.1 In d = 2 spacetime dimensions, the net {R(O)} contains a subnet O B(O) indexed by double cones O which is local, covariant under U(P +) and satisfies B(O) R(W), if O W, and B(O) R(W), if O W, for wedges W in the coherent family determined by a fixed W 0 W. Moreover, for each double cone O, B(O) is a factor of type III 1 for which Ω is a cyclic vector. he analogous assertions are also valid for the net { R(O)}. It is thus a natural question whether this is also possible for d > 2. he answer is negative. We prove the following no-go theorem, which establishes the fact that only a subset of the properties satisfied by the net {B(O)} in the previous theorem suffices to entail the triviality of each algebra B(O). heorem 7.2 Let d > 2 and O B(O) be a U(P +)-covariant net indexed by double cones such that the following two conditions hold: 21 i=1

22 (i) B(O) R(W), if O W, (ii) B(O) R(W), if O W. hen for any double cone O, the algebra B(O) is trivial, i.e. consists of multiples of the identity operator. Proof. Recall the global gauge transformation Z = ( 1) N implementing the map γ(φ(x)) = φ(x). Since γ(r(w)) = R(W) and Z commutes with U(P +), the net O B(O) γ(b(o)) satisfies the hypothesis of the theorem, as well. One may thus assume that ZB(O)Z = B(O), for all O. Let O 0 be a double cone containing the origin which is left invariant by all rotations, and let W be a wedge whose edge contains the origin. Let A B(O 0 ) be chosen so that γ(a) = A, and let r be an involutive rotation in R d such that rw = W.. hen one has ñ = 1(A ± 2 U(r)AU(r) 1 ) B(O 0 ) and γ(ã±) = A ±. here exists a wedge W 1 W such that O 0 W 1. Condition (ii) then implies that [ñ, φ + (f) + φ (f)] = 0, (7.1) for any test function f with support in W 1. On the other hand, since O 0 is rotation invariant, one also has O 0 (rw 1 ), so that Lemma 2.1 entails B(O 0 ) R(rW 1 ) R(rW 1 ). One therefore deduces [ñ, (U(r)φ + (f)u(r) 1 U(r)φ (f)u(r) 1 )Z] = 0, (7.2) for the same class of test functions f. Making use of the fact that [ñ, Z] = 0 and conjugating equation (7.2) by U(r), one obtains ±[ñ, φ + (f) φ (f)] Z = 0, since U(r)ñU(r) 1 = ±Ã±. Since Z is an involution, this yields ±[ñ, φ + (f) φ (f)] = 0. Adding, resp. subtracting, this to (7.1), one finds [ñ, φ + (f)] = 0 = [ñ, φ (f)]. But since φ (x) is a positive, resp. negative, frequency solution of the Klein- Gordon equation, the resulting analyticity properties and the Edge-of-the-Wedge heorem entail that ñ commutes with φ ± (f), and thus with φ(f), for any test function f. Since the field φ(x) is irreducible in H, A must be a multiple of the identity operator on H. Hence, the only elements of B(O 0 ) left fixed by γ are multiples of 1. Now let A, A B(O 0 ) and set A = 1 2 (A γ(a)), for any A. hen γ(a A ) = A A, so that A A = c 1. If, in particular, A 0, right multiplying this last equation by A yields A = c A, for a suitable constant c, since A A must also be a multiple of the identity. Hence, all odd (under the action of γ) elements of B(O 0 ) must be mutually proportional. 22

23 It follows that if 0 C B(O 0 ) is odd, then B(O 0 ) = span{1, C}. Now let O 1 O 0 be a double cone centered at the origin and rotation invariant. hen, since B(O 0 ) B(O 1 ), one concludes from the preceding argument that B(O 1 ) = B(O 0 ). However, the covariance of the net {B(O)} entails that U(λ)B(O 0 )U(λ) 1 B(O 1 ), for all λ in some neighborhood of the identity in P+. his implies U(λ)CU(λ) 1 = C, for all such λ and thus for all λ P+. It then follows from the uniqueness of the vacuum that CΩ = cω, for some constant c, and since Ω is separating for B(O 0 ) (it is cyclic for R(W 1 ) B(O 0 ) ), one has C = c 1. Since C is odd, c = 0. he covariance of the net completes the proof. he nets {R(O)}, { R(O)} therefore do not contain nontrivial local subnets which satisfy reasonable conditions and are indexed by bounded spacetime regions. 8 Scattering heory Given the preparations made above, we shall be able to appeal to results already in the literature and to keep this discussion brief. o begin, for d 2 if f, resp. g, is a real-valued solution of ( + m 2 )h(x) = 0 with support in W, resp. W, then since φ(f)ω and φ(g)ω are vectors in the one-particle subspace of H, φ(f) R(W) and φ(g) R(W ) are temperate polarization-free generators [6] with domains of temperateness equal to H. Following [6], asymptotic scattering states for two-body scattering can then be defined. For d > 2, in this model the resultant scattering matrix is trivial [6, hm. 3.5]. Moreover, since the algebras A(W 2 W 1 ) do not have any odd elements (heorem 6.3), the corresponding net contains no particle information at all. However, for d = 2 the situation is quite different. As already mentioned, in two dimensions the model under study here coincides with the S 2 = 1 factorizing S matrix model studied in [23, 24]. As proven in [23], the S matrix for the polarization-free generators discussed above is equal to ( 1) N(N 1). From the proof of heorem 7.1, we have X(h) B(O) and X(h)Ω = φ(h)ω is a vector in the one-particle space of H, for any function h as described in the proof of Lemma 6.1. Hence, the usual Haag-Ruelle scattering theory can be applied to the net {B(O)} and yields the same S-matrix. We summarize: heorem 8.1 For d > 2 the model admits a scattering theory resulting in a trivial S matrix. he algebras A(W 2 W 1 ) have no particle content. For d = 2 Haag-Ruelle scattering theory applied to the local net {B(O)} describes nontrivial scattering with S = ( 1) N(N 1)/2. 9 Final Comments In a number of recent papers [10, 11, 23 25, 29, 30], it has proven advantageous to consider nonlocal but weakly local fields, which admit localization into wedge 23