13,794 research outputs found
Pro-Lie Groups: A survey with Open Problems
A topological group is called a pro-Lie group if it is isomorphic to a closed
subgroup of a product of finite-dimensional real Lie groups. This class of
groups is closed under the formation of arbitrary products and closed subgroups
and forms a complete category. It includes each finite-dimensional Lie group,
each locally compact group which has a compact quotient group modulo its
identity component and thus, in particular, each compact and each connected
locally compact group; it also includes all locally compact abelian groups.
This paper provides an overview of the structure theory and Lie theory of
pro-Lie groups including results more recent than those in the authors'
reference book on pro-Lie groups. Significantly, it also includes a review of
the recent insight that weakly complete unital algebras provide a natural
habitat for both pro-Lie algebras and pro-Lie groups, indeed for the
exponential function which links the two. (A topological vector space is weakly
complete if it is isomorphic to a power of an arbitrary set of copies of
. This class of real vector spaces is at the basis of the Lie theory of
pro-Lie groups.) The article also lists 12 open questions connected with
pro-Lie groups.Comment: 19 page
Poisson spaces with a transition probability
The common structure of the space of pure states of a classical or a
quantum mechanical system is that of a Poisson space with a transition
probability. This is a topological space equipped with a Poisson structure, as
well as with a function , with certain properties. The
Poisson structure is connected with the transition probabilities through
unitarity (in a specific formulation intrinsic to the given context).
In classical mechanics, where p(\rho,\sigma)=\dl_{\rho\sigma}, unitarity
poses no restriction on the Poisson structure. Quantum mechanics is
characterized by a specific (complex Hilbert space) form of , and by the
property that the irreducible components of as a transition probability
space coincide with the symplectic leaves of as a Poisson space. In
conjunction, these stipulations determine the Poisson structure of quantum
mechanics up to a multiplicative constant (identified with Planck's constant).
Motivated by E.M. Alfsen, H. Hanche-Olsen and F.W. Shultz ({\em Acta Math.}
{\bf 144} (1980) 267-305) and F.W. Shultz ({\em Commun.\ Math.\ Phys.} {\bf 82}
(1982) 497-509), we give axioms guaranteeing that is the space of pure
states of a unital -algebra. We give an explicit construction of this
algebra from .Comment: 23 pages, LaTeX, many details adde
Locally Trivial W*-Bundles
We prove that a tracially continuous W-bundle over a
compact Hausdorff space with all fibres isomorphic to the hyperfinite
II-factor that is locally trivial already has to be globally
trivial. The proof uses the contractibility of the automorphism group
shown by Popa and Takesaki. There is no
restriction on the covering dimension of .Comment: 20 pages, this version will be published in the International Journal
of Mathematic
A note on information theoretic characterizations of physical theories
Clifton, Bub, and Halvorson [Foundations of Physics 33, 1561 (2003)] have
recently argued that quantum theory is characterized by its satisfaction of
three information-theoretic axioms. However, it is not difficult to construct
apparent counterexamples to the CBH characterization theorem. In this paper, we
discuss the limits of the characterization theorem, and we provide some
technical tools for checking whether a theory (specified in terms of the convex
structure of its state space) falls within these limits.Comment: 16 pages, LaTeX, Contribution to Rob Clifton memorial conferenc
Lattice-ordered abelian groups and perfect MV-algebras: a topos-theoretic perspective
We establish, generalizing Di Nola and Lettieri's categorical equivalence, a
Morita-equivalence between the theory of lattice-ordered abelian groups and
that of perfect MV-algebras. Further, after observing that the two theories are
not bi-interpretable in the classical sense, we identify, by considering
appropriate topos-theoretic invariants on their common classifying topos, three
levels of bi-intepretability holding for particular classes of formulas:
irreducible formulas, geometric sentences and imaginaries. Lastly, by
investigating the classifying topos of the theory of perfect MV-algebras, we
obtain various results on its syntax and semantics also in relation to the
cartesian theory of the variety generated by Chang's MV-algebra, including a
concrete representation for the finitely presentable models of the latter
theory as finite products of finitely presentable perfect MV-algebras. Among
the results established on the way, we mention a Morita-equivalence between the
theory of lattice-ordered abelian groups and that of cancellative
lattice-ordered abelian monoids with bottom element.Comment: 54 page
Algebraic Quantum Theory on Manifolds: A Haag-Kastler Setting for Quantum Geometry
Motivated by the invariance of current representations of quantum gravity
under diffeomorphisms much more general than isometries, the Haag-Kastler
setting is extended to manifolds without metric background structure. First,
the causal structure on a differentiable manifold M of arbitrary dimension
(d+1>2) can be defined in purely topological terms, via cones (C-causality).
Then, the general structure of a net of C*-algebras on a manifold M and its
causal properties required for an algebraic quantum field theory can be
described as an extension of the Haag-Kastler axiomatic framework.
An important application is given with quantum geometry on a spatial slice
within the causally exterior region of a topological horizon H, resulting in a
net of Weyl algebras for states with an infinite number of intersection points
of edges and transversal (d-1)-faces within any neighbourhood of the spatial
boundary S^2.Comment: 15 pages, Latex; v2: several corrections, in particular in def. 1 and
in sec.
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