1,720,981 research outputs found
Punctured Haag duality in locally covariant quantum field theories
No full textWe investigate a new property of nets of local algebras over 4-dimensional globally hyperbolic spacetimes, called punctured Haag duality. This property consists in the usual Haag duality for the restriction of the net to the causal complement of a point p of the spacetime. Punctured Haag duality implies Haag duality and local definiteness. Our main result is that, if we deal with a locally covariant quantum field theory in the sense of Brunetti, Fredenhagen and Verch, then also the converse holds. The free Klein-Gordon field provides an example in which this property is verified. © Springer-Verlag 2005
Homotopy of posets, net-cohomology and superselection sectors in globally hyperbolic space-times
No full textWe study sharply localized sectors, known as sectors of DHR-type, of a net of local observables, in arbitrary globally hyperbolic space-times with dimension ≥ 3. We show that these sectors define, as it happens in Minkowski space, a C*-category in which the charge structure manifests itself by the existence of a tensor product, a permutation symmetry and a conjugation. The mathematical framework is that of the net-cohomology of posets according to J. E. Roberts. The net of local observables is indexed by a poset formed by a basis for the topology of the space-time ordered under inclusion. The category of sectors, is equivalent to the category of 1-cocycles of the poset with values in the net. We succeed in analyzing the structure of this category because we show how topological properties of the space-time are encoded in the poset used as index set: the first homotopy group of a poset is introduced and it is shown that the fundamental group of the poset and one of the underlying space-time are isomorphic; any 1-cocycle defines a unitary representation of these fundamental groups. Another important result is the invariance of the net-cohomology under a suitable change of index set of the net. © World Scientific Publishing Company
Essential properties of the vacuum sector for a theory of superselection sectors
No full textAs a generalization of DHR analysis, the superselection sectors are studied in the absence of the spectrum condition for the reference representation. Considering a net of local observables in 4-dimensional Minkowski spacetime, we associate to a set of representations, that are local excitations of a reference representation fulfilling Haag duality, a symmetric tensor C*-category B(Θ) of bimodules of the net, with subobjects and direct sums. The existence of conjugates is studied introducing an equivalent formulation of the theory in terms of the presheaf associated with the observable net. This allows us to find, under the assumption that the local algebras in the reference representation are properly infinite, necessary and sufficient conditions for the existence of conjugates. Moreover, we present several results that suggest how the mentioned assumption on the reference representation can be considered essential also in the case of theories in curved spacetimes
Superselection sectors and general covariance. I
No full textThis paper is devoted to the analysis of charged superselection sectors in the framework of the locally covariant quantum field theories. We shall analyze sharply localizable charges, and use net-cohomology of J.E. Roberts as a main tool. We show that to any 4-dimensional globally hyperbolic spacetime a unique, up to equivalence, symmetric tensor C*-category with conjugates (in case of finite statistics) is attached; to any embedding between different spacetimes, the corresponding categories can be embedded, contravariantly, in such a way that all the charged quantum numbers of sectors are preserved. This entails that to any spacetime is associated a unique gauge group, up to isomorphisms, and that to any embedding between two spacetimes there corresponds a group morphism between the related gauge groups. This form of covariance between sectors also brings to light the issue whether local and global sectors are the same. We conjecture this holds that at least on simply connected spacetimes. It is argued that the possible failure might be related to the presence of topological charges. Our analysis seems to describe theories which have a well defined short-distance asymptotic behaviour
Aharonov–Bohm superselection sectors
Full text linkWe show that the Aharonov–Bohm effect finds a natural description in the setting of QFT on curved spacetimes in terms of superselection sectors of local observables. The extension of the analysis of superselection sectors from Minkowski spacetime to an arbitrary globally hyperbolic spacetime unveils the presence of a new quantum number labelling charged superselection sectors. In the present paper, we show that this “topological” quantum number amounts to the presence of a background flat potential which rules the behaviour of charges when transported along paths as in the Aharonov–Bohm effect. To confirm these abstract results, we quantize the Dirac field in the presence of a background flat potential and show that the Aharonov–Bohm phase gives an irreducible representation of the fundamental group of the spacetime labelling the charged sectors of the Dirac field. We also show that non-Abelian generalizations of this effect are possible only on spacetimes with a non-Abelian fundamental group
A New Light on Nets of C*-Algebras and Their Representations
No full textThe present paper deals with the question of representability of nets of C*-algebras whose underlying poset, indexing the net, is not upward directed. A particular class of nets, called C*-net bundles, is classified in terms of C*-dynamical systems having as group the fundamental group of the poset. Any net of C*-algebras has a canonical morphism into a C*-net bundle, the enveloping net bundle, which generalizes the notion of universal C*-algebra given by Fredenhagen to nonsimply connected posets. This allows a classification of nets; in particular, we call injective those nets such that the canonical morphism is faithful. Injectivity turns out to be equivalent to the existence of faithful representations. We further relate injectivity to a generalized ech cocycle of the net, and this allows us to give examples of nets exhausting the above classification. Using these results we have shown, in another paper, that any conformal net over S (1) is injective
Representations of Nets of C-*-Algebras over S (1)
No full textIn recent times a new kind of representations has been used to describe superselection sectors of the observable net over a curved spacetime, taking into account the effects of the fundamental group of the spacetime. Using this notion of representation, we prove that any net of C-*-algebras over S (1) admits faithful representations, and when the net is covariant under Diff(S (1)), it admits representations covariant under any amenable subgroup of Diff(S (1))
A cohomological description of connections and curvature over posets
No full textWhat remains of a geometrical notion like that of a principal bundle when the base space is not a manifold but a coarse graining of it, like the poset formed by a base for the topology ordered under inclusion? Motivated by the search for a geometrical framework for developing gauge theories in algebraic quantum field theory, we give, in the present paper, a first answer to this question. The notions of transition function, connection form and curvature form find a nice description in terms of cohomology, in general non-Abelian, of a poset with values in a group G. Interpreting a 1-cocycle as a principal bundle, a connection turns out to be a 1-cochain associated in a suitable way with this 1-cocycle; the curvature of a connection turns out to be its 2-coboundary. We show the existence of nonflat connections, and relate flat connections to homomorphisms of the fundamental group of the poset into G. We discuss holonomy and prove an analogue of the Ambrose-Singer theorem
The C(X)-algebra of a net and index theory.
No full textGiven a connected and locally compact Hausdorff space X with a good base K we assign, in a functorial way, a C(X)-algebra to any precosheaf of C*-algebras A defined over K. Afterwards we consider the representation theory and the Kasparov K-homology of A, and interpret them in terms, respectively, of the representation theory and the K-homology of the associated C(X)-algebra.
When A is an observable net over the spacetime X in the sense of algebraic quantum field theory, this yields a geometric description of the recently discovered representations affected by the topology of X
The K-homology of nets of C*-algebras
No full textLet XX be a space, intended as a possibly curved space–time, and AA a precosheaf of C∗C∗-algebras on XX. Motivated by algebraic quantum field theory, we study the Kasparov and ΘΘ-summable KK-homology of AA interpreting them in terms of the holonomy equivariant KK-homology of the associated C∗C∗-dynamical system. This yields a characteristic class for KK-homology cycles of AA with values in the odd cohomology of XX, that we interpret as a generalized statistical dimension
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