Doplicher, S. and Roberts, J.E., Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics, Comm. Math. Phys. 131, 51-107 (1990).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....weak modular covariance, and let ae be an irreducible covariant morphism with finite statistics. Then ae = s ae . Proof. If n 4, the proof is contained in [17] cf. also [21] Indeed in this case (or when ae is localized in a double cane) one may construct the Doplicher Roberts field algebra [13], and the theorem follows by the equality between the statistics operator and U(2) on such algebra. When n = 3 and ae is localized in a space like cone, the previous technique does not apply and we refer to the proof in [7] which is a natural extension of the arguments in [18] ut Conformal ....

Doplicher S., Roberts J.E., "Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics", Commun. Math. Phys. 131 (1990), 51-107. 14

....starts with an algebra of observables A, in which the gauge fields and and the charged (matter) fields themselves are obviously invisible. The role of charged matter fields as intertwining operators between various representations of A is by now well understood in theories with global symmetries [7], but local gauge theories have defied a full understanding so far. Therefore, any example in which gauge fields emerge in connection with the representation theory of an algebra of observables should be welcome. As we shall see, particle motion on G=H provides a whole class of such examples. The ....

Doplicher, S., Roberts, J.E.: Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics. Commun. Math.Phys. 131, 51-107 (1990) 31

....chiral algebras are extended by an A k primary simple current: a Bose field of isospin and conformal dimension I = k 2 and Delta I = I(I 1) k 2 = k 4 2 N: 3. 1) The inclusion of the (nets of) algebras A k in the resulting field algebras are well understood: it is of the DHR type [5, 19] with a global Z 2 gauge group which singles out the observables A k as the gauge invariant elements [20] for a recent review and further references see [21] Here we shall deal with the more interesting exceptional extensions corresponding to conformal embeddings [12] These are not of the ....

....N s given by the dimensions of the representations of the gauge group. The corresponding coefficients (e) k ij in the expansion (4.17) of X are precisely the group theoretical Clebsch Gordan coefficients. Indeed, one may rephrase the content of the Doplicher Roberts (DR) reconstruction theorem [19] as follows: every system of sectors of the observables which have finite permutation group statistics among each other, closed under composition, reduction, and conjugation, admits a solution to (4.18) with X given by (4.17) in terms of Clebsch Gordan coefficients of some compact gauge group. The ....

Doplicher, S., Roberts, J.E.: Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics. Commun. Math. Phys. 131, 51--107 (1990).

.... with a finite number of sectors) quantum groups are also ruled out, unless all sectors have integer statistical dimensions (see e.g. FrKe] for a review or [N2] for a specific discussion of q dimensions in finite quantum groups) Based on the theory of quasi Hopf algebras introduced by Drinfel d [Dr2], G. Mack and V. Schomerus [MS] have proposed the notion of weak quasi Hopf algebras G as appropriate symmetry candidates, where weak means that the tensor product of two physical representations of G may also contain unphysical subrepresentations (i.e. of q dimension 0) which have to be ....

....mentioned the Drinfel d double D(G) and the constructions of quantum chains based on a Hopf algebra G. We now proceed to generalize the above ideas to quasi Hopf algebras G. In Section 6 we give a short review of the definitions and properties of quasi Hopf algebras as introduced by Drinfel d [Dr2]. In Section 7 we propose an obvious generalization of the notion of right G coactions ae on an algebra M to the case of quasi Hopf algebras G (and similarly for left coactions ) As for the coproduct on G, the basic idea here is that (ae Omega id) ffi ae and (id Omega Delta) ffi ae are still ....

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S. Doplicher and J.E. Roberts, Why there is a Field Algebra with a compact gauge group describing the superselection structure in particle physics, Comm.Math.Phys. 131, (1990) 51

....that space like separated points form a simply connected manifold in the product of Minkowski spaces for D 3. It follows that for 3 and higher space time dimensions fields should satisfy permutation group statistics and the superselection structure is associated to a compact gauge symmetry group [DoR 91] By contrast, primary CVO in a 2D CFT have nontrivial monodromy and obey braid group exchange relations for causally disjoint arguments. As a consequence their symmetry can only be described by a quantum group in general, by a weak quasi Hopf algebra [MaS 92] If one gives up Wightman ....

S. Doplicher, J.E. Roberts, Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics, Commun. Math. Phys. 131 (1991), 51-107.

....theory satisfies a condition of geometrical modular action invented by Bisognano and Wichmann, then the scaling limit has this property, too. As a consequence, the scaling limit nets comply with the condition of (essential) Haag duality and one can apply the methods of Doplicher and Roberts [20] to determine from these nets the charged fields and the global gauge group appearing in the scaling limit. These applications will be discussed elsewhere. Another interesting consequence of this result is the insight that, excluding the case of theories with a classical scaling limit, the local ....

....structure of the scaling limit theories. Anticipating that the underlying theory complies with the condition of geometric modular action, given in Sec. 6, we have shown that the scaling limit satisfies the condition of essential Haag duality. Hence, by the fundamental work of Doplicher and Roberts [20] we know that the superselection structure of any scaling limit theory is in one to one correspondence to the spectrum (that is, the dual) of a compact group G 0 . Moreover, there exist charged Bose and Fermi fields which transform as tensors under the action of G 0 and generate the charged ....

Doplicher, S., Roberts, J.E.: Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics. Commun. Math. Phys. 131 (1990) 51

....are inverses up to a 2 isomorphism. 6 Reconstruction Theorems In this section we give a classification of symmetric 2 H algebras. Doplicher and Roberts proved a theorem which implies that connected even symmetric 2 H algebras are all equivalent to categories of compact group representations [8, 9]. Here and in all that follows, by a representation of a compact group we mean a finite dimensional continuous unitary representation. Given a compact group G, let Rep(G) denote the category of such representations of G. This becomes a connected even symmetric 2 H algebra in an obvious way. ....

