45 research outputs found
Homotopy of posets, net-cohomology and superselection sectors in globally hyperbolic spacetimes
We study sharply localized sectors, known as sectors of DHR-type, of a net of
local observables, in arbitrary globally hyperbolic spacetimes with dimension
. We show that these sectors define, has it happens in Minkowski space,
a 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 spacetime 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 to analyze the structure of this category because
we show how topological properties of the spacetime 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 the one of the underlying
spacetime 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.Comment: 4 figures. To be published in Reviews in Mathematical Physic
Representations of nets of C*-algebras over S^1
In 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 of 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)
The K-homology of nets of C*-algebras
Let X be a space, intended as a possibly curved spacetime, and A a precosheaf
of C*-algebras on X. Motivated by algebraic quantum field theory, we study the
Kasparov and Theta-summable K-homology of A interpreting them in terms of the
holonomy equivariant K-homology of the associated C*-dynamical system. This
yields a characteristic class for K-homology cycles of A with values in the odd
cohomology of X, that we interpret as a generalized statistical dimension.Comment: To appear in Journal of Geometry and Physic
A new light on nets of C*-algebras and their representations
The 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 embeds into a unique 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 having a faithful embedding
into the enveloping net bundle. Injectivity turns out to be equivalent to the
existence of faithful representations. We further relate injectivity to a
generalized Cech cocycle of the net, and this allows us to give examples of
nets exhausting the above classification. Using the results of this paper we
shall show, in a forthcoming paper, that any conformal net over S^1 is
injective
A cohomological description of connections and curvature over posets
What 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 finding 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 . 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 . We discuss
holonomy and prove an analogue of the Ambrose-Singer theorem
Causal posets, loops and the construction of nets of local algebras for QFT
We provide a model independent construction of a net of C*-algebras
satisfying the Haag-Kastler axioms over any spacetime manifold. Such a net,
called the net of causal loops, is constructed by selecting a suitable base K
encoding causal and symmetry properties of the spacetime. Considering K as a
partially ordered set (poset) with respect to the inclusion order relation, we
define groups of closed paths (loops) formed by the elements of K. These groups
come equipped with a causal disjointness relation and an action of the symmetry
group of the spacetime. In this way the local algebras of the net are the group
C*-algebras of the groups of loops, quotiented by the causal disjointness
relation. We also provide a geometric interpretation of a class of
representations of this net in terms of causal and covariant connections of the
poset K. In the case of the Minkowski spacetime, we prove the existence of
Poincar\'e covariant representations satisfying the spectrum condition. This is
obtained by virtue of a remarkable feature of our construction: any Hermitian
scalar quantum field defines causal and covariant connections of K. Similar
results hold for the chiral spacetime with conformal symmetry
General Covariance in Algebraic Quantum Field Theory
In this review we report on how the problem of general covariance is treated
within the algebraic approach to quantum field theory by use of concepts from
category theory. Some new results on net cohomology and superselection
structure attained in this framework are included.Comment: 61 pages, 3 figures, LaTe
The universal C*-algebra of the electromagnetic field II. Topological charges and spacelike linear fields
Conditions for the appearance of topological charges are studied in the
framework of the universal C*-algebra of the electromagnetic field, which is
represented in any theory describing electromagnetism. It is shown that
non-trivial topological charges, described by pairs of fields localised in
certain topologically non-trivial spacelike separated regions, can appear in
regular representations of the algebra only if the fields depend non-linearly
on the mollifying test functions. On the other hand, examples of regular vacuum
representations with non-trivial topological charges are constructed, where the
underlying field still satisfies a weakened form of "spacelike linearity". Such
representations also appear in the presence of electric currents. The status of
topological charges in theories with several types of electromagnetic fields,
which appear in the short distance (scaling) limit of asymptotically free
non-abelian gauge theories, is also briefly discussed.Comment: 24 pages, 2 figure
The universal C*-algebra of the electromagnetic field
A universal C*-algebra of the electromagnetic field is constructed. It is
represented in any quantum field theory which incorporates electromagnetism and
expresses basic features of this field such as Maxwell's equations, Poincar\'e
covariance and Einstein causality. Moreover, topological properties of the
field resulting from Maxwell's equations are encoded in the algebra, leading to
commutation relations with values in its center. The representation theory of
the algebra is discussed with focus on vacuum representations, fixing the
dynamics of the field.Comment: 17 pages, 1 figure; v2: minor corrections, version as to appear in
Lett. Math. Phys. Dedicated to the memory of D. Kastler and J.E. Roberts; v3
improvement of layou