1,012 research outputs found
Posets having continuous intervals
In this paper we consider posets in which each order interval [a,b] is a continuous poset or continuous domain. After developing some basic theory for such posets, we derive our major result: if X is a core compact space and L is a poset equipped with the Scott topology (assumed to satisfy a mild extra condition) for which each interval is a continuous sup-semilattice, then the function space of continuous locally bounded functions from X into L has intervals that are continuous sup-semilattices. This substantially generalizes known results for continuous domains. © 2004 Elsevier B.V. All rights reserved
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
Finite Approximations to Quantum Physics: Quantum Points and their Bundles
There exists a physically well motivated method for approximating manifolds
by certain topological spaces with a finite or a countable set of points. These
spaces, which are partially ordered sets (posets) have the power to effectively
reproduce important topological features of continuum physics like winding
numbers and fractional statistics, and that too often with just a few points.
In this work, we develop the essential tools for doing quantum physics on
posets. The poset approach to covering space quantization, soliton physics,
gauge theories and the Dirac equation are discussed with emphasis on physically
important topological aspects. These ideas are illustrated by simple examples
like the covering space quantization of a particle on a circle, and the
sine-Gordon solitons.Comment: 24 pages, 8 figures on a uuencoded postscript file, DSF-T-29/93,
INFN-NA-IV-29/93 and SU-4240-55
Unsharp Values, Domains and Topoi
The so-called topos approach provides a radical reformulation of quantum
theory. Structurally, quantum theory in the topos formulation is very similar
to classical physics. There is a state object, analogous to the state space of
a classical system, and a quantity-value object, generalising the real numbers.
Physical quantities are maps from the state object to the quantity-value object
-- hence the `values' of physical quantities are not just real numbers in this
formalism. Rather, they are families of real intervals, interpreted as `unsharp
values'. We will motivate and explain these aspects of the topos approach and
show that the structure of the quantity-value object can be analysed using
tools from domain theory, a branch of order theory that originated in
theoretical computer science. Moreover, the base category of the topos
associated with a quantum system turns out to be a domain if the underlying von
Neumann algebra is a matrix algebra. For general algebras, the base category
still is a highly structured poset. This gives a connection between the topos
approach, noncommutative operator algebras and domain theory. In an outlook, we
present some early ideas on how domains may become useful in the search for new
models of (quantum) space and space-time.Comment: 32 pages, no figures; to appear in Proceedings of Quantum Field
Theory and Gravity, Regensburg (2010
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 -Construction for Lattices, Spheres and Polytopes
We describe and analyze a new construction that produces new Eulerian
lattices from old ones. It specializes to a construction that produces new
strongly regular cellular spheres (whose face lattices are Eulerian). The
construction does not always specialize to convex polytopes; however, in a
number of cases where we can realize it, it produces interesting classes of
polytopes. Thus we produce an infinite family of rational 2-simplicial 2-simple
4-polytopes, as requested by Eppstein, Kuperberg and Ziegler. We also construct
for each an infinite family of -simplicial 2-simple
-polytopes, thus solving a problem of Gr\"unbaum.Comment: 21 pages, many figure
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