S. Doplicher and J. Roberts, Why there is a field algebra with a compact gauge group describing the superselection in particle physics, Comm. Math. Phys. 131 (1990), 51-107.

....C systems with a compact group. We propose a generalization of the notion of an irreducible endomorphism and study the influence of the sector structure on Z . Finally we give several characterizations of the stabilizer of A. 1 Introduction The Doplicher Roberts superselection theory [1, 2, 3, 4, 5, 6, 7, 8] starts with a C algebra A with trivial center, i.e. Z(A) Z = C 1l . A is interpreted as the algebra of quasilocal observables. The field algebra F oe A, together with the gauge group G is then constructed as a special C dynamical system fF ; Gg (cf. 9] namely as a crossed product ....

S. Doplicher and J.E. Roberts, Why there is a field algebra with compact gauge group describing the superselection structure in particle physics, Commun. Math. Phys. 131, 51-- 107 (1990).

.... 149, 22761 Hamburg, Germany, e mail: i02bku dsyibm.desy.de A field net consisiting of von Neumann algebras which generates in particular all massive parabosonic and parafermionic sectors from the vacuum and exhibits normal commutation relations has been constructed by Doplicher and Roberts [19] for any local net of observables satisfying the standard assumptions, and such a field is unique up to unitary equivalence. For the algebraic framework, the spin statistics theorem in (1 3) dimensional spacetime has been proven in [17] for charges which are localizable in bounded open sets and in ....

....violate the familiar spin statistics connection. They admit several unitary representations of g P under which they are covariant, so, a fortiori, our compactness assumption is violated. Finally, we recall the definitions and results of the Doplicher Roberts field construction performed in [19] which are used in our argument. 2.3 Definition Let (H 0 ; A; U; be as above, let H be a (not necessarily separable) Hilbert space, and let (F(C) C2 Sigma be a net 1 of von Neumann algebras. Let be a faithful representation of e A in H, and let G be a strongly compact group of unitaries ....

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Doplicher, S., Roberts, J. E.: Why There is a Field Algebra with a Compact Gauge Group Describing the Superselection Structure in Particle Physics, Commun. Math. Phys. 131, 51-107 (1990)

....relation are proven in the context of the field algebras, 2 where the formalism is close to the classical formulation. Then, in section 3, we obtain our result in the context of local observables. This is done by rephrasing the statements in terms of the Doplicher Roberts field algebra [18]. This last step has certain pedagogical advantages, but has to be avoided in order to extend our work to more general settings where the field algebra does not exists. In a forthcoming paper [23] we shall indeed provide a more intrinsic approach in terms of local observables only, that will cover ....

.... in this section we shall make the usual assumption the von Neumann algebras associated with unbounded regions are generated by additivity by the ones associated with double cones within the region, so that the Doplicher Roberts theorem on the reconstruction of the field algebras applies [18]. We shall consider morphisms of the local observable algebras localized in space like cones in a 4dimensional space time [15,11] The same techniques would prove the Spin and Statistics theorem for sectors localized in space like cones in the n dimensional Minkowski space for any n 4 and for ....

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Doplicher S., Roberts J.E., "Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics", Commun. Math. Phys. 131 (1990), 51-107.

....relations . Para fields, introduced by O.W. Greenberg [26] will not be considered. Para statistics are usually only discussed in the frame of quantum field theory of local observables, where a reduction to ordinary Bose or Fermi commutation relations has been achieved by Doplicher and Roberts in [22]. Quantum field theory of local observables (QFTLO) in the sense of Araki, Haag and Kastler [31] is concerned with C algebras A(O) associated with bounded open regions O ae R d . These algebras shall fulfill isotony, i.e. O 1 ae O 2 implies A(O 1 ) ae A(O 2 ) and locality, i.e. if O 1 and ....

....a charged sector if 0 and 1 have the same kernel and if for every O the isomorphism 0 (A(O 0 ) 1 (A(O 0 ) extends to an isomorphism of the corresponding von Neumann algebras. Here O 0 denotes the space like complement of O. With this concept Doplicher, Haag and Roberts [19, 20] [22] have worked out the details of the mentioned program. The algebra generated by the local observables and by the localized charged fields is called the field algebra and the corresponding net is usually denoted by fF(O)g. Within this setting also the concept of conjugate charge sectors has a ....

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Doplicher, S. and Roberts, J.E., Why there is a field algebra with a compact gauge group describing the superselection structure of particle physics, Commun. Math. Phys. 131 (1990), 187.

....Fermi fields commuting or anticommuting when localized in causally disjoint regions. This is the generic situation in physical spacetime dimension. In lower spacetime dimension, braid group statistics may occur and the statistics phase may take values different from Sigma1. In previous papers [25, 24] it was shown that, in Minkowski spacetime, one can construct a field net together with a unitary action of a compact (global) gauge group containing the observable net A as fixed points so that the superselection sectors correspond naturally to the equivalence classes of irreducible ....

....The existence of statistics is established in this generality. If the index set K is directed, all the other basic results known for superselection theory on Minkowski spacetime, classification of statistics, existence of charge conjugation and construction of field algebra and gauge group (cf. [25]) can again be shown to hold. Chapter 4 begins with a summary of the geometry of spacetimes with a bifurcate Killing horizon following Kay and Wald [40] We introduce a family of wedge regions R a , a 0 which are copies of the canonical right wedge shifted by a in the affine geodesic parameter ....

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S. Doplicher, J.E. Roberts: "Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics", Commun. Math. Phys. 131, 51 (1990)

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Doplicher, S. and Roberts, J.E., Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics, Comm. Math. Phys. 131, 51-107 (1990).

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